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+The Evolution of Web Search User Interfaces - An
+Archaeological Analysis of Google Search Engine Result Pages
+Bruno Oliveira
+Faculty of Engineering of the University of Porto
+Porto, Portugal
+up201605516@edu.fe.up.pt
+Carla Teixeira Lopes
+Faculty of Engineering of the University of Porto and
+INESC-TEC
+Porto, Portugal
+ctl@fe.up.pt
+ABSTRACT
+Web search engines have marked everyone’s life by transforming
+how one searches and accesses information. Search engines give
+special attention to the user interface, especially search engine
+result pages (SERP). The well-known “10 blue links” list has evolved
+into richer interfaces, often personalized to the search query, the
+user, and other aspects. More than 20 years later, the literature has
+not adequately portrayed this development. We present a study
+on the evolution of SERP interfaces during the last two decades
+using Google Search as a case study. We used the most searched
+queries by year to extract a sample of SERP from the Internet
+Archive. Using this dataset, we analyzed how SERP evolved in
+content, layout, design (e.g., color scheme, text styling, graphics),
+navigation, and file size. We have also analyzed the user interface
+design patterns associated with SERP elements. We found that
+SERP are becoming more diverse in terms of elements, aggregating
+content from different verticals and including more features that
+provide direct answers. This systematic analysis portrays evolution
+trends in search engine user interfaces and, more generally, web
+design. We expect this work will trigger other, more specific studies
+that can take advantage of our dataset.
+CCS CONCEPTS
+• Human-centered computing → Interaction design process
+and methods; Human computer interaction (HCI); • Infor-
+mation systems → Users and interactive retrieval; Web search
+engines.
+KEYWORDS
+Search engines, SERP features, Web interfaces, Web design, Evolu-
+tion
+ACM Reference Format:
+Bruno Oliveira and Carla Teixeira Lopes. 2023. The Evolution of Web Search
+User Interfaces - An Archaeological Analysis of Google Search Engine
+Result Pages. In ACM SIGIR Conference on Human Information Interaction
+and Retrieval (CHIIR’23), March 19–23, 2023, Austin, TX, USA. ACM, New
+York, NY, USA, 14 pages. https://doi.org/10.1145/3576840.3578320
+Permission to make digital or hard copies of part or all of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for third-party components of this work must be honored.
+For all other uses, contact the owner/author(s).
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+© 2023 Copyright held by the owner/author(s).
+ACM ISBN 979-8-4007-0035-4/23/03.
+https://doi.org/10.1145/3576840.3578320
+1
+INTRODUCTION
+The wealth of information available on the Web make search en-
+gines an essential tool nowadays [1]. Web search engines have
+evolved a lot, and their user interface is no exception. Web search
+engines’ front end is gaining importance in a scenario where rel-
+evance is becoming more difficult to evaluate by users, and their
+perception becomes more influenced by the user experience [5,
+p. 512].
+Search interfaces support tasks from query formulation to select-
+ing and understanding search results [5, p. 26-40]. Typically, web
+search engines have a home page containing a search entry form
+in which the user types a query. Retrieval results are usually dis-
+played as vertical lists on Search Engine Results Pages (SERP). For
+its richness, our study focuses on these pages. SERP have started
+as simple “10 blue links” pages. Although search engines have kept
+a consistent format of presenting search results, the information in
+SERP goes way beyond these links. The design of query interfaces
+and retrieval results display is an active area of research and exper-
+imentation. Although works provide an in-depth analysis of search
+user interfaces, such as the one from Hearst [22], the temporal
+evolution of SERP is understudied.
+Portraying the evolution of SERP contributes to preserving the
+history of web search user interfaces. Moreover, assuming these
+interfaces follow the general trends in web user interfaces, it con-
+tributes to the overall study of web interfaces. Broadly, it depicts
+trends in web design. This evolutionary analysis is also helpful
+for the interactive information retrieval community to understand
+better how the search interfaces have evolved in content, layout,
+and navigation and build upon this for further and deeper analysis.
+This study conducts an evolutionary analysis of Google’s SERP
+during the last two decades of Google Search, analyzing the overall
+user interfaces over time and their elements. We chose Google for
+its popularity in the web search engine landscape. For this analysis,
+we have captured from the Internet Archive 5,653 Google SERP
+from 2000 to 2020.
+This work makes several contributions. First, we present a sys-
+tematization of the elements that appear or have appeared in a SERP,
+defining each and providing visual examples. This systematization
+can be helpful in future studies that have the SERP as their focus
+and allows the establishment of common terminology. Second, we
+analyze the evolution of the overall SERP and each element. Third,
+we propose a methodology that can be used to study different types
+of web search user interfaces (e.g., mobile ones) or user interfaces
+in other contexts. Fourth, this paper makes two resources available
+arXiv:2301.08613v1  [cs.IR]  20 Jan 2023
+
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+Bruno Oliveira and Carla Teixeira Lopes
+to the community: a dataset1 containing the screenshot and files
+associated with each extracted capture and a website2 summarizing
+the analysis. These resources can also be the input of further studies,
+inclusively done by researchers without advanced technological
+skills.
+2
+RELATED WORK
+Research in search user interfaces focuses on their design and
+evaluation, either in broad or focused on query formulation, the
+presentation of search results, or even query reformulation. There
+are also works focused on personalization, information visualiza-
+tion, or domain-specific search interfaces such as mobile, social, or
+multimedia. There are whole books and monographs dedicated to
+this subject [22, 35, 43, 48, 61, 63]. Despite such research, given our
+focus, we only describe works focused on analyzing the anatomy
+of SERP and, eventually, its evolution. This section does not cover
+works proposing or evaluating search interface components.
+Höchstötter and Lewandowski [23] address SERP composition
+and count the various elements’ appearances. To the best of our
+knowledge, this was the first work to analyze the entire structure of
+the SERP. Besides, the authors examined the retrieved results, their
+sources, and types (e.g., organic results, advertisements, shortcuts).
+When the authors wrote this paper, advanced features in SERP
+were not widespread, which was not the case when Moran and
+Goray [40] studied the anatomy of SERP, defining the terminology
+for SERP elements. Nielsen Norman Group uses this terminology in
+several articles [36, 40–42]. In their ‘Search Patterns’ book, Morville
+and Callender [43], apart from addressing the anatomy of the search
+process and related behavior, also list elements and principles of
+interaction design, illustrating many user interface design patterns
+around search websites.
+To the best of our knowledge, no works systematically collect
+and analyze SERP interfaces over time.
+3
+METHODOLOGY
+The first stage of our work involves building a sample of SERP inter-
+faces over time, a process described in Section 3.1 and Section 3.2.
+After collecting this sample, our attention focused on its analysis
+and automation, as described in Section 3.3.
+3.1
+What SERP have we captured?
+Google Search currently has 91.4% of the market share [10], a lead-
+ership that goes back to 2002 [27, 46]. In this context, we decided
+to focus our analysis on this search engine. This study will address
+desktop versions of Google Search from 2000 until 2020. A compar-
+ative analysis with the seconded ranked search engine, Microsoft
+Bing, is done in another work [45] and is available on the study’s
+website.
+The Internet Archive keeps snapshots and the respective HTML
+version of web pages over time. Its collection contains 588 billion
+web pages [3]. Internet Archive provides the Wayback CDX Server
+API, which allows complex querying, filtering, and analysis of cap-
+tures. While filtering by URL, we can use a wildcard (*) at the end of
+1Available at https://doi.org/10.25747/991g-f765
+2https://bedgarone.github.io/serpevolution/
+the URL to specify the latter as a prefix and receive entries beyond
+the specified URL (e.g., www.google.com/search?q=cookies*).
+We found more than 195 thousand captures of Google SERP
+during two decades using the API. This large number of SERP and
+existing resource restrictions led us to devise a method to iden-
+tify a smaller set of SERP. To increase the likelihood of reaching
+pages with SERP element diversity, we have used a set of 129 most
+searched queries in the last 20 years, retrieved from Google Trends
+during the same period. This set3 contains the first search query
+from each available category, such as People, Health, or Electron-
+ics. These queries include relevant terms often searched by users
+and trigger features in SERP. Hence, it is highly likely that SERP
+interfaces derived from these queries are richer and, thus, more
+relevant for this study than those generated by random searches.
+We decided to append these queries with the ‘*’ wildcard while
+submitting them to the API to obtain more captures.
+We noticed that some years had no captures using the most
+searched queries, which coincided with periods in which there
+were few captures from Google Search’s domain. Hence, in those
+years, we collected all the available captures (all method in Table 1).
+We also noticed that the last two years had much more captures
+(>10 thousand). Therefore, in 2019 and 2020, we restricted the URL
+submitted to the API to those containing queries shorter than 37
+characters. Considering that an English word has, on average, about
+five characters [44, 65], the 37 characters are the equivalent of 6
+words plus spaces between them, which is more than the two to
+three words that a query typically has [6, 24, 25]. This restriction
+excludes more specific queries that are probably less useful to the
+plurality of interfaces. Our sample has 5.653 captures. The last
+column of Table 1 has ordered lists with the number of captures
+per year.
+Table 1: Method used to collect the captures, maximum
+length of the query (search URL), the width of the screen-
+shot, and the number of captures extracted per year.
+method
+max. length
+width
+#captures per year
+2000 - 2002
+queries
+-
+800px
+200, 3, 23
+2003
+queries
+-
+1024px
+231
+2004 - 2008
+all
+-
+1024px
+12, 0, 200, 0, 26
+2009
+queries
+-
+1024px
+11
+2010
+all
+-
+1024px
+78
+2011
+queries
+-
+1024px
+7
+2012 - 2018
+queries
+-
+1366px
+57, 975, 30,
+89, 172, 192, 548
+2019
+queries
+37 char
+1366px
+171
+2020
+queries
+37 char
+1920px
+2628
+3.2
+How have we captured SERP?
+We used Python and Selenium Webdriver to visit each capture
+online, check if the capture was valid, save the HTML version, and
+3Available at https://bedgarone.github.io/serpevolution/mostsearchedqueries
+
+The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+CNN
+AMAZON
+TRUMP
+...
+Organic result
+found within 
+6 seconds?
+Load capture
+in Browser
+Remove
+banners
+Extract
+source and files
+Wayback
+Machine CDX API
+FOR-EACH capture
+Set of
+queries
+Get SERP timestamp
+and original URL
+Generate SERP
+archived URL
+NO
+YES
+web.archive.org
+/web/
++ 
+TIMESTAMP
++ 
+ORIGINAL URL
+{TIMESTAMP,
+ORIGINAL URL}
+{TIMESTAMP,
+ORIGINAL URL}
+{TIMESTAMP,
+ORIGINAL URL}
+Skip capture
+Figure 1: Extracting captures procedure
+generate a screenshot. The capture process is shown in Figure 1.
+The original URL is the URL of the original SERP (e.g., google.com/
+search?q=photography), while the archived URL is the URL of its
+archived version (e.g., web.archive.org/web/20160125203434/www.
+google.com/search?q=photography).
+Some captures with an HTTP OK status code were not consid-
+ered valid. Some are inexistant, showing a contradictory message
+of URL not captured, while others are defective (e.g., incomplete
+interfaces without search results). To automatically check the va-
+lidity of each capture, we try to find a result entry, the element
+that cannot lack in a SERP. Captures from SERP tabs other than the
+general first page, identified with “tbm” in the URL, were discarded
+for being outside this work’s scope. We raise a timeout exception
+after 6 seconds, the time we empirically considered sufficient to
+load the capture in the browser. In these situations, the program
+will skip the capture. Before downloading the page, we still remove
+graphical elements from Internet Archive, such as its information
+and donation bars. Some of the captures present other distracting
+banners and overlapping parts of the interface, such as the ones
+related to cookie consent. We removed all the identified ones and
+extracted the source and associated files.
+The process concludes with generating full-height screenshots
+of every HTML version opened in another browser instance in
+headless mode. We produced screenshots considering the most
+popular screen size at the time of the capture, as stated by the
+statistics [55]. We only considered the width, shown in Table 1,
+because SERP height is highly variable. The dataset with all the
+extracted captures is available online4.
+3.3
+How have we analyzed SERP?
+The analysis process included two main stages, as shown in Figure 2.
+First, we have extracted a sample of captures from the primary
+dataset to identify SERP elements. For each month with captures in
+the primary dataset, we manually looked at the screenshots of that
+month’s captures and selected the capture with the most features.
+In the end, this set included 117 captures, with which we visually
+identified SERP elements. We manually analyzed each element’s
+source code, looking for identifiers to locate the element in a later
+4https://doi.org/10.25747/991g-f765
+automated process. Element identifiers consist of HTML classes, ids,
+tags, or a combination of these using CSS selectors (e.g., ‘.knowledge-
+panel’, ‘#tads’ or ‘#newsbox’). All the encountered identifiers were
+logged and listed on the website5.
+FOR-EACH
+element
+Identify SERP
+elements
+Build set of 
+SERP
+Seek identifiers in
+captures’ HTML
+Log detected
+identifiers
+Search element
+in the dataset
+Log element’s first timestamp
+Screenshot
+element
+Color
+interface
+Load capture
+in Broswer
+.#id
+.id
+#id > .id
+Figure 2: Detection and analysis of elements procedure
+In the second stage, we automated the detection of these elements
+over time, allowing the exploration of a more significant number of
+cases. Finding an element with these identifiers triggers a function
+that stores the date of the element’s appearance in a log file. We
+imposed no limit of captures per month to register the element’s
+appearance, as the computation permits a full dataset scan in an
+acceptable time. The function also receives the element’s upper-left
+corner coordinates, width, and height, generating and saving its
+image in the element’s folder. Contrary to the element’s timestamp,
+we imposed a limit of 15 captures per month while screenshotting
+to reduce and make the scan time feasible. We estimate that 15
+monthly samples are enough to capture the possible changes of an
+element.
+Following a similar procedure, we automatically used the iden-
+tifiers to detect and color the web page’s targeted areas. We used
+Python, Selenium Webdriver, and BeautifulSoup to scrape every
+HTML capture to identify and generate transparency-colored im-
+ages for each category of elements. We generated these images in
+a headless browser with a 1920px width, regardless of the capture’s
+width. This simplification does not affect the final result because the
+elements in the interfaces do not move dynamically as the width
+increases or decreases. We have not constrained the height in this
+generation process. Due to the size of the dataset, we imposed a
+limit of 15 elements per month.
+We overlayed all the individual images from single captures for
+each category of elements, which allows the overlay to enhance the
+most common areas while leaving the others almost unnoticeable.
+The overlaying process uses the upper-left corner as the reference
+for image alignment. Navigation & user inputs includes elements
+in and next to the footer, where common areas were not evident
+due to the page height’s variability. In this case, to generate and
+correctly overlap the footers of the interface, we considered a height
+value of 600px for the footer, cropping it from the bottoms of the
+interface. The 600px height was estimated after a visual analysis of
+SERP pages and included a margin of error to ensure we covered all
+elements under study. Thus, the orange result displayed in Figure 3
+is trimmed in the middle and combines those two capturing steps.
+We will analyze the positioning of every category of elements in
+5https://bedgarone.github.io/serpevolution/elements
+
+...
+HtmL
+FILES..
+HTMl
+FILES<>
+HTML<>
+HTMLCHIIR’23, March 19–23, 2023, Austin, TX, USA
+Bruno Oliveira and Carla Teixeira Lopes
+the next section. An animated version of each result in this Figure
+is available on the website6, permitting us to observe how the
+positioning of the elements’ categories in SERP changed over time.
+Figure 3: Transparency-colored overlaying results for each
+category
+4
+EVOLUTION OF SERP ELEMENTS
+We present each element’s description and analyze its period of
+presence and positioning in SERP (also displayed on the website7).
+Moreover, we analyze each element’s evolution regarding content,
+graphics, navigation, and their relation with user interface design
+patterns. These patterns are problem-oriented and generally repeat-
+able solutions to usability problems in interface and interaction
+design [18, 57].
+4.1
+Visual identity & search statistics
+Considering Google’s visual identity, each logo version has kept its
+position and size with rare variations. Figure 3 shows the overall
+position of the logo in red. Some interfaces are exceptions, offering
+a more significant left margin and a right-shifted logo and search
+statistics. Figure 3, in gray, shows how search statistics appear
+consistently below the search query or navigation bar, either left-
+aligned, right-aligned, or justified. Statistics included the number of
+results seen per page in the first decade. Later, Google removed this
+information and kept only the estimated number of results. Details
+about the logo evolution and the content about search statistics can
+be seen online8.
+4.2
+Navigation & User Inputs
+Figures 4 and 5 display the main stages of how the search box and its
+surroundings evolved. The left-aligned query bar has also marked
+its place at the top of the page. Yet, in the early phases, it also
+appeared at the bottom, as seen in the 2000 and 2006 screenshots of
+Figure 5. The Input Prompt design pattern has always been applied
+to User Inputs.
+We notice a change in the width of the entry form after 2006,
+which may suggest an encouragement to the formulation of longer
+queries [6, 19]. This change is in line with experimental evaluations
+where query length is positively related to effectiveness in the IR
+6https://bedgarone.github.io/serpevolution/layout
+7https://bedgarone.github.io/serpevolution/elements
+8https://bedgarone.github.io/serpevolution/design
+Figure 4: SERP headers neatly from 2000, 2001, 2006, 2012,
+2017 and 2020
+Figure 5: SERP footers neatly from 2000, 2006, 2012, 2017 and
+2020
+task [6, 14]. Since query formulation influences effectiveness more
+than algorithmic factors [14], it makes sense to encourage users
+to do so. We can also notice the appearance of a way to specify
+
+800
+1024
+1366
+1920
+800
+1024
+1366
+1920
+800
+1024
+1366
+1920
+800
+1024
+1366
+1920
+600
+768
+Visual identity
+1080
+&results statistics
+800
+010241366
+1920
+1080
+Navigation&
+Organic
+Sponsored
+Features
+user inputs
+results
+resultsGoogle
+mp3 pure
+10 results
+All Languages
+GooqleSearch
+I'm Feeling Lucky
+Search Tips
+LanguageQptions
+EmailTheseResults
+Sponsored Links
+ClickheretofindMP3files.Listen,ShareandStore themforfree!
+http://www.myplay.com
+SignupTodayforamyplayLockerandget3GBofFreeStorage!
+Music - 1o0's of songs to choose from! Stop searching. Start listening:
+www.firstlook.com
+Wheretomorrow'smusicis heardfirst!Firstlook.com
+Googleresults1-10ofabout23,997formp3pure.Searchtook 0.34seconds
+Category:
+Arts>Music>Sound Files>MP3>Link Lists
+Google
+AdvancedSearch
+Preferences
+LanguageTools
+Search Tips
+Spathiphyllum
+GoogleSearch
+I'm Feeling Lucky
+Web
+Images
+GroupsDirectory
+SearchedthewebforSpathiphyllum
+Results1-10ofabout7,160.Searchtook0.10seconds
+Category:
+Science>Biology>...>Magnoliophyta>Liliopsida>Araceae>Spathiphyllum
+Sign in
+Google
+Web
+Images
+VideoNew!
+News
+Maps
+more "
+"tsunami"
+Search
+Advanced Sesrch
+Preferences
+Web
+Results 1 - 10 of about 85,900,000 for "tsunami" [definition]. (0.28 seconds)
+Search Images Videos Maps News Shopping Gmail More
+Google
+ipod touch
+Search
+About 325,000,000 results
+Advanced search
+Search Images Maps Play YouTube News Gmail Drive More
+Google
+ sparky
+α
+All
+Images
+Videos
+News
+Shopping
+Maps
+Books
+Gougle
+scoliosis
+X
+Q
+Q All
+Images
+国News
+ Videos
+ Books
+: More
+Settings
+ToolsGoo0000000ogle
+ResultPage:
+12345678910
+Next
+Space Racer
+GooqleSearch
+Search within results
+Try your query on: AltaVista Deja Excite HotBot Infoseek Lycos Yahoo!
+Google Web Directory - Cool Jobs - Advertise with Us! - Add Google to your Site - Google in your Language - All About Google
+Result Page:
+12345678910
+Next
+"tsunami"
+Search
+Search within results I Language ToolsI Search Tips I Dissatisfied? Help us improve
+Google Home -Advertising.Programs - Business Solutions -About Google
+2345
+8910
+Next
+Search Help
+Give us feedback
+Google Home
+Advertising Programs
+Business Solutions
+Privacy &Terms
+About Google
+1 2 34 5 6 78 9 10
+Next
+Advanced search
+Search Help
+Send feedback
+Google Home
+Advertising Programs
+Business Solutions
+Privacy
+Terms
+About Google
+234567
+8910
+NextThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+the query via spoken commands in the latest screenshot. This ap-
+pearance aligns with the increasing attention conversational search
+interfaces have received [2]. Although works show promising re-
+sults in incorporating visual elements in query formulation [53],
+there is no sign of them in Google Search.
+The search box and the current query are always visible for the
+searcher, as recommended [63], since 2019. This visibility occurs
+even when the user scrolls down and the top of the SERP is no
+longer visible. Before 2019, the query was always available in the
+search box, but this box disappeared if the user scrolled down.
+Because we cannot interact with past SERP versions, we cannot
+analyze interaction features such as auto-complete or auto-correct.
+In 2000, as seen in Figure 4, the main buttons for this category
+were ‘Google Search’ and ‘I’m feeling lucky’. It was possible to
+select, directly in a dropdown near the search box, how many
+results should be shown, the language intended for the results, and
+an option to send the retrieved results by email. In the screenshots
+of 2000 and 2001 presented in Figure 4, the Category Hierarchy
+feature is also visible showing one or more categories related to the
+query, probably obtained from Google Web Directory. This feature,
+available until 2004, located these categories in more general areas
+using breadcrumbs, a design pattern that linearly specifies hierarchy
+levels leading to the current subject or page [57, 58]. At the bottom
+of the interface, there was also an option to search within the results
+and links to trigger the query on other Web search engines. Other
+lesser relevant links would point to SERP experience (e.g., language,
+search tips).
+Although nonexistent for several years, it is possible to notice in
+Figure 3, in orange, the significant presence of the left navigation
+bar during some years of SERP history. This left column in orange
+is naturally interrupted in the middle area of the interface, trimmed
+as explained in Section 3.3.
+In 2001, Google introduced the tabs bar in the shape of Module
+Tabs used when content is groupable and there is no room for
+everything. Modules of content are divided into small tabbed areas
+with only one visible at a time, allowing the user to click on tabs
+to reveal other modules [57, 58]. Tabs don’t need to be literal tabs
+and don’t have to be at the top of the stack of modules [57]. The
+modules were Web, Images, Groups, and Directory. In 2002, Google
+added the News tab. In 2006, the tabs were replaced by simple links
+above the search query. This year, the Video, Maps, Froogle, and More
+tabs were introduced. In 2008, the tabs bar went to the very top of
+the page, and the Shopping and Gmail tabs were included. In 2010,
+the left sidebar was introduced, complementing the interface with
+other tabs and information such as location and results filtering. In
+2012, Google removed the bottom query bar. In 2013, the tabs bar
+was displaced underneath the main query bar, maintaining some
+links and changing others. At the same time, in the left sidebar, only
+the results’ filtering options were available before Google removed
+this sidebar in 2014. Finally, in 2015, the tabs bar below the query
+bar became the only existing one.
+There is also a usual space for sign-in and user account informa-
+tion on the right side of the screen. At the bottom of the page, two
+areas are noticeable: pagination, aligned to the center of the results
+container, and the footer, at the very end, covering all the width.
+These areas are visible in Figure 3. The Pagination design pattern
+has been applied to Navigation since the beginning of SERP. Pagi-
+nation breaks up the long results list into different SERP, loading
+them one at a time [57].
+4.3
+Organic Results
+Organic results are retrieved based on the document’s content
+and the overall retrieval algorithm. Regarding their positioning,
+in Figure 3 (blue), there is a more substantial presence of colored
+frames in the area where results are typically included (henceforth
+called results container), with a greater focus on the visible area. By
+visible area, we mean the interface area that can be seen without
+scrolling. Information that needs scrolling to be accessed is in a
+scrolling area. Over time, it is noticeable that SERP pages have
+increased their height due to the vertical decrease of color intensity,
+revealing results in lower page regions. Part of the results is slightly
+shifted to the right, referring to interfaces with a left-side navigation
+bar. It is possible to observe two other very tenuous sets. One with
+more centered results since the interface from 2010 to 2012 adjusted
+elements’ position according to the width and central axis of the
+screen rather than a left alignment. Some frames cover the entire
+interface width, not because the content was that large, but because,
+in the early days, HTML divisions (div) typically spanned across the
+complete width of the viewport because Document Object Model
+(DOM) trees were less deep. The viewport consists of the visible
+area of an interface on a display device.
+Organic results can be regular or enriched. The structure of
+regular results, seen in Figure 6, started as a basic block of the
+page title, snippet, and URL links for similar or/and cached pages
+for the result. In 2013, Google hid these links in a dropdown, only
+visible by its arrow icon until now. In 2018, Google introduced a
+link to translate the result.
+Figure 6: Regular result from 2003 (left) and 2020 (right)
+The enriched results, seen in Figure 7, are variations of regu-
+lar results, with extra elements below the title, snippet, and URL,
+giving some additional information to the user. This element can
+have greater visibility and, in turn, a higher click rate [8, 21, 47].
+It appeared for the first time in 2008, lasting until now. Its posi-
+tioning is consistently at the top of the results container. Initially,
+the extra content included two columns of site links pointing to
+sub-pages of the result’s domain. These links, named quicklinks, are
+navigational aid that attempts to take the users to the content they
+want quickly [8]. In 2009, it started highlighting structured data,
+such as reviews and ratings for products and services, based on
+experiments that showed that users find value in this new data [29].
+In 2010, Google enhanced each site link with a short description.
+In 2016, a search bar was introduced so that the user could search
+the result’s website directly from SERP.
+In both regular and enriched results, query term highlighting
+with boldfacing has been applied since 2006 to improve the usability
+of search results listings [11, 32, 38]. The dominant colors are blue
+for the title, green for the URL, and black/grey for the snippet.
+
+Holisticopia:WyomingMassageTherapy
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+r/Coronavirus: In December 2019,a novel coronavirus strain (SARS-CoV-2)emergec
+city of Wuhan, China. This subreddit seeks to monitortheCHIIR’23, March 19–23, 2023, Austin, TX, USA
+Bruno Oliveira and Carla Teixeira Lopes
+Figure 7: Enriched result from 2009 (left) and 2016 (right)
+About 2011, the URL moved above the snippet. Google underlined
+the title until 2014. In 2020, the URL changed its color to gray and
+moved above the title. The new position of the URL may have
+been influenced by the importance of URL and domain names in
+evaluating the credibility of a result [34, 50]. In 2019, the URL started
+being displayed as a breadcrumb, which may have been motivated
+by research that concluded that long and complex URL negatively
+impact clickthroughs [11].
+In both types of organic results, the snippet length has not visibly
+changed, with most of the snippets having two lines, the suggested
+size for informing the user, and including as many results as pos-
+sible in the visible area [63]. As most of the queries used for data
+collection are associated with informational tasks, we cannot con-
+clude if Google adjusts the length of the snippet to the type of
+query (e.g., informational, navigational) as suggested by existing
+research [15].
+4.4
+Sponsored Results
+As seen in Figure 8, textual ads are short advertisements that
+appear alongside organic results. These entries are focused on com-
+mercial intent, combining relevance with revenue, and are usually
+manually crafted, a standard in the advertisement industry [16].
+This element has been present in SERP since the first day, lasting
+until now. This element’s content and evolution are identical to the
+ones of Regular Results, except these were initially marked with a
+specific tag, ‘Sponsored link’. In 2012, some of the results started
+to include quicklinks, although less noticeable than in Enriched re-
+sults. Until 2014, a different background color distinguished these
+elements from organic results. Scarce occurrences in 2017 also had
+a colored background. In 2014, the tag ‘Ad’ substituted the previous
+one, having a yellow background color, while from 2016 onwards,
+it was green. In 2020, some ads could have the shape of an actual
+Enriched result. The URL also changed its color to gray and moved
+above the title.
+Figure 8: Textual ads from 2002 (top), 2014 (left) and 2020
+(right)
+The other type of ads correspond to shopping content and are,
+therefore, called shopping ads, seen in Figure 9. These are acti-
+vated when the query is commercial [47]. These are very striking
+results, listing various information about each product. This el-
+ement appeared for the first time in 2013, lasting until now. Its
+positioning is usually at the top of the results container and, more
+frequently, at the right of that container. Shopping ads used to be
+exclusive to the right sidebar, displaying one to four results in a
+matrix. In 2018, each result was embedded in an individual card.
+Until 2019, query terms were highlighted in bold. In 2020, this ele-
+ment appeared in the results container in a carousel of cards with
+more width and less height. The Cards design pattern has been
+applied to this element since 2019, and the Carousel since 2020.
+Cards display content composed of distinct parts, generally about
+a single subject, to form one coherent piece of content designed
+to expose information efficiently [56, 57]. It is usual for cards to
+accompany other cards, carrying similar content but addressing
+different subjects. The Carousel is a horizontal strip of simple cards,
+letting the user scroll horizontally to view them and encouraging
+the inspection of the following items [57, 58]. Since the element’s
+beginning, the Thumbnail Grid has been applied, enabling a quick
+overview of images by shrinking the original ones [51, 57].
+In Figure 3, it is possible to identify two major advertisement
+areas in SERP: top ads and right ads, following a typical SERP
+layout [5, p. 489]. Elements in the right sidebar move horizontally
+over time because some interfaces force this sidebar to be responsive
+to the screen’s width. Top ads are the most common positions
+for ads within a soft vertical variation. Older interfaces placed
+advertisements on a fully colored div that occupied almost the
+entire viewport width for the same reason described in Section 4.3.
+Compared with regular results and other features, we can see in
+Figure 3 that sponsored results are more concentrated in the visible
+area, probably motivated by the fact that searchers rarely scroll [63].
+Figure 9: Shopping ads from 2013 (left), 2019 (center) and
+2020 (right)
+4.5
+Features
+SERP features complement organic and sponsored results, attempt-
+ing to provide answers to the query without just pointing to web-
+sites that might deliver that information.
+Figure 3, in green, shows features spread around the interface,
+mainly in the visible area and upper half of the scrolling area. Many
+features are similar to regular results but with more significant
+height. These features also share a place in the sidebar with adver-
+tisements, recently becoming more present than the latter. Most
+frames with large height values correspond to the well-known
+Knowledge Panel. A horizontal area is noticeable, generally assigned
+to the Carousel and the Carousel Grid.
+
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+GecnnaTHa A..The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+Due to space limitations, we selected a subset of features based
+on their lifetime and distinctiveness to be analyzed in detail. The
+Video Pack feature will not be considered for its similarity with the
+Image Pack. The same happens with Direct Answer Results, Local
+Pack, People Also Ask, Carousel, Carousel Grid, Recipe Cards, Twitter
+Pack, Category Hierarchy, and Covid-19 Left Panel for their shorter
+presence. More information about these features is available in
+another paper [45] and online9.
+Featured Snippets, seen in Figure 10 are answer boxes in which
+Google responds to a question-related query based on information
+taken from a page [59]. This element appeared for the first time in
+2016, lasting until now. Its positioning is consistently at the top of
+the results container. Although these snippets resemble enriched
+results, they differ in the type and position of extra content. In
+featured snippets, the additional content appears before the result’s
+title and not after, as happens in enriched results. The content of a
+featured snippet was initially made of a short paragraph with an-
+swering information. It evolved to a general layout, used until now,
+consisting of a larger paragraph, a thumbnail at the upper-right
+corner, and the title and link to the information’s source, to where
+it is possible to navigate. The answer also started to be returned as
+an ordered list or table. Instead of a thumbnail, a video or a carousel
+of images may also accompany it. In 2018, when available, Google
+introduced the date of the source’s publication after the information
+paragraph. The underlined style of the title was removed in 2017,
+and the green URL was changed to a gray breadcrumb URL in 2020,
+as in the organic and sponsored results. Text styling is used in both
+paragraph and title, enhancing relevant words in bold. Studies have
+found that these snippets help make more accurate decisions when
+containing the correct information [7].
+Figure 10: Featured Snippets from 2016 (top), 2017 (left) and
+2020 (right)
+The Knowledge Panel, seen in Figure 11, is perhaps the high-
+light of SERP features. It is a dynamic feature that provides direct
+information in various formats within the same panel, pointing to
+related content. The contents range from text to images, ratings,
+social profiles, factual information, and similar search topics [47],
+helping the user to understand a particular subject quickly and
+facilitating a more in-depth search [17]. This element appeared for
+the first time in 2014, lasting until now. Its positioning is always
+at the right of the results container. The basic structure consists
+of a panel with a top thumbnail of the subject, vertically followed
+by a title, a website link if applicable, a resume paragraph usually
+by Wikipedia, a structured list of direct information, and a block
+9https://bedgarone.github.io/serpevolution/elements
+of People also search for. During the following years, Google intro-
+duced other content highly dependable on the search topic and
+variable in coverage and quality [37]. It is common to have this
+panel populated by Wikipedia information [60]. The graphics of
+this element was stable over time, following the improvements
+in Google’s interface design. The Grid of Equals and Thumbnail
+Grid design patterns have been applied to this element since its
+beginning. Grid of Equals is a pattern to display items in a grid or
+matrix, each following a standard template, linking to respective
+pages [58]. The Cards, Carousel, and Module Tabs design patterns
+were applied to this element in 2018.
+Figure 11: Knowledge Panel from 2016 (left) and 2018 (right)
+The Image Pack, seen in Figure 12, presents a set of images
+taken from various sources in Google’s index in searches where
+visual content is valuable [59]. This element appeared for the first
+time in 2006, but only after 2010 did it frequently appear, lasting
+until now. Its positioning is highly variable throughout the results
+container. Most of the time, the content was a title associating
+images with the search query and a block of image thumbnails. In
+2014, Google included a link for ‘more images’ and a link to report
+pictures. In 2019, Google introduced a bar of image categories. It
+could have simple buttons or buttons with a thumbnail associated
+with its category. The graphics started with a considerable presence
+of blue colors, typical in Google’s early interfaces when images
+had a blue border. In 2014, Google removed this border, but the
+main change was in 2019 when images started to be in a carousel.
+In 2018, the layout of a matrix appeared for the first time. These
+changes increased the element’s area in the last two years. In 2020
+the title, as usual in most elements, turned dark gray. The carousel
+of image categories changed its shape to a line that expands to a
+matrix in the form of progressive disclosure using the Collapsible
+Panels pattern. The Grid of Equals design pattern has been applied
+to this element since 2018, while Thumbnail Grid, naturally, since
+its beginning. Arguello et al. [4] examined vertical results, such as
+images, in aggregated search. They found that these results had
+more clicks in more complex tasks and that users were divided in
+their preferences for vertical search displays.
+
+Kourtney Kardashian and Scott Disick were partners for 9 years (until 2015)
+Kourtney Kardashian produced and Scott Disick appears on Keeping Up with the
+Kardashians
+Closing ceremony to celebrate Brazil 2014 in
+style. Before the 2014 FIFA World Cup TM Final
+gets underway at the Maracana in Rio de
+Janeiro on Sunday, a special closing ceremony
+involvingaround1,ooopeoplewillcelebratethe
+sport as the tournament nears its unmissable
+climax .
+Space architecture, in its simplest definition, is the theory and practice of designing
+and building inhabited environments in outer space. ... Space architecture borrows
+Closing ceremony to celebrate Brazil 2014 in style - FIFA.com
+from multiple forms of niche architecture to accomplish the task of ensuring human
+2014-in-stvle-2404018.btml
+beings can live and work in space
+en.wikipedia.org>wiki>Space_architecture
+Space architecture - WikipediaCNN
+Cable channel - cnn.com
+The Cable News Network is
+an American basic cable
+and satellite television channel that is
+owned by the Turner Broadcasting System
+news channel was founded in 1980 by
+division of Time Warner. The 24-hour cable
+American media proprietor Ted Turner.
+images
+Wikipedia
+ScottGreenstein
+Customer service: 1 (404) 827-1500
+Headquarters: Atlanta, GA
+Film producer
+Founder: Ted Turner
+Founded: June 1, 1980, Atlanta, GA
+Scott Greenstein is president and chief content
+Parent organization: Turner
+Broadcasting System
+officer of Sirius XM Radio. He leads the programming
+and advertising sales of the largest radio company
+Profiles
+by revenue and one of the largest subscription media
+f
+in
+companies in the worid. Wikipedia
+Twitter
+Linkedin
+Born: 1959 (age 61 years), Freehold, NJ
+Facebook
+Employer: SiriusXM Satellite Radio
+TV shows
+Movies: The Enqlish Patient
+CW
+NEWDAY
+Profiles
+Live
+Twitter
+CNN Live
+Anthony
+New Day
+Today
+Parts Unk..
+Bourdain
+Sinoe 2013
+2001  2006
+People also searchfor
+Sinoe 2013
+View 5+ more
+riusxm!
+People also search for
+BBC
+NEWS
+James E.
+Mel
+Barry Diller
+saul
+Meyer
+Karmazin
+Zaentz
+BBC
+MSNBC
+ESPN
+ Claim this knowiedge panel
+FeedbackCHIIR’23, March 19–23, 2023, Austin, TX, USA
+Bruno Oliveira and Carla Teixeira Lopes
+Figure 12: Image Pack from 2010 (left) and 2020 (right)
+The Top Stories, seen in Figure 13, are blocks of three or more
+recent news considered relevant to the query, recently placed in
+the form of a carousel [47]. Each story is now presented with a
+thumbnail, publisher, and timestamp. This element appeared for the
+first time in 2004, but only after 2011 it started to appear frequently,
+lasting until now. Its positioning is mainly in the visible area of the
+results container. The element’s content started with a vertical list
+of at most four news titles, each followed by the source’s name and
+how long ago it was published. In 2006, a journal icon was placed at
+the left of the list, and a link to ‘today’s top stories’ was introduced.
+In 2013, the icon was substituted by a thumbnail for the first news
+result, being the most important news in the element. The latter was
+complemented with an extract of the news, while the rest stayed the
+same. In 2020, leading to a considerable increase in the element’s
+area, the graphics was majorly altered to display the results in
+a carousel of cards. However, the content was simplified to only
+present, for each result, a thumbnail, title, source, and how long ago
+it was published. As usual, the color scheme was mainly blueish
+and filled with blue borders and underlines. These were removed
+after 2020 with the softer gray colors for additional information
+but still blue titles. The Streams and Feeds design pattern has been
+applied to this element since its beginning. It defines a pattern to
+list time-sensitive items chronologically, combining the sources in
+one place [58]. However, in this element, the relation to the query
+appears to be more relevant than publication time since it is no
+longer possible to observe any chronological order. The Cards and
+Carousel design patterns have been applied since 2020.
+Figure 13: Top Stories from 2006 (left-top), 2013 (left-bottom)
+and 2020 (right)
+Related Searches, seen in Figure 14, is a common element on
+SERP pages from a very early age and offers suggestions for related
+searches, i.e., queries that are in some way related to the current
+query and may be good candidates for follow-on queries. These
+suggestions can be helpful to support exploration or provide query
+statements that express information needs in different ways [62].
+Usually, these suggestions are generated based on search log data,
+either picking queries that frequently follow the current query [28]
+or clustering queries based on results’ clicking [13]. Each link takes
+the user to the respective SERP. This element appeared for the first
+time in 2008, lasting until now, except for 2010. Its positioning is
+always at the bottom of the results container. The content was
+diversified regarding how many suggestions would appear and its
+layout. Each suggestion of search is a hyperlinked title pointing to
+its respective SERP. Initially, it was organized in a matrix of columns.
+In 2011, Google reduced this schema to two columns, which could
+be displayed in just one column for suggestions with longer text.
+Until mid-2020, suggestions were blue and, until 2014, underlined.
+Google applied a search icon to each entry in 2020. Later, a new
+version changed the graphics, making each entry a button with a
+solid gray background, a search icon, and a title in black. This latter
+change contributed to a recent increase in the element’s average
+area.
+Figure 14: Related Searches from 2008 (left - top), 2017 (left
+- bottom) and 2020 (right)
+5
+AGGREGATED ANALYSIS
+This section analyzes SERP’s evolution from an aggregated per-
+spective.
+5.1
+Elements’ lifetime
+As shown, SERP have always had a large variety of elements, each
+with its evolution and active periods, as described in Section 4. Ac-
+cording to the framework proposed by Wilson [63], input features
+were analyzed in Section 4.2, with the search box being the main
+one. To support control, besides the input features, we also iden-
+tified the related searches feature. All the other analyzed features
+are informational. It is important to note that, given the nature of
+this study, it was not possible to analyze dynamic features such as
+interactive querying or personalizable features.
+Black cells in Table 2 identify the years we detected elements
+in our dataset. For all years in-between black cells, we have con-
+ducted a manual search in other sources to avoid false negatives.
+In these cases, we have searched specific websites10 for evidence
+of elements in such years. If found, we paint the cell grey. It is
+noticeable how SERP features have emerged in the last decade,
+contributing to a matrix full of element possibilities in recent years.
+Almost every SERP element includes well-known design patterns.
+A visual timeline with screenshots of these elements’ presence in
+SERP is available online11.
+5.2
+Design pattern application
+Table 3 maps each element with the patterns proposed by Tidwell
+et al. [57], along with the start date of that appliance. Older SERP
+elements make later use of design patterns for individual improve-
+ment. In contrast, some contemporary elements may have arisen
+10https://searchengineland.com, https://googlesystem.blogspot.com and Internet
+Archive
+11https://bedgarone.github.io/serpevolution/timeline/2010
+
+ImagestorParisweather
+paris france
+weather forecast
+climate
+temperature
+eiffel tower
+average rainfall
+accuwea
+Images for"complex event processing"
+<
+<国Topstories
+News results for"american idol" - View today's top stories
+'ldol Winner Hicks Sues Former Producer - Washington Post - 9 hours ago
+LIVE
+LIVE
+LIVE
+'ldolFinalist OK After Vegas Robbery - FOX News - 14 hours ago
+Life after 'ldol' - MLive.com - 19 hours ago
+NIA
+LIVECOVERAGE
+ CNN.com
+News forfacebook
+Che New ork Cimes
+OCBSNEWS
+Facebook Surpasses $30_Ahead of Q4 Earnings
+Electionresults2020:
+Live Trump vs Biden
+Pennsylvania2020
+Wall Street Journal (blog) - 1 hour ago - 239 related articles 
+LivenewsonTrump
+Tracker: Presidential
+election results
+Shares of Facebook Inc. breached the $30 mark on Wednesday, the first
+Biden and electoral
+Election
+time in six months.
+votes
+CBC.ca
+As Facebook shares hit $30, ETFs with holdingS_gain
+MarketWatch (blog) - 2 hours ago - 2 related articles 
+10 mins ago
+24mins ago
+19 mins ago
+Facebook rolling out Timeline changes, including redesign
+Chicago Tribune - 8 hours ago - 28 related articles 》
+<
+7
+ViewallSearches related to: "logo design"
+Related searches
+online logo design
+design your own logo
+logo design tips
+logo design ideas
+who was prince henry the
+Q
+what did prince henry
+navigator
+discover
+a
+prince henry the navigator
+a
+how did prince henry the
+Searches related to amazon diversity
+school
+navigator die
+amazon diversity officer
+amazon black employee network
+a
+prince henry the navigator facts
+a
+prince henry the navigator fun
+amazon diversity 2016
+amazon workforcelogin
+facts
+amazon diversity report 2016
+amazon supplier diversity program
+a
+prince henry the navigator
+prince henry the navigator
+amazon supplier diversity registration
+amazonworkforcephonenumber
+accomplishments
+timelineThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+Table 2: Presence of SERP elements from 2000 to 2020. In black the years in which the element appeared in our dataset. In
+grey are the years in which the element’s existence is documented elsewhere.
+2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
+Visual Identity
+Search Statistics
+Navigation
+User Inputs
+Regular Results
+Enriched Results
+Textual Ads
+Shopping Ads
+Knowledge Panel
+Featured Snippets
+Direct Answer
+Local Pack
+Image Pack
+Video Pack
+Top Stories
+Carousel
+Carousel Grid
+People Also Ask
+Related Searches
+Twitter Pack
+Recipe Cards
+Category Hierarchy
+Covid-19 Left Panel
+from the need to apply a design pattern solution whose traces are
+evident from the element’s beginning. The website12 lists design
+patterns with images and elements that use them.
+5.3
+Highlights
+We identify the most relevant changes along the upper part of a
+two-decade timeline of SERP in Figures 15 and 16. These changes
+correspond to the entry of new elements or significant changes
+in input and navigation options. This timeline of changes is also
+available online13, enhanced with images and the option to filter
+the entries for navigation changes or element additions.
+As stated before, the first stage of the analysis used a visual
+selection from a monthly sample from the dataset. Figures 15 and
+16 also display, in their lower part, how Google’s SERP interfaces
+evolved in terms of design. In these timelines, we include the main
+versions of the SERP interface based on significant changes. Apart
+from how the overall interfaces visually evolved, similar timelines
+about visual identity, search statistics, and navigation are available
+on the website14.
+Google’s initial interfaces had few depth levels in their HTML
+DOM. The first interface design, traced in 2000, differentiated the
+sponsored results with a colored background. A second search
+12https://bedgarone.github.io/serpevolution/patterns
+13https://bedgarone.github.io/serpevolution/timeline
+14https://bedgarone.github.io/serpevolution/design
+query bar existed at the bottom of the page, and the user could
+change the number of results presented. Google removed this bar in
+the following interfaces. The one traced from 2000 to 2004 revealed
+a right block of results, exclusive to sponsored ones. It marked the
+appearance of the first bar with tabs directing the user to other
+types of content (e.g., images and news). The fourth interface design,
+traced from 2010 to 2012, was not left-oriented but varied in a spaced
+manner depending on the screen’s width. It introduced a sidebar
+on the left, containing tabs to manage the results, but some of these
+tabs were duplicated due to the navbar’s tabs bar mentioned in
+Section 4.2.
+Significant aesthetic changes occurred in 2012. The fifth interface
+design relates to the launch of the Knowledge Graph, with the
+right column being divided between it and sponsored results. Some
+modifications were found earlier in the dataset during those years,
+the sixth interface design. However, a design close to the current
+one began at the end of 2018, the seventh interface design. As noted
+in some elements’ graphics, this interface focused on modernizing
+its lines.
+5.4
+User interface area
+We calculated the area of all screenshots in the dataset to analyze its
+evolution over time. Figure 17 shows the development of interface
+area per month (dots) and per year (line), measured in pixels. Each
+entry in the chart corresponds to the average area per month for all
+
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+Bruno Oliveira and Carla Teixeira Lopes
+Table 3: Design Patterns and time of their appliance to SERP elements. Cells without a dash after the year represent single
+years.
+Organizing Navigation
+Layout
+Lists
+Input
+Streams
+and Feeds
+Bread-
+crumbs
+Grid of
+Equals
+Module
+Tabs
+Accor-
+dion
+Collapsi-
+ble
+Panels
+Cards
+Thumb-
+nail
+Grid
+Carousel
+Pagina-
+tion
+Input
+Prompt
+Visual Identity
+Search Statistics
+Navigation
+2000-2020
+User Inputs
+2000-2020
+Regular Results
+2019-
+Enriched Results
+2020-
+2020-
+Textual Ads
+Shopping Ads
+2019-
+2020-
+Knowledge Panel
+2014-
+2018
+2018
+2014-
+2016-
+Featured Snippets
+2016-
+2020-
+Direct Answer
+2018-
+2018-
+Local Pack
+Image Pack
+2018-
+2019-
+2006-
+2019-
+Video Pack
+2015-
+2015-
+Top Stories
+2004-
+2020-
+2020-
+Carousel
+2015-
+2016-
+2015-
+2015-
+Carousel Grid
+2017-
+2017-
+2017-
+People Also Ask
+2016-
+Related Searches
+Twitter Pack
+2017-
+2017-
+Recipe Cards
+2020-
+2020-
+Category Hierarchy
+2000-2004
+Covid-19 Left Panel
+Right sidebar with sponsored results
+Top Stories introduced
+SERP tabs bar introduced (Images, Groups, Directory) 
+Tabs bar placed above the query bar
+Image Pack introduced
+Left sidebar introduced
+Local Pack introduced
+Related Searches introduced
+Tabs bar placed on navbar
+News tab added
+Videos and Maps tabs added
+Shopping and Gmail tabs added
+2010
+2009
+2008
+2007
+2006
+2005
+2004
+2003
+2002
+2001
+2000
+Figure 15: Highlights of SERP overall evolution (top) and Interfaces’ visual evolution (bottom) from 2000 to 2010
+captures in the dataset. Months without values are months without
+captures in the dataset, as indicated in Table 1. Results show an
+increase close to exponential due to the appearance of SERP fea-
+tures that have added extra content to SERP, thus, making them
+
+Googlem
+GoogleThe Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+Shopping ads introduced
+Bottom query bar removed
+Carousel introduced
+Left sidebar removed
+Knowledge panel introduced
+Ads loose their colored background
+Tabs bar placed under search query
+Direct Answers introduced
+Feature Snippets introduced
+Covid-19 left panel
+2017
+2016
+2015
+2014
+2013
+2012
+2011
+2018
+2019
+2020
+Figure 16: Highlights of SERP overall evolution (top) and Interfaces’ visual evolution (bottom) from 2011 to 2020
+more extensive over time. We can also notice this evolution in the
+animated overlaying of elements on the website15.
+px (103)
+Figure 17: Interface area evolution, measured in squared
+pixel units
+5.5
+Files’ size and number
+We made a similar approach to study the variation in file size re-
+garding the entire dataset. Figure 18 represents the SERP captures’
+file size evolution and the number of files in its folder. For each
+capture, we summed the size of each associated file and averaged it
+per month and, consequently, year. We did not consider embedded
+files in the HTML but files linked through the associated files folder.
+Size results accompany the development of the interface area,
+as seen in Figure 17, expressing a steep rise in the last few years.
+This increase cannot be related to a surge in the associated files,
+as later values share similar values with the first years of SERP
+existence. Nevertheless, results suggest that SERP sought to reduce
+the number of related files, achieving this aim during the first
+decade. This number started to rise again because interfaces evolved
+15Available at https://bedgarone.github.io/serpevolution/layout
+and demanded more images and graphics, which can increase the
+files needed to load a SERP. Besides, protocol advancements such
+as HTTP pipelining or SPDY may have contributed to the increase
+of these associated assets.
+Bytes (103)
+Average nr. of fles
+Figure 18: Source code size (left - line - circumference),
+Source code + Files folder size (left - dashed - plus sign) and
+average number of files associated (right) over time
+6
+DISCUSSION
+SERP are no longer pages with “10 blue links”. We have shown that
+the range of SERP elements has been growing continuously but
+steeper in the last few years. Although this increase might con-
+tribute to cluttering the page and falling into the “more is less” trap,
+we expect SERP to continue evolving quickly [5, p. 512]. Interfaces
+have kept track of web development’s evolution, as shown by the
+regular adoption of design patterns.
+The Category Hierarchy stands out as the only feature used in the
+first years and later discontinued. The decrease in web directories’
+popularity may have been the reason for this.
+Although organic results have been keeping a relatively stable
+format, SERP have become more diversified over time, providing
+increasingly sophisticated navigational aids to enhance results in-
+terpretation, relevance assessment, and user satisfaction [8, 21, 54].
+
+Goole m
+Google
+cnn
+sgin
+at Archi
+YouTube
+NNG
+Z>
+Gogle
+G
+CNN
+ND
+CNN
+The 10 Best Paris Tou
+CNN(@CN)ITVCHIIR’23, March 19–23, 2023, Austin, TX, USA
+Bruno Oliveira and Carla Teixeira Lopes
+These features complement organic and sponsored results and at-
+tempt to infer users’ needs and quickly satisfy them even if not ex-
+plicitly mentioned in their search queries [8] following a “universal
+search” vision [39]. As shown, most SERP elements are informa-
+tional, providing information about results.
+The growth in SERP elements almost exponentially increased
+interface area since modern pages need more vertical space. These
+pages are getting heavier. Surprisingly, given the number of current
+features that use non-textual content, the average number of files
+is not much more significant than in 2000. Notwithstanding, we
+noticed a higher dispersion in the average number of files in more
+recent years, as seen in Figure 18.
+Aggregating results from heterogeneous sources - verticals - and
+presenting them in a single interface – aggregated search – has
+become standard practice [66]. We could notice this in the Image
+Pack, Video Pack, Local Pack, and Top Stories features. Other features
+like the Featured Snippet assemble information in different formats
+extracted from one source. Since 2020, information on the Web
+has dramatically increased in quantity and diversity. Videos are
+nowadays much more popular than they were at the beginning
+of the century, and content from new platforms such as location
+technology (e.g., Google Maps) or social media (e.g., Twitter) has
+emerged. This evolution naturally affected the need for the change
+of the SERP elements. The fact that graphical information is pro-
+cessed before the textual information [23] might also explain the
+recurrent appearance of images and videos in features. On the other
+hand, the evolution of SERP is strongly informed and influenced
+by users’ search behavior. Users scarcely look at results other than
+the first ones [26] and tend to reformulate the query if they cannot
+find promising results at the top of the list [20]. They rarely look at
+results, such as videos or news, in their respective SERP ‘tabs’ [52].
+This behavior may also motivate search engines to include aggre-
+gations of other types of results [23], not for the diversity, because
+most users will not reach them outside the first results page.
+The SERP are also affected by the interests of the search engine
+providers who provide users not only with relevant results but
+also with results of their interest. This reality gained prominence
+when the European Commission concluded that Google abused its
+market dominance by the way it presented sponsored results [12].
+Although not a focus of analysis of this work, it is frequent to see
+Google showing results from its maps service, YouTube results in
+the video container, shopping results from its shopping ads service,
+and blurring the lines between organic and sponsored results. These
+decisions have a higher impact on users with less search engine
+knowledge, who are more likely to trust and use Google [49].
+SERP features often allow the user to interact with the contents
+of a web page directly from the SERP [21, 43]. This cannibalizes
+clicks [9] and might mean that users get satisfied without clicking
+on search results, which was defined by Li et al. [33] as “good
+abandonment”. Studies have found that features that provide direct
+answers improve user engagement on SERP, reduce user effort,
+and promote user satisfaction [64]. Besides contributing to user
+satisfaction, these features also encourage user engagement and,
+thus, revenue [21, 31].
+Our results reinforce the idea that evaluation measures solely
+based on the list of “10 blue links” must be rethought based on
+the SERP we have today. The standard practices of aggregating
+results from heterogeneous verticals and including features that
+provide direct answers on SERP have implications for how users
+interact with search systems and, therefore, on their evaluation.
+The cannibalization of clicks requests evaluations that consider
+other types of interactions with the SERP. Challenges emerge in the
+way users’ feedback is explored, either explicitly from user studies
+or implicitly from weblogs. Work has already been conducted to
+rethink evaluation in the context of aggregated search pages [66]
+and good abandonment scenarios [30].
+7
+CONCLUSIONS AND FUTURE WORK
+Using Google as a case study, we studied how SERP user interfaces
+evolved over two decades. While existing research has relied on
+the actual states of these interfaces, we have updated and improved
+the analysis with an evolution perspective, addressing old and new
+elements, their positioning, size, and patterns. We extracted and
+provide a dataset with 5,000+ SERP captures, including HTML
+versions and screenshots.
+We showed that SERP are becoming more diverse in terms of
+elements, aggregating content from different verticals and including
+more features that provide direct answers. These changes affect
+user behavior that, more often, abandon the page satisfied, the
+so-called “good abandonment”.
+In the future, we want to analyze other web search engines’ SERP
+and compare results. We would also like to explore the evolution
+of SERP in mobile environments. Here, we would like to know if
+the increase in the SERP area found in this work results in a more
+significant differentiation of user interfaces between desktop and
+mobile environments. As stated previously, features that require
+interaction with the SERP were not analyzed here because our page
+captures from the Internet Archive don’t allow such interaction.
+Given the importance of such features, we would like to explore
+them in current SERP versions. Studies that analyze SERP charac-
+teristics by types of queries (e.g., informational, navigational) and
+user studies comparing old and contemporary SERP would also be
+interesting.
+ACKNOWLEDGMENTS
+The Master in Informatics and Computing Engineering and the
+Department of Informatics Engineering of the Faculty of Engineer-
+ing of the University of Porto supported this work by funding the
+registration fee.
+REFERENCES
+[1] Khalil Amjad and Fadi Alrub. 2013. A comparison of search engine’s features and
+mechanisms. Advanced Database Systems 2131-8070580-sec1 (December 2013),
+1–8.
+[2] Avishek Anand, Lawrence Cavedon, Hideo Joho, Mark Sanderson, and Benno
+Stein. 2020. Conversational Search (Dagstuhl Seminar 19461). Dagstuhl Reports
+9, 11 (2020), 34–83. https://doi.org/10.4230/DagRep.9.11.34
+[3] The Internet Archive. 2021. The Wayback Machine’s First Crawl 1996. https:
+//archive.org/details/wayback-machine-1996.
+[4] Jaime Arguello, Wan-Ching Wu, Diane Kelly, and Ashlee Edwards. 2012. Task
+Complexity, Vertical Display and User Interaction in Aggregated Search. In
+Proceedings of the 35th International ACM SIGIR Conference on Research and
+Development in Information Retrieval (Portland, Oregon, USA) (SIGIR ’12). As-
+sociation for Computing Machinery, New York, NY, USA, 435–444.
+https:
+//doi.org/10.1145/2348283.2348343
+[5] Ricardo Baeza-Yates and Berthier Ribeiro-Neto. [n.d.]. Modern Information Re-
+trieval: The Concepts and Technology behind Search (2nd ed.). Addison-Wesley
+Publishing Company, USA.
+
+The Evolution of Web Search User Interfaces - An Archaeological Analysis of Google SERP
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+[6] N. J. Belkin, D. Kelly, G. Kim, J.-Y. Kim, H.-J. Lee, G. Muresan, M.-C. Tang, X.-J.
+Yuan, and C. Cool. 2003. Query Length in Interactive Information Retrieval. In
+Proceedings of the 26th Annual International ACM SIGIR Conference on Research and
+Development in Informaion Retrieval (Toronto, Canada) (SIGIR ’03). Association
+for Computing Machinery, New York, NY, USA, 205–212.
+https://doi.org/10.
+1145/860435.860474
+[7] Markus Bink, Steven Zimmerman, and David Elsweiler. 2022. Featured Snippets
+and Their Influence on Users’ Credibility Judgements. In ACM SIGIR Conference
+on Human Information Interaction and Retrieval (Regensburg, Germany) (CHIIR
+’22). Association for Computing Machinery, New York, NY, USA, 113–122. https:
+//doi.org/10.1145/3498366.3505766
+[8] Deepayan Chakrabarti, Ravi Kumar, and Kunal Punera. 2009. Quicklink Selection
+for Navigational Query Results. In Proceedings of the 18th International Conference
+on World Wide Web (Madrid, Spain) (WWW ’09). Association for Computing Ma-
+chinery, New York, NY, USA, 391–400. https://doi.org/10.1145/1526709.1526762
+[9] Lydia B. Chilton and Jaime Teevan. 2011. Addressing People’s Information Needs
+Directly in a Web Search Result Page. In Proceedings of the 20th International
+Conference on World Wide Web (Hyderabad, India) (WWW ’11). Association for
+Computing Machinery, New York, NY, USA, 27–36.
+https://doi.org/10.1145/
+1963405.1963413
+[10] Alex Chris. 2020. Top 10 Search Engines In The World (2021 Update). https:
+//www.reliablesoft.net/top-10-search-engines-in-the-world/.
+[11] Charles L. A. Clarke, Eugene Agichtein, Susan Dumais, and Ryen W. White. 2007.
+The Influence of Caption Features on Clickthrough Patterns in Web Search. In
+Proceedings of the 30th Annual International ACM SIGIR Conference on Research
+and Development in Information Retrieval (Amsterdam, The Netherlands) (SIGIR
+’07). Association for Computing Machinery, New York, NY, USA, 135–142. https:
+//doi.org/10.1145/1277741.1277767
+[12] European Commission. 2017. Antitrust: Commission fines Google €2.42 billion
+for abusing dominance as search engine by giving illegal advantage to own
+comparison shopping service. https://ec.europa.eu/commission/presscorner/
+detail/en/IP_17_1784.
+[13] Nick Craswell and Martin Szummer. 2007. Random Walks on the Click Graph. In
+Proceedings of the 30th Annual International ACM SIGIR Conference on Research
+and Development in Information Retrieval (Amsterdam, The Netherlands) (SIGIR
+’07). Association for Computing Machinery, New York, NY, USA, 239–246. https:
+//doi.org/10.1145/1277741.1277784
+[14] J. Shane Culpepper, Guglielmo Faggioli, Nicola Ferro, and Oren Kurland. 2021.
+Topic Difficulty: Collection and Query Formulation Effects. ACM Trans. Inf. Syst.
+40, 1, Article 19 (sep 2021), 36 pages. https://doi.org/10.1145/3470563
+[15] Edward Cutrell and Zhiwei Guan. 2007. What Are You Looking for? An Eye-
+Tracking Study of Information Usage in Web Search. In Proceedings of the SIGCHI
+Conference on Human Factors in Computing Systems (San Jose, California, USA)
+(CHI ’07). Association for Computing Machinery, New York, NY, USA, 407–416.
+https://doi.org/10.1145/1240624.1240690
+[16] Cristian Danescu-Niculescu-Mizil, Andrei Z. Broder, Evgeniy Gabrilovich, Vanja
+Josifovski, and Bo Pang. 2010. Competing for Users’ Attention: On the Inter-
+play between Organic and Sponsored Search Results. In Proceedings of the 19th
+International Conference on World Wide Web (Raleigh, North Carolina, USA)
+(WWW ’10). Association for Computing Machinery, New York, NY, USA, 291–300.
+https://doi.org/10.1145/1772690.1772721
+[17] Danny Sullivan. 2020. Google’s Knowledge Graph and Knowledge Panels. https://
+blog.google/products/search/about-knowledge-graph-and-knowledge-panels/.
+[18] Eelke. 2012.
+Interaction Design Patterns.
+https://www.interaction-design.
+org/literature/book/the-glossary-of-human-computer-interaction/interaction-
+design-patterns.
+[19] Kristofer Franzén and Jussi Karlgren. 2000. Verbosity and Interface Design. Techni-
+cal Report. Human Computer Interaction and Language Engineering Laboratory.
+[20] Zhiwei Guan and Edward Cutrell. 2007. An Eye Tracking Study of the Effect of
+Target Rank on Web Search. In Proceedings of the SIGCHI Conference on Human
+Factors in Computing Systems (San Jose, California, USA) (CHI ’07). Association
+for Computing Machinery, New York, NY, USA, 417–420.
+https://doi.org/10.
+1145/1240624.1240691
+[21] Kevin Haas, Peter Mika, Paul Tarjan, and Roi Blanco. 2011. Enhanced Results
+for Web Search. In Proceedings of the 34th International ACM SIGIR Conference on
+Research and Development in Information Retrieval (Beijing, China) (SIGIR ’11).
+Association for Computing Machinery, New York, NY, USA, 725–734.
+https:
+//doi.org/10.1145/2009916.2010014
+[22] Marti A. Hearst. 2009. Search User Interfaces (1st ed.). Cambridge University
+Press, USA.
+[23] Nadine Höchstötter and Dirk Lewandowski. 2009. What users see - Structures
+in search engine results pages. Information Sciences 179, 12 (2009), 1796–1812.
+https://doi.org/10.1016/j.ins.2009.01.028
+[24] Bernard J. Jansen, Amanda Spink, and Sherry Koshman. 2007. Web Searcher
+Interaction with the Dogpile.com Metasearch Engine. J. Am. Soc. Inf. Sci. Technol.
+58, 5 (mar 2007), 744–755.
+[25] Bernard J. Jansen, Amanda Spink, and Jan Pedersen. 2005. A temporal comparison
+of AltaVista Web searching. Journal of the American Society for Information Science
+and Technology 56, 6 (2005), 559–570. https://doi.org/10.1002/asi.20145
+[26] Thorsten Joachims, Laura Granka, Bing Pan, Helene Hembrooke, and Geri
+Gay. 2005. Accurately Interpreting Clickthrough Data as Implicit Feedback.
+In Proceedings of the 28th Annual International ACM SIGIR Conference on Re-
+search and Development in Information Retrieval (Salvador, Brazil) (SIGIR ’05).
+Association for Computing Machinery, New York, NY, USA, 154–161.
+https:
+//doi.org/10.1145/1076034.1076063
+[27] Joseph Johnson. 2022.
+Global market share of search engines 2010-
+2022. https://www.statista.com/statistics/216573/worldwide-market-share-of-
+search-engines/.
+[28] Rosie Jones, Benjamin Rey, Omid Madani, and Wiley Greiner. 2006. Generating
+Query Substitutions. In Proceedings of the 15th International Conference on World
+Wide Web (Edinburgh, Scotland) (WWW ’06). Association for Computing Ma-
+chinery, New York, NY, USA, 387–396. https://doi.org/10.1145/1135777.1135835
+[29] Othar Hansson Kavi Goel, Ramanathan V. Guha. 2009.
+Introducing Rich
+Snippets. https://developers.google.com/search/blog/2009/05/introducing-rich-
+snippets. Accessed: 2022-10-04.
+[30] Madian Khabsa, Aidan Crook, Ahmed Hassan Awadallah, Imed Zitouni, Tasos
+Anastasakos, and Kyle Williams. 2016. Learning to Account for Good Abandon-
+ment in Search Success Metrics. In Proceedings of the 25th ACM International on
+Conference on Information and Knowledge Management (Indianapolis, Indiana,
+USA) (CIKM ’16). Association for Computing Machinery, New York, NY, USA,
+1893–1896. https://doi.org/10.1145/2983323.2983867
+[31] Ron Kohavi, Alex Deng, Brian Frasca, Toby Walker, Ya Xu, and Nils Pohlmann.
+2013. Online Controlled Experiments at Large Scale. In Proceedings of the 19th
+ACM SIGKDD International Conference on Knowledge Discovery and Data Mining
+(Chicago, Illinois, USA) (KDD ’13). Association for Computing Machinery, New
+York, NY, USA, 1168–1176. https://doi.org/10.1145/2487575.2488217
+[32] Michael Lesk. 1997. Practical Digital Libraries: Books, Bytes, and Bucks. Morgan
+Kaufmann Publishers Inc., San Francisco, CA, USA.
+[33] Jane Li, Scott Huffman, and Akihito Tokuda. 2009. Good Abandonment in Mobile
+and PC Internet Search. In Proceedings of the 32nd International ACM SIGIR
+Conference on Research and Development in Information Retrieval (Boston, MA,
+USA) (SIGIR ’09). Association for Computing Machinery, New York, NY, USA,
+43–50. https://doi.org/10.1145/1571941.1571951
+[34] Eric Lin, Saul Greenberg, Eileah Trotter, David Ma, and John Aycock. 2011. Does
+Domain Highlighting Help People Identify Phishing Sites?. In Proceedings of
+the SIGCHI Conference on Human Factors in Computing Systems (Vancouver, BC,
+Canada) (CHI ’11). Association for Computing Machinery, New York, NY, USA,
+2075–2084. https://doi.org/10.1145/1978942.1979244
+[35] Chang Liu, Ying-Hsang Liu, Jingjing Liu, and Ralf Bierig. 2021. Search Interface
+Design and Evaluation. Foundations and Trends® in Information Retrieval 15, 3-4
+(2021), 243–416. https://doi.org/10.1561/1500000073
+[36] Feifei Liu. 2020. How Search Engines Shape Gaze Patterns During Information
+Seeking: Google vs. Baidu. https://web.archive.org/web/20220511211235/https:
+//www.nngroup.com/articles/google-baidu-serp-comparison/.
+[37] Emma Lurie and Eni Mustafaraj. 2018. Investigating the Effects of Google’s Search
+Engine Result Page in Evaluating the Credibility of Online News Sources. In
+Proceedings of the 10th ACM Conference on Web Science (Amsterdam, Netherlands)
+(WebSci ’18). Association for Computing Machinery, New York, NY, USA, 107–116.
+https://doi.org/10.1145/3201064.3201095
+[38] Gary Marchionini. 1995. Information Seeking in Electronic Environments. Cam-
+bridge University Press, USA.
+[39] Marissa Mayer. 2017. Universal search: The best answer is still the best answer
+. https://googleblog.blogspot.com/2007/05/universal-search-best-answer-is-still.
+html. Accessed: 2022-10-04.
+[40] Kate Moran and Cami Goray. 2019.
+The Anatomy of a Search-Results
+Page. https://web.archive.org/web/20220511210638/https://www.nngroup.com/
+articles/anatomy-search-results-page/.
+[41] Kate Moran and Cami Goray. 2020. Good Abandonment on Search Results
+Pages. https://web.archive.org/web/20220511211722/https://www.nngroup.com/
+articles/good-abandonment/.
+[42] Kate Moran and Cami Goray. 2020. Three Key SERP Features: Featured Snip-
+pets, People Also Ask, and Knowledge Panels. https://web.archive.org/web/
+20220511203205/https://www.nngroup.com/articles/key-serp-features/.
+[43] Peter Morville and Jeffery Callender. 2010. Search Patterns: Design for Discovery
+(1st ed.). O’Reilly Media, Inc., USA.
+[44] Peter Norvig. 2012. English Letter Frequency Counts: Mayzner Revisited. http:
+//norvig.com/mayzner.html.
+[45] Bruno Oliveira and Carla Teixeira Lopes. 2023. From 10 Blue Links Pages to
+Feature-Full Search Engine Results Pages - Analysis of the Temporal Evolution
+of SERP Features. In Proceedings of the 2023 Conference on Human Information
+Interaction and Retrieval (Austin, TX, USA) (CHIIR ’23). Association for Computing
+Machinery, New York, NY, USA. https://doi.org/10.1145/3576840.3578307
+[46] James Reynolds. 2022. A Brief History of Link Buying - The Ultimate Link Buyers
+Guide for SEO (2022). https://seosherpa.com/buy-backlinks/.
+[47] Larisa Rosu. 2020. The Anatomy of a Google Search Results Page in 2020 -
+Advanced Web Ranking. https://www.advancedwebranking.com/serp/.
+
+CHIIR’23, March 19–23, 2023, Austin, TX, USA
+Bruno Oliveira and Carla Teixeira Lopes
+[48] Tony Russell-Rose and Tyler Tate. 2012. Designing the Search Experience: The
+Information Architecture of Discovery (1st ed.). Morgan Kaufmann Publishers Inc.,
+San Francisco, CA, USA.
+[49] Sebastian Schultheiß and Dirk Lewandowski. 0. Misplaced trust? The relationship
+between trust, ability to identify commercially influenced results and search
+engine preference. Journal of Information Science 0, 0 (0), 01655515211014157.
+https://doi.org/10.1177/01655515211014157
+[50] Julia Schwarz and Meredith Morris. 2011. Augmenting Web Pages and Search
+Results to Support Credibility Assessment. In Proceedings of the SIGCHI Conference
+on Human Factors in Computing Systems (Vancouver, BC, Canada) (CHI ’11).
+Association for Computing Machinery, New York, NY, USA, 1245–1254. https:
+//doi.org/10.1145/1978942.1979127
+[51] Bongwon Suh, Haibin Ling, Benjamin B. Bederson, and David W. Jacobs. 2003.
+Automatic Thumbnail Cropping and Its Effectiveness. In Proceedings of the 16th
+Annual ACM Symposium on User Interface Software and Technology (Vancouver,
+Canada) (UIST ’03). Association for Computing Machinery, New York, NY, USA,
+95–104. https://doi.org/10.1145/964696.964707
+[52] Danny Sullivan. 2003.
+Searching With Invisible Tabs.
+https://www.
+searchenginewatch.com/2003/12/02/searching-with-invisible-tabs/.
+[53] Tanja Svarre and Tony Russell-Rose. 2022. Think outside the search box: A
+comparative study of visual and form-based query builders. https://doi.org/10.
+48550/ARXIV.2205.04212
+[54] Jaime Teevan, Edward Cutrell, Danyel Fisher, Steven M. Drucker, Gonzalo Ramos,
+Paul André, and Chang Hu. 2009. Visual Snippets: Summarizing Web Pages for
+Search and Revisitation. In Proceedings of the SIGCHI Conference on Human Factors
+in Computing Systems (Boston, MA, USA) (CHI ’09). Association for Computing
+Machinery, New York, NY, USA, 2023–2032.
+https://doi.org/10.1145/1518701.
+1519008
+[55] Teoalida. 2019. Screen resolution statistics. https://www.teoalida.com/webdesign/
+screen-resolution/.
+[56] Paul Thomas, Alistair Moffat, Peter Bailey, Falk Scholer, and Nick Craswell. 2018.
+Better Effectiveness Metrics for SERPs, Cards, and Rankings. In Proceedings of
+the 23rd Australasian Document Computing Symposium (Dunedin, New Zealand)
+(ADCS ’18). Association for Computing Machinery, New York, NY, USA, Article
+1, 8 pages. https://doi.org/10.1145/3291992.3292002
+[57] Jenifer Tidwell, Charles Brewer, and Aynne Valencia-Brooks. 2020. Designing
+Interfaces, 3rd Edition. O’Reilly Media, Inc, USA. 500 pages.
+https://learning.
+oreilly.com/library/view/designing-interfaces-3rd/9781492051954/
+[58] Anders Toxboe. 2022. Design patterns. http://ui-patterns.com/patterns.
+[59] Kristen Vaughn. 2019. SERP Features in 2019: The Complete Guide | KoMarketing.
+https://komarketing.com/blog/serp-features/.
+[60] Nicholas Vincent and Brent Hecht. 2021. A Deeper Investigation of the Impor-
+tance of Wikipedia Links to Search Engine Results. Proc. ACM Hum.-Comput.
+Interact. 5, CSCW1, Article 4 (apr 2021), 15 pages. https://doi.org/10.1145/3449078
+[61] Ryen White and Resa Roth. 2013. Exploratory Search: Beyond the Query-Response
+Paradigm. Morgan & Claypool Publishers, USA.
+[62] Ryen W. White. 2016. Interactions with Search Systems. Cambridge University
+Press, Cambridge. https://doi.org/10.1017/CBO9781139525305
+[63] Max L. Wilson. 2011. Search User Interface Design. Synthesis Lectures on Infor-
+mation Concepts, Retrieval, and Services 3, 3 (2011), 1–143. https://doi.org/10.
+2200/S00371ED1V01Y201111ICR020
+[64] Zhijing Wu, Mark Sanderson, B. Barla Cambazoglu, W. Bruce Croft, and Falk
+Scholer. 2020. Providing Direct Answers in Search Results: A Study of User
+Behavior. In Proceedings of the 29th ACM International Conference on Information
+and Knowledge Management (Virtual Event, Ireland) (CIKM ’20). Association for
+Computing Machinery, New York, NY, USA, 1635–1644. https://doi.org/10.1145/
+3340531.3412017
+[65] Ann Wylie. 2021. What’s the best length of a word online? https://web.archive.
+org/web/20221012095143/https://www.wyliecomm.com/2021/11/whats-the-
+best-length-of-a-word-online/.
+[66] Ke Zhou, Ronan Cummins, Mounia Lalmas, and Joemon M. Jose. 2012. Eval-
+uating Aggregated Search Pages. In Proceedings of the 35th International ACM
+SIGIR Conference on Research and Development in Information Retrieval (Portland,
+Oregon, USA) (SIGIR ’12). Association for Computing Machinery, New York, NY,
+USA, 115–124. https://doi.org/10.1145/2348283.2348302
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+Draft version January 12, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Detecting exomoons from radial velocity measurements of self-luminous planets: application to
+observations of HR 7672 B and future prospects
+Jean-Baptiste Ruffio
+,1, 2 Katelyn Horstman
+,1 Dimitri Mawet
+,1, 3 Lee J. Rosenthal
+,1
+Konstantin Batygin
+,4 Jason J. Wang (王劲飞)
+,5 Maxwell Millar-Blanchaer
+,6 Ji Wang (王吉)
+,7
+Benjamin J. Fulton
+,8 Quinn M. Konopacky
+,2 Shubh Agrawal
+,1, 9 Lea A. Hirsch
+,10
+Andrew W. Howard
+,1 Sarah Blunt
+,1 Eric Nielsen
+,11 Ashley Baker,1 Randall Bartos,3
+Charlotte Z. Bond,12 Benjamin Calvin,1, 13 Sylvain Cetre,14 Jacques-Robert Delorme
+,14, 1 Greg Doppmann,14
+Daniel Echeverri
+,1 Luke Finnerty
+,13 Michael P. Fitzgerald
+,13 Nemanja Jovanovic
+,1 Ronald L´opez,13
+Emily C. Martin
+,15 Evan Morris,15 Jacklyn Pezzato,1 Garreth Ruane
+,1, 3 Ben Sappey
+,16
+Tobias Schofield,1 Andrew Skemer,15 Taylor Venenciano,17 J. Kent Wallace,3 Nicole L. Wallack
+,18
+Peter Wizinowich,14 and Jerry W. Xuan
+1
+1Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
+2Center for Astrophysics and Space Science, University of California, San Diego; La Jolla, CA 92093, USA
+3Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA
+4Division of Geological and Planetary Sciences California Institute of Technology, Pasadena, CA 91125, USA
+5Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy,
+Northwestern University, Evanston, IL 60208, USA
+6Department of Physics, University of California, Santa Barbara, Santa Barbara, California, USA
+7Department of Astronomy, The Ohio State University, 100 W 18th Ave, Columbus, OH 43210 USA
+8NASA Exoplanet Science Institute, Caltech/IPAC-NExScI, 1200 East California Boulevard, Pasadena, CA 91125, USA
+9Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
+10Department of Chemical & Physical Sciences, University of Toronto Mississauga, 3359 Mississauga Road Mississauga , Ontario L5L
+1C6m, Canada
+11Department of Astronomy, New Mexico State University, P.O. Box 30001, MSC 4500, Las Cruces, NM 88003, USA
+12UK Astronomy Technology Centre, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom
+13Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA
+14W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA
+15Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA
+16Center for Astrophysics and Space Sciences, University of California, San Diego, La Jolla, CA 92093
+17Physics and Astronomy Department, Pomona College, 333 N. College Way, Claremont, CA 91711, USA
+18Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015, USA
+Submitted to AJ
+ABSTRACT
+The detection of satellites around extrasolar planets, so called exomoons, remains a largely unex-
+plored territory. In this work, we study the potential of detecting these elusive objects from radial
+velocity monitoring of self-luminous directly imaged planets. This technique is now possible thanks
+to the development of dedicated instruments combining the power of high-resolution spectroscopy and
+high-contrast imaging. First, we demonstrate a sensitivity to satellites with a mass ratio of 1 − 4% at
+separations similar to the Galilean moons from observations of a brown-dwarf companion (HR 7672 B;
+Kmag=13; 0.7′′ separation) with the Keck Planet Imager and Characterizer (KPIC; R ∼ 35, 000 in K
+band) at the W. M. Keck Observatory. Current instrumentation is therefore already sensitive to large
+unresolved satellites that could be forming from gravitational instability akin to binary star formation.
+Using end-to-end simulations, we then estimate that future instruments such as MODHIS, planned for
+Corresponding author: Jean-Baptiste Ruffio
+jruffio@caltech.edu
+arXiv:2301.04206v1  [astro-ph.EP]  10 Jan 2023
+
+ID2
+Ruffio et al.
+the Thirty Meter Telescope, should be sensitive to satellites with mass ratios of ∼ 10−4. Such small
+moons would likely form in a circumplanetary disk similar to the Jovian satellites in the solar system.
+Looking for the RossiterMcLaughlin effect could also be an interesting pathway to detecting the small-
+est moons on short orbital periods. Future exomoon discoveries will allow precise mass measurements
+of the substellar companions that they orbit and provide key insight into the formation of exoplanets.
+They would also help constrain the population of habitable Earth-sized moons orbiting gas giants in
+the habitable zone of their stars.
+Keywords: Natural satellites (Extrasolar) (483) — Direct imaging (387) — Radial velocity (1332) —
+Exoplanet detection methods (489)
+1. INTRODUCTION
+1.1. Exomoon formation pathways
+Moons similar to those around Jupiter are expected
+to form in circumplanetary disks (CPD) as a by-product
+of planet formation (Batygin & Morbidelli 2020). The
+typical CPD total dust mass relative to the planet is
+around 10−4 (Canup & Ward 2006) commensurate with
+the mass ratios of solar system satellites around the gas
+giants listed in Table 1. This is consistent with the mea-
+sured value of the CPD around PDS 70 c from ALMA
+observations (Benisty et al. 2021), which is 0.031MEarth
+and corresponds to a mass ratio of about 5 × 10−5,
+assuming a 2MJup planet (Benisty et al. 2021).
+It is
+also possible to form larger moons from the merger of
+Galilean-like multiple systems. This is the proposed sce-
+nario to explain the high-eccentricity and large mass of
+Saturn’s moon Titan (Asphaug & Emsenhuber 2018).
+Alternative formation pathways include the capture of
+satellites (e.g., Neptune’s moon Triton, Agnor & Hamil-
+ton (2006)), collisions with protoplanets (e.g., the Moon,
+Canup & Asphaug (2001)]), or even gravitational insta-
+bility like in the formation of brown dwarf binaries (Laz-
+zoni et al. 2020). The detection of satellites that formed
+in a CPD remains challenging with current instrumenta-
+tion, but binary planets and brown dwarfs are already
+accessible with various techniques depending on their
+separation (Lazzoni et al. 2022). Characterizing the dif-
+ferent populations with different mass ratios, such as
+binary brown-dwarf companions (ie, triple systems) and
+smaller CPD moons, could help inform the formation
+pathways of directly imaged planets.
+Indeed, binary
+companions could only occur in a top-down scenario,
+while CPD formation could occur in all cases. An im-
+portant lesson from early exoplanet discoveries is that
+the solar system is not a good predictor of exoplanet de-
+mographics. Planet formation theories also often strug-
+gle to account for the diversity of new discoveries. For
+example, the first discoveries of hot Jupiters and the
+ubiquity of super-Earths and mini Neptunes were ini-
+tially a surprise to the community (Batalha 2014). As
+a corollary, it would be unwise to assume that exomoon
+searches should be any different (Kipping et al. 2015).
+It is therefore important that we keep pushing the dis-
+covery space with new observational methods.
+1.2. Status of exomoon searches
+For the past decade, transiting surveys have unequiv-
+ocally dominated the landscape of exomoon searches
+(Kipping et al. 2012) through the analysis of transit
+timing variations and additional transit signal from the
+moon. They have placed the first constraints on exo-
+moon occurrence rates and shown that high-mass ratio
+satellites are not common around short-period exoplan-
+ets (Kipping et al. 2015; Teachey et al. 2018). Other
+detection techniques have been used to look for exo-
+moons around directly imaged exoplanets such as as-
+trometry or direct imaging. While these terms also re-
+fer to planet detection techniques, in this context, direct
+imaging means to spatially resolve the satellite from the
+planet.
+Astrometric detections refers to the measure-
+ment of the astrometric wobble of a planet caused by
+orbiting moons with precise interferometric instruments
+such as VLTI/GRAVITY (Gravity Collaboration et al.
+2021). To date, only a handful of exomoon candidates
+have been proposed: for example two around transit-
+ing planets (Teachey & Kipping 2018; Kipping et al.
+2022), one orbiting a directly-imaged brown dwarf (Laz-
+zoni et al. 2020), and another around an isolated plan-
+etary mass object (Limbach et al. 2021).
+None have
+been confirmed. Most notably, the exomoon candidate
+Kepler 1708 b-i is a transiting 2.6 Earth radii object or-
+biting a Jupiter-sized planet, which, if confirmed, would
+be several orders of magnitude larger than the Galilean
+moons in terms of mass ratio (Kipping et al. 2022).
+Transiting planets generally have short orbital peri-
+ods and smaller Hill spheres. These conditions could be
+less favorable to moon formation and retention, while
+observed transits of long period exoplanets (> 1 au) are
+intrinsically rare. For example, it has been suggested
+
+RV detection of exomoons
+3
+that planets could lose their satellites as they migrate
+inward (Spalding et al. 2016).
+The detection of exo-
+moons around imaged planets with astrometry or direct
+imaging is more sensitive to longer-period moons (Laz-
+zoni et al. 2022). These two techniques might not be
+well suited for satellites with mass ratios and separa-
+tions (< 30RJup) similar to the ones orbiting the solar
+system gas giants.
+1.3. A promising alternative: RV detections around
+directly imaged planets
+Another technique has been proposed to look for
+moons around directly imaged planets using RV mea-
+surements of the planet itself (Vanderburg et al. 2018;
+Vanderburg & Rodriguez 2021). By measuring the wob-
+ble of planets caused by orbiting satellites, planetary RV
+surveys is a promising alternative for finding Galilean
+moons analogs. Directly-imaged companions are likely
+to have a different formation and migration history com-
+pared to transiting exoplanets.
+They are also gener-
+ally more massive and further away resulting in much
+more extended Hill spheres. Models suggest that larger
+planets form even larger moons following the scaling
+m ∝ M 3/2, with m and M the masses of the moon and
+the planet respectively (based on equation 43 in Baty-
+gin & Morbidelli (2020)). As another formation path-
+way, binary systems forming as the tail end of stellar
+formation through gravitational instability of the proto-
+stellar cloud or the protoplanetary disk could lead to a
+population of easily detectable high-mass ratio satellites
+and binary companions (Lazzoni et al. 2022). In sum-
+mary, directly imaged companions could be more likely
+to host larger moons, which could be detected from RV
+monitoring of the planets themselves.
+1.4. Recent technology developments
+There are two aspects to optimizing the choice of a
+target for exomoon searches: the RV precision that can
+be achieved and the probability of the object to host a
+moon. In this work, we focus on the former and study
+the detectable mass ratios for exomoons as a function of
+the instrument, the telescope, and the planet or brown-
+dwarf properties.
+Although the RV detection of exo-
+moons remains challenging due to the intrinsic faintness
+of planets and the light contamination from the glare
+of the host star, the expertise obtained from 30 years
+of stellar RV exoplanet detections is an invaluable as-
+set.
+Indeed, stable high-resolution spectrographs and
+data analysis techniques are already demonstrating sta-
+bility and performance in excess of the level of preci-
+sion needed for exomoons.
+Vanderburg & Rodriguez
+(2021) derived the first exomoon mass upper limits with
+this technique around the HR 8799 planets based on the
+planetary RV time series reported by Ruffio et al. (2021)
+using OSIRIS, an R = 4, 000 spectrograph, at the W.
+M. Keck observatory. This study ruled out moons with
+mass greater than a Jupiter mass and period less than
+one day that would be orbiting the 7 Jupiter mass planet
+HR 8799 c.
+Recent
+technological
+advances
+in
+infrared
+high-
+resolution spectroscopy for high-contrast companions
+are enabling the first planetary RV searches for exo-
+moons (Snellen et al. 2015; Jovanovic et al. 2017; De-
+lorme et al. 2021; Otten et al. 2021). The Keck Planet
+Imager and Characterizer (KPIC) recently demon-
+strated R = 35, 000 K-band spectroscopy of directly-
+imaged exoplanets, including the HR 8799 system, mea-
+suring their RVs and obtaining spin measurements for
+the first time (Wang et al. 2021b). KPIC is the first
+implementation of a new class of spectrographs that
+combines the power of the Keck II adaptive optics sys-
+tems, the stability and starlight suppression of single
+mode fibers, and the high spectral resolution of the NIR-
+SPEC spectrograph for the detailed study of directly
+imaged planets (Delorme et al. 2021). We observed a
+bright brown dwarf companion (HR 7672 B, K = 13.04,
+∼ 0.7′′, Liu et al. (2002); Boccaletti et al. (2003)) as part
+of the commissioning and science verification of KPIC
+(Delorme et al. 2021; Wang et al. 2022a). While it is
+at the boundary of the stellar regime, HR 7672 B is an
+interesting benchmark companion, because it has a dy-
+namically measured mass of 72.7±0.8MJup (Crepp et al.
+2012; Brandt et al. 2019) and its composition should be
+similar to that of the star due to its assumed formation
+history from gravitational instability. Using this target,
+Wang et al. (2022a) showed that accurate atmospheric
+compositions could be retrieved using KPIC’s high re-
+solving power and angular resolution by demonstrating
+a 1.5σ consistency between the composition of HR 7672
+B and its host star (see also Xuan et al. 2022).
+HR
+7672 B was also observed for a full night with KPIC as
+a test case for variability studies. This time series can
+be used to put the deepest limits to date on the mass
+of an orbiting satellite around the sub-stellar compan-
+ion, which we are demonstrating in this work. KPIC
+is already undergoing several upgrades including a laser
+frequency comb which will enable precise RV science (Yi
+et al. 2016; Jovanovic et al. 2020). The expected dou-
+bling of the instrumental throughput will significantly
+improve its sensitivity (Jovanovic et al. 2020; Echev-
+erri et al. 2022). The next generation of high-contrast
+high-resolution spectrograph such as HISPEC at the
+W. M. Keck Observatory and MODHIS on the future
+Thirty Meter Telescope (TMT) will undoubtedly open
+
+4
+Ruffio et al.
+new frontiers in this field by allowing 0.98 − 2.46 µm si-
+multaneous coverage at an average spectral resolution
+R > 100, 000 (Mawet et al. 2019; Mawet et al. 2022).
+1.5. Outline
+In section 2, we present exomoon RV detection lim-
+its for the brown dwarf companion HR 7672 B using
+KPIC. In section 3, we then simulate observations of the
+same brown dwarf and the planet HR 8799 c with next
+generation facilities and compare their sensitivity to the
+moons in the solar system. In section 4, we explore the
+parameter space of satellites that could be detected with
+TMT/MODHIS as a function of planet properties. Fi-
+nally, we conclude on the prospects for RV detections of
+exomoons in section 5.
+2. EXOMOON LIMITS AROUND HR 7672 B WITH
+KPIC
+2.1. Observations and data reduction
+The brown dwarf companion HR 7672 B was observed
+three times in 2020 and then for a full night on July 4,
+2021 with KPIC (R ∼ 35, 000) in K band (1.9−2.4 µm)
+(Mawet et al. 2017; Delorme et al. 2021). These obser-
+vations are detailed in Table 2. The first three epochs
+included one to two hours of on-target exposures per
+night and were already published in Wang et al. (2022a)
+and Delorme et al. (2021).
+Unfortunately, the condi-
+tions on July 4, 2021 were well below average with the
+companion undetectable in some individual 5-minute ex-
+posures. During this one night specifically, we used an
+ABAB pattern to nod the companion between two KPIC
+fibers, fiber 1 and 2, to limit or identify any fiber-specific
+biases. There was no nodding during the other epochs.
+The data was reduced with the KPIC data reduction
+pipeline (DRP)1 following the same approach described
+in Wang et al. (2021b, 2022a). The first steps include
+background subtraction, bad pixel correction, and the
+calibration of the fiber trace location and width on the
+detector for each NIRSPEC spectroscopic order. Opti-
+mal extraction is then used to extract the spectra and
+the wavelength solution is derived from the telluric and
+stellar lines of a M giant, namely HIP 81497, taken on
+the same night. For this purpose, the telluric model is
+generated with the Planetary Spectrum Generator (Vil-
+lanueva et al. 2018) and star is modeled by a Phoenix
+model (log(g/[1 cm.s−2]) = 1; Teff = 3600 K Husser
+et al. (2013)).
+2.2. Forward model and likelihood
+1 https://github.com/kpicteam/kpic pipeline
+We use a forward modelling approach similar to Wang
+et al. (2021b) and Ruffio et al. (2021) to measure the RV
+of HR 7672 B, which includes a joint modelling of the
+starlight and the companion signal. Wang et al. (2021c)
+showed that the continuum could be included in the for-
+ward model with a fourth order polynomial, therefore
+not requiring the data to be high-pass filtered nor contin-
+uum normalized. In this work, we model the continuum
+using a spline-based linear model, which can be ana-
+lytically marginalized using the general purpose python
+module breads2 (Broad Repository for Exoplanet Anal-
+ysis, Discovery, and Spectroscopy) based on the formal-
+ism in Ruffio et al. (2019).
+The spline forward mod-
+eling has the advantage of being more robust to bad
+pixels than a Fourier based high-pass filter and avoids
+the non-linearity of a sliding-window median filter. The
+spline parameters are also easier to optimize than the
+coefficients of a high order polynomial for example.
+We define the forward model as,
+d = MRVφ + n,
+(1)
+where d is the data vector of size Nd, MRV is the linear
+model, φ are the linear parameters, and n is a random
+vector of the noise with a diagonal covariance matrix Σ.
+A scaling factor for the noise is also fitted to model any
+underestimation of the noise. Off-diagonal elements in
+the covariance matrix are neglected here, but subsequent
+data processing steps would correct for this inaccuracy.
+The different column vectors of the linear model are il-
+lustrated in Figure 1. The data vector and the standard
+deviation of the noise used to define Σ0 are direct out-
+puts of the KPIC data reduction pipeline. The variance
+of the noise is multiplied by a free parameter scaling fac-
+tor s2 to account for any underestimation of the noise.
+KPIC includes four single mode fibers separated by
+0.8′′ on a line. We can therefore acquire simultaneous
+spectra of the companion and the host star, more specif-
+ically the speckle field, by rotating the field of view using
+the Keck II adaptive optics system front-end K-mirror
+rotator. The observations of the star are used to derive
+simultaneous empirical models of the transmission and
+the starlight spectra used in the forward model. The
+starlight is used to model the speckle noise leaking into
+the fiber at the position of the companion. The wave-
+length calibration is different in each fiber so the spectra
+are linearly interpolated to match the sampling of the
+science fiber. The planet model is defined as the spin-
+broadened best fit model from Wang et al. (2022a) using
+petitRADTRANS (Molli`ere et al. 2019) multiplied by
+2 https://github.com/jruffio/breads
+
+RV detection of exomoons
+5
+Moon
+Planet
+Mass
+Mass ratio
+Semi-Major axis
+Period
+RV semi-amplitude
+(M⊕)
+(RJup)
+(day)
+(m/s)
+Io
+Jupiter
+1.50 × 10−2
+4.71 × 10−5
+5.90
+1.77
+0.82
+Europa
+Jupiter
+8.04 × 10−3
+2.53 × 10−5
+9.39
+3.55
+0.35
+Ganymede
+Jupiter
+2.48 × 10−2
+7.81 × 10−5
+14.97
+7.15
+0.85
+Callisto
+Jupiter
+1.80 × 10−2
+5.67 × 10−5
+26.33
+16.69
+0.46
+Titan
+Saturn
+2.25 × 10−2
+2.37 × 10−4
+17.09
+15.95
+1.32
+Titania
+Uranus
+5.73 × 10−4
+3.94 × 10−5
+6.10
+8.71
+0.14
+Oberon
+Uranus
+4.82 × 10−4
+3.32 × 10−5
+8.16
+13.46
+0.10
+Triton
+Neptune
+3.58 × 10−2
+2.09 × 10−4
+4.96
+-5.88
+0.92
+Kepler-1708 b-i
+Kepler-1708 b
+< 37
+< 0.11(2σ)
+Table 1. Properties of the largest satellites orbiting the solar system gas giants from the NASA Space Science Data Coordinated
+Archive (https://nssdc.gsfc.nasa.gov/planetary/). The negative period of Triton is indicating its retrograde orbit. Kepler-1708
+b-i is a transiting exomoon candidate (Kipping et al. 2022). The period and RV semi-amplitude for these moons can also be
+found in Vanderburg et al. (2018).
+Object
+Date
+Exposure time
+Seeing
+Throughput
+HR 7672 B
+2020-06-08
+11 × 10 min
+0.4′′
+1%
+HR 7672 B
+2020-06-09
+10 × 10 min
+0.6′′
+1.5%
+HR 7672 B
+2020-09-28
+7 × 10 min
+0.4′′
+2.7%
+HR 7672 B
+2021-07-04
+61 × 5 min
+1′′
+2%
+Table 2. K-band observations of HR 7672 A and B with KPIC. The quoted throughput is the end-to-end from the top of the
+atmosphere, which is a better proxy of performance than the seeing for KPIC.
+the empirical telluric and instrument transmission pro-
+file. The continuum of both the planet and the speckle
+are modulated by a 3rd order spline model. Ten spline
+nodes are used in each spectral order (∆λ ∼ 0.05 µm) for
+the planet model to manage any inaccuracies in the con-
+tinuum due to imperfections in the atmosphere model
+fit. This number of nodes is analogous to a 200 pixel-
+wide high-pass filter. The number of nodes was chosen
+as a trade-off between the number of additional param-
+eters and the optimal high-pass filter scale of 100 pixels
+found in Xuan et al. (2022). The speckle continuum is
+modeled with three spline nodes to model any speckle
+crossing the fiber location as the wavelength changes.
+This results in 13 linear parameters per spectral order
+representing the values of the continua at the location of
+the nodes (See Figure 1). This defines the linear model
+MRV with dimensions Nd × 13, which is also a function
+of the RV of the planet, the only non-linear parameter
+fitted for here.
+KPIC data features strong spectral fringing due to
+the FabryP´erot cavities formed by the transmissive op-
+tics inside the NIRSPEC spectrograph (Hsu et al. 2021)
+and within the KPIC fiber injection unit (Finnerty et al.
+2022). This effect is made worse by the high spatial co-
+herence of the wavefront in KPIC. We therefore apply a
+Fourier filter to the data and the forward model by zero-
+ing frequencies corresponding to the fringes. A physical
+model of the fringing such as Cale et al. (2019) could be
+explored in the future.
+The likelihood function is defined from a multivariate
+Gaussian distribution as,
+L(RV, φ, s2) =
+1
+�
+(2π)Nd|Σ0|s2Nd
+exp
+�
+− 1
+2s2 (d − MRVφ)⊤Σ−1
+0 (d − MRVφ)
+�
+.
+(2)
+The likelihood is maximized using a linear least square
+solver on a grid of RV values from −400 to 400 km/s in
+steps of 0.2 km/s. The 1σ RV uncertainties are derived
+from the RV posterior calculated analytically according
+to Equation 10 in Ruffio et al. (2021) on this RV sam-
+pling.
+This method analytically marginalized the RV
+posterior for the modulation of the continuum and the
+noise scaling factor. The linear spline parameters used
+to fit the continuum are forced to be positive. This is
+theoretically inconsistent with the framework, which as-
+sumes unconstrained parameters, but it does not appear
+to significantly impact the RV time series.
+Only the three reddest orders, out of nine in K band,
+are used in this analysis. The bluest three orders (num-
+bered 39-37; 1.94 − 2.09 µm) were discarded because
+they feature strong saturated CO2 telluric lines that are
+generally harder to model, but also make for an unsta-
+ble fit due to overlapping frequencies with the fringing
+
+6
+Ruffio et al.
+and the simple Fourier filter. The middle three orders
+(2.10 − 2.27 µm) lack sufficient stellar and telluric spec-
+tral lines to calibrate the wavelength precisely enough.
+Thus, only the remaining three orders are used in this
+analysis: 2.29−2.34 µm (order 33), 2.36−2.41 µm (order
+32), and 2.44 − 2.49 µm (order 31). Order 33 includes
+the carbon monoxide bandhead and therefore results in
+the strongest signal-to-noise ratio (S/N) and the most
+precise radial velocity measurement.
+Each NIRSPEC
+spectral order is fitted separately resulting in three RV
+estimates for each exposure.
+2.3. RV measurements
+The barycentric corrected RV measurements for the
+four epochs and three orders are shown in Figure 2.
+Following the method described in subsection 2.2, the
+median RV uncertainties in five minute exposures are
+2.5 km.s−1,4.0 km.s−1,4.9 km.s−1 for order 6, 7, and 8
+respectively. We overplot the predicted radial velocity of
+the brown dwarf from orbital fits to the relative astrom-
+etry from Crepp et al. (2012) and RV measurements of
+the host star (Crepp et al. 2012; Rosenthal et al. 2021).
+The orbit fits were done with orbitize! (Blunt et al.
+2020) following its RV tutorial3 and using the emcee
+(Foreman-Mackey et al. 2013) sampler to obtain a pos-
+terior of allowed orbits. This orbital RV of the compan-
+ion in each epoch is predicted from this orbit fit and is
+subsequently subtracted from the estimated RV of the
+planet when running the exomoon search. Similarly to
+fitting the centroid of a Gaussian (King 1983), the RV
+precision goes as the typical linewidth in the spectrum
+divided by the total S/N of the detection. In the case of
+HR 7672 B, the large spin with v sin i = 45.0±0.5 km.s−1
+(Wang et al. 2022a) is a limiting factor in deriving more
+precise RVs. The impact on the exomoon sensitivity of
+other fundamental parameters such as the brightness,
+age, mass, and separation from the star are discussed in
+section 4 in the context of TMT/MODHIS.
+2.4. Exomoon sensitivity
+The open-source Python package RVSearch4 (Rosen-
+thal et al. 2021) is used to look for possible exomoons
+around HR 7672 B and derive the sensitivity of our
+KPIC RV time series. RVSearch is a planet search algo-
+rithm that was developed by the California Legacy Sur-
+vey for high-precision radial velocity surveys (Howard &
+Fulton 2016; Rosenthal et al. 2021; Fulton et al. 2021).
+Planets are detected from periodograms, which are ex-
+3 https://orbitize.readthedocs.io/en/latest/tutorials/
+RV MCMC Tutorial.html
+4 https://github.com/California-Planet-Search/rvsearch
+pressed as the difference in Bayesian Information Cri-
+terion (BIC) between a model including the planet and
+a model without it (Rosenthal et al. 2021). The ∆BIC
+can be used to select the model that best represents the
+data, or, in other words, determine if a planet is neces-
+sary to explain the observations. Planet candidates are
+detected by iteratively adding additional planet signal
+to the model (Rosenthal et al. 2021). For each iterative
+search, the algorithm fits a detection threshold to the
+periodogram using the power law noise model described
+in Howard & Fulton (2016). To characterize the search
+completeness of a dataset, RVSearch performs injection-
+recovery tests, drawing many synthetic planet signals,
+injecting them in the data, and checking whether their
+signals surpass the last detection threshold. The sim-
+ulated signals were injected as described in (Rosenthal
+et al. 2021) with period and M sin i from log-uniform
+distributions, and eccentricity from an empirically cali-
+brated beta distribution (Kipping 2013).
+RVSearch is directly applicable to the search for ex-
+omoons by replacing the properties of the star by the
+ones of the planet. We assume that each spectral order
+in NIRSPEC has a different zero RV point due to pos-
+sible inconsistencies between them. This can be done
+with RVSearch, which linearly solves for offsets between
+subsets of RVs, and uses a wide, Gaussian, uninforma-
+tive prior on white noise for each subset.
+This fea-
+ture is usually used to fit data from different instru-
+ments. Two analyses are performed, first only using the
+long night of observations (07/04/2021) and then all the
+available data. The latter provides a longer time base-
+line.
+The resulting periodograms and exomoon com-
+pleteness are shown in Figure 3. By combining the four
+epochs, the observations are sensitive to satellites with
+a mass ratio of 1% at semi-major axes similar to that
+of Io (6RJup) around Jupiter or 4% at the distance of
+Callisto (15RJup). While these are encouraging results,
+the smallest detectable satellites would be as large as
+Jupiter due to the already large mass of HR 7672 B. As
+shown in section 4, targeting smaller brown dwarfs and
+planets does not generally allow the detection of moons
+with smaller absolute masses, because the S/N drops
+faster than the mass of the object due to the decreas-
+ing brightness. If satellites around HR 7672 B were to
+orbit within ∼ 10RJup of the brown-dwarf, they would
+likely fall within the Roche radius (See Figure 4). Such
+satellites would be tidally disrupted and likely result in
+the formation of rings around the planet. It is possible
+that this issue would prevent the formation of a reso-
+nant chain of satellites if the inner edge of the decretion
+disk falls within the Roche limit. This is for example
+cited as a possibility to explain the difference between
+
+RV detection of exomoons
+7
+2.30
+2.31
+2.32
+2.33
+200
+0
+200
+400
+600
+Data number
+Data
+Combined model
+Planet model
+Starlight model
+Residuals
+Data uncertainty
+2.30
+2.31
+2.32
+2.33
+200
+0
+200
+400
+600
+Data number
+Planet model
+Sub-components
+Single sub-component
+2.30
+2.31
+2.32
+2.33
+ ( m)
+20
+0
+20
+40
+60
+80
+100
+Data number
+Starlight model
+Sub-components
+Single sub-component
+Figure 1. Illustration of the forward model used to derive the RV of HR 7672 B. This figure shows a single NIRSPEC order
+overlapping with the CO bandhead. (Top panel) A planet and a starlight model are jointly fitted to the data to account for
+the diffracted starlight contamination at the location of the companion. The data uncertainty measured by the KPIC DRP
+(shaded grey) slightly underestimates the amplitude of the residuals. (Center panel) The planet model is itself made of a linear
+combination of ten spline modes to model the continuum of the companion spectrum. (Bottom panel) The starlight intensity
+is also fitted with a spline using three nodes to account for speckles crossing at the location of the fiber. This flexible model of
+the continuum is an alternative to high-pass filtering and continuum normalization of high-resolution spectra.
+the Galilean and the Saturnian satellite systems in Baty-
+gin & Morbidelli (2020). At the other end of possible
+satellite semi-major axes, stable orbits can generally ex-
+ist up to one half of the Hill sphere for prograde orbits
+(Shen & Tremaine 2008). The Hill sphere of HR 7672
+B being rH ≈ 5.6 au = 1.2 × 104RJup, time series like
+these ones will not be sensitive to the vast majority of
+possible orbits without observations spanning years or
+decades.
+3. FUTURE PROSPECTS FOR HR 7672 B AND HR
+8799 C
+3.1. Simulations
+In this section, we simulate observations from current
+and future instrumentation at the Keck observatory and
+TMT to estimate the properties of putative satellites
+that should be detectable using planetary RVs. We use
+an instrument and observation simulator called PSIsim
+5, which was first developed for the Planet Systems Im-
+ager (PSI Fitzgerald et al. 2022) instrument concept for
+TMT, and then expanded to include other instruments
+and telescopes. PSIsim is first used to estimate the RV
+precision. Then, RV times series are simulated assuming
+6 full nights of observations over 25 days, and the ex-
+omoon sensitivity is finally computed using RVSearch.
+These simulations are meant to represent an ideal sce-
+nario in terms of instrument performance and telescope
+time allocation.
+We simulate observations of two substellar compan-
+ions, the brown-dwarf companion HR 7672 B and the
+planet HR 8799 c, with four generations of instru-
+ments.
+An exhaustive analysis of all directly imaged
+companions is beyond the scope of this work so HR
+5 https://github.com/planetarysystemsimager/psisim
+
+8
+Ruffio et al.
+0.0
+0.5
+1.0
+1.5
+2.0
+time (h)
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+RV (km/s)
+06-08-2020 MJD=59399.308804+t/24
+0.0
+0.5
+1.0
+1.5
+time (h)
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+RV (km/s)
+06-09-2020 MJD=59008.421342+t/24
+0.00
+0.25
+0.50
+0.75
+1.00
+time (h)
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+RV (km/s)
+09-28-2020 MJD=59009.452114+t/24
+0
+1
+2
+3
+4
+5
+6
+7
+time (h)
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+RV (km/s)
+07-04-2021 MJD=59120.258465+t/24
+Prediction
+fiber 1 order 6
+fiber 2 order 6
+fiber 1 order 7
+fiber 2 order 7
+fiber 1 order 8
+fiber 2 order 8
+Figure 2. Measured RVs of HR 7672 B with KPIC. The grey lines are predicted RVs from one hundred posterior samples of
+the orbital motion of the brown dwarf.
+8799 c was chosen as a representative example of the
+field with a planetary mass.
+HR 8799 is also the
+only other high-contrast system with published RV time
+series and exomoon upper limits (Vanderburg & Ro-
+driguez 2021). The four instruments considered in this
+work are Keck/KPIC I, Keck/KPIC II, Keck/HISPEC,
+and TMT/MODHIS. KPIC I corresponds to obser-
+vations carried out pre-2022A (Delorme et al. 2021).
+KPIC II refers to the series of upgrades started dur-
+ing the first semester of 2022 with the primary goal
+of doubling the instrument throughput Jovanovic et al.
+(2020); Echeverri et al. (2022).
+The High-resolution
+Infrared Spectrograph for Exoplanet Characterization
+(HISPEC) is expected to provide Y-K (0.98 − 2.46 µm)
+spectroscopy at a spectral resolution of R > 100, 000
+(Mawet et al. 2019).
+The Multi-Object Diffraction-
+limited High-resolution Infrared Spectrograph (MOD-
+HIS) is a similar instrument to HISPEC planned for the
+future TMT. A broader range of exoplanet masses is
+explored in section 4 for this latter TMT instrument.
+PSIsim includes full budgets of the throughput and
+thermal background for each instrument, telescope, and
+the Earth atmosphere.
+The Strehl ratio is calculated
+based on a empirically calibrated model of the adaptive
+optics’ performance under median seeing conditions for
+Maunakea. For KPIC I and KPIC II, we assumed Keck
+AO’s current performance with the infrared Pyramid
+Wavefront Sensor described in Bond et al. (2020). For
+HISPEC, we assumed extreme-AO performance as pre-
+dicted for the upcoming HAKA high-density deformable
+mirror upgrade (W.M. Keck Observatory, private com-
+munication). The star is modeled with a PHOENIX
+model (Husser et al. 2013) and the substellar companion
+with a BT-Settl atmospheric model grid6 (Allard et al.
+2012a). Table 3 includes the input parameters and the
+predicted RV precision for these simulations. The simu-
+lations include a level of systematics at 1% of the contin-
+uum, which is modeled by an additional white Gaussian
+noise. Otherwise, the estimated RV precision assumes a
+perfect data reduction.
+6 https://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011c/
+
+RV detection of exomoons
+9
+10
+2
+10
+1
+100
+101
+102
+Period (day)
+10
+8
+6
+4
+2
+BIC
+All data, BICthresh = 13.1
+One night, BICthresh = 10.5
+(a) Periodogram
+100
+101
+102
+Semi-major axis (RJup)
+10
+3
+10
+2
+10
+1
+100
+MMoon sini/MBD
+Injection recovered
+Injection missed
+50% completeness - One night
+50% completeness - All data
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Probability of detection
+(b) Completeness
+Figure 3. Exomoon detection limits around HR 7672 B with the Keck Planet Imager and Characterizer (KPIC) using the
+open-source python module RVsearch (Rosenthal et al. 2021). (Left) Periodogram of the RV times series shown in Figure 2
+expressed a ∆BIC comparing a model with and a model without a planet. The empirical detection threshold is indicated in the
+legend. (Right) Exomoon completeness derived from injection and recovery tests. The periodogram and the completeness are
+shown for two cases: the single full night of observations on 07/04/2022 and all the available data including three additional
+epochs with 1-2 hours of data each. The variable conditions on 07/04/2022 led to HR 7672 B to not be detected during portions
+of the night, or in the RV precision to get significantly worse. By simulating RV time series, we estimate that the lost data only
+affected the final sensitivity by 20%.
+The predictions from PSI-sim are about a factor two
+more sensitive than existing measurements with KPIC I
+(See Table 3). This difference can first be explained by
+uncorrected wavefront errors reducing the throughput,
+both non-common path aberrations and uncorrected at-
+mospheric turbulence.
+Then, our current data analy-
+sis framework remains limited in its ability to model
+KPIC systematics. As explained in subsection 2.2, only
+the redder orders of NIRSPEC are being reduced due
+to strong telluric lines in the bluer orders, and an im-
+perfect Fourier filtering is used to remove the fringing.
+The gap between the simulations and the measurements
+should decrease as observing strategies and data reduc-
+tion frameworks are improved.
+The final expected exomoon sensitivity of the four in-
+struments is shown for the two companions in Figure 4.
+For a fixed time sampling of the RV series, the minimum
+detectable mass ratio is approximately proportional to
+the RV semi amplitude of the signal, which is also pro-
+portional to the RV precision of the instrument, so the
+improvement for each generation of instrument can be
+read from the simulated RV precision shown at the bot-
+tom of Table 3. These simulations are compared to other
+detection techniques in Appendix A, specifically astro-
+metric monitoring of the companion or spatially resolv-
+ing the moon through imaging. We separately discuss
+the possibility of detecting transiting exomoons using
+the Rossiter-McLaughlin (RM) effect in subsection 5.2.
+3.2. Comparing to solar system moons
+The mass ratios of the largest gas giant satellites in
+the solar system are also shown in Figure 4 for compar-
+ison. The higher planet masses, M, of directly imaged
+planets and brown dwarfs compared to the solar sys-
+tem could yield significantly bigger moons, so we also
+include scaled-up mass ratios, q, according to q ∝
+√
+M
+(Batygin & Morbidelli 2020). While the CPD does scale
+with the Hill Sphere, we do not expect the semi-major
+axis of satellites to depend on this parameter. Indeed,
+young moons are thought to migrate toward the planet
+during their formation due to the interaction with the
+gas. The migration is stopped at the inner radius of the
+CPD which is set by the magnetic field of the planet
+(Batygin & Morbidelli 2020). In this work, we therefore
+keep the semi-major axis of the solar system satellites
+constant. A caveat is that large moons could be suscep-
+tible to tidal forces if they form or migrate too close to
+the planet within the Roche limit. The Roche limit is
+calculated using the mass-radius relationship from Chen
+
+10
+Ruffio et al.
+& Kipping (2017) and their associated Python package7.
+However, this relationship does not account for the fact
+that young objects are likely inflated.
+Parameters
+Star - Phoenix model
+HR 7672
+HR 8799
+Apparent K mag
+4.4a
+5.2a
+Effective temperature (Teff)
+6000 Kb
+7400 Kc
+Surface gravity (log(g))
+4.5b
+4.5c
+Spin (vsin(i); km/s)
+5.6d
+49e
+Companion - BTsettl model
+HR 7672 B
+HR 8799 c
+Mass
+73MJupb
+7MJupf
+Apparent K mag
+13.0b
+16.1g
+Effective temperature (Teff)
+1800 Kb
+1200 Kh
+Surface gravity (log(g))
+5.5b
+4.0h
+Spin (vsin(i))
+45 km.s−1b
+10 km.s−1i
+Separation
+0.72′′j
+0.95′′j
+Telescope and instrument
+airmass
+1.2
+water vapor column
+1.5 mm
+integration time (tint)
+5 min
+Predicted RV sensitivity (m.s−1)
+assuming 0.6′′ − 1.0′′ seeing
+HR 7672 B
+HR 8799 c
+Keck/KPIC I (measured)
+∼ 2, 000k
+∼ 7, 000i
+Keck/KPIC I (simulated)
+800-1400
+3,000-5,000
+Keck/KPIC II
+500-800
+2,000-3,000
+Keck/HISPEC
+∼ 200
+100-200
+TMT/MODHIS
+30-40
+10-20
+Table 3. Radial velocity (RV) precision simulations of cur-
+rent and future instrumentation for two substellar compan-
+ions: HR 7672 B and HR 8799 c. (Top) Representative pa-
+rameters for the telescope, instrument, star, and companions
+used in the PSIsim simulations. (Bottom) Predicted RV sen-
+sitivity for values of seeing ranging from 0.6′′ to 1.0′′.
+References—a: Cutri et al. (2003), b: Wang et al.
+(2022a), c: Wang et al. (2020), d: Luck (2017), e: Royer
+et al. (2007), f: Wang et al. (2018), g: Currie et al. (2011),
+h: Wang et al. (2018), i: Wang et al. (2021b), j:
+http://whereistheplanet.com/ (Wang et al. 2021a), k: This
+work
+4. FUTURE EXOMOON SENSITIVITY OF
+TMT/MODHIS
+Looking to the future, we expect substantial gains
+in RV precision by using the next generation of high-
+resolution spectrographs on large telescopes.
+These
+7 https://github.com/chenjj2/forecaster
+gains in RV precision will lead to enhanced sensitivity
+to systems with lower mass, close in exomoons, which
+would form in a similar way to the Galilean moons
+around Jupiter.
+Using the same framework as in section 3, we calcu-
+late the RV sensitivity for a variety of simulated planets
+that could exist around a host star with the properties
+of HR 8799 referenced in Table 3. We modeled planets
+with varying effective temperatures and apparent mag-
+nitudes, fixing the separation between the planet and
+star to 700 mas and the surface gravity of the planet to
+log(g) = 4.5cm.s−2, and used PSIsim to calculate the
+RV sensitivity. The effect of the starlight contamination
+on RV sensitivity can be neglected for the type of di-
+rectly imaged planets that are known today and would
+be observed with TMT. The RV sensitivity vary by less
+than 20 percent for planets that lie beyond 500 mas and
+have a flux ratio greater than ∼ 3 × 10−6.
+On aver-
+age, for every 0.5 dex change in surface gravity on the
+planet, the RV sensitivity changes by ±0.7 m/s. Fig-
+ure 5 (a) shows the RV sensitivity MODHIS could have
+for a single, two hour exposure, for planets of varying
+effective temperatures and apparent magnitudes around
+an HR 8799 like star. The RV sensitivity of MODHIS
+driven by the brightness of the planet more than than its
+temperature. However, the RV sensitivity is decreased
+for planets with temperatures between 1500 and 1700
+K using the BT-settl model grid due to the L-T transi-
+tion. At these temperatures, clouds form in the upper
+layers of the atmosphere, shrouding detectable spectral
+lines. For a given planet temperature and magnitude,
+the RV precision of TMT/MODHIS Figure 5 (a) can
+be compared to the RV semi amplitude in Figure 5 (b)
+as a function of the planet mass, the mass ratio, and
+the period of the satellite. However, such a comparison
+assumes multiple epochs of observations with a given
+sensitivity in order to detect a moon with a similar RV
+semi amplitude.
+In the following, the surface gravity, temperature, and
+mass of the planet are treated more self-consistently us-
+ing BT-Settl evolutionary grids (Allard et al. 2012b).
+The dependence of the exomoon sensitivity to the num-
+ber of observations is also made explicit by using simu-
+lated RV time series. We therefore express the RV pre-
+cision and exomoon sensitivity as a function of planet
+mass and distance to the Sun in Figure 6.
+We fixed
+the age of the system to different values to represent
+the parameter space occupied by different populations
+of stars. The 3 Myr age group is representative of the
+youngest stars, such as those found in star forming re-
+gions (e.g. Ophiuchus, Taurus, etc). The 30 Myr age
+group is representative of young moving groups, such
+
+RV detection of exomoons
+11
+100
+101
+102
+103
+104
+Semi-major axis (RJup)
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+(MMoon/Mplanet) sini
+Hill sphere
+Roche limit
+Io
+Europa
+Ganymede
+Callisto
+Titan
+Triton
+Titania
+Oberon
+KPIC I (Data)
+KPIC I
+KPIC II
+HISPEC
+MODHIS
+1h
+1d
+1wk 1mo
+1yr
+10yr
+100yr
+Period
+100
+101
+102
+103
+104
+(MMoon/MEarth) sini
+(a) HR 7672 B
+100
+101
+102
+103
+104
+Semi-major axis (RJup)
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+(MMoon/Mplanet) sini
+Hill sphere
+Roche limit
+Io
+Europa
+Ganymede
+Callisto
+Titan
+Triton
+Titania
+Oberon
+KPIC I
+KPIC II
+HISPEC
+MODHIS
+1h
+1d
+1wk 1mo
+1yr
+10yr
+100yr
+Period
+10
+1
+100
+101
+102
+103
+(MMoon/MEarth) sini
+(b) HR 8799 c
+Figure 4.
+Future prospects for exomoon detections around the brown dwarf companion HR 7672 B (left) and planet HR 8799
+c (right). Simulated sensitivity for Keck/KPIC I, Keck/KPIC II, Keck/HISPEC, and TMT/MODHIS are shown in colored
+curves assuming 6 nights of observations over 25 days. The sensitivity demonstrated in this work from ∼ 1.5 nights of KPIC
+observations is labeled as KPIC I (Data). The mass ratios of the Galilean satellites are shown as black dots for comparison.
+Their predicted scaled-up mass ratios, q, accounting for the larger mass, M, of the brown dwarf compared to Jupiter are shown
+as grey crosses (q ∝
+√
+M; Batygin & Morbidelli (2020)). The Roche limit is computed for both a rigid and a fluid satellite
+shown as the inner and outer greyed region respectively.
+as Beta Pictoris Moving Group and the Tucana and
+Horologium Associations.
+The 300 Myr age group is
+representative of the oldest directly imaged substellar
+companions. The RV sensitivity decreases the further
+the system is away at each distinct age. For younger
+systems, there is larger decrease in sensitivity as the
+mass of the planet decreases below ∼ 13 MJup.
+The
+large decrease in RV sensitivity once the object is below
+∼ 13 MJup is due to the onset of deuterium burning for
+brown dwarfs, which makes them much more luminous
+than a planet of a similar mass. Another interesting fea-
+ture in Figure 6 (a) is the apparent independence of the
+RV precision to the brown dwarf mass above ∼ 13 MJup
+at 30 Myr. This can be explained by the facts that the
+RV precision is mostly driven by the brightness of the
+object, and that brown dwarfs have a similar brightness
+over a range of masses around this age. Indeed, larger
+brown dwarfs cool faster than smaller ones resulting in
+the different cooling curves to meet over a small range
+of brightness around 30 Myr as illustrated in Figure 7
+in Burrows et al. (1997).
+Figure 6 (b) shows the moons that could be detected
+around a planet from Figure 6 (a) if they were placed
+at the distance of Callisto. For each planetary mass and
+distance, we create an RV time series assuming six full
+eight-hour nights of observations over 25 days with er-
+ror bars that represent the RV sensitivity calculated by
+PSIsim.
+The detection threshold was computed from
+simulated data created by RVsearch as in section 3.
+For more massive planets and brown dwarfs, we expect
+TMT/MODHIS to reach the RV sensitivity needed to
+look for close in moons with mass ratios smaller than
+10−4 around brown-dwarfs, similar to the ones found in
+the solar system for a median age of 30 Myr. However, to
+detect moons around lower mass, directly imaged plan-
+ets of the same age, we are sensitive to mass ratios of
+10−3 or larger.
+5. DISCUSSION
+5.1. The viability of exomoon RV searches
+Using KPIC, we derive the most sensitive upper limits
+on the mass ratio of satellites orbiting a high-contrast
+substellar companion.
+We rule out satellites larger
+than 1-4% the mass of the brown dwarf HR 7672 B
+at separations similar to the Galilean moons.
+Based
+on end-to-end simulations, we predict that instruments
+such as TMT/MODHIS could be two orders of mag-
+nitude more sensitive. This would be sufficient to de-
+tect moons forming in the CPD of a planet with mass
+ratios of ∼ 10−4, albeit with a substantial investment
+in observing time.
+If the satellite to planet mass ra-
+tio grows as q ∝
+√
+M, with M the mass of the planet,
+
+12
+Ruffio et al.
+5
+10
+15
+20
+25
+Planet Magnitude
+500
+1000
+1500
+2000
+2500
+Teff (K)
+0.5 m/s
+1 m/s
+2 m/s
+5 m/s
+20 m/s
+100 m/s
+500 m/s
+1 km/s
+3 km/s
+100
+101
+102
+103
+RV Sensitivity (m/s)
+(a) RV sensitivity
+100
+101
+102
+1 day
+1 m/s
+10 m/s
+100 m/s
+1 km/s
+10 km/s
+1 mo
+1 m/s
+10 m/s
+100 m/s
+1 km/s
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Mass ratio (q)
+100
+101
+102
+Planet mass (MJup)
+1 yr
+0.1 m/s
+1 m/s
+10 m/s
+100 m/s
+1 km/s
+10
+4
+10
+3
+10
+2
+10
+1
+100
+10 yr
+0.1 m/s
+1 m/s
+10 m/s
+100 m/s
+1 km/s
+(b) RV semi-amplitude
+Figure 5. RV precision of MODHIS. (Left) The RV sensitivity of MODHIS for model planets around a HR 8799-like host star
+using BT-Settl models (Allard et al. 2012a). The RV sensitivity was predicted using PSIsim for a single, two hour exposure.
+Both the contour curves and color map indicate the RV sensitivity for a specified effective temperature and apparent magnitude
+of the model planet. The RV sensitivity relies more on the brightness of the planet than its effective temperature. However, the
+RV sensitivity decreases for planets with temperatures between 1500 and 1700 K due to the L-T transition. (Right) The RV
+semi-amplitude for different planet masses and mass ratios. Note, increasing the exposure time will increase the RV sensitivity.
+the Keck/HISPEC should be sensitive to these objects
+around brown dwarfs. Any detection with HISPEC, or
+lack thereof, will therefore already be capable of con-
+straining CPD formation models. In order to validate
+our instrument simulations, we compared them with
+existing observations.
+The gap in sensitivity can be
+explained by imperfections in the data reductions. A
+continued investment in more accurate data processing
+algorithms or observing strategies is therefore required
+in order to realize these predictions. Planet variability
+will also be a challenge to overcome using the different
+timescales and the wavelength dependence of the vari-
+ability compared an exomoon signal for example (Van-
+derburg et al. 2018). Measuring the variability of sub-
+stellar companions would in fact be an important result
+of exomoon surveys to better understand the physics of
+their atmospheres (Biller 2017).
+Binary formation processes favor high-mass ratios so
+they would be more easily detectable than the smaller
+satellites forming by accretion in the CPD. The ma-
+jority of multiplicity surveys for isolated brown dwarfs
+(Fontanive et al. 2018), or companion brown dwarfs
+(Burgasser et al. 2005; Lazzoni et al. 2020), have
+searched for visual companions, leaving the separation
+regime of < 1 au underexplored.
+Figure 5 (b) shows
+that unresolved binary substellar companion would be
+detectable with RV precision between 0.1 − 1 km.s−1,
+which is already routinely achieved with KPIC. As an
+example, the measured dynamical mass of the brown
+dwarf companion HD 47127 B suggest that it could be
+a binary (Bowler et al. 2021), but this specific compan-
+ion is too faint (K ∼ 18.4) to be a practical target for
+KPIC.
+From Appendix A and Figure 7, we conclude that the
+different detection techniques are sensitive to distinct
+regions of the parameter space, and therefore comple-
+mentary, not unlike exoplanet searches.
+If exomoons
+follow the model of solar system gas giant satellites, RV
+searches could be the most promising approach due to
+its sensitivity to short period moons. However, unless
+the theoretical prediction that bigger planets form even
+bigger moons hold true, small satellites with mass ratios
+∼ 10−4 might only be detectable around brown dwarfs.
+5.2. Detections of moons using the
+Rossiter-McLaughlin effect
+An alternative strategy to look for exomoons around
+directly imaged planets using RV measurements could
+be to look for transiting moons through the Rossiter-
+McLaughlin (RM;
+Gaudi & Winn 2007) effect on the
+planet. Precise photometric calibration and stability of
+high-constrast instrument is notoriously difficult (Wang
+et al. 2022b), so detecting a RM event during a transit
+
+RV detection of exomoons
+13
+2
+5
+10
+20
+30
+50
+70
+2 m/s
+3 m/s
+4 m/s
+5 m/s
+10 m/s
+50 m/s
+4
+10
+20
+30
+50
+70
+Planet Mass (MJup)
+2 m/s
+3 m/s
+4 m/s
+5 m/s
+10 m/s
+50 m/s
+100 m/s
+20
+40
+60
+80
+100
+120
+140
+Distance (pc)
+11
+20
+30
+40
+50
+60
+70
+2 m/s
+3 m/s
+4 m/s
+5 m/s
+20 m/s
+30 m/s
+40 m/s
+50 m/s
+100 m/s
+200 m/s
+100
+101
+102
+103
+RV Sensitivity (m/s)
+3 Myr
+30 Myr
+300 Myr
+(a) RV sensitivity
+2
+5
+10
+20
+30
+50
+70
+2.5e-5
+5e-5
+1e-4
+5e-4
+1e-3
+4
+10
+20
+30
+50
+70
+Planet Mass (MJup)
+2.5e-5
+5e-5
+1e-4
+5e-4
+1e-3
+5e-3
+1e-2
+20
+40
+60
+80
+100
+120
+140
+Distance (pc)
+11
+20
+30
+40
+50
+60
+70
+2.5e-5
+5e-5
+1e-4
+5e-4
+1e-3
+5e-3
+1e-2
+10
+5
+10
+4
+10
+3
+10
+2
+(MMoon/MBD) sin i
+3 Myr
+30 Myr
+300 Myr
+(b) Detectable mass ratio
+Figure 6. RV precision and detectable mass ratio of MODHIS similar to Figure 5, but as a function of planet mass, distance,
+and age of the system. (Left) BT-Settl evolutionary models (Allard et al. 2012b) were used to infer the mass of the planet and
+distance to the system at an age of 3, 30, and 300 Myr. The minor contour lines cover an evenly spaced, 50 step log scale from
+0 to 1 km/s. RV sensitivity decreases the further the system is away and the lower in mass the planet is. The large decrease in
+RV sensitivity when the companion mass is below ∼ 13 MJup for young systems is due to the difference in cooling rates between
+brown dwarfs and planets over time. (Right) The mass ratio detectable by MODHIS assuming a fixed semi major axis for the
+moon equal to that of Callisto (≈ 26RJup). For each planetary mass and distance from panel (a), we create an RV time series
+assuming six nights of observations over 25 days with error bars that represent the RV sensitivity calculated by PSIsim.
+could be easier than detecting its photometric counter-
+part.
+An RM event consists of the subsequent masking of a
+portion of the blue and red-shifted areas of the surface
+of a spinning object, therefore leading to large and very
+distinct deviations of the measured RV. The amplitude
+of the RV signal can be hundreds of times larger than
+the RV semi amplitude due to the orbital motion of the
+moon. Its amplitude is proportional to the spin of the
+planet, which could make it an interesting alternative
+to detect the smallest moons around rapidly rotating
+planets and brown dwarfs. Indeed, the RV uncertainties
+scale with the spin of the object so detecting the orbital
+signal of small exomoons could be more challenging.
+The Galilean moons have rather small orbital periods
+from days to weeks. Assuming a random inclination dis-
+
+14
+Ruffio et al.
+tribution, the transit probability of a moon (P) is given
+by the ratio of the planet radius (Rp) and the moon
+semi-major axis (dm), P = Rp/dm (Borucki & Sum-
+mers 1984). Therefore, the probability of a transit of
+a moon at the separation of Io around Jupiter is 1:6,
+and 1:27 for the farthest Galilean moon Callisto. As-
+suming a full 8 hour night of observations, we estimate
+the probability of observing an RM event for Galilean-
+like moons around a Jupiter like planet to be around 3%
+for Io, 1% for Europa, 0.3% for Ganymede, and 0.07%
+for Callisto. However, the orbital periods of the moons
+would be even shorter around larger substellar compan-
+ions, which would increase the probabilities up to 17%
+for Io, 8% for Europa, 2.6% for Ganymede, and 0.6% for
+Callisto . The transits would last between ∼2-5 hours
+for the Galilean moons around Jupiter, but they would
+only last 15-30 minutes for similar moons around HR
+7672 B.
+As an example, a satellite around HR 7672 B with a
+mass of 1M⊕ (q = 5×10−5) would generate a RM signal
+of ∼ 300 m.s−1 compared to the ∼ 0.5 m.s−1 generated
+by the orbital motion (Gaudi & Winn 2007). The am-
+plitude would be ∼ 5 km.s−1 for a Neptune-size moon.
+Multiple satellite systems would increase the probability
+of a detection. Given the low detection probability, RM
+searches could be carried out in synergy with other sci-
+ence cases such as brown dwarf variability (Biller 2017).
+For example, Doppler spectroscopy also favors long ob-
+servations of rapidly rotating objects, which would make
+for ideal datasets for exomoon RM searches.
+5.3. Searching for Pandora: Habitable exomoons
+Estimating the occurrence rate of Earth-sized exo-
+planets in the habitable zone (HZ) of Sun-like star,
+called η⊕, has been an important goal of exoplanet sur-
+veys. While such planets remain challenging to detect,
+the best estimates of η⊕ range between 5 − 50% to date
+(Gaudi et al. 2021). However, these are not the only
+Earth-sized objects that could harbor life in the HZ of
+their stars. Any rocky satellites orbiting HZ gas giant
+planets could also provide suitable conditions for life.
+Close-in exomoons can be protected from stellar radia-
+tion by the strong magnetic field of Jovian mass planets
+(Heller & Zuluaga 2013).
+Integrating the distribution of gas giants with an inci-
+dent flux between 0.3−1.5 times the solar irradiance on
+Earth for an optimistic habitable zone, or 0.3 − 1 for a
+conservative habitable zone (Kasting & Harman 2013),
+yields about 5 − 7 giant planets per hundred FGKM
+stars. This is using the giant planet (30 − 6000M⊕ sin i)
+occurrence rates derived from the California Legacy Sur-
+vey as a function of stellar irradiation (figure 11; Fulton
+et al. 2021). Given that each planet can have multiple
+satellites, this could represent a significant number of
+habitable Earth-size moons that are not accounted for
+in η⊕. The occurence rate of habitable exomoons could
+be constrained by measuring the population of satellites
+around more distant directly imaged planets and brown-
+dwarfs.
+6. CONCLUSION
+In this work, we aimed at evaluating the prospects
+for radial velocity (RV) detections of exomoons around
+self-luminous directly-imaged planets. We used real ob-
+servations as well as end-to-end simulations of future
+facilities at the Keck observatory and the Thirty Meter
+Telescope (TMT). Using data from KPIC, we were able
+to derive upper limits for satellites orbiting the brown
+dwarf companion HR 7672 B at a mass ratio of 1−4% for
+separations similar to the Galilean moons. Current in-
+strumentation is already sensitive to unresolved binary
+companions that could form through gravitational in-
+stability. We demonstrate that future thirty-meter class
+telescopes will likely push the sensitivity down to the
+mass ratios of solar system satellites (∼ 10−4), which are
+thought to form in a circumplanetary disk. We note that
+second generation instruments like Keck/HISPEC on
+current ten meter class telescopes might be sufficient to
+detect these moons if theoretical predictions that larger
+planets form even larger moons hold true. Everything
+else being equal and considering the RV signal from the
+orbital motion of the moon, the deepest exomoon sen-
+sitivity will be reached for the brightest substellar com-
+panions with the smallest spin. Small moons could also
+be detected from their Rossiter-McLaughlin (RM) ef-
+fect on the planetary RV signal. An RM event can be
+orders of magnitude larger than the orbital signal, albeit
+with percents level detection probability assuming a full
+night of observation. We conclude that the detection of
+exomoons from planetary RV surveys is now becoming
+a reality thanks to the development of high-resolution
+spectrographs dedicated to directly imaged planets.
+
+RV detection of exomoons
+15
+J.-B. R. acknowledges support from the David and Ellen
+Lee Prize Postdoctoral Fellowship.
+1
+2
+Funding for KPIC has been provided by the California
+Institute of Technology, the Jet Propulsion Laboratory,
+the Heising-Simons Foundation through grants #2019-
+1312 and #2015-129, the Simons Foundation, and the
+United States National Science Foundation Grant No.
+AST-1611623.
+3
+4
+5
+6
+7
+8
+Ji Wang acknowledges the support by the National
+Science Foundation under Grant No. 2143400.
+9
+10
+The W. M. Keck Observatory is operated as a scien-
+tific partnership among the California Institute of Tech-
+nology, the University of California, and NASA. The
+Keck Observatory was made possible by the generous
+financial support of the W. M. Keck Foundation. We
+also wish to recognize the very important cultural role
+and reverence that the summit of Maunakea has always
+had within the indigenous Hawaiian community. We are
+most fortunate to have the opportunity to conduct ob-
+servations from this mountain.
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+Facilities: Keck II (KPIC)
+Software:
+astropy8 (Astropy Collaboration et al.
+2013), Matplotlib9 (Hunter 2007), PSIsim10, RVSearch11
+(Rosenthal et al. 2021), KPIC Data Reduction Pipeline12
+(Delorme et al. 2021), BREADS13 (Ruffio et al. 2021;
+Agrawal 2022),
+APPENDIX
+8 http://www.astropy.org
+9 https://matplotlib.org
+10 https://github.com/planetarysystemsimager/psisim
+11 https://github.com/California-Planet-Search/rvsearch
+12 https://github.com/kpicteam/kpic pipeline
+13 https://github.com/jruffio/breads
+
+16
+Ruffio et al.
+100
+101
+102
+103
+104
+Semi-major axis (RJup)
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+(MMoon/Mplanet) sini
+Hill sphere
+Roche limit
+10 as astrometry
+100 as astrometry
+Io
+Europa
+Ganymede
+Callisto
+Titan
+Triton
+Titania
+Oberon
+Keck
+TMT
+VLTI
+KPIC I (Data)
+KPIC I
+KPIC II
+HISPEC
+MODHIS
+1h
+1d
+1wk 1mo
+1yr
+10yr
+100yr
+Period
+100
+101
+102
+103
+104
+(MMoon/MEarth) sini
+(a) HR 7672 B
+100
+101
+102
+103
+104
+Semi-major axis (RJup)
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+(MMoon/Mplanet) sini
+Hill sphere
+Roche limit
+10 as astrometry
+100 as astrometry
+Io
+Europa
+Ganymede
+Callisto
+Titan
+Triton
+Titania
+Oberon
+Keck
+TMT
+VLTI
+KPIC I
+KPIC II
+HISPEC
+MODHIS
+1h
+1d
+1wk 1mo
+1yr
+10yr
+100yr
+Period
+10
+1
+100
+101
+102
+103
+(MMoon/MEarth) sini
+(b) HR 8799 c
+Figure 7.
+Similar to Figure 4, but including idealized exomoon sensitivities of alternative detection techniques. The diagonal
+dashed black lines represent the simplified sensitivity of VLTI/Gravity through astrometry. The vertical gray scale bars represent
+the diffraction limit of different telescopes for direct imaging of satellites, namely the W. M. Keck observatory, the future Thirty
+Meter Telescope (TMT), and the Very Large Telescope Interferometer (VLTI).
+APPENDIX
+A. COMPARING TO OTHER DETECTION METHODS
+Alternative exomoon detection techniques include astrometry and direct imaging of imaged planets. Figure 7 shows
+their idealized detection limits to be compared to the RV sensitivity originally presented in Figure 4.
+With an
+astrometric precision of 10 − 100 µas (Gravity Collaboration et al. 2021), interferometry with VLTI/GRAVITY could
+be sensitive to moons further away than radial velocity, but remains limited by the orbital period of the satellite at
+the furthest separations. The simplified detection limits are computed by matching the astrometric precision (σastro)
+of VLTI/GRAVITY with the amplitude of the planet astrometric displacement in the sky around the center of mass.
+The smallest detectable mass ratio (q) is given by
+q =
+�
+2 ∗
+�sma
+1 au
+� �1 pc
+d
+� � 1 as
+σastro
+�
+− 1
+�−1
+,
+(A1)
+with d the distance of the star to the Sun, and sma the semi major axis of the moon. We use the diffraction limit
+of the telescope to illustrate the parameter space that might be accessible to direct imaging. More specifically, the
+detection threshold is taken at twice the spatial resolution of the telescope (∼ 2λ/D) with D the diameter of the
+telescope and λ = 2 µm. Unfortunately, estimating the brightness of low mass objects (< 1MJup) remains challenging
+and will depend on the age of the system, so we arbitrarily chose a lower limit of one Jupiter mass for Keck and VLTI,
+and a mass similar to the solar system ice giants for TMT. Direct imaging would be sensitive to the longest periods
+and largest moons.
+REFERENCES
+Agnor, C. B., & Hamilton, D. P. 2006, Nature, 441, 192,
+doi: 10.1038/nature04792
+Agrawal, S. 2022, Senior thesis (Major), California Institute
+of Technology, doi: 10.7907/17sv-vf40
+
+RV detection of exomoons
+17
+Allard, F., Homeier, D., & Freytag, B. 2012a, Philosophical
+Transactions of the Royal Society of London Series A,
+370, 2765, doi: 10.1098/rsta.2011.0269
+Allard, F., Homeier, D., Freytag, B., & Sharp, C. M. 2012b,
+in EAS Publications Series, Vol. 57, EAS Publications
+Series, ed. C. Reyl´e, C. Charbonnel, & M. Schultheis,
+3–43, doi: 10.1051/eas/1257001
+Asphaug, E., & Emsenhuber, A. 2018, in European
+Planetary Science Congress, EPSC2018–569
+Astropy Collaboration, Robitaille, T. P., Tollerud, E. J.,
+et al. 2013, A&A, 558, A33,
+doi: 10.1051/0004-6361/201322068
+Batalha, N. M. 2014, Proceedings of the National Academy
+of Science, 111, 12647, doi: 10.1073/pnas.1304196111
+Batygin, K., & Morbidelli, A. 2020, ApJ, 894, 143,
+doi: 10.3847/1538-4357/ab8937
+Benisty, M., Bae, J., Facchini, S., et al. 2021, ApJL, 916,
+L2, doi: 10.3847/2041-8213/ac0f83
+Biller, B. 2017, The Astronomical Review, 13, 1,
+doi: 10.1080/21672857.2017.1303105
+Blunt, S., Wang, J. J., Angelo, I., et al. 2020, AJ, 159, 89,
+doi: 10.3847/1538-3881/ab6663
+Boccaletti, A., Chauvin, G., Lagrange, A. M., & Marchis,
+F. 2003, A&A, 410, 283, doi: 10.1051/0004-6361:20031216
+Bond, C. Z., Cetre, S., Lilley, S., et al. 2020, Journal of
+Astronomical Telescopes, Instruments, and Systems, 6,
+039003, doi: 10.1117/1.JATIS.6.3.039003
+Borucki, W. J., & Summers, A. L. 1984, Icarus, 58, 121,
+doi: 10.1016/0019-1035(84)90102-7
+Bowler, B. P., Endl, M., Cochran, W. D., et al. 2021, ApJL,
+913, L26, doi: 10.3847/2041-8213/abfec8
+Brandt, T. D., Dupuy, T. J., & Bowler, B. P. 2019, AJ,
+158, 140, doi: 10.3847/1538-3881/ab04a8
+Burgasser, A. J., Kirkpatrick, J. D., & Lowrance, P. J.
+2005, AJ, 129, 2849, doi: 10.1086/430218
+Burrows, A., Marley, M., Hubbard, W. B., et al. 1997, ApJ,
+491, 856, doi: 10.1086/305002
+Cale, B., Plavchan, P., LeBrun, D., et al. 2019, AJ, 158,
+170, doi: 10.3847/1538-3881/ab3b0f
+Canup, R. M., & Asphaug, E. 2001, Nature, 412, 708
+Canup, R. M., & Ward, W. R. 2006, Nature, 441, 834,
+doi: 10.1038/nature04860
+Chen, J., & Kipping, D. 2017, ApJ, 834, 17,
+doi: 10.3847/1538-4357/834/1/17
+Crepp, J. R., Johnson, J. A., Fischer, D. A., et al. 2012,
+ApJ, 751, 97, doi: 10.1088/0004-637X/751/2/97
+Currie, T., Burrows, A., Itoh, Y., et al. 2011, ApJ, 729,
+128, doi: 10.1088/0004-637X/729/2/128
+Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003,
+VizieR Online Data Catalog, II/246
+Delorme, J.-R., Jovanovic, N., Echeverri, D., et al. 2021,
+Journal of Astronomical Telescopes, Instruments, and
+Systems, 7, 035006, doi: 10.1117/1.JATIS.7.3.035006
+Echeverri, D., Jovanovic, N., Delorme, J.-R., et al. 2022, in
+Ground-based and Airborne Instrumentation for
+Astronomy IX, ed. C. J. Evans, J. J. Bryant, &
+K. Motohara, Vol. 12184, International Society for Optics
+and Photonics (SPIE), 121841W, doi: 10.1117/12.2630518
+Finnerty, L., Schofield, T., Delorme, J.-R., et al. 2022, in
+Ground-based and Airborne Instrumentation for
+Astronomy IX, ed. C. J. Evans, J. J. Bryant, &
+K. Motohara, Vol. 12184, International Society for Optics
+and Photonics (SPIE), 121844Y, doi: 10.1117/12.2630276
+Fitzgerald, M. P., Sallum, S., Millar-Blanchaer, M. A., et al.
+2022, in Ground-based and Airborne Instrumentation for
+Astronomy IX, ed. C. J. Evans, J. J. Bryant, &
+K. Motohara, Vol. 12184, International Society for Optics
+and Photonics (SPIE), 1218426, doi: 10.1117/12.2630410
+Fontanive, C., Biller, B., Bonavita, M., & Allers, K. 2018,
+MNRAS, 479, 2702, doi: 10.1093/mnras/sty1682
+Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman,
+J. 2013, PASP, 125, 306, doi: 10.1086/670067
+Fulton, B. J., Rosenthal, L. J., Hirsch, L. A., et al. 2021,
+ApJS, 255, 14, doi: 10.3847/1538-4365/abfcc1
+Gaudi, B. S., Meyer, M., & Christiansen, J. 2021, in
+ExoFrontiers; Big Questions in Exoplanetary Science, ed.
+N. Madhusudhan, 2–1, doi: 10.1088/2514-3433/abfa8fch2
+Gaudi, B. S., & Winn, J. N. 2007, ApJ, 655, 550,
+doi: 10.1086/509910
+Gravity Collaboration, Abuter, R., Amorim, A., et al. 2021,
+A&A, 647, A59, doi: 10.1051/0004-6361/202040208
+Heller, R., & Zuluaga, J. I. 2013, ApJL, 776, L33,
+doi: 10.1088/2041-8205/776/2/L33
+Howard, A. W., & Fulton, B. J. 2016, PASP, 128, 114401,
+doi: 10.1088/1538-3873/128/969/114401
+Hsu, C.-C., Burgasser, A. J., Theissen, C. A., et al. 2021,
+ApJS, 257, 45, doi: 10.3847/1538-4365/ac1c7d
+Hunter, J. D. 2007, Computing In Science & Engineering,
+9, 90, doi: 10.1109/MCSE.2007.55
+Husser, T. O., Wende-von Berg, S., Dreizler, S., et al. 2013,
+A&A, 553, A6, doi: 10.1051/0004-6361/201219058
+Jovanovic, N., Guyon, O., Kotani, T., et al. 2017, arXiv
+e-prints, arXiv:1712.07762.
+https://arxiv.org/abs/1712.07762
+Jovanovic, N., Calvin, B., Porter, M., et al. 2020, in Society
+of Photo-Optical Instrumentation Engineers (SPIE)
+Conference Series, Vol. 11447, Society of Photo-Optical
+Instrumentation Engineers (SPIE) Conference Series,
+114474U, doi: 10.1117/12.2563107
+
+18
+Ruffio et al.
+Kasting, J. F., & Harman, C. E. 2013, Nature, 504, 221,
+doi: 10.1038/504221a
+King, I. R. 1983, PASP, 95, 163, doi: 10.1086/131139
+Kipping, D., Bryson, S., Burke, C., et al. 2022, Nature
+Astronomy, 6, 367, doi: 10.1038/s41550-021-01539-1
+Kipping, D. M. 2013, MNRAS, 434, L51,
+doi: 10.1093/mnrasl/slt075
+Kipping, D. M., Bakos, G. ´A., Buchhave, L., Nesvorn´y, D.,
+& Schmitt, A. 2012, ApJ, 750, 115,
+doi: 10.1088/0004-637X/750/2/115
+Kipping, D. M., Schmitt, A. R., Huang, X., et al. 2015,
+ApJ, 813, 14, doi: 10.1088/0004-637X/813/1/14
+Lazzoni, C., Desidera, S., Gratton, R., et al. 2022, MNRAS,
+doi: 10.1093/mnras/stac2081
+Lazzoni, C., Zurlo, A., Desidera, S., et al. 2020, A&A, 641,
+A131, doi: 10.1051/0004-6361/201937290
+Limbach, M. A., Vos, J. M., Winn, J. N., et al. 2021, ApJL,
+918, L25, doi: 10.3847/2041-8213/ac1e2d
+Liu, M. C., Fischer, D. A., Graham, J. R., et al. 2002, ApJ,
+571, 519, doi: 10.1086/339845
+Luck, R. E. 2017, AJ, 153, 21,
+doi: 10.3847/1538-3881/153/1/21
+Mawet, D., Delorme, J. R., Jovanovic, N., et al. 2017, in
+Society of Photo-Optical Instrumentation Engineers
+(SPIE) Conference Series, Vol. 10400, Society of
+Photo-Optical Instrumentation Engineers (SPIE)
+Conference Series, ed. S. Shaklan, 1040029,
+doi: 10.1117/12.2274891
+Mawet, D., Fitzgerald, M., Konopacky, Q., et al. 2019, in
+Bulletin of the American Astronomical Society, Vol. 51,
+134. https://arxiv.org/abs/1908.03623
+Mawet, D., Fitzgerald, M. P., Konopacky, Q., et al. 2022, in
+Ground-based and Airborne Instrumentation for
+Astronomy IX, ed. C. J. Evans, J. J. Bryant, &
+K. Motohara, Vol. 12184, International Society for Optics
+and Photonics (SPIE), 121841R, doi: 10.1117/12.2630142
+Molli`ere, P., Wardenier, J. P., van Boekel, R., et al. 2019,
+arXiv e-prints, arXiv:1904.11504.
+https://arxiv.org/abs/1904.11504
+Otten, G. P. P. L., Vigan, A., Muslimov, E., et al. 2021,
+A&A, 646, A150, doi: 10.1051/0004-6361/202038517
+Rosenthal, L. J., Fulton, B. J., Hirsch, L. A., et al. 2021,
+ApJS, 255, 8, doi: 10.3847/1538-4365/abe23c
+Royer, F., Zorec, J., & G´omez, A. E. 2007, A&A, 463, 671,
+doi: 10.1051/0004-6361:20065224
+Ruffio, J.-B., Macintosh, B., Konopacky, Q. M., et al. 2019,
+AJ, 158, 200, doi: 10.3847/1538-3881/ab4594
+Ruffio, J.-B., Konopacky, Q. M., Barman, T., et al. 2021,
+AJ, 162, 290, doi: 10.3847/1538-3881/ac273a
+Shen, Y., & Tremaine, S. 2008, AJ, 136, 2453,
+doi: 10.1088/0004-6256/136/6/2453
+Snellen, I., de Kok, R., Birkby, J. L., et al. 2015, A&A, 576,
+A59, doi: 10.1051/0004-6361/201425018
+Spalding, C., Batygin, K., & Adams, F. C. 2016, ApJ, 817,
+18, doi: 10.3847/0004-637X/817/1/18
+Teachey, A., & Kipping, D. M. 2018, Science Advances, 4,
+eaav1784, doi: 10.1126/sciadv.aav1784
+Teachey, A., Kipping, D. M., & Schmitt, A. R. 2018, AJ,
+155, 36, doi: 10.3847/1538-3881/aa93f2
+Vanderburg, A., Rappaport, S. A., & Mayo, A. W. 2018,
+AJ, 156, 184, doi: 10.3847/1538-3881/aae0fc
+Vanderburg, A., & Rodriguez, J. E. 2021, ApJL, 922, L2,
+doi: 10.3847/2041-8213/ac33b4
+Villanueva, G. L., Smith, M. D., Protopapa, S., Faggi, S., &
+Mandell, A. M. 2018, JQSRT, 217, 86,
+doi: 10.1016/j.jqsrt.2018.05.023
+Wang, J., Wang, J. J., Ma, B., et al. 2020, AJ, 160, 150,
+doi: 10.3847/1538-3881/ababa7
+Wang, J., Kolecki, J. R., Ruffio, J.-B., et al. 2022a, AJ, 163,
+189, doi: 10.3847/1538-3881/ac56e2
+Wang, J. J., Kulikauskas, M., & Blunt, S. 2021a,
+whereistheplanet: Predicting positions of directly imaged
+companions, Astrophysics Source Code Library, record
+ascl:2101.003. http://ascl.net/2101.003
+Wang, J. J., Graham, J. R., Dawson, R., et al. 2018, AJ,
+156, 192, doi: 10.3847/1538-3881/aae150
+Wang, J. J., Ruffio, J.-B., Morris, E., et al. 2021b, AJ, 162,
+148, doi: 10.3847/1538-3881/ac1349
+Wang, J. J., Delorme, J.-R., Ruffio, J.-B., et al. 2021c, in
+Society of Photo-Optical Instrumentation Engineers
+(SPIE) Conference Series, Vol. 11823, Society of
+Photo-Optical Instrumentation Engineers (SPIE)
+Conference Series, 1182302, doi: 10.1117/12.2596484
+Wang, J. J., Gao, P., Chilcote, J., et al. 2022b, arXiv
+e-prints, arXiv:2208.05594.
+https://arxiv.org/abs/2208.05594
+Xuan, J. W., Wang, J., Ruffio, J.-B., et al. 2022, ApJ, 937,
+54, doi: 10.3847/1538-4357/ac8673
+Yi, X., Vahala, K., Li, J., et al. 2016, Nature
+Communications, 7, 10436, doi: 10.1038/ncomms10436
+

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+ 
+1 
+Improve the Field Strength by Adding Soft Iron  
+in the Hybrid Permanent magnet  
+Quanling Peng1,2, Jianxin Zhou1,  Saike Tian1, Yingzhe Wang1 
+1. Institute of High Energy Physics, Chinese Academy of sciences, Beijing, 100049, China 
+2. School of electronics, Nanchang institute of Technology, Jiangxi, 330013, China. 
+ 
+Abstract—Permanent magnet has a small and compact structure, is especially suitable for a narrow space. With the aid of soft 
+iron, the magnetic field can be increased much more and the field uniformity can be well controlled. Most Permanent magnets 
+have a symmetry structure; the soft iron can be selected as the float magnet pole to keep its constant scalar potential, and take 
+roles as media to collect the direct flux from the permanent blocks, then release indirect flux in magnet aperture and to the 
+nearby return yoke. This paper presents the magnetic flux method to design and fabricate a hybrid permanent dipole by using 
+axially and radially magnetized permanent blocks. A variable gradient permanent quadrupole and a variable gradient sextupole 
+are designed as the extend design examples . They all consist of two nested hybrid permanent rings, where the iron poles are 
+used to control the field quality, collect the magnetic flux from the outer ring, block the skew quadrupole and high order 
+harmonics.    
+Index Terms—Pure permanent magnet, hybrid permanent magnet, direct flux, indirect flux, variable gradient 
+quadrupole.  
+ 
+1. Introduction 
+Electromagnet or superconducting magnets, with field strength varies along with the beam energy, are widely used in 
+modern particle accelerators to bend or focus the particle beams. In some circumstances, where the beam energy is fixed or only 
+a little adjustment, permanent magnet will be a better selection, since it has advantages of small space occupation, no cooling 
+system, and no operation costs, one time investment can maintain a long time operation.  
+Permanent magnet can be made of all permanent blocks or permanent blocks mixed with soft iron, where the former called 
+as pure permanent magnet, later called as the hybrid permanent magnet. For pure permanent magnet design, K. Halbach had 
+given the design principle in 1980s [1-2]. Each permanent block can be treated as air with the current loops surrounding the 
+magnet or as magnetic charges on its surfaces along the easy axis [3-4]. Field strength in magnet aperture is collected the 
+contribution in each permanent block by its position and easy axis orientation.  
+An important issue is that field strength from the pure permanent magnet is weak compared with electromagnets or 
+superconducting magnets. According to reference [1], even with the highest residual field strength, maximum field of a dipole 
+that built with pure permanent blocks is less than the remnant field of Br, say about 1.4 T for the highest NdFeB magnetic 
+materials. On the other hand, pure permanent magnet needs permanent blocks with different easy axis orientation, that will bring 
+fabrication difficulties and increase the cost [5-9]. For a hybrid permanent magnet, on the other hand, with the nonlinear material 
+such as iron poles to collect the surrounding magnetic flux and release it into a small compact space, it can reach to a higher 
+magnetic field. Another way to increase the field strength is to add more permanent blocks surround the iron pole to add more 
+magnetic flux. Variable field permanent magnets can be built by adding auxiliary coils around the iron pole or adding an outer 
+concentric permanent ring surround the inner permanent ring [10].   
+ 
+2. Principle of flux method  
+Assume a permanent magnet has an infinite length, the 3D case can be treated in 2D case. In 2D analysis, a homogeneously 
+magnetized permanent block can be treated as an air space with the surface charge density =Br on its upper and lower 
+surfaces or a surface current distribution of density i = Hc around the other four surfaces [3-4]. Hear Br and Hc are the remnant 
+field and the coercively of the permanent block respectively, with Br=0Hc. For the hard permanent magnetic material, the slope 
+of demagnetized curve in the second quarter is near 1, or μr=1, no extra field contribution from the material magnetization. 
+In 2D non-current space, magnetic field can be expressed as the negative gradient of scalar potential as
+V
+
+
+B
+. As shown 
+in Fig. 1, in a hybrid permanent magnet, the permanent block can be modeled by charge sheets at the top and bottom surfaces. 
+The iron pole has large relative permeability, all the surface of the magnet pole keep a constant scalar potential of V2=V0, the 
+return yoke and mid-plane are all kept in zero scalar potential. Since V1 and V2 are in different scalar potential, magnetic field 
+will be produced between the magnet pole to the mid-plane or to the return yoke, which call as useful field and stray field 
+respectively.  
+@Manuscript received Jan 2, 2023. Work supported by the accelerator research program of the Chinese Academy of Sciences Grant No. Y5294104TD. 
+Author email address: pengql@ihep.ac.cn. 
+
+ 
+2 
+ 
+Fig. 1 Different scalar potentials in a quarter of the hybrid permanent dipole, the shade region represent the iron yoke and 
+iron pole. 
+2.1 Direct flux φd 
+Direct flux is magnetic flux coming from the permanent magnet blocks and deposits on the iron pole, it is the source to 
+maintain iron pole in a higher scalar potential. The direct flux to the iron pole is  
+        ，   
+     
+  .                (1) 
+c is the fraction of magnetic charge σ deposited on the surface of the iron pole, it equals to the scalar potential V at the 
+magnetic charges with respect to the potential V0 at the iron pole, D is the width of the permanent block. Assume the scalar 
+potential changes uniformly in the permanent block, then   
+  
+ .In Fig. 1, the direct flux includes the contributions from the 
+positive and the negative charges, which can be expressed as  
+                       ,      (2) 
+here    
+      
+  
+  ,    
+      
+  
+ 
+  
+ . If magnet blocks directly touch on the iron pole and leave some space h1 between the 
+top yoke, eq. (2) can be written as: 
+      (  
+  
+ )     
+  
+  .            (3) 
+If the permanent block fully occupies the space between the iron pole and top yoke, then the direct flux on the iron pole is: 
+     .                                      (4) 
+In 3d case, the direct flux deposited on the iron pole is: 
+     ,                                      (5) 
+S is the surface area of the permanent block.                 
+                                      
+2.2 Indirect flux φi  
+Indirect flux escapes from the iron pole faces to the nearby zero scalar potential areas. As shown in Fig. 2, add permanent 
+magnet between the magnet pole and the side yoke, the scalar potential of the iron pole will rise up to V0, the back yoke, the top 
+yoke and the mid-plane still keeps zero scalar potential, parasite magnetic field will be produced between the different scalar 
+potential areas. Here φi1 is the expected field, φi2 is the demagnetization field for the permanent magnet, φi3 is the nearby leakage 
+field, total indirect flux is φi =φi1+φi2 +φi3.                  
+Assume the expected field B0 keeps constant in the magnet gap, with     
+  
+    
+  
+  , h0 is half-length of the magnet gap. 
+The indirect flux on the mid-plane is 
+    
+  
+  ∬          
+  
+   .           (6) 
+In general, indirect flux to nearby zero scalar potential is written as: 
+      
+  
+  .                              (7) 
+S’ is the surface area of the iron pole, h is the distance between the iron pole and the zero potential. Consider the corner 
+effect, the scale factor f>1,  
+                                                  
+2.3 Total magnetic flux 
+From ∯       , total magnetic flux around the float iron pole is zero, that is the direct flux deposits on iron pole equals 
+to the indirect flux leaves away from the iron pole, which can be expressed as:  
+0
+
+
+i
+d
+
+
+.                        (8) 
+  
+
+V1=0
+h1
+Br
+h2
+h
+++
++++
+V2=VO
+ho
+V1=0 
+3 
+ 
+Fig. 2. The indirect flux calculation model. Indirect flux goes from the iron pole to the nearby zero scalar potential area. 
+ 
+3. H type hybrid permanent dipole design   
+A hybrid permanent dipole was fabricated for magnetic material processing, aimed to produce the field higher than 2.4 T in 
+a 7 mm gap. Assume the half gap as h0, the expected magnetic field at the central plane is B0，then the scalar potential on the 
+surface of the iron pole is V0=B0h0。Fig. 3 shows the cross section of the upper half magnet, the iron yokes are displayed in 
+hatch. The top permanent block is magnetized along the negative y, whereas the side permanent block is radially magnetized 
+inward. For the lower half magnet, the side permanent block is radially magnetized outward.      
+The direct fluxes come from the top magnetized block and the side radially magnetized permanent ring, which can be written 
+as: 
+       
+                  .            (9) 
+The indirect fluxes scatter from the iron pole to the nearby zero scalar potential faces, which includes parts to the mid-plane, 
+to the top yoke, to the side yoke and to the upper and lower corners. Here selects factor f as 1.9 to contain the corner effects. 
+Then total indirect flux is  
+           
+    
+      
+    
+      
+           
+     
+   .   (10) 
+ NdFeB N44H material is selected for permanent blocks, the remnant field Br=1.36 T. Other related parameters are: half 
+magnet gap h0=3.5 mm, pole tip length h1=10 mm, distance from pole top to mid-plane h2=20 mm, distance between the top of 
+the side permanent blocks to the mid-plane h3=30 mm, side yoke height h4=40 mm, pole radius R2=14 mm, pole tip radius R1=5 
+mm, radius of the magnet gap R3=32 mm, return yoke thickness 8 mm. Vanadium Iron is select as the magnetic pole, since it 
+has high saturated field as Bs=2.2 T, the return yokes are made of DT4 soft iron. By eq. 10, magnetic field produced in the 
+mid-plane can reach 2.42 T.  OPERA-3d [13] software is used to check the field strength, the calculated peak field on the 
+mid-plane is 2.45 T. 
+In comparison, three cases were calculated when the top permanent blocks removed or replaced with soft iron. Fig 4 shows 
+field differences along the central mid-plane. When the top permanent block was removed, total direct fluxes were reduced, 
+field strength on the mid-plane will drop accordingly. What’s more, when the top permanent blocks are replaced with iron, part 
+of magnetic flux will directly return the top yoke, the field in the magnet gap will decrease much more.  
+ 
+Fig 3. Cross section of the upper half hybrid permanent dipole. The material of the magnet pole and the side yoke are Vanadium 
+and DT4 iron respectively.  
+     
+
+中i2
+:
+i3
+Qil
+DR2.
+4
+5
+Y
+R1
+R3 
+4 
+ 
+Fig 4. Field differences on the mid-plane when the top permanent magnet replaced with air or DT4 iron.  
+ 
+4.  Magnet fabrication and field measurement 
+For technical limitation, the radial magnetized block was replaced by 6 tile-liked blocks. Fig. 5 shows the lower half magnet 
+assebly. In order to protect the permanent blocks, they are covered by a G10 board. Fig. 6 shows the whole magnet assembly. 
+ 
+ 
+Fig.5. Lower half of the hybrid permanent magnet assembly 
+ 
+Field measurement was done by a Hall probe along the slot in the G10 board, Fig. 7 shows the field measurement result, it 
+has a little difference compare with the 3D field calculation. 
+ 
+ 
+ 
+Fig. 6. Whole Hybrid permanent magnet assembly 
+-10
+-5
+0
+5
+10
+0
+0.5
+1
+1.5
+2
+2.5
+3
+x(mm)
+Bz(T)
+ 
+ 
+Top PM
+Top air
+Top iron
+
+ 
+5 
+ 
+ 
+ Fig. 7. Calculated and measured field distribution along the central line in the magnet mid-plane  
+ 
+5. Design variable gradient permanent quadrupole by two nested permanent rings  
+Variable gradient quadrupole can be built with pure or hybrid permanent magnets. Fig 8 shows a kind of pure permanent 
+magnet design, where the variable gradient was realized by the relative rotation between the inner and outer permanent rings, 
+field gradient varies from G1-G2 to G1+G2, here G1 and G2 are field gradient of the inner and outer permanent rings respectively. 
+The nested pure permanent rings have two disadvantages, they are the lower efficiency and the accompanied skew quadrupole 
+components.  
+First, permanent blocks in outer rings are much away from the inner, its field contribution are greatly reduced, which needs 
+larger size and the cost will increase accordingly. On the other hand, since permanent blocks is similar as air, a skew quadrupole 
+component will produce during rotation and cannot be canceled. Skew quadrupole component will give rise to work point drift, 
+increase the beam emittance and will eventually affect the beam life time. Same problem exists in reference [15] for two sets of 
+nest permanent rings that made of cylindrical permanent rods.  
+Another plan is using the hybrid permanent quadrupole, where several permanent blocks are replaced by iron poles, by which 
+to control the field quality and concentrate the magnetic flux. Fig. 9 shows a hybrid permanent quadrupole that consists of two 
+sets of permanent rings, the variable gradient is realized by the relative rotation between the inner and outer ring.  
+In a circular particle accelerator, the ramping period is in a few seconds, gradient changes for a quadrupole can go along with 
+that of the beam energy. According to the design idea for the conventional electromagnet, the inner surface of the iron pole in a 
+hybrid permanent magnet is selected as a part of hyperbola to increase the field uniformity [14]. The outer circular surface of the 
+iron pole is selected as wide enough to collect the magnetic flux from the outer ring. Relative rotation between the two 
+permanent rings does not bring extra skew quadrupole components, since the stray field is blocked by the iron pole.  
+ 
+          
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig 8. A variable gradient quadrupole that consists of two pure permanent rings 
+ 
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+x 10
+4
+r(mm)
+By(G)
+ 
+ 
+calculaled
+measured
+
+22
+21
+20
+23
+19
+24
+18
+8
+2
+17
+9
+33
+25
+10
+16
+11
+15
+26
+12
+32
+13
+14
+27
+31
+28
+29
+30 
+6 
+ 
+ 
+Fig. 9. Variable gradient quadrupole consists of two nest hybrid permanent rings，the dash regions are made of iron.  
+ 
+Fig. 10 shows the 3D field calculation when the outer ring rotated at 60 degrees, The design parameters are: magnet aperture 
+40 mm, outer diameter 320 mm, magnet length is 100 mm. Taking the suitable shimmed on the iron pole surface and end plate, 
+all the high order harmonics can be reduced less than 5 units at different rotation angle. Using FFT function in OPERA-3d, the 
+quadrupole gradient and field harmonics at the reference radius of 13 mm can be found. The calculated gradient varies from 21 
+T/m to 64 T/m in a 90 degrees rotation period, maximum torque is 240 N.m, which can be realized by motors with the reduction 
+gearbox.    
+Fig. 12 shows the normalized high order harmonics along the beam line, all the integral harmonics are less than 5 units. Table 
+1 shows the normal and skew quadrupole values at different rotation angles, where all skew quadrupoles is near to zero.  
+ 
+                             
+Fig. 10. Calculation example of a variable gradient quadrupole that consists of two nest hybrid permanent rings when the 
+outer ring rotation at 60 degrees.  
+ 
+ 
+Fig 11  High order harmonics along the beam line (@r=13mm )  when the outer ring rotaed at 60 degrees, all data are normalized 
+with the integral quadrupole strength.  
+-50
+0
+50
+100
+150
+200
+250
+300
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+2
+z（mm）
+unit of 10 -4
+ 
+ 
+A3
+B3
+A4
+B4
+A5
+B5
+
+ 
+7 
+ 
+Table 1 Normal and skew quadrupole components changes at difference rotation angles 
+Rotation 
+angles 
+0 
+30 
+60 
+90 
+B2(T/m) 
+64.21 
+55.12 
+35.07 
+25.47 
+A2(T/m) 
+2.0E-004 
+-0.0060 
+0.016 
+0.027 
+ 
+6. Design variable gradient sextupole for small-angle neutron scattering detector  
+A variable gradient hybrid permanent sextupole was designed for the Very Small-angle Neutron Scattering instrument 
+(VSANS) in the China Spallation Neutron Source Science (CSNS). As shown in Fig. 12, inner permanent ring has 12 permanent 
+blocks and 6 Vanadium Iron poles to collect the magnetic fluxes from the inner and outer permanent rings. Magnetization angle 
+for each permanent block is 60 degrees relative to its central symmetrical axis, which can contribute 20% field strength compare 
+with the 90 degrees from the calculation. For the 12 blocks outer ring, easy axis orientation for each permanent block is parallel 
+or perpendicular to its central symmetrical axis. The calculated gradient varies from 7188 T/m2 to19968T/m2 in a rotation recyle 
+from 0 to 60 degrees. Fig. 13 shows the 3D simulation field when the outer ring rotated at 60 degrees relative to the inner ring.  
+ 
+    Fig. 12 , Schematic layout for the magnetic angles for inner and outer permanent ring when the outer ring at 0 degrees. The 
+dashed areas are the iron poles.   
+ 
+ 
+Fig 13. 3D simulated magnetic filed when the outer permanent ring rotated at 60 degrees. 
+   For the machnical design, the inner ring is fixed on the support seat by the connected flanges at both ends, the outer ring 
+rotates relatve to the inner ring by a set of high speed motors with reduction gearboxes. Each iron pole is a set of 5 mm sliced 
+lamated Vanadium Irons with water cooling wholes to get rid of eddy current overheating during the 1.5 kHz high speed rotation. 
+From 3d calculation, maximum torque is 220 N.m at 45 degrees rotation for the 200 mm long nested permanent sextupole 
+prototype. The 200m long variable gradient sextupole has been fabricated and tested successfully.     
+ 
+7 . Conclusion 
+In a symmetrical hybrid permanent magnet, the mid-plane can be treated as the reference zero scalar potential, whereas the 
+iron pole is looked as high scalar potential to collect the magnetic flux and release to the low potential area. This paper presents 
+how it possible to produce the expected field by using magnetic fluxes method for hybrid permanent magnet design. Through 
+
+40
+20
+40
+20
+20
+40
+20
+-40 
+8 
+theoretical calculation and 3D field simulation, a permanent dipole with field strength higher than 2.4 T was fabricated and tested. 
+Variable gradient nested permanent quadrupole or sextupole can also be realized by using iron poles to collect the magnetic flux 
+and block the high order harmonics from the outer ring. For its small, compact and low operation cost, hybrid permanent magnet 
+can find more applications in areas such as particle accelerator, motor, medical equipment and material research. 
+ 
+References 
+[1] K. Halbach, Design of permanent multipole magnet with oriented rare earth cobalt material, Nucl. Instr.and Meth. 169(1980), 1-8. 
+[2] Quanling Peng, S. M. McMurry, and J. M. D. Coey, Cylindrical Permanent-Magnet Structures Using Images in an Iron Shield, IEEE TRANSACTIONS ON 
+MAGNETICS, 39 (2003), 1983-1989. 
+[3] Quanling Peng, S. M McMurry, J.M.D. Coey, Axial Magnetic Field Produced by Axially and Radially Magnetized Permanent Rings, Journal of magnetism 
+and magnetic materials 268 (2004) 165-169. 
+[4] Quanling PENG, 2D Field Calculation of Pure Permanent Magnet by Using Current Pair Model, journal of magnetism and magnetic materials , 309 (2007) 
+126-131. 
+[5] Ross D. Schlueter, Field errors in hybrid insertion devices, LBL-36839, USA. 
+[6] Q. L. Peng, Z. S. Yin, et al, Construction and Tuning of BEPC mini- Permanent Quadrupoles Prototype, Nucl. Instr. and Meth. in Phys. Res. A 406 (1998), 
+53~57. 
+[7] Vikas Teotia, Sanjay Malhotra, Elina Mishra, Prashant Kumar, Design, development and characterization of tunable Permanent Magnet Quadrupole for 
+Drift Tube Linac, Nuclear Inst. and Methods in Physics Research, A 982 (2020) 164528. 
+[8] M. Kumada, Development of High Field Permanent Magnet, IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 12, NO. 1, 
+MARCH 2002. 
+[9] Wending Zhong, Ferromagnetics(II), Science press, 1998 (in Chinese). 
+[10]  P.P. Sanchez, T.S. do Espirito Santo, E. Conforti,, G. Tosin, Concepts of tunable magnets using permanent magnetic material for synchrotron radiation 
+sources, Nuclear Instruments and Methods in Physics Research A 778 (2015) 67–76. 
+[11]  Y. Iwashita, Y. Tajima, M. Ichikawa, S. Nakamura, T. Ino, S. Muto, H.M. Shimizu, Variable permanent magnet sextupole lens for focusing of pulsed cold 
+neutrons, Nuclear Instruments and Methods in Physics Research Section A, 586 (2008) 73-76 
+[12]  Junghoon Lee, Jeonghoon Yoo，Topology optimization of the permanent magnet type MRI considering the magnetic field homogeneity [J], Journal of 
+Magnetism and Magnetic Materials, 2010, 322:1651.  
+[13]  Opera Manager User Guide, Version 15R1, Vector Fields Software, July 2013  
+[14]  Jack Tanabe, Iron Dominated Electromagnets Design, Fabrication, assembly and Measurements, SLAC-R-754, 2005. 
+[15]  Gautam Sinha, Conceptual design of a compact high gradient quadrupole magnet of varying strength using permanent magnets, PHYSICAL REVIEW 
+ACCELERATORS AND BEAMS 21, 022401 (2018).
+
+ 
+9 
+ 
+ 
+

7dAyT4oBgHgl3EQfpvi2/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf,len=256
+page_content='1 Improve the Field Strength by Adding Soft Iron in the Hybrid Permanent magnet Quanling Peng1,2, Jianxin Zhou1,  Saike Tian1, Yingzhe Wang1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Institute of High Energy Physics, Chinese Academy of sciences, Beijing, 100049, China 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' School of electronics, Nanchang institute of Technology, Jiangxi, 330013, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Abstract—Permanent magnet has a small and compact structure, is especially suitable for a narrow space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' With the aid of soft iron, the magnetic field can be increased much more and the field uniformity can be well controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Most Permanent magnets have a symmetry structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' the soft iron can be selected as the float magnet pole to keep its constant scalar potential, and take roles as media to collect the direct flux from the permanent blocks, then release indirect flux in magnet aperture and to the nearby return yoke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' This paper presents the magnetic flux method to design and fabricate a hybrid permanent dipole by using axially and radially magnetized permanent blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' A variable gradient permanent quadrupole and a variable gradient sextupole are designed as the extend design examples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' They all consist of two nested hybrid permanent rings, where the iron poles are used to control the field quality, collect the magnetic flux from the outer ring, block the skew quadrupole and high order harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Index Terms—Pure permanent magnet, hybrid permanent magnet, direct flux, indirect flux, variable gradient quadrupole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Introduction Electromagnet or superconducting magnets, with field strength varies along with the beam energy, are widely used in modern particle accelerators to bend or focus the particle beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' In some circumstances, where the beam energy is fixed or only a little adjustment, permanent magnet will be a better selection, since it has advantages of small space occupation, no cooling system, and no operation costs, one time investment can maintain a long time operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Permanent magnet can be made of all permanent blocks or permanent blocks mixed with soft iron, where the former called as pure permanent magnet, later called as the hybrid permanent magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' For pure permanent magnet design, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Halbach had given the design principle in 1980s [1-2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Each permanent block can be treated as air with the current loops surrounding the magnet or as magnetic charges on its surfaces along the easy axis [3-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Field strength in magnet aperture is collected the contribution in each permanent block by its position and easy axis orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' An important issue is that field strength from the pure permanent magnet is weak compared with electromagnets or superconducting magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' According to reference [1], even with the highest residual field strength, maximum field of a dipole that built with pure permanent blocks is less than the remnant field of Br, say about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='4 T for the highest NdFeB magnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  On the other hand, pure permanent magnet needs permanent blocks with different easy axis orientation, that will bring fabrication difficulties and increase the cost [5-9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' For a hybrid permanent magnet, on the other hand, with the nonlinear material such as iron poles to collect the surrounding magnetic flux and release it into a small compact space, it can reach to a higher magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Another way to increase the field strength is to add more permanent blocks surround the iron pole to add more magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Variable field permanent magnets can be built by adding auxiliary coils around the iron pole or adding an outer concentric permanent ring surround the inner permanent ring [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Principle of flux method Assume a permanent magnet has an infinite length, the 3D case can be treated in 2D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' In 2D analysis, a homogeneously magnetized permanent block can be treated as an air space with the surface charge density \uf073\uf0b1=\uf0b1Br on its upper and lower surfaces or a surface current distribution of density i = Hc around the other four surfaces [3-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Hear Br and Hc are the remnant field and the coercively of the permanent block respectively, with Br=\uf06d0Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' For the hard permanent magnetic material, the slope of demagnetized curve in the second quarter is near 1, or μr=1, no extra field contribution from the material magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' In 2D non-current space, magnetic field can be expressed as the negative gradient of scalar potential as V \uf02d\uf0d1 \uf03d B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 1, in a hybrid permanent magnet, the permanent block can be modeled by charge sheets at the top and bottom surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The iron pole has large relative permeability, all the surface of the magnet pole keep a constant scalar potential of V2=V0, the return yoke and mid-plane are all kept in zero scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Since V1 and V2 are in different scalar potential, magnetic field will be produced between the magnet pole to the mid-plane or to the return yoke, which call as useful field and stray field respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' @Manuscript received Jan 2, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Work supported by the accelerator research program of the Chinese Academy of Sciences Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Y5294104TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Author email address: pengql@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 1 Different scalar potentials in a quarter of the hybrid permanent dipole, the shade region represent the iron yoke and iron pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='1 Direct flux φd Direct flux is magnetic flux coming from the permanent magnet blocks and deposits on the iron pole, it is the source to maintain iron pole in a higher scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The direct flux to the iron pole is ，  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (1) c is the fraction of magnetic charge σ deposited on the surface of the iron pole, it equals to the scalar potential V at the magnetic charges with respect to the potential V0 at the iron pole, D is the width of the permanent block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Assume the scalar potential changes uniformly in the permanent block, then  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 1, the direct flux includes the contributions from the positive and the negative charges, which can be expressed as ,      (2) here  ,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' If magnet blocks directly touch on the iron pole and leave some space h1 between the top yoke, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (2) can be written as: (  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (3) If the permanent block fully occupies the space between the iron pole and top yoke, then the direct flux on the iron pole is: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (4) In 3d case, the direct flux deposited on the iron pole is: ,                                      (5) S is the surface area of the permanent block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='2 Indirect flux φi Indirect flux escapes from the iron pole faces to the nearby zero scalar potential areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 2, add permanent magnet between the magnet pole and the side yoke, the scalar potential of the iron pole will rise up to V0, the back yoke, the top yoke and the mid-plane still keeps zero scalar potential, parasite magnetic field will be produced between the different scalar potential areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Here φi1 is the expected field, φi2 is the demagnetization field for the permanent magnet, φi3 is the nearby leakage field, total indirect flux is φi =φi1+φi2 +φi3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Assume the expected field B0 keeps constant in the magnet gap, with  , h0 is half-length of the magnet gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The indirect flux on the mid-plane is  ∬  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (6) In general, indirect flux to nearby zero scalar potential is written as:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (7) S’ is the surface area of the iron pole, h is the distance between the iron pole and the zero potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Consider the corner effect, the scale factor f>1,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='3 Total magnetic flux From ∯       , total magnetic flux around the float iron pole is zero, that is the direct flux deposits on iron pole equals to the indirect flux leaves away from the iron pole, which can be expressed as: 0 \uf03d \uf02b i d \uf066 \uf066 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (8)  V1=0  h1  Br  h2  h  ++  +++  V2=VO  ho  V1=0  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The indirect flux calculation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Indirect flux goes from the iron pole to the nearby zero scalar potential area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' H type hybrid permanent dipole design A hybrid permanent dipole was fabricated for magnetic material processing, aimed to produce the field higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='4 T in a 7 mm gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Assume the half gap as h0, the expected magnetic field at the central plane is B0，then the scalar potential on the surface of the iron pole is V0=B0h0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 3 shows the cross section of the upper half magnet, the iron yokes are displayed in hatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The top permanent block is magnetized along the negative y, whereas the side permanent block is radially magnetized inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' For the lower half magnet, the side permanent block is radially magnetized outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The direct fluxes come from the top magnetized block and the side radially magnetized permanent ring, which can be written as:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (9) The indirect fluxes scatter from the iron pole to the nearby zero scalar potential faces, which includes parts to the mid-plane, to the top yoke, to the side yoke and to the upper and lower corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Here selects factor f as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='9 to contain the corner effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Then total indirect flux is  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' (10) NdFeB N44H material is selected for permanent blocks, the remnant field Br=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='36 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Other related parameters are: half magnet gap h0=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 mm, pole tip length h1=10 mm, distance from pole top to mid-plane h2=20 mm, distance between the top of the side permanent blocks to the mid-plane h3=30 mm, side yoke height h4=40 mm, pole radius R2=14 mm, pole tip radius R1=5 mm, radius of the magnet gap R3=32 mm, return yoke thickness 8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Vanadium Iron is select as the magnetic pole, since it has high saturated field as Bs=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='2 T, the return yokes are made of DT4 soft iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' By eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 10, magnetic field produced in the mid-plane can reach 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='42 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  OPERA-3d [13] software is used to check the field strength, the calculated peak field on the mid-plane is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='45 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' In comparison, three cases were calculated when the top permanent blocks removed or replaced with soft iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Fig 4 shows field differences along the central mid-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' When the top permanent block was removed, total direct fluxes were reduced, field strength on the mid-plane will drop accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' What’s more, when the top permanent blocks are replaced with iron, part of magnetic flux will directly return the top yoke, the field in the magnet gap will decrease much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Cross section of the upper half hybrid permanent dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The material of the magnet pole and the side yoke are Vanadium and DT4 iron respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  中i2  :  i3  Qil  DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  4  5  Y  R1  R3  4  Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Field differences on the mid-plane when the top permanent magnet replaced with air or DT4 iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Magnet fabrication and field measurement For technical limitation, the radial magnetized block was replaced by 6 tile-liked blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 5 shows the lower half magnet assebly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' In order to protect the permanent blocks, they are covered by a G10 board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 6 shows the whole magnet assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Lower half of the hybrid permanent magnet assembly  Field measurement was done by a Hall probe along the slot in the G10 board, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 7 shows the field measurement result, it has a little difference compare with the 3D field calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Whole Hybrid permanent magnet assembly -10 -5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 3 x(mm) Bz(T)  Top PM  Top air  Top iron  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Calculated and measured field distribution along the central line in the magnet mid-plane  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Design variable gradient permanent quadrupole by two nested permanent rings Variable gradient quadrupole can be built with pure or hybrid permanent magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Fig 8 shows a kind of pure permanent magnet design, where the variable gradient was realized by the relative rotation between the inner and outer permanent rings, field gradient varies from G1-G2 to G1+G2, here G1 and G2 are field gradient of the inner and outer permanent rings respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The nested pure permanent rings have two disadvantages, they are the lower efficiency and the accompanied skew quadrupole components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' First, permanent blocks in outer rings are much away from the inner, its field contribution are greatly reduced, which needs larger size and the cost will increase accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' On the other hand, since permanent blocks is similar as air, a skew quadrupole component will produce during rotation and cannot be canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Skew quadrupole component will give rise to work point drift, increase the beam emittance and will eventually affect the beam life time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Same problem exists in reference [15] for two sets of nest permanent rings that made of cylindrical permanent rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Another plan is using the hybrid permanent quadrupole, where several permanent blocks are replaced by iron poles, by which to control the field quality and concentrate the magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  9 shows a hybrid permanent quadrupole that consists of two sets of permanent rings, the variable gradient is realized by the relative rotation between the inner and outer ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' In a circular particle accelerator, the ramping period is in a few seconds, gradient changes for a quadrupole can go along with that of the beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' According to the design idea for the conventional electromagnet, the inner surface of the iron pole in a hybrid permanent magnet is selected as a part of hyperbola to increase the field uniformity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The outer circular surface of the iron pole is selected as wide enough to collect the magnetic flux from the outer ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Relative rotation between the two permanent rings does not bring extra skew quadrupole components, since the stray field is blocked by the iron pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' A variable gradient quadrupole that consists of two pure permanent rings  20  15  10  5 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='6 x 10 4 r(mm) By(G)  calculaled  measured  22  21  20  23  19  24  18  8  2  17  9  33  25  10  16  11  15  26  12  32  13  14  27  31  28  29  30  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Variable gradient quadrupole consists of two nest hybrid permanent rings，the dash regions are made of iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 10 shows the 3D field calculation when the outer ring rotated at 60 degrees, The design parameters are: magnet aperture 40 mm, outer diameter 320 mm, magnet length is 100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Taking the suitable shimmed on the iron pole surface and end plate, all the high order harmonics can be reduced less than 5 units at different rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Using FFT function in OPERA-3d, the quadrupole gradient and field harmonics at the reference radius of 13 mm can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The calculated gradient varies from 21 T/m to 64 T/m in a 90 degrees rotation period, maximum torque is 240 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='m, which can be realized by motors with the reduction gearbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 12 shows the normalized high order harmonics along the beam line, all the integral harmonics are less than 5 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Table 1 shows the normal and skew quadrupole values at different rotation angles, where all skew quadrupoles is near to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Calculation example of a variable gradient quadrupole that consists of two nest hybrid permanent rings when the outer ring rotation at 60 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig 11  High order harmonics along the beam line (@r=13mm )  when the outer ring rotaed at 60 degrees, all data are normalized with the integral quadrupole strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' -50 0 50 100 150 200 250 300 -2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 -1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 2 z（mm） unit of 10 -4  A3  B3  A4  B4  A5  B5  7  Table 1 Normal and skew quadrupole components changes at difference rotation angles Rotation angles 0 30 60 90 B2(T/m) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='21 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='12 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='07 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='47 A2(T/m) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='0E-004 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='027  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Design variable gradient sextupole for small-angle neutron scattering detector A variable gradient hybrid permanent sextupole was designed for the Very Small-angle Neutron Scattering instrument (VSANS) in the China Spallation Neutron Source Science (CSNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 12, inner permanent ring has 12 permanent blocks and 6 Vanadium Iron poles to collect the magnetic fluxes from the inner and outer permanent rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Magnetization angle for each permanent block is 60 degrees relative to its central symmetrical axis, which can contribute 20% field strength compare with the 90 degrees from the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' For the 12 blocks outer ring, easy axis orientation for each permanent block is parallel or perpendicular to its central symmetrical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The calculated gradient varies from 7188 T/m2 to19968T/m2 in a rotation recyle from 0 to 60 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 13 shows the 3D simulation field when the outer ring rotated at 60 degrees relative to the inner ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 12 , Schematic layout for the magnetic angles for inner and outer permanent ring when the outer ring at 0 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The dashed areas are the iron poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  Fig 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 3D simulated magnetic filed when the outer permanent ring rotated at 60 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' For the machnical design, the inner ring is fixed on the support seat by the connected flanges at both ends, the outer ring rotates relatve to the inner ring by a set of high speed motors with reduction gearboxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Each iron pole is a set of 5 mm sliced lamated Vanadium Irons with water cooling wholes to get rid of eddy current overheating during the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='5 kHz high speed rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' From 3d calculation, maximum torque is 220 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='m at 45 degrees rotation for the 200 mm long nested permanent sextupole prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' The 200m long variable gradient sextupole has been fabricated and tested successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Conclusion In a symmetrical hybrid permanent magnet, the mid-plane can be treated as the reference zero scalar potential, whereas the iron pole is looked as high scalar potential to collect the magnetic flux and release to the low potential area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' This paper presents how it possible to produce the expected field by using magnetic fluxes method for hybrid permanent magnet design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Through  40 20 40 20 20 40 20 -40 8 theoretical calculation and 3D field simulation, a permanent dipole with field strength higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='4 T was fabricated and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Variable gradient nested permanent quadrupole or sextupole can also be realized by using iron poles to collect the magnetic flux and block the high order harmonics from the outer ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' For its small, compact and low operation cost, hybrid permanent magnet can find more applications in areas such as particle accelerator, motor, medical equipment and material research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Halbach, Design of permanent multipole magnet with oriented rare earth cobalt material, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='and Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 169(1980), 1-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [2] Quanling Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' McMurry, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Coey, Cylindrical Permanent-Magnet Structures Using Images in an Iron Shield, IEEE TRANSACTIONS ON MAGNETICS, 39 (2003), 1983-1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [3] Quanling Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' M McMurry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Coey, Axial Magnetic Field Produced by Axially and Radially Magnetized Permanent Rings, Journal of magnetism and magnetic materials 268 (2004) 165-169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [4] Quanling PENG, 2D Field Calculation of Pure Permanent Magnet by Using Current Pair Model, journal of magnetism and magnetic materials , 309 (2007) 126-131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [5] Ross D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Schlueter, Field errors in hybrid insertion devices, LBL-36839, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [6] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Peng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Yin, et al, Construction and Tuning of BEPC mini-\uf062 Permanent Quadrupoles Prototype, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' and Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' A 406 (1998), 53~57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [7] Vikas Teotia, Sanjay Malhotra, Elina Mishra, Prashant Kumar, Design, development and characterization of tunable Permanent Magnet Quadrupole for Drift Tube Linac, Nuclear Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' and Methods in Physics Research, A 982 (2020) 164528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Kumada, Development of High Field Permanent Magnet, IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 12, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' 1, MARCH 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [9] Wending Zhong, Ferromagnetics(II), Science press, 1998 (in Chinese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [10]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Sanchez, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  do Espirito Santo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Conforti,, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Tosin, Concepts of tunable magnets using permanent magnetic material for synchrotron radiation sources, Nuclear Instruments and Methods in Physics Research A 778 (2015) 67–76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [11]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Iwashita, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Tajima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Ichikawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Nakamura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Ino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Muto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' Shimizu, Variable permanent magnet sextupole lens for focusing of pulsed cold neutrons, Nuclear Instruments and Methods in Physics Research Section A, 586 (2008) 73-76 [12]  Junghoon Lee, Jeonghoon Yoo，Topology optimization of the permanent magnet type MRI considering the magnetic field homogeneity [J], Journal of Magnetism and Magnetic Materials, 2010, 322:1651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [13]  Opera Manager User Guide, Version 15R1, Vector Fields Software, July 2013 [14]  Jack Tanabe, Iron Dominated Electromagnets Design, Fabrication, assembly and Measurements, SLAC-R-754, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content=' [15]  Gautam Sinha, Conceptual design of a compact high gradient quadrupole magnet of varying strength using permanent magnets, PHYSICAL REVIEW ACCELERATORS AND BEAMS 21, 022401 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfpvi2/content/2301.00532v1.pdf'}

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+Mode selection in concentric jets. The
+steady-steady 1:2 resonant mode interaction with
+O(2) symmetry.
+A. Corrochano1, J. Sierra-Aus´ın2,3, J. A. Martin1,
+D. Fabre2, and S. Le Clainche ∗1
+1School of Aerospace Engineering, Universidad Polit´ecnica de
+Madrid, Madrid 28040, Spain
+2Institut de M´ecanique des Fluides de Toulouse (IMFT), Toulouse
+31400, France
+3DIIN, Universit´a degli Studi di Salerno, Via Giovanni Paolo II,
+84084 Fisciano (SA), Italy
+Abstract
+In this article, a thorough characterization of the configuration com-
+posed by two concentric jets at a low Reynolds number is presented. The
+analysis comprises a layout with a wide range for the velocity ratio be-
+tween the inner and outer jets, defined within the interval [0, 2], and also
+details the influence of the distance between jets, where the wall thick-
+nesses separating the two jets is [0.5, 4].
+Global linear stability analy-
+sis identifies the most significant modes driving the changes in the flow
+dynamics. The neutral lines revealing the critical Reynolds number con-
+nected to the presence of the main (steady and unsteady) flow bifurcations,
+which are presented by global azimuthal modes, show the high complexity
+of the problem under study, where hysteresis and other types of complex
+cycles are pointed out. Finally, the mode interaction is analysed, high-
+lighting the presence of travelling waves emerging from the interaction
+of steady states, and the existence of robust heteroclinic cycles that are
+asymptotically stable. The high level of detail in the results presented,
+makes this work as a reference for future research development in the field
+of concentric jets.
+1
+Introduction
+Double concentric jets is a configuration enhancing the turbulent mixing of two
+jets, which is used in several industrial applications where the breakup of the jet
+∗Email address for correspondence: soledad.leclainche@upm.es
+1
+arXiv:2301.04429v1  [physics.flu-dyn]  11 Jan 2023
+
+Rin
+Rout
+Uin
+Uout
+INITIAL MERGING 
+ZONE
+r
+TRANSITIONAL
+ZONE
+MERGED ZONE
+Outer
+potential core
+Inner
+potential core
+Outer mixing
+region
+Inner mixing
+region
+Reattachment
+point
+z
+Figure 1: Sketch representing the three flow regimes in the near field of double
+concentric jets. Figure based on the sketch presented in [12, 31].
+into droplets due to flow instabilities is presented as the key technology. Com-
+bustion (i.e.: combustion chamber of rocket engines, gas turbine combustion,
+internal combustion engines, etc.) and noise reduction (e.g.: in turbofan en-
+gines) are the two main applications of this geometry, although the annular jets
+can also be found in some other relevant applications such as ink-jet printers or
+spray coating.
+The qualitative picture emerging from this type of flow divides the inner
+field of concentric jets in three different regions: (i) initial merging zone, (ii)
+transitional zone and (iii) merged zone, as presented in fig. 1, that follows the
+initial sketch presented by [12]. In the initial merging zone (i), just at the exit
+of the two jets, two axisymmetric shear layers (inner and outer boundary layer)
+develop and start to merge. In this region, we distinguish the inner and outer
+shear layers, related with the inner and outer jet stream. Then, most of the
+mixing occurs in the transitional zone (ii), that extends until the external shear
+layer reaches the centreline. Finally, in the merged zone (iii), the two jets are
+totally merged, modelling a single jet flow.
+Several parameters define the characteristic of this flow: the inner and outer
+jet velocities, the jet diameters, the shape and thickness of the wall separating
+both jets, the Reynolds number, the boundary layer state and thickness at
+the jet exit and the free stream turbulence. Based on these parameters, it is
+possible to identify several types of flow behaviour, which can be related with
+the presence of flow instabilities.
+Ko & Kwan (1976) [12] postulated that the double concentric jet configura-
+tion could be considered as a combination of single jets. Nevertheless, Dahm et
+al. (1992) [7] revealed by means of flow visualizations, several topology patterns
+as function of the outer/inner jet velocity ratio, reflecting that the dynamics of
+2
+
+the inner and outer jet shear layers were different from that in a single jet. More-
+over, this study exhibited a complex interaction between vortices identified in
+both shear layers, affecting the instability mechanism of the flow. Buresti et
+al. (1992) [4] found that the outer shear layer dominated the flow dynamics for
+cases in which the outer velocity was much larger than the inner velocity. These
+authors also detected the presence of an alternate vortex shedding when the
+wall thickness between the two jets was sufficiently large. The same mechanism
+was recognised by other authors [7, 20]. Rehab et al. (1997) [25] studied in
+detail the flow differences as function of the outer/inner velocity ratio, finding
+two different flow regimes when the external jet diameter is much larger than
+the internal jet one. When the outer/inner velocity ratio was larger than a crit-
+ical value, the authors spotted a low frequency recirculation bubble at the jet
+outlet. On the contrary, for outer/inner velocity ratio smaller than such critical
+value, the outer (still fast) jet excites the inner jet, which ends oscillating at the
+same frequency as the external jet. This is known as the lock-in phenomenon.
+Moreover, the oscillation frequency detected was similar to the one defined by a
+Kelvin-Helmholtz flow instability, which is generally encountered in single jets.
+This lock-in phenomenon was also identified by other authors [7, 6].
+Following previous works [4, 7, 20] and paying especial attention to the sep-
+arating wall thickness and the vortex shedding located behind the wall, Wallace
+& Redekopp (1992) [33] showed that the wall thickness and sharpness change
+the characteristic of the jet. Segalini & Talamelli (2011) [26] performed exper-
+iments to inspect in detail the effects of the outer/inner velocity ratio and the
+wall thickness in double concentric jets. These authors found that for small
+outer/inner velocity ratios, the inner jet presents its own flow instability in the
+shear layer, while a different flow instability was identified in the outer jet. On
+the contrary, for large outer/inner velocity ratios, the outer shear layer drives the
+flow dynamics, forcing the inner shear layer to oscillate with the same frequency,
+occurring in the lock-in phenomenon previously mentioned. Finally, for similar
+outer/inner velocity ratios, a Von K´arm´an vortex street was detected near the
+separating wall, as also depicted by other authors [4, 7, 20]. A wake instability
+affected the inner and outer shear layers, reversing the lock-in phenomenon.
+Different configurations can also be found, changing the velocity ratio be-
+tween jets. Williams et al. (1969) [34] worked on the influence of the exte-
+rior/interior velocity ratio on noise attenuation, which was analysed experimen-
+tally. It was observed that for some given configurations, more noise attenuation
+was present than for the others, with a maximum between 12 and 15dB.
+Talamelli & Gavarini (2006) [31] performed a local linear stability analysis,
+finding that for specific wall thickness, the vortex shedding identified behind
+the wall, can be related with an absolute instability that exists for some specific
+outer/inner velocity ratios. The authors explained that this absolute instabil-
+ity may trigger the destabilization of the flow field. This theoretical work was
+verified experimentally by [21]. These authors showed once more that the wake
+behind the wall separating the two jets creates a vortex shedding driving the
+frequency of the external shear layer also controlling the evolution of the inner
+shear layer, which can be the mechanism that triggers a global absolute insta-
+3
+
+bility. This passive mechanism can be considered as a potential tool for flow
+control, delaying the transition to turbulence by means of controlling the near
+field of the jet. Recently, Canton et al. (2017) [5] performed a global linear sta-
+bility analysis to study more in detail this vortex shedding mechanism behind
+the wall. They examined a concentric jet configuration with a very small wall
+thickness (0.1Di, with Di the inner jet diameter), but the authors selected an
+outer/inner velocity ratios where it was known that the alternate vortex shed-
+ding behind the wall was driving the flow. A global unstable mode (absolute
+instability) with azimuthal wavenumber m = 0 was found, confirming that the
+primary instability was axisymmetric (the modes with m = 1, 2 were stable at
+the flow conditions at which the study was carried out). The highest intensity
+of the global mode was located in the wake of the jet, composed by an array
+of counter-rotating vortex rings. The shape of the mode changes when moving
+along its neutral curve, revealing through the numerical simulations a Kelvin-
+Helmholtz instability over the shear-layer between the two jets and in the outer
+jet at high Reynolds numbers. Nevertheless, the authors showed that the wave-
+maker was located in the bubble formed upstream the separating wall, in good
+agreement with the results presented by [32], who performed a similar stability
+analysis in a two-dimensional configuration (wakes with co-flow).
+The stability of annular jets, a limit case where the inner jets have zero
+velocity, has also been investigated. In different analysis of annular jets [3, 17],
+it has been illustrated that this type of axisymmetric configuration does not
+behave as it appears. The m = 0 modes studied have been shown to be stable,
+and the dominant mode found by both studies is helical (m = 1). In addition,
+to characterise the annular jet, these investigations analyse the behaviour of
+the case by adding an azimuthal component to the inflow velocity, making the
+discharge of the annular jet eddy-like, comparing the evolution of the frequency
+and growth rate of this m = 1 mode.
+This paper expands on the work done by [5], where they use a specific ge-
+ometry and vary the outer/inner velocity ratio. This paper presents a complete
+characterisation of the main global modes identified in two concentric jets. The
+wall thicknesses separating the two jets are defined in the interval L ∈ [0.5, 4],
+and the flow is simulated for different inside/outside velocity ratios in the in-
+terval Ui/Uo ∈ [0, 2], where the case with Ui/Uo = 0 represents an annular
+jet. Global modes with azimuthal wavenumber m = 0 (axisymmetric modes),
+m = 1 and m = 2 will be searched for. As identified in the literature [17, 3],
+no axisymmetric modes (m = 0) could be identified for any of the distances, as
+this is a helical case. This paper expands the conclusions found in these two
+previous works, extending the results to different wall thicknesses between jets.
+This part of the paper studies in detail the configuration of two concentric jets
+at low Reynolds numbers. Using a linear approximation of the equations that
+model the flow, the base flows will be obtained on which to apply the linear
+stability analysis, by means of which it is possible to identify the most relevant
+modes that influence the flow dynamics.
+This work also performs a study of mode selection, as some configurations
+presents interactions between different modes.
+Different analysis have been
+4
+
+done to know the different coherent structures when there is an interaction
+between modes. [30] conducted the study on the flow past a rotating sphere,
+finding different coherent structures on a triple-Hopf bifurcation. Some of these
+configurations are steady states, travelling waves or rotating waves.
+To the authors’ knowledge, this is the first time that the characterisation
+of two concentric jets is presented with such level of detail, presenting neutral
+curves for a wide range of different configurations, as well as providing a deep
+understanding of the flow physics through the interaction between the different
+modes.
+The article is organized as follows. Section 2 defines the problem and the
+governing equations for the double concentric jets, as well as the linear sta-
+bility equations and the methodology for mode selection. The axisymmetric
+steaty-state is characterised in Section 3. In Section 4, we perform a parametric
+exploration in terms of the velocity ratio between the jets and the jet distance
+in order to determine the neutral curves of global stability. The results about
+the mode selection are discussed in Section 5. Finally, Section 6 summarises the
+main conclusions.
+2
+Problem formulation
+2.1
+Computational domain and general equations
+The computational domain, represented in fig. 2, models a coaxial flow configu-
+ration, which is composed of two inlet regions, an inner and outer pipe, both of
+diameter D and length 5D, i.e. zmin = −5D. The computational domain has
+an extension of zmax = 50D and rmax = 25D. The distance between the pipes
+is equal to L, measured from the inner face of the outer tube to the face of the
+inner jet.
+The governing equations of the flow within the domain are the incompressible
+Navier–Stokes equations. These are written in cylindrical coordinates (r, θ, z),
+which are made dimensionless by considering D as the reference length scale
+and Wo,max as the reference velocity scale, which is the maximum velocity in
+the outer pipe at z = zmin.
+∂U
+∂t + U · ∇U = −∇P + ∇ · τ(U),
+∇ · U = 0,
+(1a)
+with τ(U) = 1
+Re (∇U + ∇UT ),
+Re = Wo,maxD
+ν
+.
+(1b)
+The dimensionless velocity vector U = (U, V, W) is composed of the radial,
+azimuthal and axial components, P is the dimensionless-reduced pressure, the
+dynamic viscosity ν and the viscous stress tensor τ(U).
+The incompressible Navier–Stokes equations eq. (1) are complemented with
+the following boundary conditions
+U = (0, 0, Wi) on Γin,i and U = (0, 0, Wo) on Γin,o,
+(2)
+5
+
+Figure 2: Computational domain of the configuration of two concentric jets,
+used in StabFem.
+where
+Wi = δu tanh
+�
+bi(1 − 2r)
+�
+and Wo = tanh
+�
+bo
+�
+1 +
+����
+r − (Router,1 + Router,2)
+D
+����
+��
+.
+The parameter δu corresponds to the velocity ratio between the two jets, defined
+as δu = Wi,max/Wo,max. The parameters bo and bi represent the boundary layer
+thickness within the nozzle, which are fixed equal to 5 (as in [5]). There is a
+weak influence of the boundary layer thickness on the stability properties of
+the jet, and it is related to the vortex shedding regime developed upstream the
+separation wall (more details may be found in [31]). Finally, no-slip boundary
+condition is set on Γwall and stress-free (
+� 1
+Reτ(U) − P
+�
+· n = 0) boundary
+condition is set on Γtop and Γout, as shown in fig. 2.
+In the sequel, Navier–Stokes equations eq. (1) and the associated boundary
+conditions will be written symbolically under the form
+B∂Q
+∂t = F(Q, ) ≡ LQ + N(Q, Q) + G(Q, η),
+(3)
+with the flow state vector Q = [U, P]T , η = [Re, δu]T . Such a form of the
+governing equations takes into account a linear dependency on the state variable
+Q through L. And a quadratic dependency on the parameters and the state
+variable through operators G(·, ·) and N(·, ·).
+2.2
+Asymptotic stability
+2.2.1
+Linear stability analysis
+In this study, the authors attempt to characterize the stable asymptotic state
+from the spectral properties of the Navier–Stokes equations eq. (1). First, let us
+6
+
+Tmar
+Router,?
+D
+out
+Router,1
+r=0
+Z=Zmin
+z=0
+z=Zmarconsider the stability of an axisymmetric steady-state solution named Q0, which
+will be also referred to as trivial steady-state. For that purpose, let evaluate
+a solution of eq. (1) in the neighbourhood of the trivial steady state, i.e., a
+perturbed state as follows,
+Q(x, t) = Q0(x, t) + εˆq(r, z)e−i(ωt−mθ).
+(4)
+The next step consists in the characterization of the dynamics of small-amplitude
+perturbations around this base flow by expanding them over the basis of linear
+eigenmodes (4). If there is a pair [iωℓ, ˆqℓ] with Im(ωℓ) > 0 (resp. the spectrum
+is contained in the half of the complex plane with negative real part) there ex-
+ists a basin of attraction in the phase space where the trivial steady-state Q0 is
+unstable (resp. stable) [11]. The eigenpair [iωℓ, ˆqℓ] is determined as a solution
+of the following eigenvalue problem,
+J(ωℓ,mℓ)ˆq(zℓ) =
+�
+iωℓB − ∂F
+∂q |q=Q0,η=0
+�
+ˆq(zℓ),
+(5)
+where
+�
+∂F
+∂q |q=Q0,η=0
+�
+ˆq(zℓ) = Lmℓ ˆq(zℓ) + Nmℓ(Q0, ˆq(zℓ)) + Nmℓ(ˆq(zℓ), Q0). The
+subscript mℓ indicates the azimuthal wavenumber used for the evaluation of the
+operator. In the following, we account for eigenmodes ˆq(zℓ)(r, z) that have been
+normalised in such a way ⟨ˆu(zℓ), ˆu(zℓ)⟩L2 = 1.
+2.2.2
+Methodology for the study of mode selection
+In the following, we briefly outline the main aspects of the methodology em-
+ployed in the study of mode interaction, a comprehensive explanation is left to
+appendix A. The determination of the attractor or coherent structure is explored
+within the framework of equivariant bifurcation theory. The trivial steady-state
+is axisymmetric, i.e. the symmetry group is the orthogonal group O(2). Near
+the onset of the instability, dynamics can be reduced to those of the centre
+manifold. Particularly, due to the non-uniqueness of the manifold one can al-
+ways look for its simplest polynomial expression, which is known as the normal
+form of the bifurcation. The reduction to the normal form is carried out via a
+multiple scales expansion of the solution Q of eq. (3). The expansion considers
+a two scale development of the original time t �→ t + ε2τ, here ε is the order of
+magnitude of the flow disturbances, assumed to be small ε ≪ 1. In this study
+we carry out a normal form reduction via a weakly non-linear expansion, where
+the small parameters are
+ε2
+δu = δu,c − δu ∼ ε2 and ε2
+ν =
+�
+νc − ν
+�
+=
+�
+Re−1
+c
+− Re−1�
+∼ ε2.
+A fast timescale t of the self-sustained instability and a slow timescale of the
+evolution of the amplitudes zi(τ) are also considered in eq. (10), for i = 1, 2, 3.
+The ansatz of the expansion is as follows
+Q(t, τ) = Q0 + εq(ε)(t, τ) + ε2q(ε2)(t, τ) + O(ε3).
+(6)
+7
+
+Herein, we evaluate the mode interaction between two steady symmetry break-
+ing states with azimuthal wave number m1 = 1 and m2 = 2, that is,
+q(ε)(t, τ)
+=
+�
+z1(τ)ˆq(z1)(r, z)e−im1θ + c.c.
+�
++
+�
+z2(τ)ˆq(z2)(r, z)e−im2θ + c.c.
+�
+.
+(7)
+Note that the expansion of the LHS of eq. (3) up to third order is as follows
+εB∂q(ε)
+∂t
++ ε2B∂q(ε2)
+∂t
++ ε3�
+B∂q(ε3)
+∂t
+�
++ O(ε4),
+(8)
+and the RHS respectively,
+F(q, η) = F(0) + εF(ε) + ε2F(ε2) + ε3F(ε3) + O(ε4).
+(9)
+Then, the problem up to third order in z1 and z2 can be reduced to [1]
+˙z1
+= λ1z1 + e3z1z2 + z1
+�
+c(1,1)|z1|2 + c(1,2)|z2|2�
+,
+˙z2
+= λ2z2 + e4z2
+1 + z2
+�
+c(2,1)|z1|2 + c(2,2)|z2|2�
+.
+(10)
+An exhaustive analysis of the nonlinear implications of this normal form on
+dynamics is left to section 5. The procedure followed for the determination of
+the coefficients c(i,j) for i, j = 1, 2 and e3 and e4 is left to Appendix A.
+2.2.3
+Numerical methodology for stability tools
+Results presented herein follow the same numerical approach adopted by [9,
+28, 27, 30], where a comparison with DNS can be found. The calculation of
+the steady-state, the eigenvalue problem and the normal form expansion are
+implemented in the open-source software FreeFem++. Parametric studies and
+generation of figures are collected by StabFem drivers, an open-source project
+available in https://gitlab.com/stabfem/StabFem. For steady-state, stabil-
+ity and normal form computations, we set the stress-free boundary condition at
+the outlet, which is the natural boundary condition in the variational formula-
+tion.
+The resolution of the steady nonlinear Navier-Stokes equations is tackled
+by means of the Newton method. While, the generalised eigenvalue problem
+(eq. (24)) is solved following the Arnoldi method with spectral transformations.
+The normal form reduction procedure of section 2.2.2 only requires to solve a
+set of linear systems, which is also carried out within StabFem. On a standard
+laptop, every computation considered below can be attained within a few hours.
+3
+Characterisation of the axisymmetric steady-
+state
+3.1
+Velocity ratio effects
+We begin by characterizing the development of the axisymmetric steady-state
+with varying δu at a constant Reynolds number fixed to Re = 100. Figure 3
+8
+
+0
+0
+1
+r
+5
+2
+4
+3
+W0
+-0.4
+-0.5
+-1
+-0
+0
+1
+r
+5
+2
+4
+3
+3
+2.5
+2
+1.5
+0.5
+1
+Lr
+min(W0)
+δu
+δu
+-0.1
+-0.15
+-0.2
+-0.25
+-0.3
+-0.35
+-0.4
+0.5       1      1.5      2
+a
+b
+c
+c
+b
+a
+d
+e
+f
+a
+b
+d
+f
+e
+c
+δ1
+u δ2
+u
+0
+2          4
+0
+0.1    0.2   0.3   0.4    0.5
+0
+z
+0
+0
+1
+z
+r
+2          4
+5
+2
+4
+3
+Figure 3: Evolution of the recirculation length (Lr) of the recirculating bubble
+with respect to the velocity ratio δu between the inner and outer jet.
+The
+diagram of the second row on the left displays the minimum value within the
+domain of the axial velocity. It is spatially localised within the recirculating
+region for δu < 0.5 and near the middle wall for larger values of the velocity
+ratio. Meridional projections of the axisymmetric streamfunction isolines and
+the axial velocity contour in a range of (z, r) ∈ [−1, 5] × [0, 5].
+synthesises the main topological changes experiences by the steady-state. At
+δu = 0, the solution (point (a) in fig. 3) represents an annular jet, which diffuses
+as it travels downstream and enters the ambient fluid. This figure illustrates
+that the solution curve can be divided into three segments. The first segment
+comprised between 0 ≤ δu < δ1
+u is characterised by an inner jet nearly trapped
+by a large recirculation region with a characteristic length Lr, which remains
+almost constant with the velocity ratio.
+In the second region, which ranges between δ1
+u < δu < δ2
+u and it is represented
+as a shaded area in the figure, the recirculating region rapidly reduces its size.
+In this region, the axial velocity of the inner jet is comparable with the axial
+velocity observed in the recirculating region, which promotes mixing between
+both regions.
+As the velocity ratio is increased, the inner jet is sufficiently
+9
+
+energetic to break the recirculating region, which occurs between point (c) and
+(d) in fig. 3. The final segment, that ranges between δu > δ2
+u, is characterised
+by two quasi-planar jets that rapidly mix to form a larger one at around z ≈ 5.
+4
+Linear stability analysis
+We explore the parameter space (Re, δu, L). Herein, we examine the velocity
+ratio between the jets (0 < δu < 2) and the distance between the jets (0.5 <
+L < 4). Within this range of parameters, we have analysed the linear stability
+properties of the flow configuration. For this purpose, we first investigate the
+influence of the jet distance on the stability for the case of the annular jet
+(δu = 0).
+These findings are summarized in fig. 4 which displays the evolution of the
+critical Reynolds number with respect to the distance (L) for the four most un-
+stable modes: two steady modes with azimuthal wavenumber m = 1 and m = 2,
+hereinafter referred to as modes S1 and S2, respectively. A cross-section view at
+z = 1 is displayed in fig. 4 (a-b). The other two unsteady modes, named F1 and
+F2 have respectively azimuthal wavenumbers m = 1 and m = 2. A cross-section
+view of these two modes is displayed in fig. 4 (c-d). Please note that for the
+chosen set of parameters the axisymmetric unsteady mode F0, is always found
+at larger Reynolds numbers than the aforementioned modes. This is one of the
+major differences with the case studied by [5], for small values of the jet distance
+L, the dominant instability is an unsteady axisymmetric one, which would be
+named F0 with our nomenclature. Thus, in the following, we only include the
+results for the S1, S2, F1 and F2 modes. The primary instability of the annular
+jet is then a steady symmetry-breaking bifurcation that leads to a jet flow with
+a single symmetry plane, displayed in fig. 4 (a). On the contrary, bifurcations
+that lead to the mode S2 possess two orthogonal symmetry planes, see fig. 4 (b).
+As indicated in fig. 4 (g-h), these two stationary modes S1 and S2 are localised
+within the recirculation bubble. For jet distances L < 2, the second mode that
+bifurcates is F1 mode, depicted in fig. 4 (i). This situation corresponds to a
+bifurcation scenario similar to other axisymmetric flow configurations, such as
+the flow past a sphere or a disk [2, 14]. For larger distances between jets, the
+scenario changes. The second bifurcation from the axisymmetric steady-state is
+the F2, displayed in fig. 4 (j). Other configurations where the primary or sec-
+ondary instability involves modes with azimuthal component m = 2 are swirling
+jets [15] and the wake flow past a rotating sphere [29]. The unsteady modes
+F1 and F2 possess a much larger spatial support than S1 and S2. They are
+formed by an array of counter-rotating vortex spirals developing in the wake of
+the separating duct wall. For the mode F2 the amplitude of these structures
+grows downstream of the nozzle, in the axial direction, with a maximum around
+z ≈ 10, after which they slowly decay. The mode F1 grows further downstream,
+with a maximum around z ≈ 50. The spatial structure of these eigenmodes
+resembles the axisymmetric mode of Figure 9 in [5]. Thus, the steady modes
+and unsteady modes differ in their spatial support, that is, even though both
+10
+
+-1
+0
+1
+x
+-1
+-0.5
+0
+0.5
+1
+y
+-0.5 0.5
+(a)
+-1
+0
+1
+x
+-1
+-0.5
+0
+0.5
+1
+y
+-0.5 0.5
+(b)
+-4
+-2
+0
+2
+4
+x
+-4
+-2
+0
+2
+4
+y
+-0.5 0.5
+(c)
+-4
+-2
+0
+2
+4
+x
+-4
+-2
+0
+2
+4
+y
+-0.5 0.5
+(d)
+1
+2
+3
+4
+L
+0
+100
+200
+300
+400
+500
+Re
+(e)
+1
+2
+3
+4
+L
+0
+0.1
+0.2
+0.3
+0.4
+!
+(f)
+-1
+0
+1
+2
+3
+z
+r
+-0.2
+0
+0.2
+(g)
+-1
+0
+1
+2
+3
+z
+r
+-0.2
+0
+0.2
+(h)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+z
+0
+5
+10
+r
+-0.05
+0
+0.05
+(i)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+z
+0
+5
+10
+r
+-0.05
+0
+0.05
+(j)
+Figure 4: Cross-section view at z = 1 of the four unstable modes at criticality
+for the annular jet case (δu = 0). The streawise component of the vorticity
+vector ϖz is visualised by colours. (a) Mode S1 for L = 0.5, (b) Mode S2 for
+L = 0.5, (c) Mode F1 for L = 3 and (d) Mode F2 for L = 3. (e) Linear stability
+boundaries for the annular jet (δu = 0). (f) Frequency evolution of the unsteady
+modes. Legend: S1 mode is displayed with a solid black line, S2 with a solid
+red line and F1 and F2 modes are depicted with dashed black and red lines,
+respectively. Streamwise velocity of the neutral modes for L = 3 and δu = 0 (i)
+F1, (h) F2 .
+11
+
+0
+0.5
+1
+1.5
+2
+/u
+0
+200
+400
+600
+800
+Re
+S1 ! S2
+(a)
+0
+0.5
+1
+1.5
+2
+/u
+0
+100
+200
+300
+400
+500
+600
+Re
+S1 ! S2
+(b)
+0
+2
+4
+6
+z
+0
+1
+2
+3
+4
+5
+6
+r
+-0.2
+0
+0.2
+0.4
+(c)
+0
+2
+4
+6
+z
+0
+1
+2
+3
+4
+5
+6
+r
+-0.5
+0
+0.5
+1
+1.5
+(d)
+Figure 5: Linear stability boundaries for the concentric jets (a) L = 0.5 and (b)
+L = 1. Same legend as fig. 4. Visualizations of real part of the streamwise axial
+velocity of the critical modes (c) S1 and (d) S2.
+steady and unsteady modes are localised in space, the support of the steady
+ones is confined within the recirculation bubble. Instead, the unsteady modes
+are convected far downstream until they reach a maximum. This latter char-
+acteristic is classical of modes with a large transient growth, as it was noticed
+by [5]. On the other hand, the nature of the steady modes is similar to the
+symmetry-breaking instabilities behind the disk or the sphere. These modes are
+far less sensitive to transient growth and are observable with direct numerical
+simulations and experiments.
+12
+
+4.1
+Fixed distance between jets and variable velocity ratio
+δu
+In the following, we focus on the influence of the velocity ratio δu between jets
+for fixed jet distances L. Figure 5 displays the neutral curve of stability for
+jet distances (a) L = 0.5 and (b) L = 1. One may observe that the primary
+bifurcation is not always associated to the mode S1 as it is the case for δu = 0.
+For sufficiently large velocity ratios, the primary instability leads to a non-
+axisymmetric steady-state with a double helix.
+Another interesting feature,
+which could motivate a control strategy, is the occurrence of vertical asymptotes.
+This sudden change in the critical Reynolds number is due to the retraction
+and disappearance of the recirculating region. For L = 0.5, this sudden change
+occurs for δu ≈ 0.25, and for higher values of δu the critical Reynolds number
+is around twice larger than the one of the annular jet (δu = 0). The case of jet
+distance L = 1 was discussed in section 3. The sudden change in the stability
+of the branch S1 occurs between δu ∈ [0.25, 0.5]. Within this narrow interval,
+the primary branch of instability is the F1. At around δu = 0.4, the primary
+bifurcation is again the branch F1, which becomes secondary at around δu ≈ 0.8
+in favour of the branch S2. In fig. 5 we have highlighted the codimension two
+point interaction between the S1 −S2 modes, whose modes are depicted in fig. 5
+(c-d), which will be analysed in detail in section 5. Around this point, we can
+observe the largest stabilisation ratio between the annular jet (δu = 0) and a
+configuration of concentric jets (δu ̸= 0).
+4.2
+Fixed velocity ratio δu and variable distance between
+jets
+Figure 6 compares the results obtained for a constant velocity ratio when vary-
+ing the distance between jets. As observed before, the solution becomes more
+unstable by increasing the distance between jets. The largest critical Reynolds
+number is found at δu = 0.
+The critical Reynolds decreases with the dis-
+tance between jets L. The points of codimension two, i.e., the points where
+mode switching occurs, are highlighted in fig. 6. We can appreciate that the
+interaction between the branch S1 and S2 happens for every velocity ratio δu
+explored, and it is the mode interaction associated to the smallest distance be-
+tween jets. Additionally, for a velocity ratio δu = 0.5 there exists two points
+where the branches of the linear modes S1 and F1 intersect. Another feature
+of the neutral curves is the existence of turning points, which are associated
+to the existence of saddle node bifurcations of the axisymmetric steady-state.
+The saddle node bifurcations of the steady-state induces the existence of regions
+in the neutral curves with a tongue shape. These saddle node bifurcations are
+also responsible for the formation of the vertical asymptotes observed in fig. 5.
+Finally, it is of interest the transition of the modes S1 and S2, which induce
+the symmetry breaking of the axisymmetric steady state to slow low frequency
+spiralling structures. These modes have been identified for δu = 0.5 for m = 1,
+δu = 1 for m = 2, and δu = 2 for both m = 1 and m = 2. As it will be clari-
+13
+
+a)
+b)
+1
+2
+3
+4
+L
+0
+200
+400
+600
+800
+1000
+Re
+LS1!S2
+LS1!F1 LF1!S1
+1
+2
+3
+4
+L
+0
+0.5
+1
+1.5
+!
+c)
+d)
+1
+2
+3
+4
+L
+0
+200
+400
+600
+800
+1000
+Re
+LS1!S2
+2
+3
+4
+40
+60
+80
+100
+120
+1
+2
+3
+4
+L
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+!
+e)
+f)
+1
+2
+3
+4
+L
+0
+200
+400
+600
+800
+1000
+Re
+LS1!S2
+1
+2
+3
+4
+L
+0
+0.1
+0.2
+0.3
+0.4
+!
+Figure 6: Neutral lines of the four modes found studying the configuration of
+two concentric jets fixing the velocity ratio. (a-b) δu = 0.5, (c-d) δu = 1, (e-f)
+δu = 2. Black lines: modes with m = 1, red lines: modes with m = 2. Straight
+lines: steady modes, dashed lines: unsteady modes.
+14
+
+fied in section 5, these oscillations are issued from the non-linear interaction of
+modes, emerging simultaneously for a specific Reynolds number, and changing
+their position as the most unstable global mode of the flow.
+5
+Mode interaction between two steady states.
+Resonance 1 : 2
+5.1
+Normal form, basic solutions and their properties
+The linear diagrams of section 4 have shown the existence of the mode in-
+teraction between the modes S1 and S2. It corresponds roughly to the mode
+interaction that occurs at the largest critical Reynolds number for any value of
+L herein explored. In this section, we analyse the dynamics near the S1 : S2
+organizing centre. We perform a normal form reduction, which allows us to
+predict non-axisymmetric steady, periodic, quasiperiodic and heteroclinic cy-
+cles between non-axisymmetric states.
+The mode interaction that is herein analysed corresponds to a steady-steady
+bifurcation with O(2) symmetry and with strong resonance 1 : 2. Such a bifur-
+cation scenario has been extensively studied in the past by [8, 10, 23, 1] and the
+reflection symmetry breaking case (SO(2)) by [24]. In order to unravel the exis-
+tence and the stability of the nonlinear states near the codimension two point,
+let write the flow field as
+q
+= Q0 + Re
+�
+r1(τ)eiφ1(τ)e−iθˆqs,1
+�
++ Re
+�
+r2(τ)eiφ2(τ)e−2iθˆqs,2
+�
+(11)
+in polar coordinates for the complex amplitudes z1 = r1eiφ1 and z2 = r2eiφ2
+where rj and φj for j = 1, 2 are the amplitude and phase of the symmetry-
+breaking modes m = 1 and m = 2, respectively. The complex-amplitude normal
+form eq. (10) is expressed in this reduced polar notation as follows,
+˙r1 = e3r1r2 cos(χ) + r1
+�
+λ(s,1) + c(1,1)r2
+1 + c(1,2)r2
+2
+�
+,
+(12a)
+˙r2 = e4r2
+1 cos(χ) + r2
+�
+λ(s,2) + c(2,1)r2
+1 + c(2,2)r2
+2
+�
+,
+(12b)
+˙χ = −
+�
+2e3r2 + e4
+r2
+1
+r2
+�
+sin(χ),
+(12c)
+where the phase χ = φ2 −2φ1 is coupled with the amplitudes r1 and r2 because
+of the existence of the 1 : 2 resonance. The individual phases evolve as
+˙φ1
+= e3r2 sin(χ),
+˙φ2
+= −e4
+r2
+1
+r2 sin(χ).
+(13)
+Before proceeding to the analysis of the basic solutions of eq. (12), we can
+simplify these equations by the rescaling
+�
+r1
+|e3e4|1/2 , r2
+e3
+�
+→ (r1, r2),
+15
+
+which yields the following equivalent system
+˙r1 = r1r2 cos(χ) + r1
+�
+λ(s,1) + c11r2
+1 + c12r2
+2
+�
+,
+(14a)
+˙r2 = sr2
+1 cos(χ) + r2
+�
+λ(s,2) + c21r2
+1 + c22r2
+2
+�
+,
+(14b)
+˙χ = − 1
+r2
+�
+2r2
+2 + sr2
+1
+�
+sin(χ),
+(14c)
+where the coefficients
+s = sign(e3e4),
+c11 = c(1,1)
+|e3e4|,
+c12 = c(1,2)
+e2
+3
+,
+c21 = c(2,1)
+|e3e4|,
+c22 = c(2,2)
+e2
+3
+.
+Finally, we consider a third normal form equivalent to the previous ones but
+which removes the singularity of eqs. (12) and (14) when r2 = 0. Standing
+waves (sin χ = 0) naturally encounter this type of artificial singularity, which
+manifests as in eq. (14) as an instantaneous jump from one standing subspace
+to the other by a π-translation.
+This is the case of the heteroclinic cycles,
+previously studied by [1, 23]. The third normal form, which we shall refer to as
+reduced Cartesian normal form, takes advantage of the simple transformation
+x = r2 cos(χ), y = r2 sin(χ) [24]:
+˙r1 = r1
+�
+λ(s,1) + c11r2
+1 + c12(x2 + y2) + x
+�
+,
+(15a)
+˙x = sr2
+1 + 2y2 + x
+�
+λ(s,2) + c21r2
+1 + c22(x2 + y2)
+�
+,
+(15b)
+˙y = −2xy + y
+�
+λ(s,2) + c21r2
+1 + c22(x2 + y2)
+�
+,
+(15c)
+In this final representation standing wave solutions are contained within the
+invariant plane y = 0, and due to the invariance of eq. (15) under the reflection
+y �→ −y, one can restrict attention, without loss of generality, to solutions with
+y ≥ 0, cf [23].
+The system eq. (14) possess four types of fixed points, which are listed in
+table 1.
+First, the axisymmetric steady state (O) is represented by (r1, r2) = (0, 0), so
+it is the trivial steady-state of the normal form. The second steady-state is what
+it is denoted as pure mode (P). In the original coordinates, it corresponds to the
+symmetry breaking structure associated to the mode S2. This state bifurcates
+from the axisymmetric steady state (O) when λ(s,2) = 0. The third fixed point
+is the mixed mode state (MM), which is listed in table 1. It corresponds to
+the reflection symmetry preserving state associated to the mode S1. It may
+bifurcate directly from the trivial steady state O, when λ(s,1) = 0 or from P
+whenever σ+ = 0 or σ− = 0, where σ± is defined as
+σ± ≡ λ(s,1) − −λ(s,2)c12
+c22
+±
+�
+−λ(s,2)
+c22
+.
+(16)
+16
+
+Name
+Definition
+Bifurcations
+Comments
+O
+r1,O = r2,O = 0
+−
+Steady axisymmetric state
+P
+r2
+2,P =
+−λ(s,2)
+c22
+, r1,P = 0
+λ(s,2) = 0
+Bifurcation from O
+r1,MM = −
+λ(s,1)±r2,MM+c12r2
+2,MM
+c11
+λ(s,1) = 0
+Bifurcation from O
+MM
+PMM(r2,MM cos(χMM)) = 0
+σ± = 0
+Bifurcation from P
+cos(χMM) = ±1
+cos(χT W ) =
+(2c11+c12)λ(s,2)−(2c21+c22)λ(s,1)
+ΣT W (2λ(s,1)+λ(s,2))
+TW
+r2
+2,T W =
+−(2λ(s,1)+λ(s,2))
+ΣT W
+cos(χT W ) = ±1
+Bifurcation from MM
+r2
+1,T W = 2r2
+2,T W
+Table 1: Definition of the fixed points of the reduced polar normal form eq. (14).
+σ± is defined in eq. (16), the polynomial PMM is defined in eq. (17) and ΣT W ≡
+4c11 + 2(c12 + c21) + c22.
+Name
+Bifurcation condition
+Comments
+SW
+sr2
+1 − 2c11r2
+1r2,MM cos(χMM) − 2c22r3
+2,MM cos(χMM)3 = 0
+Bif. from MM
+MTW
+DT W − TT W IT W = 0, IT W > 0
+Bif. from TW
+Table 2: Definition of the limit cycles of the reduced polar normal form eq. (14).
+The representation in the reduced polar form is
+r1,MM = −
+λ(s,1) ± r2,MM + c12r2
+2,MM
+c11
+,
+cos(χMM) = ±1,
+and the condition PMM(r2,MM cos(χMM)) = 0, where PMM is defined as
+PMM(x) ≡ sµ1+(s+c21λ(s,1)−c(1,1)λ(s,2))x+(c21+sc12)x2+(c12c21−c11c22)x3.
+(17)
+Finally, the fourth fixed point of the system are travelling waves (TW). It is
+surprising that the interaction between two steady-states causes a time-periodic
+solution. The travelling wave emerges from MM in parity-breaking pitchfork
+bifurcation that breaks the reflection symmetry when cos(χT W ) = ±1. The
+TW drifts at a steady rotation rate ωT W along the group orbit, i.e., the phases
+˙φ1 = r2,T W sin(χT W ) and ˙φ2 = −s
+r2
+1,T W
+r2,T W sin(χT W ) are non-null.
+Mixed modes and travelling waves may further bifurcate into standing waves
+(SW) and modulated travelling waves (MTW), respectively. These are generic
+features of the 1 : 2 resonance for small values of λ(s,1) and λ(s,2), when s =
+−1. In the original coordinates, SW are periodic solutions, whereas MTW are
+quasiperiodic. Standing waves emerge via a Hopf bifurcation from MM when
+the conditions PSW
+�
+r2,MM cos(χMM)
+�
+> 0 for
+PSW(x) ≡ (2c22x3 − sr2
+1)c11 − (2c12x + 1)(c21x + s)x,
+17
+
+Name
+Condition
+Comments
+Ht AGH
+λ(s,1) > 0, λ(s,2) > 0, c22 < 0
+Existence
+σ+ > 0, σ− < 0
+Asymptotic stability
+Table 3: Definition of the conditions for the existence of the Ht AGH (robust
+heteroclinic cycles connecting pure modes) of the reduced polar normal form
+eq. (14).
+O
+P
+MM
+Ht AGH
+SW
+TW
+MTW
+Condition Tab. 2
+Condition Tab. 2
+Cond.
+Tab. 3
+Figure 7: Schematic representation of the basic solutions of eq. (12) and their
+bifurcation path.
+and the one listed in table 2 are satisfied.
+MTW are created when a torus
+bifurcation happens on the travelling wave branch when the conditions listed in
+table 2 are satisfied.
+Another remarkable feature of eq. (12) is the existence of robust heteroclinic
+cycles that are asymptotically stable. When s = −1, there are open sets of
+parameters (see table 3) where the reduced polar normal form exhibits struc-
+turally stable connections between π−translations on the circle of pure modes,
+cf [1]. These structures are robust and have been observed in a large variety
+of systems, [19, 18, 16, 22, 13]. In addition to these robust heteroclinic cycles
+connecting pure modes, there exists more complex limit cycles connecting O,
+P, MM and SW, cf [23]. These cycles are located for larger values of λ(s,1) and
+λ(s,2), with possibly chaotic dynamics (Shilnikov type). In this study, we have
+not identified any of these. Finally, a summary of the basic solutions and the
+bifurcation path is sketched in fig. 7.
+5.2
+Results of the steady-steady 1 : 2 mode interaction
+Section 4.2 reported the location of mode interaction points for discrete values
+of the velocity ratio δu. The location of the mode interaction between S1 and
+S2 is depicted in fig. 8. It shows that the mode switching between the modes S1
+18
+
+a)
+b)
+0
+0.5
+1
+1.5
+2
+/u
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+L
+0
+0.5
+1
+1.5
+2
+/u
+100
+200
+300
+400
+500
+600
+700
+800
+Re
+Figure 8: Evolution of the codimension two interaction S1 − S2 in the space of
+parameters (Re, L, δu). Grey points denote the points that were computed and
+the red point denotes the transition from steady to unsteady with low frequency
+as reported in section 4.2.
+and S2 is indeed stationary only for δu < 1.5 and L < 1.3. For larger values of
+the velocity ratio and the jet distance, the interaction is not purely stationary;
+at least one of the linear modes oscillates with a slow frequency. It implies that
+the mode selection for large velocity ratios near the codimension two points is
+similar to the one reported by [15] for swirling jets. However, even when the two
+primary bifurcations are non-oscillating (S1 and S2), the 1 : 2 resonance of the
+azimuthal wavenumbers induces a slow frequency, what we denote as travelling
+wave solutions (TW).
+We consider the bifurcation sequence for δu = 1.0 and L = 1.15, which
+is qualitatively similar to transitions in the range 0.5 < δu < 1.5, near the
+codimension two points, which are depicted in fig. 8. At the codimension two
+points for δu < 0.5, at least one of the two bifurcations is sub-critical and a
+normal form reduction up to fifth order is necessary. Subcritical transition was
+also noticed for a distance between jets L = 0.1 by [5], who reported high
+levels of the linear gain associated to transient growth mechanisms. This last
+case is out of the scope of the present manuscript. Figure 9 displays the phase
+portrait of the stable attractors near the S1 : S2 interaction.
+For values of
+δu > 1.0, the axisymmetric steady-state loses its axisymmetry leading to a
+new steady-state with symmetry m = 2, herein denoted as pure mode (P). A
+reconstruction of the fluctuating flow field of such a state is performed at the
+bottom right of fig. 9, which shows that the state P possesses two orthogonal
+planes of symmetry. Near the codimension two point, for values of the velocity
+ratio δu < 1.1, the state P is only observable, that is non-linearly stable, within
+a small interval with respect to the Reynolds number.
+For larger values of
+the velocity ratio, the state P remains stable within the analysed interval of
+Reynolds numbers. For values of the velocity ratio δu < 1.0, the bifurcation
+19
+
+w' = ± 0.025 
+w' = ± 0.01 
+w' = ± 0.01 
+0.02
+0.01
+0
+-0.01
+-0.02
+w' 
+170
+180
+190
+200
+210
+Re
+𝛿u
+Rotating
+Non-rotating
+Non-rotating
+TW
+MM
+P
+Ht AGH
+MM
+TW
+MTW
+P
+0.8
+0.9
+1.0
+1.1
+1.2
+z=1/2
+z=1
+z=1
+z=1
+z=2
+t-T/3
+t+T/3
+t
+Figure 9: Phase portrait at the codimension two point S1 : S2 for parameter
+values (L, δu) = (1.15, 1.0). Visualisations of blue and red surfaces in the iso-
+metric views represent the respective positive and negative isocontour values of
+the perturbative axial velocity indicated in the figure.
+180
+185
+190
+195
+200
+Re
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+r2
+MM
+TW
+PM
+BifTW!MTW
+BifMM!TW
+0
+-0.1
+0.05
+y
+x
+0
+0.1
+0
+0.05
+r1
+0.1
+0.1
+170
+175
+180
+185
+190
+195
+200
+Re
+Figure 10: Bifurcation diagram with respect to the Reynolds number for L =
+1.15 and δu = 0.8. The left diagram reports the evolution of r2 for the fixed
+point solutions of the normal form. The right diagram displays the bifurcation
+diagram in the Cartesian coordinates. Solid lines and dashed lines denote stable
+attractors and unstable attractors, respectively.
+20
+
+diagram is more complex. Figure 10 displays the bifurcation diagram of the
+fixed-point solutions of eq. (15) on the left diagram and the full set of solutions
+of the normal form in the right diagram. The axisymmetric steady-state first
+bifurcates towards a Mixed-Mode solution, which is the solution in the y = 0
+plane for the right diagram of fig. 10. A solution with a non-symmetric wake
+has been reconstructed in fig. 9. The Mixed-Mode solution is only stable within
+a small interval of the Reynolds number.
+A secondary bifurcation, denoted
+BifMM−T W , gives raise to a slowly rotating wave of the wake. The TW and the
+MM solutions are identical at the bifurcation point. The phase speed is zero
+at the bifurcation, thus this is not a Hopf bifurcation. It corresponds to a drift
+instability that breaks the azimuthal symmetry, i.e. it starts to slowly drift.
+This unusual feature, that travelling waves bifurcate from a steady solution at a
+steady bifurcation, is a generic feature of the 1 : 2 resonance. A reconstruction
+of the travelling wave solution is depicted on the top of fig. 9. It corresponds
+to the line with non-zero y component in the right diagram of fig. 10. The TW
+solution loses its stability in a tertiary bifurcation, denoted as BifT W −MT W . It
+conforms to a Hopf bifurcation of the TW solution, which gives birth to a quasi-
+periodic solution name Modulated Travelling Wave (MTW). A representation
+of this kind of solution in the Cartesian coordinates (r1, x, y) is depicted on the
+right image of fig. 10.
+Eventually, the Modulated Travelling Wave experiences a global bifurcation.
+That occurs when the periodic MTW solution, in the (r1, x, y) coordinates,
+nearly intersects the invariant r1 = 0 and y = 0 planes. The transition sequence
+is represented in the right image of fig. 10 in the Cartesian coordinates (r1, x, y).
+The amplitude of the MTW limit cycle increases until the MTW arising at
+the tertiary bifurcation BifT W −MT W are destroyed by meeting a heteroclinic
+cycle at BifMT W −Ht.
+The locus of BifMT W −Ht is reported in fig. 9 and in
+good agreement with [1]. The conditions for the existence of the heteroclinic
+cycles are listed in table 3. When σ− becomes negative, the cycle is attracting
+and robust heteroclinic cycles are observed. It is destroyed when σ+ becomes
+negative, in that case the pure modes are no longer saddles which breaks the
+heteroclinic connection. Figure 11 displays the instantaneous fluctuation field
+from a heteroclinic orbit connecting P and its conjugate solution P’, which is
+obtained by a rotation of π/2, for parameter values Re = 200 and δu = 0.8. The
+dynamics of the cycle takes place in two phases. Figure 11 depicts the motion
+of the coherent structure associated to the heteroclitic cycle. Starting from the
+conjugated pure mode P’, the cycle leaves the point (a), located in the vicinity
+of P’, along the unstable eigenvector y, which is the stable direction of P. The
+first phase consists in a rapid rotation by π/2 of the wake, it corresponds to the
+sequence a-b-c-d-e displayed in fig. 11. Then it is followed by a slow approach
+following the direction y and departure from the pure mode state P along the
+direction r1.
+The second phase consists in a rapid horizontal motion of the
+wake, which is an evolution from P to P’ that takes place by the breaking of
+the reflectional symmetry with respect to the vertical axis; it constitutes the
+sequence e-f-g-h-i-a. Please note that equivalent motions are also possible. The
+first phase of rapid counter-clockwise rotation by π/2 can be performed in the
+21
+
+0.1
+0.1
+0.15
+0.05
+0.05
+0
+0
+-0.1
+-0.1
+0.1
+0.05
+-0.05
+0
+0.1
+r1
+y
+x
+c
+e
+d
+f
+h
+h
+i
+g
+f
+b
+a
+c
+d
+e
+2000
+1500
+500
+1000
+r1
+x≡r2cos(χ)
+y≡r2sin(χ)
+0
+0
+-0.1
+0.1
+0.2
+t
+sin(χ) = 0
+P'
+P
+b
+a
+g
+i
+w'
+w' = ± 0.05
+w' = ± 0.05
+r1=0
+Figure 11: Heteroclinic cycle solution for parameter values Re = 200, δu = 0.8.
+The top and bottom image sequences along the heteroclinic cycle show (from
+left to right) an axial slice plane at z = 1 of the instantaneous fluctuations
+of the axial velocity of the flow field as viewed from downstream, along with
+a three-dimensional isometric view (d on the top and g on the bottom).The
+middle diagram displays the heteroclinic cycle in the coordinates (r1, x, y).
+22
+
+opposite sense. It corresponds to a motion in the Cartesian coordinates along
+the plane r1 along negative values of y. The sequence e-f-g-h-i-a can be replaced
+by a horizontal movement in the opposite sense, which adjusts to connect the
+plane y = 0 corresponding to negative values of r1,
+6
+Discussion & Conclusions
+This article achieves a complete description of the configuration consisting of
+two coaxial jets, broadly found in industrial processes, covering a wide range
+of applications such as noise reduction, mixing enhancement, or combustion
+control.
+The numerical procedure herein employed has been validated with
+the existing literature in the case of the stability analysis (see B for a detailed
+overview), and compared to DNS results, as done in [30]. The analysis comprises
+a layout with a wide range of the velocity ratio (δu = Ui/Uo) between the jets,
+from δu = 0 to δu = 2, as well as the distance between jets (L) enclosing values
+from L = 0.5 to L = 4, substantially expanding the work of [5].
+A linear stability analysis reveals the most significant modes, consisting of
+two steady modes (S1 and S2, located within the recirculation bubble) and two
+unsteady ones (F1 and F2, evolving as a transient growth in the downstream
+direction).
+The critical Reynolds number is determined for a wide range of
+velocity and distance ratios, starting with the influence of the velocity ratio.
+As the relation between inner and outer velocities grows, the flow is stabilised,
+increasing the critical Reynolds number. The primary instability swaps from
+mode S1, characterised with one symmetry plane, to mode S2 that possesses
+two symmetry planes. An abrupt divergence in the critical Reynolds number is
+captured, associated with the vanishing of the recirculation region, that could
+suggest a stability control strategy.
+Subsequently, the effect of the distance
+L between jets is analysed. The primary effect of increasing this distance is
+a decrease in the critical Reynolds number for all values of δu investigated.
+Additionally, the existence of saddle node bifurcations, that swap the most
+unstable mode of the flow, generates turning points in the neutral curve.
+The investigated bifurcation scenario starts from the codimension point, with
+an axisymmetric steady state located at a velocity ratio δu = 1.0 and distance
+between jets of L = 1.15. It is qualitatively equivalent to transitions found in
+the range 0.5 < δu < 1.5. It reveals a break of the axisymmetry for values higher
+than δu = 1.0, presenting a steady state as a pure mode P with two orthogonal
+planes of symmetry. For values lower than δu = 1.0, the bifurcation diagram
+exhibits a slightly complicated path. Firstly, it drives into a Mixed-Mode (MM)
+solution presenting a non-symmetric wake, that is only stable for a small range
+of the Reynolds number.
+Subsequently, a slowly rotating wake is triggered
+in the form of a Travelling Wave (TW). This unusual feature, an unsteady
+state emerging from a steady state, corresponds to a drift instability commonly
+found at 1 : 2 resonance. Then, the TW solution encounters a Hopf bifurcation,
+developing a quasi-periodic solution in the form of a Modulated Travelling Wave
+(MTW). Finally, the MTW solution undergoes a global bifurcation meeting a
+23
+
+heteroclinic cycle (Ht).
+This heteroclinic orbit links the solution P with its
+conjugate solution P’, spinning the wake from P’ to P, and moving it horizontally
+from P to P’.
+24
+
+Acknowledgments
+A.C., J.A.M. and S.L.C. acknowledge the grant PID2020-114173RB-I00 funded
+by MCIN/AEI/ 10.13039/501100011033. S.L.C., J.A.M. and A.C. acknowledge
+the support of Comunidad de Madrid through the call Research Grants for
+Young Investigators from Universidad Polit´ecnica de Madrid. AC also acknowl-
+edges the support of Universidad Polit´ecnica de Madrid, under the programme
+‘Programa Propio’.
+A
+Normal form reduction
+In this section we provide a detail explanation of the normal form reduction to
+obtain the coefficients of eq. (10), we define the terms of the compact notation
+of the governing equations eq. (3), which is reminded here, for the sake of
+conciseness,
+B∂Q
+∂t = F(Q, η) ≡ LQ + N(Q, Q) + G(Q, η).
+(18)
+The nonlinear convective operator N(Q1, Q2) = U1 · ∇U2 accounts for the
+quadratic interaction on the state variable. The linear operator on the state
+variable is LQ = [∇P, ∇·U]T . The remaining term accounts for the linear vari-
+ations in the state variable and the parameter vector. It is defined as G(Q, η) =
+G(Q, [η1, 0]T ) + G(Q, [0, η2]T ) where G(Q, [η1, 0]T ) = η1∇ · (∇U + ∇UT ) and
+G(Q, [0, η2]T ). The former operator shows the dependency on the parameter
+η1, which accounts for the viscous effects. The latter operator depends on the
+parameter η2, which accounts for the velocity ratio between jets and it is used
+to impose the boundary condition U = (0, η2 tanh
+�
+bi(1 − 2r)
+�
+, 0) on Γin,i. In
+addition, we consider the following splitting of the parameters η = ηc + ∆η.
+Here ηc denotes the critical parameters ηc ≡ [Re−1
+c , δu,c]T attained when the
+spectra of the Jacobian operator posses at least an eigenvalue whose real part
+is zero. The distance in the parameter space to the threshold is represented by
+∆η = [Re−1
+c
+− Re−1, δu,c − δu]T .
+A.1
+Multiple scales ansatz
+The multiple scales expansion of the solution q of eq. (3) is
+q(t, τ) = Q0 + εq(ε)(t, τ) + ε2q(ε2)(t, τ) + O(ε3),
+(19)
+where ε ≪ 1 is a small parameter. The distance in the parameter space to
+the critical point ∆η = [Re−1
+c
+− Re−1, δu,c − δu]T is assumed to be of second
+order, i.e. ∆ηi = O(ε2) for i = 1, 2. The expansion eq. (19) considers a two
+scale expansion of the original time t �→ t + ε2τ. A fast timescale t and a slow
+timescale of the evolution of the amplitudes zi(τ) in eq. (19), for i = 1, 2. Note
+that the expansion of the LHS eq. (3) up to third order is as follows
+εB∂q(ε)
+∂t
++ ε2B∂q(ε2)
+∂t
++ ε3�
+B∂q(ε3)
+∂t
++ B∂q(ε)
+∂τ
+�
+,
+(20)
+25
+
+and the RHS respectively,
+F(q, η) = F(0) + εF(ε) + ε2F(ε2) + ε3F(ε3).
+(21)
+The expansion eq. (21) will be detailed at each order.
+A.1.1
+Order ε0
+The zeroth order Q0 of the multiple scales expansion eq. (19) is the steady state
+of the governing equations evaluated at the threshold of instability, i.e. η = ηc,
+0 = F(Q0, ηc).
+(22)
+A.1.2
+Order ε1
+The first order q(ε)(t, τ) of the multiple scales expansion of eq. (19) is composed
+of the eigenmodes of the linearized system
+q(ε)(t, τ) ≡
+�
+z1(τ)e−im1θˆq1 + z2(τ)ei−m2θˆq2 + c. c.
+�
+.
+(23)
+in our case, m1 = 1 and m2 = 2. Each term ˆqℓ of the first order expansion
+eq. (23) is a solution of the following linear equation
+J(ωℓ,mℓ)ˆqℓ =
+�
+iωℓB − ∂F
+∂q |q=Q0,η=ηc
+�
+ˆqℓ,
+(24)
+where ∂F
+∂q |q=Q0,η=ηc ˆqℓ = Lmℓ ˆqℓ + Nmℓ(Q0, ˆqℓ) + Nmℓ(ˆqℓ, Q0). The subscript
+mℓ indicates the azimuthal wavenumber used for the evaluation of the operator.
+A.1.3
+Order ε2
+The second order expansion term q(ε2)(t, τ) is determined from the resolution
+of a set of forced linear systems, where the forcing terms are evaluated from first
+and zeroth order terms. The expansion in terms of amplitudes zi(τ) (i = 1, 2)
+of q(ε2)(t, τ) is assessed from term-by-term identification of the forcing terms at
+the second order. Non-linear second order terms in ε are
+F(ε2)
+≡
+2
+�
+j,k=1
+�
+zjzkN(ˆqj, ˆqk)e−i(mj+mk)θ + c.c.
+�
++
+2
+�
+j,k=1
+�
+zjzkN(ˆqj, ˆqk)e−i(mj−mk)θ + c.c.
+�
++
+2
+�
+ℓ=0
+ηℓG(Q0, eℓ),
+(25)
+where the terms proportional to zjzk are named ˆF(zjzk)
+(ϵ2)
+and eℓ is an element of
+the orthonormal basis of R2.
+26
+
+Then, we look for a second order term expanded as follows
+q(ε2) ≡
+2
+�
+j,k=1
+k≤j
+�
+zjzkˆqzjzk + zjzkˆqzjzk + c.c
+�
++
+2
+�
+ℓ=1
+ηℓQ(ηℓ)
+0
+.
+(26)
+Terms ˆqz2
+j are azimuthal harmonics of the flow. The terms ˆqzjzk with j ̸= k are
+coupling terms, and ˆq|zj|2 are harmonic base flow modification terms. Finally,
+Q(ηℓ)
+0
+are base flow corrections due to a variation of the parameter ηℓ from the
+critical point.
+At this order, there exists two resonant terms, the terms proportional to z1z2
+and z2
+1, which are associated with the singular Jacobian J(0,mk) for k = 1, 2. To
+ensure the solvability of these terms, we must enforce compatibility conditions,
+i.e. the Fredholm alternative. The resonant terms are then determined from the
+resolution of the following set of bordered systems
+�J(0,mk)
+ˆqk
+ˆq†
+k
+0
+� �ˆq(z(R))
+e
+�
+=
+�
+ˆF(z(R))
+(ε2)
+0
+�
+, z(R) ∈ [z1z2, z2
+1]T ,
+(27)
+where e = e3 for z(R) = z1z2 and e = e4 for z(R) = z2
+1. The non-resonant terms
+are computed by solving the following non-degenerated forced linear systems
+J(0,mj+mk)ˆqzjzk = ˆF(zjzk)
+(ϵ2)
+,
+(28)
+and
+J(0,0)Q(ηℓ)
+0
+= G(Q0, eℓ).
+(29)
+A.1.4
+Order ε3
+At third order, there exists six degenerate terms. In our case, we are not inter-
+ested in solving for terms of third-order, instead, we will determine the linear
+and cubic coefficients of the third order normal form eq. (10) from a set of
+compatibility conditions.
+The linear terms λ(s,1) and λs,2 and cubic terms c(i,j) for i = 1, 2 are deter-
+mined as follows
+λ(s,1) =
+⟨ˆq†
+1, ˆF(z1)
+(ε3)⟩
+⟨ˆq†
+z, Bˆqz⟩
+, λ(s,2) =
+⟨ˆq†
+2, ˆF(z2)
+(ε3)⟩
+⟨ˆq†
+2, Bˆq2⟩
+, c(i,j) =
+⟨ˆq†
+2, ˆF(zi|zj|2)
+(ε3)
+⟩
+⟨ˆq†
+i, Bˆqi⟩
+.
+(30)
+The forcing terms for the linear coefficient are
+ˆF(zj)
+(ε3) ≡
+2
+�
+ℓ=1
+ηℓ
+��
+N(ˆqj, Q(ηℓ))
+0
++ N(Q(ηℓ)
+0
+, ˆqj)
+�
++ G(ˆqj, eℓ)
+�
+.
+(31)
+which allows the decomposition of λ(s,ℓ) = λ(s,ℓ),Re(Re−1
+c Re−1)+λ(s,ℓ),δu(δu,c −
+δu) for ℓ = 1, 2.
+27
+
+Canton et al. (2017) [5]
+Present work
+Rec
+1420
+1405
+ω
+5.73
+5.72
+Table 4: Comparison of Rec and ω between previous work and the present one.
+The forcing terms for the cubic coefficients are
+ˆF(zj|zk|2)
+(ε3)
+≡
+�
+N(ˆqj, ˆq|zk|2) + N(ˆq|zk|2, ˆqj)
+�
++
+�
+N(ˆq−k, ˆqzjzk) + N(ˆqj,k, ˆq−k)
+�
++
+�
+N(ˆqk, ˆqzjzk) + N(ˆqzjzk, ˆqk)
+�
+.
+(32)
+if j ̸= k and
+ˆF(zj|zj|2)
+(ϵ3)
+≡
+�
+N(ˆqj, ˆq|z|2
+j) + N(ˆq|z|2
+j, ˆqj)
+�
++
+�
+N(ˆq−j, ˆqz2
+j ) + N(ˆqz2
+j , ˆq−j)
+�
+,
+(33)
+for the diagonal forcing terms.
+B
+Validation of the code - Comparison with the
+literature
+The calculations made in StabFem in the sections at the main manuscript are
+validated comparing the leading global mode in the geometry proposed by [5].
+Moreover, the critical Reynolds number and the frequency associated are also
+analysed. In the cited work, the authors use an analogous geometry with the
+following parameters:
+• Radious of the inner jet Rinner = 0.5
+• Diameter of the outer jet D = 0.4
+• Distance between jets L = 0.1
+• Ratio between velocities δu = 1
+The linear stability analysis has been carried out imposing m = 0, as done
+by [5], so the leading global mode will be axisymmetric. The critical Reynolds
+number Rec and the frequency ω of the leading global mode are compared in
+Tab. 4. As seen, few differences can be found on the critical Reynolds number
+and the frequency. The relative error in the Rec calculation is 1.06% and the
+one of the frequency is 0.17%.
+The global mode is now calculated using StabFem and compared with the
+one calculated by [5]. This mode can be found in figures 9, 10 and 11 on the
+cited paper. As it can be seen, there are not substantial differences between the
+direct modes, being both of them a vortex street with their biggest amplitude
+28
+
+0
+5
+10
+15
+20
+25
+z
+0
+1
+2
+r
+-0.5 0
+0.5
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+z
+0
+0.5
+1
+1.5
+r
+-2
+0
+2
+0
+0.2
+0.4
+0.6
+z
+0.4
+0.5
+0.6
+0.7
+r
+0
+0.5
+1
+Figure 12: Direct mode, adjoint mode and sensitivity of the leading global mode
+studied by [5] calculated using StabFem.
+situated 10 units downstream the exit of the jets. The adjoint mode is con-
+centrated within the nozzle, with its biggest amplitude situated on the sharp
+corners. There is not any difference between the adjoint mode calculated with
+StabFem and the one in [5]. Finally, the structural sensitivity is similar to the
+one computed by [5]. It is composed by two lobes in the space between the exit
+of the two jets.
+References
+[1] Dieter Armbruster, John Guckenheimer, and Philip Holmes. Heteroclinic
+cycles and modulated travelling waves in systems with o (2) symmetry.
+Physica D: Nonlinear Phenomena, 29(3):257–282, 1988.
+[2] F. Auguste, D. Fabre, and J. Magnaudet. Bifurcations in the wake of a thick
+circular disk. Theoretical and Computational Fluid Dynamics, 24:305–313,
+2010.
+[3] A. Bogulawski and K. Wawrzak. Absolute instability of an annular jet:
+local stability analysis. Meccanica, 55:2179–2198, 2020.
+[4] G. Buresti, G. A. Talamelli, and P. Petagna. Experimental characterization
+of the velocity field of a coaxial jet configuration. Exp. Therm. Fluid Sci.,
+9:135, 1994.
+[5] J. Canton, F. Auteri, and M. Carini. Linear global stability of two incom-
+pressible coaxial jets. J. Fluid Mech., 824:886–911, 2017.
+[6] C. B. da Silva, G. Balarac, and O. M´etais. Transition in high velocity ratio
+coaxial jets analysed from direct numerical simulations. J. Turbul., 24:1,
+2003.
+29
+
+[7] W. J. A. Dahm, C. E. Frieler, and G. Tryggvason. Vortex structure and
+dynamics in the near field of a coaxial jet. J. Fluid Mech., 241:371, 1992.
+[8] Gerhard Dangelmayr. Steady-state mode interactions in the presence of 0
+(2)-symmetry. Dynamics and Stability of Systems, 1(2):159–185, 1986.
+[9] D. Fabre, V. Citro, D. Ferreira Sabino, P. Bonnefis, J. Sierra, F. Giannetti,
+and M. Pigou. A Practical Review on Linear and Nonlinear Global Ap-
+proaches to Flow Instabilities. Applied Mechanics Reviews, 70(6), 02 2019.
+060802.
+[10] CA Jones and MRE Proctor. Strong spatial resonance and travelling waves
+in b´enard convection. Physics Letters A, 121(5):224–228, 1987.
+[11] Todd Kapitula and Keith Promislow. Spectral and dynamical stability of
+nonlinear waves, volume 457. Springer, 2013.
+[12] N.W.M. Ko and A.S.H. Kwan. The initial region of subsonic coaxial jets.
+J. Fluid Mech., 73:305, 1976.
+[13] Paolo Maria Mariano and Furio Lorenzo Stazi. Computational aspects of
+the mechanics of complex materials. Archives of Computational Methods
+in Engineering, 12(4):391–478, 2005.
+[14] Philippe Meliga, Jean-Marc Chomaz, and Denis Sipp. Global mode in-
+teraction and pattern selection in the wake of a disk: a weakly nonlinear
+expansion. Journal of Fluid Mechanics, 633:159–189, 2009.
+[15] Philippe Meliga, Fran¸cois Gallaire, and Jean-Marc Chomaz.
+A weakly
+nonlinear mechanism for mode selection in swirling jets. Journal of Fluid
+Mechanics, 699:216–262, 2012.
+[16] Isabel Mercader, Joana Prat, and Edgar Knobloch. Robust heteroclinic
+cycles in two-dimensional rayleigh–b´enard convection without boussinesq
+symmetry. International Journal of Bifurcation and Chaos, 12(11):2501–
+2522, 2002.
+[17] A. Michalke. Absolute inviscid instability of a ring jet with back-flow and
+swirl. Eur. J. Mech. B/Fluids, 18:3–12, 1999.
+[18] C Nore, F Moisy, and L Quartier.
+Experimental observation of near-
+heteroclinic cycles in the von k´arm´an swirling flow.
+Physics of Fluids,
+17(6):064103, 2005.
+[19] Caroline Nore, Laurette S Tuckerman, Olivier Daube, and Shihe Xin. The
+1 [ratio] 2 mode interaction in exactly counter-rotating von k´arm´an swirling
+flow. Journal of Fluid Mechanics, 477:51–88, 2003.
+[20] W. Olsen and A. Karchmer. Lip noise generated by flow separation from
+nozzle surfaces. AIAA J., 76:3, 1976.
+30
+
+[21] R. ¨Orl¨u, A. Segalini, P. H. Alfredsson, and A. Talamelli.
+On the pas-
+sive control of the near-field of coaxial jets by means of vortex shedding.
+Proceedings of the International Conference on Jets, Wakes and Separated
+Flows, ICJWSF-2008, Technical University of Berlin, Berlin, Germany,
+Sept. 16–19, 2008.
+[22] Antonio Palacios, Gemunu H Gunaratne, Michael Gorman, and Kay A
+Robbins. Cellular pattern formation in circular domains. Chaos: An In-
+terdisciplinary Journal of Nonlinear Science, 7(3):463–475, 1997.
+[23] J Porter and E Knobloch. New type of complex dynamics in the 1: 2 spatial
+resonance. Physica D: Nonlinear Phenomena, 159(3-4):125–154, 2001.
+[24] J Porter and E Knobloch. Dynamics in the 1: 2 spatial resonance with bro-
+ken reflection symmetry. Physica D: Nonlinear Phenomena, 201(3-4):318–
+344, 2005.
+[25] H. Rehab,
+E. Villermaux,
+and E. J. Hopfinger.
+Flow regimes of
+largevelocity-ratio coaxial jets. J. Fluid Mech., 345:357, 1997.
+[26] A. Segalini and A. Talamelli. Experimental analysis of dominant instabili-
+ties in coaxial jets. Phys. Fluids, 23:024103, 2011.
+[27] J. Sierra, D. Fabre, V. Citro, and F. Giannetti. Bifurcation scenario in the
+two-dimensional laminar flow past a rotating cylinder. Journal of Fluid
+Mechanics, 905:A2, 2020.
+[28] Javier Sierra, David Fabre, and Vincenzo Citro. Efficient stability anal-
+ysis of fluid flows using complex mapping techniques. Computer Physics
+Communications, 251:107100, 2020.
+[29] J. Sierra-Ausin, Lorite-Diez M., V. Citro, Jimenez J.I., and D. Fabre. Ro-
+tating sphere. Journal of Fluid Mechanics, -:–, 2022.
+[30] J. Sierra-Aus´ın, M. Lorite-D´ıez, J.I. Jim´enez-Gonz´alez, V. Citro, and
+D. Fabre. Unveiling the competitive role of global modes in the pattern
+formation of rotating sphere flows. Journal of Fluid Mechanics, 942:A54,
+2022.
+[31] A. Talamelli and I. Gavarini. Linear instability characteristics of incom-
+pressible coaxial jets. Flow Turbulence Combust., 76:221–240, 2006.
+[32] O. Tammisola. Oscillatory sensitivity patterns for global modes in wakes.
+J. Fluid Mech., 701:251–277, 2012.
+[33] D. Wallace and L.G. Redekopp. Linear instability characteristics of wake-
+shear layers. Phys. Fluids, 4:189—-191, 1992.
+[34] T.J. Williams, M.R.M.H. Ali, and J.S. Anderson. Noise and flow charac-
+teristics of coaxial jets. J. Mech. Eng. Sci., 1:2, 1969.
+31
+

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+Learning Support and Trivial Prototypes for Interpretable Image Classification
+Chong Wang1
+Yuyuan Liu1
+Yuanhong Chen 1
+Fengbei Liu1
+Yu Tian2
+Davis J. McCarthy3
+Helen Frazer4
+Gustavo Carneiro5
+1 Australian Institute for Machine Learning, University of Adelaide
+2 Harvard University 3 St Vincent’s Institute of Medical Research
+4 St Vincent’s Hospital Melbourne 5 CVSSP, University of Surrey
+Abstract
+Prototypical part network (ProtoPNet) methods have
+been designed to achieve interpretable classification by
+associating predictions with a set of training prototypes,
+which we refer to as trivial (i.e., easy-to-learn) prototypes
+because they are trained to lie far from the classification
+boundary in the feature space.
+Note that it is possible
+to make an analogy between ProtoPNet and support vec-
+tor machine (SVM) given that the classification from both
+methods relies on computing similarity with a set of train-
+ing points (i.e., trivial prototypes in ProtoPNet, and support
+vectors in SVM). However, while trivial prototypes are lo-
+cated far from the classification boundary, support vectors
+are located close to this boundary, and we argue that this
+discrepancy with the well-established SVM theory can re-
+sult in ProtoPNet models with suboptimal classification ac-
+curacy. In this paper, we aim to improve the classification
+accuracy of ProtoPNet with a new method to learn support
+prototypes that lie near the classification boundary in the
+feature space, as suggested by the SVM theory. In addition,
+we target the improvement of classification interpretabil-
+ity with a new model, named ST-ProtoPNet, which exploits
+our support prototypes and the trivial prototypes to pro-
+vide complementary interpretability information. Experi-
+mental results on CUB-200-2011, Stanford Cars, and Stan-
+ford Dogs datasets demonstrate that the proposed method
+achieves state-of-the-art classification accuracy and pro-
+duces more visually meaningful and diverse prototypes.
+1. Introduction
+Deep convolutional neural networks (CNN) [14, 26, 27]
+have made remarkable achievements in various visual tasks,
+e.g., image recognition [14] and object detection [34]. De-
+spite the excellent feature extraction and discrimination
+ability, CNNs are generally treated as black-box models
+due to their complex architectures, high-dimensional fea-
+Features
+lct
+lsp
+lct
+lsp
+dct
+dsp
+Classification boundary
+Classification boundary
+(a) The learning of trivial prototypes
+(b)
+(b) The learning of support prototypes (ours)
+(d)
+Prototypes
+Feature vectors
+Clustering
+Separation 
+Our loss 
+Classification boundary
+Classification boundary
+Figure 1. The difference between the learning of trivial and sup-
+port prototypes. (a) Trivial prototypes: the separation loss pushes
+the prototypes of different classes as far as possible from the clas-
+sification boundary. (b) Support prototypes: our new closeness
+loss enforces the prototypes of different classes to be as close as
+possible to the classification boundary.
+ture spaces, and an enormous number of learnable param-
+eters. Such lack of interpretability hinders their successful
+application in fields that require understandable and trans-
+parent decisions [35], such as disease diagnosis [42], finan-
+cial risk assessment [30], and autonomous driving [21].
+Recently, increasing attention has been dedicated to the
+development of interpretable deep-learning models [1, 3, 4,
+23].
+A particularly interesting strategy is the prototype-
+based interpretable models, e.g., prototypical part network
+(ProtoPNet) [4, 10]. These methods are inherently inter-
+pretable since they can explain the model’s decisions by
+showing image classification activation maps associated
+with a set of class-specific image prototypes. These proto-
+types are automatically learned from training samples, with
+classification score being computed by comparing testing
+image parts to the learned training prototypes.
+ProtoPNet [4] is trained to learn a classifier from a set of
+class-specific prototypes by minimising the cross-entropy
+classification loss and two additional regularisation losses,
+namely: 1) a clustering loss that pulls together training im-
+1
+arXiv:2301.04011v1  [cs.CV]  8 Jan 2023
+
+DIWGU2IOUX
+X noiengmid
+V
+★
+B
+B
+V
+口
+V
+GSLUIua
+1
+-
+WGILC
+■
+cages to prototypes of their own class; and 2) a separation
+loss that pushes apart training images from all prototypes
+of other classes. More specifically, the clustering loss min-
+imises the distance of each image patch to at least one pro-
+totype of its own class, while the separation loss maximises
+the distance between all image patches and all other class
+prototypes. The combination of these two losses pushes the
+prototypes as far as possible from the classification bound-
+ary, but still within the class distribution, so we call them
+trivial (i.e., easy-to-learn) prototypes, as shown in Fig. 1(a).
+We also display these trivial prototypes in Fig. 2(a), where
+we present the ProtoPNet learning results for the two-moon
+problem, depicting the training points (red and blue points)
+and prototypes (green and black stars) in both the data
+and feature spaces, learned with a feed-forward neural net-
+work1. Notice that the trivial prototypes are located as far
+as possible from the classification boundary in the feature
+space. Similar to ProtoPNet, the support vector machine
+(SVM) [6] classifier is trained by minimising a loss func-
+tion that learns a set of support vectors. Different from the
+ProtoPNet’s prototypes, these support vectors are located
+as close as possible to the classification boundary, as shown
+in Fig. 2(c). Given that the prototypes in ProtoPNet and
+support vectors in SVM play similar roles in classification
+problems, we argue that the ProtoPNet’s loss function may
+lead to suboptimal classification results because of the triv-
+ial prototypes being learned.
+In this paper, we propose an alternative learning strat-
+egy of ProtoPNet from the SVM perspective, to force the
+learned prototypes to resemble support vectors of SVM and
+be located as close as possible to the classification bound-
+ary with the goal of increasing classification accuracy. The
+strategy consists of a new closeness loss that minimises the
+distance between prototypes of different classes. As shown
+in Fig. 1(b), our new loss enforces the prototypes to move
+closer to the classification boundary, as also demonstrated
+by Fig. 2(b) that reveals the support prototypes produced by
+the introduction of our new closeness loss are indeed more
+similar to the support vectors. Furthermore, in order to im-
+prove interpretability, we propose a new ProtoPNet classi-
+fier that integrates the support and trivial prototypes (named
+ST-ProtoPNet), where the goal is to produce two distinc-
+tive and complementary sets of prototypes to obtain more
+meaningful classification explanations. Due to the different
+natures of the two sets of prototypes, they can also enable
+further improvements in terms of classification accuracy.
+The major contributions of this work are:
+1. We provide the first study that makes an analogy be-
+tween the prototype learning from ProtoPNet methods
+and support vector learning from SVM, where we in-
+1The network has an input layer of 2 nodes, a hidden layer of 256 nodes
+(activated by tanh), and an output layer of 2 nodes (activated by sigmoid).
+Figure 2. Two-moon classification results from ProtoPNet and
+SVM classifiers. (a) Trivial prototypes (stars) and training sam-
+ples (circles) in the feature (top) and data (bottom) spaces. (b)
+Support prototypes (stars) and training samples (circles) in the fea-
+ture (top) and data (bottom) spaces. Note that in (a) and (b), each
+learned prototype is projected onto the nearest training sample in
+the feature space. (c) Support vectors (stars) and training samples
+(circles) from a Radial Basis Function (RBF) kernel based SVM.
+vestigate if by following SVM’s support vector learn-
+ing strategy and pulling the prototypes to be as close
+as possible to the classification boundary, it is possible
+to improve ProtoPNet’s classification accuracy.
+2. We present a new ST-ProtoPNet method to exploit both
+support and trivial prototypes for the interpretable im-
+age classification, where the two sets of prototypes can
+provide complementary information to improve both
+interpretability and classification accuracy.
+3. We conduct extensive experiments on three benchmark
+datasets which show that our ST-ProtoPNet method
+outperforms current state-of-the-art (SOTA) methods
+in terms of classification accuracy.
+In our experiments, we also demonstrate that the combi-
+nation of the two types of prototypes contributes to richer
+interpretability, where trivial prototypes tend to focus on
+both local parts of the visual object of interest and the back-
+ground, while support prototypes mainly focus on visually
+similar object parts of different classes.
+2
+
+0.9
+0.8
+0.8
+0.7
+0.6
+0.6
+0.5
+0.4
+0.4
+0.3
+0.2
+0.2
+0.1
+★
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.2
+1.2
+0.8
+0.8
+0.4
+0.4
+0.0
+0.0
+-0.4
+-0.4 
+0.8
+-0.8
+-1.2
+0.0
+0.6
+1.2
+-0.6
+1.8
+2.4
+-1.2
+-0.6
+0.0
+0.6
+1.2
+1.8
+2.4
+(a) Trivial prototypes
+(b) Support prototypes
+1.2
+0.8
+0.4
+Class A: red, green
+Class B: blue, black
+0.0
+0.4
+-0.8
+-1.2
+-0.6
+0.0
+0.6
+1.2
+1.8
+2.4
+(c) Support vectors from SVM2. Related Work
+In this section, we first review relevant studies on classi-
+fication interpretability where we focus on prototype-based
+methods, and then we briefly review support vector machine
+(SVM) classification. Finally, we provide a short survey on
+ensemble classification for interpretability.
+2.1. Classification Interpretability
+The interpretation of classification results produced by
+deep neural networks can be achieved by a variety of
+post-hoc explanation techniques, such as explanatory surro-
+gates [31,39,50], counterfactual examples [13,17,41], and
+saliency visualisation [38,40,49,53].
+In comparison with post-hoc explanations, prototype-
+based interpretability is directly present in the model’s in-
+ner computations. ProtoPNet [4] is the original work that
+uses class-specific prototypes for interpretable image clas-
+sification tasks.
+Similar to ProtoPNet, TesNet [48] con-
+structs class-specific transparent basis concepts on Grass-
+mann manifold for the interpretable classification.
+De-
+rived from ProtoPNet, Deformable ProtoPNet [10] employs
+spatially-flexible and deformable prototypes to adaptively
+capture meaningful object features. In ProtoPShare [37], a
+data-dependent merge-pruning method is presented to share
+prototypes among classes, which can reduce the number
+of prototypes used for classification.
+Alternatively, Pro-
+toPool [36] introduces a fully differentiable prototype as-
+signment strategy to reduce the number of prototypes. In
+Proto2Proto [18], a knowledge distillation method is de-
+signed to transfer interpretability from a teacher ProtoPNet
+to a shallow student ProtoPNet. ProtoTree [32] integrates
+the prototype learning into a binary neural decision tree
+that can explain its predictions by tracing a decision path
+throughout the tree. ViT-NeT [22] further establishes the
+prototype neural tree structure on visual transformers [11].
+Because of the ability to self-explain classification re-
+sults, prototype-based interpretability (e.g., ProtoPNet) has
+been widely utilised not only in the computer vision appli-
+cations above, but also in medical imaging [2, 20, 46] and
+face recognition [43]. However, an open question faced by
+these methods is if the prototypes being learned are the op-
+timal ones in terms of classification and interpretability.
+2.2. SVM vs Prototype-based Classification
+To better understand the optimality of prototypes, we
+consider the support vector machine (SVM) [6] classifier
+that finds support vectors to represent classes. More specif-
+ically, SVM learns the maximum-margin classifier defined
+by a classification boundary that maximises the distance to
+the closest training samples, which are the support vectors
+for the classes. The testing of SVM classifiers consists of
+computing a weighted similarity between a testing sample
+and the support vectors. It is interesting to note that the
+testing of prototype-based classifiers is also based on mea-
+suring the similarity between a testing image and a set of
+class-specific prototypes learned from the training process.
+Although the testing of SVM and prototype-based classi-
+fiers are similar, their training procedures are quite differ-
+ent. First, the training of a prototype-based classifier learns
+a fixed number of prototypes [4, 10], while the SVM clas-
+sifier learns to weight a variable number of support vectors
+from the training set. Second, in prototype-based classi-
+fiers [4,10], the learned prototypes tend to be as far as pos-
+sible from the classification boundary, which is contrary to
+the SVM training objective mentioned above.
+The study of deep learning methods from an SVM the-
+oretical perspective is a rich area of research [5,8,33] with
+huge potential given the impact that both techniques have
+had in the whole society for the last decades. However,
+there are many practical questions that need to be addressed,
+e.g., how to scale the kernel computation for large-scale
+datasets, how to shorten the training process [33], and how
+to integrate the learning of the SVM classifier with deep-
+learning features. In this paper, we focus on adapting the
+learning of ProtoPNet’s prototypes to make them similar to
+SVM’s support vectors, by forcing prototypes to be as close
+as possible to the classification boundary.
+2.3. Ensemble Classification
+Ensemble classification [9] is a traditional machine
+learning approach that combines the results from multiple
+classifiers, with the goals of improving learning generalisa-
+tion and classification calibration. The use of interpretable
+ensemble strategy to improve the classification accuracy has
+been explored in [4,10,48], which is achieved by summing
+the classification logits of multiple prototype-based clas-
+sifiers (e.g., ProtoPNets trained with different CNN back-
+bones). In this work, we obtain the interpretable ensemble
+classification by combining the predictions of two ProtoP-
+Nets with highly distinctive prototypes (i.e., support pro-
+totypes and trivial ones), which is different from previous
+studies [4,10,48] where the type of prototypes each individ-
+ual classifier produces is very similar given that the same
+objective function is employed for each classifier. More
+specifically, the ensemble classification used in this paper
+targets the utilisation of two complementary sets of pro-
+totypes, particularly when the prototypes are learned from
+quite different objective functions, such as the ones for
+learning the support and trivial prototypes.
+3. Preliminaries
+We assume to have a training set D = {(xn, yn)}|D|
+n=1,
+where x ∈ X ⊂ RH×W ×R is an image with R colour
+channels and y ∈ Y ⊂ {0, 1}C is a one-hot vector rep-
+resentation of the image class label.
+The interpretable
+3
+
+Input image
+Ensemble
+Trivial ProtoPNet
+Add-on
+layers
+Add-on 
+layers
+CNN
+Backbone 
+fθ 
+Non-cancer
+g
+FC layers k
+Support PPN
+0.42
+0.56
+0.96
+0.96
+ 
+ 
+95.31
+122.58 Cancer
+Similarity score
+Logits
+Support ProtoPNet
+Prototypes
+Feature vectors
+FC layer
+FC layer
+Output
+logits
+Logits
+Logits
+fω (s)
+fω (t)
+fΦ (t)
+fΦ (s)
+P (s) 
+P (t) 
+Figure 3. The architecture of our proposed ST-ProtoPNet method
+for the interpretable image classification.
+ProtoPNet [4, 10] is trained to learn a set of prototypes
+P = {pm}M
+m=1, where pm ∈ Rρ1×ρ2×D, with each of
+the C classes containing M/C prototypes. Without loss
+of generality, we assume ρ1 = ρ2 = 1, but the extension
+to general values is trivial. A typical ProtoPNet comprises
+four components: a CNN backbone, add-on layers, a pro-
+totype layer, and a fully connected (FC) layer. An input
+image x is fed to the CNN backbone fθ : X → F (pa-
+rameterised by θ ∈ Θ, where F ⊂ R ¯
+H× ¯
+W × ¯
+D) and then
+passed on to the add-on layers, denoted by fω : F →
+V (parameterised by ω ∈ Ω), to produce a feature map
+V ∈ V ⊂ R ¯
+H× ¯
+W ×D.
+The prototype layer computes
+the similarity between the feature map V and the M D-
+dimensional prototypes {pm}M
+m=1 to generate M similarity
+maps S(i,j)
+m
+= sim(V(i, j, :), pm), where i ∈ {1, ..., ¯H},
+j ∈ {1, ..., ¯W}, and sim(·, ·) represents a similarity mea-
+sure, e.g., cosine similarity [10] and projection metric [48].
+The prototype layer outputs M similarity scores from max-
+pooling S =
+�
+max
+i∈{1,..., ¯
+H},j∈{1,..., ¯
+W } S(i,j)
+m
+�M
+m=1, which are
+fed to the FC layer fφ : S → ∆, parameterised by φ ∈ Φ,
+to produce the classification prediction ˆy ∈ ∆ ⊂ [0, 1]C,
+where ∆ denotes the probability space for C classes.
+4. ST-ProtoPNet
+An overview of our proposed ST-ProtoPNet method
+is illustrated in Fig. 3, which comprises a shared CNN
+backbone fθ(·), two interpretable ProtoPNet classification
+branches, namely: 1) the support ProtoPNet represented by
+add-on layers fω(s)(·), prototype layer with support proto-
+types P(s), and FC layer fφ(s)(·) which outputs the classi-
+fication probability distribution ˆy(s) ∈ ∆; and 2) the trivial
+ProtoPNet branch with its add-on layers fω(t)(·), trivial pro-
+totypes P(t), and FC layer fφ(t)(·) that generates probabil-
+ity predictions ˆy(t) ∈ ∆. The final classification is obtained
+by combining the classification logits from both the support
+and trivial ProtoPNets. In our implementation, we construct
+the support and trivial ProtoPNet mainly based on the orig-
+inal ProtoPNet [4] and TesNet [48], as explained below.
+4.1. Support ProtoPNet
+The support ProtoPNet is designed to produce support
+(i.e., hard-to-learn) prototypes for classification that are as
+close as possible to the classification boundary, as shown in
+Fig. 1 and Fig. 2. The loss function to optimise the support
+ProtoPNet branch is defined as:
+θ∗,ω(s)∗, P(s)∗, φ(s)∗ =
+arg
+min
+θ,ω(s),P(s),φ(s)
+�
+(x,y)∈D
+ℓspt(x, y, θ, ω(s), P(s), φ(s)).
+(1)
+The loss for each training sample (x, y) ∈ D in Eq. (1)
+above is represented by:
+ℓspt(x, y, θ, ω(s), P(s), φ(s)) = ℓce(x, y, θ, ω(s), P(s), φ(s))
+− λ1ℓct(x, y, θ, ω(s), P(s))
++ λ2ℓsp(x, y, θ, ω(s), P(s))
+− λ3ℓcls(P(s))
++ λ4ℓort(P(s)),
+(2)
+where λ1, λ2, λ3, and λ4 are hyper-parameters to balance
+each term, ℓce(·) denotes the cross-entropy classification
+loss, ℓct(·) and ℓsp(·) represent the clustering and separa-
+tion losses, respectively, which are introduced to regularise
+the ProtoPNet’s training, as follows:
+ℓct(x, y, θ, ω(s), P(s)) = max
+p∈P(s)
+y
+max
+v∈V(s) sim(v, p),
+(3)
+ℓsp(x, y, θ, ω(s), P(s)) = max
+p/∈P(s)
+y
+max
+v∈V(s) sim(v, p),
+(4)
+where V(s) = fω(s)(fθ(x)) is the feature map extracted
+from the input image x, v represents one of the ¯H × ¯W
+feature vectors in V(s) obtained by matrix vectorisation, p
+is a normalised prototype (i.e., unit vector) in P(s), sim(·, ·)
+is one of the similarity functions defined in Section 3, and
+P(s)
+y
+denotes the set of prototypes of class y. The clustering
+loss in Eq. (3) and separation loss in Eq. (4) aim to learn
+a meaningful feature space in which the image features of
+a certain class are clustered around the prototypes of the
+class, and also well separated from those of other classes.
+As mentioned in Section 1, the effect of the clustering
+and separation losses above tend to push the prototypes
+of different classes as far as possible from the classifica-
+tion boundary, resulting in trivial prototypes, as displayed
+in Fig. 1(a) and Fig. 2(a). In order to learn the proposed
+support prototypes, we introduce the following novel close-
+ness loss ℓcls to explicitly enforce the prototypes of different
+classes to be close to each other, which is formulated as:
+ℓcls(P(s)) =
+C−1
+�
+c1=1
+C
+�
+c2=c1+1
+min
+pm∈Pc1,pn∈Pc2
+p⊤
+mpn.
+(5)
+4
+
+DIWGU2IOUX
+X noiengmid
+V
+★
+B
+B
+V
+口
+V
+GSLUIua
+1
+-
+WGILC
+■
+cb(²)ogeml bolodelnunogininT-e
+lsdel28loInnigino
+Wgbbru
+CGIGLUIGg CJ2IGL2During training, the closeness loss ℓcls maximises the
+pair-wise prototype similarity, in the form of dot product
+p⊤
+mpn between different classes in Eq. (5), with the goal of
+pulling the prototypes close to the classification boundary.
+As the prototypes move gradually towards the classification
+boundary, they are able to capture harder and harder visual
+features from training samples. On the other hand, since
+the prototypes are located near the classification boundary,
+this can put pressure on the support ProtoPNet’s learning
+and enforce it to learn highly discriminative feature repre-
+sentations to achieve accurate classification, which is bene-
+ficial to extract more meaningful semantic information from
+training samples.
+Ideally, each prototype of a class should focus on unique
+object parts of the training images (e.g., head, tail, and claw
+of birds), so that the prototypes can represent rich and di-
+verse visual patterns. However, there is no particular con-
+straints to guarantee such prototype diversity and the issue
+of prototype duplication [10] often occurs in the ProtoPNet
+family of models. To encourage the intra-class prototype di-
+versity, we employ an orthonormality loss [48] so that pro-
+totypes within a class can represent dissimilar visual pat-
+terns of training samples, which is defined as:
+ℓort(P(s)) =
+C
+�
+c=1
+∥Pc
+⊤Pc − IM/C∥2
+F ,
+(6)
+where ∥·∥2
+F represents Frobenius norm, Pc ∈ D ×R(M/C)
+stands for a matrix composed of the prototypes of class
+c (prototypes in each column of Pc are normalised), and
+IM/C is an identity matrix of size M/C × M/C.
+4.2. Trivial ProtoPNet
+As described in Section 4.1, the support ProtoPNet is
+developed to learn support (i.e., hard-to-learn) prototypes
+by forcing them to be close to the classification boundary.
+Considering that training samples contain not only hard vi-
+sual features but also important trivial ones that the support
+prototypes cannot fully capture, we propose to also learn
+trivial prototypes to provide complementary classification
+information, and exploit both the support and trivial proto-
+types for improved interpretable classification.
+The loss objective to train the trivial ProtoPNet branch is
+defined as follows:
+θ∗,ω(t)∗, P(t)∗, φ(t)∗ =
+arg
+min
+θ,ω(t),P(t),φ(t)
+�
+(x,y)∈D
+ℓtrv(x, y, θ, ω(t), P(t), φ(t)).
+(7)
+The loss for each training image (x, y) ∈ D in Eq. (7) above
+is represented by:
+ℓtrv(x, y, θ, ω(t), P(t), φ(t)) = ℓce(x, y, θ, ω(t), P(t), φ(t))
+− λ1ℓct(x, y, θ, ω(t), P(t))
++ λ2ℓsp(x, y, θ, ω(t), P(t))
++ λ3ℓdsc(P(t))
++ λ4ℓort(P(t)),
+(8)
+where λ1, λ2, λ3, and λ4 are hyper-parameters, ℓce(·) is the
+cross-entropy loss, the clustering loss ℓct, separation loss
+ℓsp, and orthonormality loss ℓort are the same as in the sup-
+port ProtoPNet defined in Eq. (3), (4) and (6), respectively.
+The trivial ProtoPNet targets the learning of easy proto-
+types that are far from the classification boundary and have
+a good discrimination ability. To help achieve this, we in-
+troduce a new discrimination loss ℓdsc to facilitate the inter-
+class separability between prototypes of different classes.
+This is formulated by minimising the pair-wise prototype
+similarities of different classes, as follows:
+ℓdsc(P(t)) =
+C−1
+�
+c1=1
+C
+�
+c2=c1+1
+max
+pm∈Pc1,pn∈Pc2
+p⊤
+mpn.
+(9)
+4.3. Training and Testing
+Training. Following the training strategies in the Pro-
+toPNet family [4, 10], the training procedure of our ST-
+ProtoPNet consists of 3 stages: 1) stochastic gradient de-
+scent (SGD) optimisation of the CNN backbone, add-on
+layers, and prototype layer, using a fixed FC layer initialised
+with +1.0 and -0.5 for correct and incorrect connection
+weights, respectively; 2) prototype projection by updating
+each prototype with its nearest latent training image patch;
+and 3) optimisation of the FC layer, with an additional L1
+regularisation on the incorrect connection weights (initially
+fixed at -0.5). In each stage, we alternate the optimisation
+of each branch of the ST-ProtoPNet between mini-batches.
+Testing.
+To exploit the complementary results from
+both branches of the ST-ProtoPNet, the final classification
+is obtained from the summed logits predicted by the two
+branches. It is worth noticing that this ensemble strategy
+introduces no loss of interpretablity but improved accuracy.
+5. Experiments
+We perform experiments on three fine-grained classifi-
+cation benchmark datasets: CUB-200-2011 [45], Stanford
+Cars [25], and Stanford Dogs [19]. To achieve fair com-
+parison, we follow previous studies [4, 48] by applying of-
+fline data augmentations (e.g., random rotation, skew, shear,
+and left-right flip) on the cropped CUB and cropped Cars
+datasets (using bounding boxes provided). We also validate
+5
+
+our method on the full CUB and full Dogs datasets, and
+employ the same online data augmentation methods (e.g.,
+random affine transformation and left-right flip) as used
+in Deformable ProtoPNet [10]. All images are resized to
+224 × 224 pixels as network input.
+5.1. Experimental Settings
+The proposed ST-ProtoPNet method is evaluated on the
+following CNN architectures: VGG-16, VGG-19, ResNet-
+34, ResNet-50, ResNet-152, DenseNet-121, and DenseNet-
+161. All CNN backbones are pre-trained on ImageNet [7],
+except for ResNet-50, which is pre-trained on iNatural-
+ist [44] for the experiment on full CUB [10]. The add-on
+layers include two 1 × 1 convolutional layers. For simplic-
+ity, we utilise the same prototype dimension D = 64 for all
+CNN backbones on the three datasets. For cropped CUB
+and Cars datasets, following [48], we use 10 prototypes (5
+support and 5 trivial) per class and the projection metric
+in the similarity function sim(·, ·). In full CUB and Dogs
+datasets, to ensure comparison fairness with Deformable
+ProtoPNet [10] that uses 10 2 × 2 (full CUB) and 10 3 × 3
+(full Dogs) deformable prototypes per class, we utilise the
+same total number of prototypes, i.e., 40 1 × 1 (20 support
+and 20 trivial) for full CUB and 90 1×1 (45 support and 45
+trivial) for full Dogs. Also, we employ the cosine similar-
+ity in sim(·, ·) and obtain 14 × 14 ( ¯H = ¯W = 14) feature
+maps by upsampling the original 7 × 7 feature maps via a
+bi-linear interpolation step, as in [10]. In our implemen-
+tation, we empirically set λ1 = 0.8, λ2 = 0.48 and 0.08
+for the support and trivial ProtoPNet branches respectively,
+λ3 = 1.0, and λ4 = 0.001. More details about the experi-
+mental setup can be found in the supplementary material.
+5.2. Performance Comparison
+Table 1 presents the classification accuracy (averaged
+across 5 runs) of our proposed ST-ProtoPNet on cropped
+CUB and cropped Cars datasets, where the Baseline is rep-
+resented by non-interpretable black-box CNN models. As
+can be seen, our ST-ProtoPNet outperforms other compet-
+ing methods across all backbone architectures for the task
+of bird species classification. Also, our method achieves the
+best results for the car model identification task when using
+VGG and DenseNet architectures as the CNN backbone.
+In particular, our VGG19-based ST-ProtoPNet reaches an
+average accuracy of 83.2% and 91.7% on CUB and Cars,
+respectively, surpassing other methods with the most im-
+provements across all backbones. Moreover, the support
+ProtoPNet generally performs better than methods utilising
+only trivial prototypes (e.g., ProtoPNet, TesNet, and Triv-
+ial ProtoPNet), showing the importance of learning support
+prototypes for the interpretable image classification. It is
+worth noting that our proposed ST-ProtoPNet produces su-
+perior performance over the support ProtoPNet method, in-
+dicating that both support and trivial prototypes are useful
+and can provide complementary information for achieving
+accurate and interpretable classification.
+Table 2 shows the classification results of different meth-
+ods on full CUB and full Dogs datasets. In both datasets,
+the classification accuracy of the original ProtoPNet method
+is generally worse than the non-interpretable counterpart
+(Baseline) under many CNN backbones. On the other hand,
+the accuracy by the trivial ProtoPNet and support ProtoP-
+Net are substantially better than those by Baseline, ProtoP-
+Net, and Deformable ProtoPNet. However, the proposed
+ST-ProtoPNet achieves more significant performance gains
+that result in the best classification accuracy across most
+CNN backbones, particularly when using a large number of
+prototypes (i.e., 40 1 × 1 prototypes per class for CUB and
+90 1×1 prototypes per class for Dogs), which demonstrates
+the effectiveness of utilising both the trivial and support
+prototypes for the interpretable image classification. Ad-
+ditionally, when using a smaller number of prototypes, i.e,
+10 1×1 prototypes per class, our ST-ProtoPNet method still
+exhibits competitive classification accuracy across multiple
+CNN backbones.
+We further compare our ST-ProtoPNet with other deep-
+learning methods that can provide different levels of inter-
+pretability on CUB dataset, with results shown in Table 3,
+where * and ** denote ensembled models trained with dif-
+ferent CNN backbones or hyper-parameters. As can be ob-
+served, an ensemble of three ST-ProtoPNets can achieve
+high classification accuracy (87.9% on cropped images,
+88.2% on full images), outperforming competing methods
+that are also based on an ensemble of three models (e.g.,
+ProtoTree, TesNet, and ProtoPool). Moreover, the ensemble
+of five ST-ProtoPNets exceeds all other competing methods
+and obtains the best classification accuracy of 88.1% and
+88.4% on cropped and full CUB images, respectively. Ex-
+perimental results on Stanford Cars and Stanford Dogs are
+presented in the supplementary material.
+5.3. Visualisation Analysis
+To better highlight the differences between the support
+and trivial prototypes, we select 8 categories of birds with
+visually similar features from cropped CUB-200-2011 to
+form a subset, and show the learned prototypes of our
+VGG19-based ST-ProtoPNet in Fig. 4, where each proto-
+type is visualised by projecting onto its nearest training im-
+age patch in the latent feature space [4]. We can see in
+Fig. 4(a) that the support prototypes can capture subtle and
+fine visual features of different classes. Importantly, notice
+that the support prototypes only focus on relevant bird parts,
+such as the head and belly. This is reasonable since our
+learning algorithm is designed to produce prototypes that
+are as close as possible to the classification boundary, where
+the image prototypical parts should be discriminative but at
+6
+
+Method
+CUB
+Cars
+VGG16
+VGG19
+ResNet34 ResNet152 Dense121 Dense161
+VGG16
+VGG19
+ResNet34 ResNet152 Dense121 Dense161
+Baseline
+73.3 ± 0.2 74.7 ± 0.4 82.2 ± 0.3 80.8 ± 0.4 81.8 ± 0.1 82.1 ± 0.2 87.3 ± 0.4 88.5 ± 0.3 92.6 ± 0.3 92.8 ± 0.4 92.0 ± 0.3 92.5 ± 0.3
+ProtoPNet [4]
+77.2 ± 0.2 77.6 ± 0.2 78.6 ± 0.1 79.2 ± 0.3 79.0 ± 0.2 80.8 ± 0.3 88.3 ± 0.2 89.4 ± 0.2 88.8 ± 0.1 88.5 ± 0.3 87.7 ± 0.1 89.5 ± 0.2
+TesNet [48]
+81.3 ± 0.2 81.4 ± 0.1 82.8 ± 0.1 82.7 ± 0.2 84.8 ± 0.2 84.6 ± 0.3 90.3 ± 0.2 90.6 ± 0.2 90.9 ± 0.2 92.0 ± 0.2 91.9 ± 0.3 92.6 ± 0.3
+Trivial ProtoPNet
+80.8 ± 0.2 81.2 ± 0.2 82.5 ± 0.2 83.1 ± 0.3 83.9 ± 0.3 84.6 ± 0.3 90.1 ± 0.2 90.7 ± 0.2 91.1 ± 0.2 91.5 ± 0.2 91.4 ± 0.3 92.4 ± 0.3
+Support ProtoPNet
+81.7 ± 0.2 81.8 ± 0.3 83.0 ± 0.1 83.6 ± 0.2 84.7 ± 0.2 85.2 ± 0.3 90.9 ± 0.2 90.8 ± 0.2 91.0 ± 0.2 91.8 ± 0.2 91.7 ± 0.2 92.7 ± 0.3
+ST-ProtoPNet (ours) 82.9 ± 0.2 83.2 ± 0.2 83.5 ± 0.1 84.1 ± 0.2 85.4 ± 0.2 86.1 ± 0.2 91.1 ± 0.2 91.7 ± 0.2 91.4 ± 0.1 92.0 ± 0.2 92.3 ± 0.3 92.7 ± 0.2
+Table 1. Classification accuracy (%) on cropped CUB-200-2011 and Stanford Cars by competing methods using different CNN backbones.
+Method
+Prototype
+CUB
+Prototype
+Dogs
+VGG16 VGG19 ResNet34 ResNet50 ResNet152 Dense121 Dense161
+VGG16 VGG19 ResNet34 ResNet50 ResNet152 Dense121 Dense161
+Baseline
+–
+70.9
+71.3
+76.0
+78.7
+79.2
+78.2
+80.0
+–
+75.6
+77.3
+81.1
+83.1
+85.2
+81.9
+84.1
+ProtoPNet [4]
+1×1p, 10pc
+70.3
+72.6
+72.4
+81.1
+74.3
+74.0
+75.4
+1×1p, 10pc
+70.7
+73.6
+73.4
+76.4
+76.2
+72.0
+77.3
+ProtoPNet [4]
+1×1p, 40pc
+72.9
+74.2
+74.1
+84.8
+76.0
+76.6
+78.5
+1×1p, 90pc
+73.9
+75.3
+76.1
+78.1
+79.7
+75.4
+78.8
+TesNet [48]
+1×1p, 10pc
+75.8
+77.5
+76.2
+86.5
+79.0
+80.2
+79.6
+1×1p, 10pc
+74.3
+77.1
+80.1
+82.4
+83.8
+80.3
+83.8
+TesNet [48]
+1×1p, 40pc
+77.6
+79.2
+76.5
+87.3
+80.1
+80.9
+81.3
+1×1p, 90pc
+78.5
+79.6
+81.2
+83.3
+84.5
+82.1
+85.2
+Deformable ProtoPNet [10] 2×2p, 10pc
+75.7
+76.0
+76.8
+86.4
+79.6
+79.0
+81.2
+3×3p, 10pc
+75.8
+77.9
+80.6
+82.2
+86.5
+80.7
+83.7
+Trivial ProtoPNet
+1×1p, 40pc
+80.0
+79.5
+77.5
+87.2
+80.8
+81.1
+82.1
+1×1p, 90pc
+78.6
+80.4
+82.6
+85.0
+87.0
+82.3
+85.9
+Support ProtoPNet
+1×1p, 40pc
+80.4
+80.0
+78.4
+87.5
+80.2
+81.5
+82.4
+1×1p, 90pc
+79.0
+80.6
+83.0
+85.1
+87.3
+82.6
+86.2
+ST-ProtoPNet (ours)
+1×1p, 10pc
+76.8
+77.6
+77.4
+86.6
+78.7
+78.6
+80.6
+1×1p, 10pc
+74.2
+77.2
+80.8
+84.0
+85.3
+79.4
+84.4
+ST-ProtoPNet (ours)
+1×1p, 40pc
+81.0
+80.2
+78.2
+88.0
+81.2
+81.8
+82.7
+1×1p, 90pc
+79.1
+80.9
+83.4
+85.7
+87.2
+82.9
+86.6
+Table 2. Classification accuracy (%) on full CUB-200-2011 and Stanford Dogs datasets by competing approaches using different CNN
+backbones, where kpc represents k prototypes per class and ρ1×ρ2p denotes the spatial shape of the prototypes.
+(a) Support prototypes
+(b) Trivial prototypes
+Laysan Albatross
+Sooty Albatross
+Crested Auklet
+Crested Auklet
+Parakeet Auklet
+Rhinoceros Auklet
+Brewer Blackbird
+Redwinged Blackbird
+Figure 4. Visual comparison between the support (a) and trivial (b) prototypes from cropped CUB-200-2011, where each row exhibits
+prototypes of the same class. In each pair, the first column shows the original image with a prototype indicated in a yellow bounding box,
+the second column demonstrates the prototype’s corresponding activation map.
+the same time visually similar among different classes. By
+contrast, it is observed in Fig. 4(b) that the trivial prototypes
+tend to focus not only on the relevant bird parts but also the
+background regions. For example, some trivial prototypes
+of the Laysan Albatross and Sooty Albatross classes cap-
+ture the sea surface as they often appear with the sea back-
+ground. We argue that this is because the trivial ProtoPNet
+may treat the background as the most easy-to-learn pattern,
+focusing less on the target’s visual parts of the class.
+Fig. 5 displays an example of the interpretable reasoning
+process of our ST-ProtoPNet method in classifying a testing
+bird image. As can be seen, each ProtoPNet branch calcu-
+lates its own classification logits (weighted sum of similar-
+ity scores), which is then combined to generate the final pre-
+dictions. To be specific, when classifying a Brewer Black-
+bird, the support prototypes are quite active on the belly or
+lower surface of the bird. Meanwhile, the trivial prototypes
+mostly have high activations on the bird’s head and tail. In
+this case, the support ProtoPNet branch obtains a relatively
+higher similarity score (22.913), compared with the trivial
+branch (20.248). Note that our ST-ProtoPNet exploits both
+the support and trivial prototypes to capture much richer
+representations of the target object from different perspec-
+tives, which enables the production of the final decision in
+a complementary way. Notably, the ensemble classification
+improves interpretability and accuracy using richer repre-
+sentations for the target object. More examples of visual
+prototypes and interpretable reasoning on Cars and Dogs
+are provided in the supplementary material.
+7
+
+Interpretability
+Method
+Accuracy (%)
+None
+B-CNN [29]
+85.1 (b) 84.1 (f)
+Object-level attention
+CAM [53]
+70.5 (b) 63.0 (f)
+CSG [28]
+82.6 (b) 78.5 (f)
+Part-level attention
+PA-CNN [24]
+82.8 (b)
+–
+MG-CNN [47]
+83.0 (b) 81.7 (f)
+MA-CNN [51]
+–
+86.5 (f)
+RA-CNN [12]
+–
+85.3 (f)
+TASN [52]
+–
+87.0 (f)
+Part-level attention + Prototypes
+Region [15]
+81.5 (b) 80.2 (f)
+ProtoPNet* [4]
+84.8 (b) 81.1 (f)
+ProtoTree* [32]
+–
+86.6 (f)
+TesNet* [48]
+86.2 (b) 83.5 (f)
+ProtoPool* [36]
+87.5 (b)
+–
+ST-ProtoPNet* (ours)
+87.9 (b) 88.2 (f)
+ProtoTree** [32]
+–
+87.2 (f)
+Deformable ProtoPNet** [10]
+–
+87.8 (f)
+ProtoPool** [36]
+87.6 (b)
+–
+ST-ProtoPNet** (ours)
+88.1 (b) 88.4 (f)
+Table 3. Classification accuracy and interpretability of different
+methods on CUB-200-2011.
+“b” and “f” denote the model is
+trained and tested on cropped and full images, respectively. *:
+Ensembled of three models. **: Ensembled of five models.
+Testing image
+Prototype
+Training image where 
+the prototype derives
+Activation map
+Similarity score
+Connection weight
+Individual logits
+Combined logits
+5.035
+0.981
+×
+4.939
+=
+4.891
+0.964
+×
+4.715
+=
+4.321
+0.979
+×
+4.230
+=
+4.206
+0.972
+×
+4.088
+=
+Support ProtoPNet
+Trivial ProtoPNet
+...
+...
+...
+...
+43.161
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+22.913
+20.248
+Figure 5. An example of the interpretable reasoning of our ST-
+ProtoPNet for classifying a testing Brewer Blackbird image.
+5.4. Ablation Study
+The Closeness and Discrimination Losses. In order to
+validate the effectiveness of our proposed closeness loss in
+Eq. (5) and discrimination loss in Eq. (9), we first conduct
+ablation studies on full CUB and full Dogs datasets using
+ResNet34 as the CNN backbone, with results provided in
+Table 4. We can notice that both the closeness and discrim-
+ination losses can improve the classification accuracy, com-
+pared with the Baseline ProtoPNet trained with only clus-
+tering, separation, and orthonormality losses. Importantly,
+the closeness loss introduces a larger performance improve-
+ment since it aims at learning support (i.e., hard-to-learn)
+prototypes that lie near the classification boundary.
+Combining Support and Trivial Prototypes. We also
+investigate the importance of integrating the two comple-
+mentary sets of support and trivial prototypes for achieving
+improved classification. To achieve this, we first train a two-
+branch model where both branches learn the same type of
+Method
+ℓct
+ℓsp
+ℓort
+ℓdsc
+ℓcls
+Accuracy (%)
+CUB
+Dogs
+Baseline ProtoPNet
+✓
+✓
+✓
+76.7
+80.9
+Trivial ProtoPNet
+✓
+✓
+✓
+✓
+77.5
+82.6
+Support ProtoPNet
+✓
+✓
+✓
+✓
+78.4
+83.0
+Table 4. Ablation analysis of the closeness loss ℓcls in Eq. (5) to
+learn support prototypes, and discrimination loss ℓdsc in Eq. (9) to
+learn trivial prototypes on full CUB-200-2011 and Stanford Dogs.
+Method
+VGG16
+VGG19
+ResNet34 ResNet152 Dense121 Dense161
+Trivial Ensemble
+81.4 ± 0.3 81.8 ± 0.2 82.7 ± 0.2 83.2 ± 0.3 84.4 ± 0.2 85.0 ± 0.3
+Support Ensemble
+82.1 ± 0.2 82.4 ± 0.3 83.0 ± 0.2 83.7 ± 0.3 84.8 ± 0.2 85.5 ± 0.2
+Trivial Branch
+81.0 ± 0.2 81.1 ± 0.3 82.4 ± 0.2 82.9 ± 0.3 84.1 ± 0.3 84.8 ± 0.3
+Support Branch
+81.5 ± 0.3 81.8 ± 0.3 82.8 ± 0.2 83.4 ± 0.3 84.6 ± 0.2 85.4 ± 0.2
+ST-ProtoPNet (ours) 82.9 ± 0.2 83.2 ± 0.2 83.5 ± 0.1 84.1 ± 0.2 85.4 ± 0.2 86.1 ± 0.2
+Table 5. Ablation study of the combination of support and trivial
+prototypes for improved classification on cropped CUB-200-2011.
+prototypes and the final result is produced by the ensemble
+of them (Trivial Ensemble and Support Ensemble). Besides,
+for our ST-ProtoPNet, we also provide results of its individ-
+ual branches (Trivial Branch and Support Branch). Table 5
+gives the experimental results on cropped CUB dataset. We
+can notice that combining the two different types of pro-
+totypes (ST-ProtoPNet) achieves superior performance over
+combining only the same type of prototypes (Trivial Ensem-
+ble and Support Ensemble), indicating that our performance
+improvements are from not only the ensemble strategy but
+also the two complementary sets of prototypes. Also, ST-
+ProtoPNet indeed exhibits higher accuracy than its individ-
+ual branches, further verifying that the results from the two
+branches are complementary, and the combination of them
+are effective to improve the final classification accuracy.
+6. Conclusion and Future Work
+In this paper, we propose the ST-ProtoPNet method to
+exploit both support (i.e., hard-to-learn) and trivial (i.e.,
+easy-to-learn) prototypes for the interpretable image clas-
+sification, where the two distinctive sets of prototypes
+can provide complementary results. In addition, our ST-
+ProtoPNet is a general approach that can be easily applied
+to existing prototype-based interpretable models. One limi-
+tation for our method is that we empirically adopt the same
+number of support and trivial prototypes and the same to-
+tal number of prototypes for each class. Considering the
+different learning difficulties and imbalanced training sam-
+ples among classes in other real-world datasets, e.g., Ima-
+geNet [7], a better way to adaptively learn a flexible num-
+ber of support and trivial prototypes is needed and deserves
+to be further investigated in our future work. Furthermore,
+given that we mimic the behaviour of the support vectors of
+SVM classifier to obtain the support prototypes by forcing
+them to be as close as possible to the classification bound-
+ary, we plan to develop new methods to learn prototypes
+with gradient-based kernel computations, e.g., neural tan-
+gent kernel [16] and path kernel [8].
+8
+
+References
+[1] David Alvarez Melis and Tommi Jaakkola. Towards robust
+interpretability with self-explaining neural networks.
+Ad-
+vances in Neural Information Processing Systems, 31, 2018.
+1
+[2] Alina Jade Barnett, Fides Regina Schwartz, Chaofan Tao,
+Chaofan Chen, Yinhao Ren, Joseph Y Lo, and Cynthia
+Rudin. A case-based interpretable deep learning model for
+classification of mass lesions in digital mammography. Na-
+ture Machine Intelligence, 3(12):1061–1070, 2021. 3
+[3] Moritz B¨ohle, Mario Fritz, and Bernt Schiele. B-cos net-
+works: Alignment is all we need for interpretability. In Pro-
+ceedings of the IEEE/CVF Conference on Computer Vision
+and Pattern Recognition, pages 10329–10338, 2022. 1
+[4] Chaofan Chen, Oscar Li, Daniel Tao, Alina Barnett, Cynthia
+Rudin, and Jonathan K Su. This looks like that: deep learn-
+ing for interpretable image recognition. Advances in Neural
+Information Processing Systems, 32, 2019. 1, 3, 4, 5, 6, 7, 8
+[5] Yilan Chen, Wei Huang, Lam Nguyen, and Tsui-Wei Weng.
+On the equivalence between neural network and support vec-
+tor machine.
+Advances in Neural Information Processing
+Systems, 34:23478–23490, 2021. 3
+[6] Corinna Cortes and Vladimir Vapnik. Support-vector net-
+works. Machine Learning, 20(3):273–297, 1995. 2, 3
+[7] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li,
+and Li Fei-Fei. Imagenet: A large-scale hierarchical image
+database. In 2009 IEEE Conference on Computer Vision and
+Pattern Recognition, pages 248–255. Ieee, 2009. 6, 8
+[8] Pedro Domingos.
+Every model learned by gradient de-
+scent is approximately a kernel machine.
+arXiv preprint
+arXiv:2012.00152, 2020. 3, 8
+[9] Xibin Dong, Zhiwen Yu, Wenming Cao, Yifan Shi, and
+Qianli Ma.
+A survey on ensemble learning.
+Frontiers of
+Computer Science, 14(2):241–258, 2020. 3
+[10] Jon Donnelly, Alina Jade Barnett, and Chaofan Chen. De-
+formable protopnet: An interpretable image classifier using
+deformable prototypes.
+In Proceedings of the IEEE/CVF
+Conference on Computer Vision and Pattern Recognition,
+pages 10265–10275, 2022. 1, 3, 4, 5, 6, 7, 8
+[11] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov,
+Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner,
+Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl-
+vain Gelly, et al. An image is worth 16x16 words: Trans-
+formers for image recognition at scale. In International Con-
+ference on Learning Representations, 2020. 3
+[12] Jianlong Fu, Heliang Zheng, and Tao Mei. Look closer to
+see better: Recurrent attention convolutional neural network
+for fine-grained image recognition. In Proceedings of the
+IEEE Conference on Computer Vision and Pattern Recogni-
+tion, pages 4438–4446, 2017. 8
+[13] Yash Goyal, Ziyan Wu, Jan Ernst, Dhruv Batra, Devi Parikh,
+and Stefan Lee. Counterfactual visual explanations. In In-
+ternational Conference on Machine Learning, pages 2376–
+2384. PMLR, 2019. 3
+[14] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep residual learning for image recognition. In Proceed-
+ings of the IEEE Conference on Computer Vision and Pattern
+Recognition, pages 770–778, 2016. 1
+[15] Zixuan Huang and Yin Li. Interpretable and accurate fine-
+grained recognition via region grouping. In Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition, pages 8662–8672, 2020. 8
+[16] Arthur Jacot, Franck Gabriel, and Cl´ement Hongler. Neu-
+ral tangent kernel: Convergence and generalization in neural
+networks. Advances in Neural Information Processing Sys-
+tems, 31, 2018. 8
+[17] Eoin M Kenny and Mark T Keane. On generating plausible
+counterfactual and semi-factual explanations for deep learn-
+ing.
+In Proceedings of the AAAI Conference on Artificial
+Intelligence, volume 35, pages 11575–11585, 2021. 3
+[18] Monish Keswani, Sriranjani Ramakrishnan, Nishant Reddy,
+and Vineeth N Balasubramanian.
+Proto2proto: Can you
+recognize the car, the way i do?
+In Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition, pages 10233–10243, 2022. 3
+[19] Aditya Khosla, Nityananda Jayadevaprakash, Bangpeng
+Yao, and Fei-Fei Li. Novel dataset for fine-grained image
+categorization: Stanford dogs. In Proc. CVPR Workshop on
+Fine-grained Visual Categorization (FGVC), volume 2. Cite-
+seer, 2011. 5
+[20] Eunji Kim, Siwon Kim, Minji Seo, and Sungroh Yoon. Xpro-
+tonet: diagnosis in chest radiography with global and local
+explanations. In Proceedings of the IEEE/CVF Conference
+on Computer Vision and Pattern Recognition, pages 15719–
+15728, 2021. 3
+[21] Jinkyu Kim and John Canny. Interpretable learning for self-
+driving cars by visualizing causal attention. In Proceedings
+of the IEEE International Conference on Computer Vision,
+pages 2942–2950, 2017. 1
+[22] Sangwon Kim, Jaeyeal Nam, and Byoung Chul Ko.
+Vit-
+net: Interpretable vision transformers with neural tree de-
+coder. In International Conference on Machine Learning,
+pages 11162–11172. PMLR, 2022. 3
+[23] Pang Wei Koh and Percy Liang. Understanding black-box
+predictions via influence functions. In International Confer-
+ence on Machine Learning, pages 1885–1894. PMLR, 2017.
+1
+[24] Jonathan Krause, Hailin Jin, Jianchao Yang, and Li Fei-Fei.
+Fine-grained recognition without part annotations. In Pro-
+ceedings of the IEEE Conference on Computer Vision and
+Pattern Recognition, pages 5546–5555, 2015. 8
+[25] Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei.
+3d object representations for fine-grained categorization. In
+Proceedings of the IEEE International Conference on Com-
+puter Vision Workshops, pages 554–561, 2013. 5
+[26] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.
+Imagenet classification with deep convolutional neural net-
+works. Communications of the ACM, 60(6):84–90, 2017. 1
+[27] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep
+learning. Nature, 521(7553):436–444, 2015. 1
+[28] Haoyu Liang, Zhihao Ouyang, Yuyuan Zeng, Hang Su, Zi-
+hao He, Shu-Tao Xia, Jun Zhu, and Bo Zhang. Training in-
+terpretable convolutional neural networks by differentiating
+9
+
+class-specific filters. In European Conference on Computer
+Vision, pages 622–638. Springer, 2020. 8
+[29] Tsung-Yu Lin, Aruni RoyChowdhury, and Subhransu Maji.
+Bilinear cnn models for fine-grained visual recognition. In
+Proceedings of the IEEE International Conference on Com-
+puter Vision, pages 1449–1457, 2015. 8
+[30] Rong Liu, Feng Mai, Zhe Shan, and Ying Wu.
+Predict-
+ing shareholder litigation on insider trading from financial
+text: An interpretable deep learning approach. Information
+& Management, 57(8):103387, 2020. 1
+[31] Scott M Lundberg and Su-In Lee.
+A unified approach to
+interpreting model predictions. Advances in Neural Infor-
+mation Processing Systems, 30, 2017. 3
+[32] Meike Nauta, Ron van Bree, and Christin Seifert. Neural
+prototype trees for interpretable fine-grained image recogni-
+tion. In Proceedings of the IEEE/CVF Conference on Com-
+puter Vision and Pattern Recognition, pages 14933–14943,
+2021. 3, 8
+[33] Garima Pruthi, Frederick Liu, Satyen Kale, and Mukund
+Sundararajan. Estimating training data influence by tracing
+gradient descent. Advances in Neural Information Process-
+ing Systems, 33:19920–19930, 2020. 3
+[34] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.
+Faster r-cnn: Towards real-time object detection with region
+proposal networks.
+Advances in Neural Information Pro-
+cessing Systems, 28, 2015. 1
+[35] Cynthia Rudin. Stop explaining black box machine learn-
+ing models for high stakes decisions and use interpretable
+models instead. Nature Machine Intelligence, 1(5):206–215,
+2019. 1
+[36] Dawid Rymarczyk, Łukasz Struski, Michał G´orszczak, Ko-
+ryna Lewandowska, Jacek Tabor, and Bartosz Zieli´nski. In-
+terpretable image classification with differentiable proto-
+types assignment. In European Conference on Computer Vi-
+sion, pages 351–368. Springer, 2022. 3, 8
+[37] Dawid Rymarczyk, Łukasz Struski, Jacek Tabor, and Bartosz
+Zieli´nski. Protopshare: Prototypical parts sharing for simi-
+larity discovery in interpretable image classification. In Pro-
+ceedings of the 27th ACM SIGKDD Conference on Knowl-
+edge Discovery & Data Mining, pages 1420–1430, 2021. 3
+[38] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das,
+Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra.
+Grad-cam:
+Visual explanations from deep networks via
+gradient-based localization. In Proceedings of the IEEE In-
+ternational Conference on Computer Vision, pages 618–626,
+2017. 3
+[39] Vivswan Shitole, Fuxin Li, Minsuk Kahng, Prasad Tadepalli,
+and Alan Fern. One explanation is not enough: structured
+attention graphs for image classification. Advances in Neural
+Information Processing Systems, 34:11352–11363, 2021. 3
+[40] Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman.
+Deep inside convolutional networks:
+Visualising image
+classification models and saliency maps.
+arXiv preprint
+arXiv:1312.6034, 2013. 3
+[41] Damien Teney, Ehsan Abbasnedjad, and Anton van den Hen-
+gel. Learning what makes a difference from counterfactual
+examples and gradient supervision. In European Conference
+on Computer Vision, pages 580–599. Springer, 2020. 3
+[42] Erico Tjoa and Cuntai Guan.
+A survey on explainable
+artificial intelligence (xai):
+Toward medical xai.
+IEEE
+Transactions on Neural Networks and Learning Systems,
+32(11):4793–4813, 2020. 1
+[43] Loc Trinh, Michael Tsang, Sirisha Rambhatla, and Yan
+Liu.
+Interpretable and trustworthy deepfake detection via
+dynamic prototypes. In Proceedings of the IEEE/CVF Win-
+ter Conference on Applications of Computer Vision, pages
+1973–1983, 2021. 3
+[44] Grant Van Horn, Oisin Mac Aodha, Yang Song, Yin Cui,
+Chen Sun, Alex Shepard, Hartwig Adam, Pietro Perona, and
+Serge Belongie.
+The inaturalist species classification and
+detection dataset. In Proceedings of the IEEE Conference
+on Computer Vision and Pattern Recognition, pages 8769–
+8778, 2018. 6
+[45] Catherine Wah, Steve Branson, Peter Welinder, Pietro Per-
+ona, and Serge Belongie. The caltech-ucsd birds-200-2011
+dataset. 2011. 5
+[46] Chong Wang, Yuanhong Chen, Yuyuan Liu, Yu Tian, Feng-
+bei Liu, Davis J McCarthy, Michael Elliott, Helen Frazer,
+and Gustavo Carneiro.
+Knowledge distillation to ensem-
+ble global and interpretable prototype-based mammogram
+classification models. In International Conference on Med-
+ical Image Computing and Computer-Assisted Intervention,
+pages 14–24. Springer, 2022. 3
+[47] Dequan Wang, Zhiqiang Shen, Jie Shao, Wei Zhang, Xi-
+angyang Xue, and Zheng Zhang. Multiple granularity de-
+scriptors for fine-grained categorization.
+In Proceedings
+of the IEEE International Conference on Computer Vision,
+pages 2399–2406, 2015. 8
+[48] Jiaqi Wang, Huafeng Liu, Xinyue Wang, and Liping Jing.
+Interpretable image recognition by constructing transparent
+embedding space. In Proceedings of the IEEE/CVF Inter-
+national Conference on Computer Vision, pages 895–904,
+2021. 3, 4, 5, 6, 7, 8
+[49] Matthew D Zeiler and Rob Fergus. Visualizing and under-
+standing convolutional networks. In European Conference
+on Computer Vision, pages 818–833. Springer, 2014. 3
+[50] Quanshi Zhang, Ruiming Cao, Feng Shi, Ying Nian Wu, and
+Song-Chun Zhu. Interpreting cnn knowledge via an explana-
+tory graph. In Proceedings of the AAAI Conference on Arti-
+ficial Intelligence, volume 32, 2018. 3
+[51] Heliang Zheng, Jianlong Fu, Tao Mei, and Jiebo Luo. Learn-
+ing multi-attention convolutional neural network for fine-
+grained image recognition. In Proceedings of the IEEE Inter-
+national Conference on Computer Vision, pages 5209–5217,
+2017. 8
+[52] Heliang Zheng, Jianlong Fu, Zheng-Jun Zha, and Jiebo Luo.
+Looking for the devil in the details: Learning trilinear atten-
+tion sampling network for fine-grained image recognition. In
+Proceedings of the IEEE/CVF Conference on Computer Vi-
+sion and Pattern Recognition, pages 5012–5021, 2019. 8
+[53] Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva,
+and Antonio Torralba. Learning deep features for discrimi-
+native localization. In Proceedings of the IEEE Conference
+on Computer Vision and Pattern Recognition, pages 2921–
+2929, 2016. 3, 8
+10
+

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+Pseudo-entropy for descendant operators in two-dimensional
+conformal field theories
+Song Hea,c,1, Jie Yangb,2, Yu-Xuan Zhanga,3, Zi-Xuan Zhaoa,b,4
+aCenter for Theoretical Physics and College of Physics, Jilin University,
+Changchun 130012, People’s Republic of China
+bSchool of Mathematical Sciences, Capital Normal University,
+Beijing 100048, People’s Republic of China
+cMax Planck Institute for Gravitational Physics (Albert Einstein Institute),
+Am M¨uhlenberg 1, 14476 Golm, Germany
+Abstract
+We study the late-time properties of pseudo-(R´enyi) entropy of locally excited states in rational
+conformal field theories (RCFTs). The two non-orthogonal locally excited states used to construct
+the transition matrix are generated by acting different descendant operators on the vacuum. We
+prove that for the cases where two descendant operators are generated by a single Virasoro generator
+respectively acting on a primary operator, the late-time excess of pseudo-entropy and pseudo-R´enyi
+entropy always coincides with the logarithmic of the quantum dimension of the corresponding
+primary operator. Furthermore, we consider two linear combination operators generated by the
+generic summation of Virasoro generators. We find their pseudo-R´enyi entropy and pseudo-entropy
+may get additional contributions, as the mixing of holomorphic and anti-holomorphic parts of
+the correlation function enhances the entanglement. Finally, we assert the pseudo-R´enyi entropy
+and pseudo-entropy are still the logarithmic quantum dimension of the primary operator when
+the correlation function of linear combination operators can be divided into the product of its
+holomorphic part and anti-holomorphic part. We offer some examples to illustrate the phenomenon.
+1hesong@jlu.edu.cn
+2yangjie@cnu.edu.cn
+3yuxuanz18@jlu.edu.cn
+4zhaozixuan@cnu.edu.cn
+arXiv:2301.04891v1  [hep-th]  12 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Setup in 2D CFTs
+3
+2.1
+Replica method with local operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+Convention and useful formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3
+Second pseudo-R´enyi entropy ∆S(2)
+A
+for descendent operators
+5
+3.1
+∆S(2)
+A
+for Vα = L−1O, Vβ = O
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+3.2
+∆S(2)
+A
+for Vα = L−nO, Vβ = O
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.3
+∆S(2)
+A
+for Vα = L−nO, Vβ = L−mO . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+4
+k-th pseudo-R´enyi entropy for generic descendent states
+9
+4.1
+∆S(k)
+A
+for Vα = L−nO, Vβ = L−mO . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.2
+∆S(k)
+A
+for Linear combination of descendent operators
+. . . . . . . . . . . . . . . . . .
+11
+5
+Conclusion and prospect
+14
+A Reduction of ⟨O(−n)(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+16
+B Reduction of⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1
+16
+1
+Introduction
+The discovery of AdS/CFT correspondence [1–3] has motivated much research related to quantum
+information theory in the high-energy physics community in recent years. Among them, quantum
+entanglement, as a carrier of quantum information, play an increasingly significant role in probing
+the structure of quantum field theories (QFTs) [4–10], the emergence of geometry [11–13], black hole
+information paradox [14–18].
+Recently, a new entanglement measure, called pseudo-entropy, was proposed in [19] as a generaliza-
+tion of entanglement entropy. Specifically, pseudo-entropy is a two-state vector version of entanglement
+entropy, defined as follows. Given two non-orthogonal states |ψ⟩ and |ϕ⟩ in the Hilbert space HS of a
+composed quantum system S = A ∪ B, we first constructs an operator called transition matrix acting
+on HS,
+T ψ|ϕ ≡ |ψ⟩⟨ϕ|
+⟨ϕ|ψ⟩ =
+ρψρϕ
+tr[ρψρϕ].
+(1)
+The pseudo-entropy of subsystem A, then, is obtained by calculating the von Neumann entropy of the
+reduced transition matrix T ψ|ϕ
+A
+≡ trB[T ψ|ϕ],
+S(T ψ|ϕ
+A
+) = −tr[T ψ|ϕ
+A
+log T ψ|ϕ
+A
+].
+(2)
+1
+
+In general, the reduced transition matrix is non-Hermitian and pseudo-entropy can be complex-valued.
+When |ϕ⟩ = |ψ⟩, pseudo-entropy reduces to entanglement entropy. Like entanglement entropy, in
+practice, especially in QFTs, one usually computes a quantity called pseudo-R´enyi entropy,
+S(n)
+A
+=
+1
+1 − n log tr
+��
+T ψ|ϕ
+A
+�n�
+,
+(3)
+instead of pseudo-entropy to avoid the computation of the logarithm of the matrix. The limit n → 1
+gives back the pseudo-entropy.
+Pseudo-entropy is originally proposed from the study of the generalization of holography entangle-
+ment entropy [19]. In the AdS/CFT context, the pseudo-entropy of a boundary subsystem is proposed
+to be dual to the area of a minimal surface in the Euclidean time-dependent AdS space [19]. In ad-
+dition, it is found that pseudo-entropy is closely related to the postselection experiments in quantum
+information (i.e., in addition to the initial state, the system’s final state is also specified [20]). The
+first is that the input of the pseudo-entropy—transition matrix (1) plays the role of density matrix
+when one computes the weak values [21, 22] of observables in the post-selected system. Secondly,
+pseudo-entropy is considered to characterize the averaged number of EPR pairs that could have been
+distilled in the post-selected systems [19, 23]. There are also many research interests and prospects
+driving the study of pseudo-entropy in QFTs [24–30]. See [31–36] for other related developments of
+pseudo-entropy.
+The present paper aims to study the properties of pseudo-entropy of locally descendant excited
+states in two-dimensional (2D) conformal field theories (CFTs). Our study can be traced back to the
+research on entanglement entropy in local quantum quenches in 2D CFTs [37–52].5 It is found that
+the excess of R´enyi entropy of the local primary or descendant excited states in rational conformal
+field theories (RCFTs) saturates to a constant equal to the logarithm of the quantum dimension [62]
+of the local operator’s conformal family [39, 44, 45]. Such saturation is well explained by the picture
+of quasiparticle pairs propagation [38]. The related research is extended to the pseudo-entropy in
+parallel [30]. When considering the real-time evolution of the pseudo-R´enyi entropy of locally primary
+excited states in RCFTs, the early-time behavior of the excess of pseudo-R´enyi entropy depends on the
+respective spatial positions of two identical primary operators, which is not universal. Nevertheless,
+its late-time behavior is universal, which only depends on the quantum dimension of the primary
+operator, just like the entanglement entropy. The result suggests that the picture of quasiparticle
+pairs propagation is preserved in the pseudo-entropy. We generalize the previous study [30] on the
+pseudo-(R´enyi) entropy to descendant operators in this paper. Specifically, we would like to explore
+the late-time behavior of pseudo-R´enyi entropy of two descendant operators in RCFTs. We construct
+the transition matrix using two locally excited states created by the operator
+Vα(x) =
+�
+{ni},{¯nj}
+α{ni}{¯nj} ·
+�
+i,j
+L−ni ¯L−¯njO(x)
+(4)
+5See [53–61] for studies on other information quantities (such as information metric, negativity, reflected entropy, etc)
+in local or global quantum quenches in CFTs.
+2
+
+and evaluate the pseudo-R´enyi entropy using the replica method [5] and conformal mapping. In (4),
+O(x) is a primary operator in Schr¨odinger picture with chiral and anti-chiral conformal dimension
+∆, L−n (¯L−n) are holomorphic (anti-holomorphic) Virasoro generators, and α{ni},{¯nj} ∈ C are su-
+perposition coefficients. Since the two-point function between descendant operators of different levels
+does not vanish, the transition matrices we are permitted to construct have more degrees of freedom
+than the cases of the primary operator. It is interesting to see whether the late-time behavior of the
+pseudo-(R´enyi) entropy of subsystems corresponding to these transition matrices has contributions
+other than the quantum dimension.
+The rest of this paper is organized as follows. In section 2, we briefly review the replica method for
+locally excited states in 2D CFTs and provide our convention and some useful formulae for the later
+calculations. In section 3, we mainly focus on the late-time behavior of the 2nd pseudo-R´enyi entropy
+of locally descendant excited states. For simplicity, we study the cases that a single holomorphic
+Virasoro generator generates the descendants. More general and complicated situations are discussed
+in section 4, where we derive the late-time behavior of the k-th pseudo-R´enyi entropy for the generic
+descendant states. We end with conclusions and prospects in section 5. Some calculation details are
+presented in the appendices.
+2
+Setup in 2D CFTs
+2.1
+Replica method with local operators
+Our focus is the pseudo-R´enyi entropy of locally excited states created by acting the operator Vα (4)
+on the ground state in RCFTs, which can be formulated in the path integral formalism using the
+replica method. W can consider a RCFT that lives on a plane and has a vacuum state |Ω⟩. We firstly
+prepare two locally excited states using Vα to construct a real-time evolved transition matrix T 1|2(t),
+|ψ1⟩ ≡ e−ϵHVα(x1)|Ω⟩,
+|ψ2⟩ ≡ e−ϵHVβ(x2)|Ω⟩,
+T 1|2(t) ≡ e−iHt |ψ1⟩⟨ψ2|
+⟨ψ2|ψ1⟩ eiHt.
+(5)
+Notice that an infinitesimally small parameter ϵ has been introduced to suppress the high energy
+modes [63]. We can obtain the reduced transition matrix of subsystem A at time t by tracing out the
+degrees of freedom of Ac (the complement of A), T 1|2
+A (t) = trAc[T 1|2(t)]. It turns out that the excess
+of the n-th pseudo-R´enyi entropy of A with respect to the ground state, defined as ∆S(n)(T 1|2
+A (t)) :=
+S(n)(T 1|2
+A (t)) − S(n)�
+trAc�
+|Ω⟩⟨Ω|
+��
+, is of the form [30]
+∆S(n)(T 1|2
+A (t)) =
+1
+1 − n
+�
+log⟨
+n
+�
+k=1
+Vα(w2k−1, ¯w2k−1)V †
+β (w2k, ¯w2k)⟩Σn
+−n log⟨Vα(w1, ¯w1)V †
+β (w2, ¯w2)⟩Σ1
+�
+(6)
+using the replica method. In (6), Σn denotes a n-sheeted Riemann surface with cuts on each copy
+corresponding to A, and (w2k−1, ¯w2k−1)and (w2k, ¯w2k) are coordinates on the kth-sheet surface. The
+term in the first line is given by a 2n-point correlation function on Σn, while a two-point function
+3
+
+gives the one in the second line on Σ1. We can have
+w2k−1 = x1 + t − iϵ,
+w2k = x2 + t + iϵ,
+¯w2k−1 = x1 − t + iϵ,
+¯w2k = x2 − t − iϵ.
+(7)
+2.2
+Convention and useful formulae
+The 2n-point correlation function on Σn in Eq.(6) can be evaluated with the help of a conformal
+mapping of Σn to the complex plane Σ1. The subsystem is A = [0, ∞) hereafter for convenience. We
+can then map Σn to Σ1 using the simple conformal mapping
+w = zn.
+(8)
+Let us first focus on the case of n = 2. The calculation of ∆S(2)(T 1|2
+A (t)) is related to the four-point
+function known pretty well for exactly solvable CFTs. In our convention, using Eq.(8), the 4 points
+z1, z2, z3, z4 in the complex plane are given by
+z1 = −z3 = i
+√
+−x1 − t + iϵ,
+¯z1 = −¯z3 = −i
+√
+−x1 + t − iϵ,
+z2 = −z4 = i
+√
+−x2 − t − iϵ,
+z2 = −z4 = −i
+√
+−x2 + t + iϵ.
+(9)
+The key point is that one should treat t±iϵ as a pure imaginary number in all algebraic calculations and
+take t to be real only in the final expression of the pseudo-R´enyi entropy. To evaluate the four-point
+correlation function, it is useful to focus on the cross ratios
+η := z12z34
+z13z24
+= (x1 + x2 + 2t) + 2
+�
+(x1 + t)(x2 + t) + ϵ2 + iϵ(x1 − x2)
+4
+�
+(x1 + t)(x2 + t) + ϵ2 + iϵ(x1 − x2)
+,
+¯η := ¯z12¯z34
+¯z13¯z24
+= (x1 + x2 − 2t) + 2
+�
+(x1 − t)(x2 − t) + ϵ2 − iϵ(x1 − x2)
+4
+�
+(x1 − t)(x2 − t) + ϵ2 − iϵ(x1 − x2)
+,
+(10)
+where zij = zi − zj, and a useful relation is
+1 − η = z14z23
+z13z24
+.
+(11)
+Since we are mainly interested in the late-time (t → ∞) behavior of pseudo-R´enyi entropy, one can
+find some useful late-time formulae from (9)
+lim
+t→∞ z1 ∼ lim
+t→∞ z4 ∼ −
+√
+t,
+lim
+t→∞ z2 ∼ lim
+t→∞ z3 ∼
+√
+t,
+lim
+t→∞ z12 ∼ lim
+t→∞ z13 ∼ −
+√
+t,
+lim
+t→∞ z24 ∼ lim
+t→∞ z34 ∼
+√
+t,
+lim
+t→∞ z14 ∼ lim
+t→∞ z23 ∼
+�
+1
+t .
+(12)
+For the cross ratios (η, ¯η), as shown in [30], we can have
+lim
+t→∞(η, ¯η) = (1 + (x2 − x1 + 2iϵ)2
+16t2
+, −(x2 − x1 − 2iϵ)2
+16t2
+) ≃ (1, 0),
+∂iη ∼ 1
+t
+3
+2
+,
+∂i∂jη ∼ 1
+t ,
+∂i∂j∂kη ∼ 1
+t
+5
+2
+,
+∂i∂j∂k∂lη ∼ 1
+t2 (i ̸= j ̸= k ̸= l).
+(13)
+4
+
+For general n-th pseudo-R´enyi entropy, the 2n points z1, z2, ..., z2n in the z-coordinates are given by
+z2k+1 =e2πi k+1/2
+n
+(−x1 − t + iϵ)
+1
+n ,
+¯z2k+1 = e−2πi k+1/2
+n
+(−x1 + t − iϵ)
+1
+n ,
+z2k+2 =e2πi k+1/2
+n
+(−x2 − t − iϵ)
+1
+n ,
+¯z2k+2 = e−2πi k+1/2
+n
+(−x2 + t + iϵ)
+1
+n ,
+(k = 0, ..., n − 1).
+(14)
+3
+Second pseudo-R´enyi entropy ∆S(2)
+A
+for descendent operators
+The pseudo-R´enyi entropy for locally excited states can be regarded as a generalization of the R´enyi
+entropy for locally excited states [19]. In RCFTs, it is known that the excess of the R´enyi entropy
+saturates to a constant equal to the logarithm of the quantum dimension of the inserted primary
+operator [39]. A similar result for pseudo-R´enyi entropy is found in [30], and the result also holds
+for R´enyi entropy constructed by two descendent operators in [45]. However, [45] only considers the
+late-time behavior of R´enyi entropy established by two descendent operators with the same Virasoro
+generators and at the same insertion spatial coordinates. The pseudo-R´enyi entropy with two descen-
+dent operators at different levels is still unknown. This section will explore the 2nd pseudo-R´enyi
+entropy for some specific descendent operators.
+3.1
+∆S(2)
+A
+for Vα = L−1O, Vβ = O
+We first consider the simplest case, which is different from the previous studies [30]: Vα(x1) =
+L−1O(x1), Vβ(x2) = O(x2). The 2nd pseudo-R´enyi entropy, which, according to (6), is related to
+a four-point function on Σ2,
+exp{−∆S(2)(T 1|2
+A (t))} =⟨L−1O(w1, ¯w1)O†(w2, ¯w2)L−1O(w3, ¯w3)O†(w4, ¯w4)⟩Σ2
+⟨L−1O(w1, ¯w1)O†(w2, ¯w2)⟩2
+Σ1
+.
+(15)
+For the first descendant operators, the transformation law of them under the conformal mapping
+w = z2 is given by
+∂O(wi, ¯wi) =(w′
+i)−∆( ¯w′
+i)−∆
+�
+(w′
+i)−1∂O(zi, ¯zi) − ∆ w′′
+i
+(w′
+i)2 O(zi, ¯zi)
+�
+,
+(16)
+where the prime denotes the derivative with respect to z or ¯z. Then the four-point function in (15)
+can be written in the light of correlators on the plane as
+⟨L−1O(w1, ¯w1)O†(w2, ¯w2)L−1O(w3, ¯w3)O†(w4, ¯w4)⟩Σ2
+=
+�
+4
+�
+i=1
+|w′
+i|−2∆�
+·
+� ∂z1
+∂w1
+∂z3
+∂w3
+⟨∂O(1)O†(2)∂O(3)O†(4)⟩Σ1 + ∆2� ∂z1
+∂w1
+�2 ∂2w1
+∂z2
+1
+� ∂z3
+∂w3
+�2 ∂2w3
+∂z2
+3
+⟨O(1)O†(2)O(3)O†(4)⟩Σ1
+− ∆ ∂z1
+∂w1
+� ∂z3
+∂w3
+�2 ∂2w3
+∂z2
+3
+⟨∂O(1)O†(2)O(3)O†(4)⟩Σ1 − ∆ ∂z3
+∂w3
+� ∂z1
+∂w1
+�2 ∂2w1
+∂z2
+1
+⟨O(1)O†(2)∂O(3)O†(4)⟩Σ1
+�
+,
+(17)
+5
+
+where we use the notation O(i) ≡ O(zi, ¯zi). Due to the conformal symmetry, we can express the
+four-point functions involved in (17) as follows
+⟨O(1)O†(2)O(3)O†(4)⟩Σ1 =|z13z24|−4∆G(η, ¯η),
+⟨∂O(1)O†(2)O(3)O†(4)⟩Σ1 =|z13z24|−4∆∂z1G(η, ¯η) − 2∆
+z13
+|z13z24|−4∆G(η, ¯η),
+⟨O(1)O†(2)∂O(3)O†(4)⟩Σ1 =|z13z24|−4∆∂z3G(η, ¯η) + 2∆
+z13
+|z13z24|−4∆G(η, ¯η),
+⟨∂O(1)O†(2)∂O(3)O†(4)⟩Σ1 =|z13z24|−4∆∂z1∂z3G(η, ¯η) + 2∆
+z13
+|z13z24|−4∆(∂z1 − ∂z3)G(η, ¯η)
++ −2∆(2∆ + 1)
+z2
+13
+|z13z24|−4∆G(η, ¯η),
+(18)
+where
+G(η, ¯η) := lim
+z→∞ |z|4∆⟨O(z, ¯z)O(1, 1)O(η, ¯η)O(0, 0)⟩Σ1.
+(19)
+Under the conformal mapping between Σ2 and Σ1, we can have
+⟨L−1O(w1, ¯w1)O†(w2, ¯w2)L−1O(w3, ¯w3)O†(w4, ¯w4)⟩Σ2
+=2−8∆|z1z2z3z4|−2∆|z13z24|−4∆ ·
+�
+1
+4z1z3
+�
+∂z1∂z3 + 2∆
+z13
+(∂z1 − ∂z3) − 2∆(2∆ + 1)
+z2
+13
+�
+G(η, ¯η)
++
+∆2
+4z2
+1z2
+3
+G(η, ¯η) −
+∆
+4z1z2
+3
+�
+∂z1 − 2∆
+z13
+�
+G(η, ¯η) −
+∆
+4z2
+1z3
+�
+∂z3 + 2∆
+z13
+�
+G(η, ¯η)
+�
+.
+(20)
+At the late times (t → ∞), as shown in [30], η and ¯η approach to 1 and 0, respectively, which leads to
+the following late time behavior of G(η, ¯η) for RCFTs
+lim
+t→∞ G(η, ¯η) ≃ d−1
+O (1 − η)−2∆¯η−2∆,
+(21)
+where dO is the quantum dimension of the operator O. Hence we can obtain
+lim
+t→∞ ∂z1G(η, ¯η) ≃2∆∂z1η
+1 − η d−1
+O (1 − η)−2∆¯η−2∆,
+lim
+t→∞ ∂z3G(η, ¯η) ≃ 2∆∂z3η
+1 − η d−1
+O (1 − η)−2∆¯η−2∆,
+lim
+t→∞ ∂z1∂z3G(η, ¯η) ≃2∆∂z1∂z3η
+1 − η
+d−1
+O (1 − η)−2∆¯η−2∆ + 2∆(2∆ + 1)∂z1η∂z3η
+(1 − η)2
+d−1
+O (1 − η)−2∆¯η−2∆. (22)
+On the other hand, the two-point function in (15) is
+⟨L−1O(w1, ¯w1)O†(w2, ¯w2)⟩Σ1 = ∂w1
+1
+|w12|4∆ = −2∆
+w12
+·
+1
+|w12|4∆ .
+(23)
+Substituting (20), (22) and (23) into (15) and setting z3 = −z1, z4 = −z2, we obtain
+lim
+t→∞ exp{−∆S(2)(T 1|2
+A (t))}
+≃ w2
+12
+4∆2 η2∆(1 − ¯η)2∆
+� −1
+4z2
+1
+� 2∆ z2
+1+z2
+2
+8z3
+1z2
+(1 − η)dO
+−
+2∆(2∆ + 1) (z2
+1−z2
+2)2
+64z4
+1z2
+2
+(1 − η)2dO
++
+2∆2 z2
+2−z2
+1
+4z2
+1z2
+z1(1 − η)dO
+− ∆(2∆ + 1)
+2z2
+1dO
+�
++
+∆2
+4z4
+1dO
+− ∆
+4z3
+1
+� 2∆ z2
+2−z2
+1
+8z2
+1z2
+(1 − η)dO
+−
+∆
+z1dO
+�
++ ∆
+4z3
+1
+� 2∆ z2
+1−z2
+2
+8z2
+1z2
+(1 − η)dO
++
+∆
+z1dO
+��
+≃d−1
+O .
+(24)
+6
+
+In going from the second to the third line, we use Eq.(9) and perform the Laurent expansion at infinity.
+The late-time limit of the 2nd pseudo-R´enyi entropy is thus given by
+lim
+t→∞ ∆S(2)�
+T 1|2(t)
+�
+= log dO.
+(25)
+In this simplest case, the late-time behavior of the 2nd pseudo-R´enyi entropy of L−1O with O is the
+same as that of the primary operator O.
+3.2
+∆S(2)
+A
+for Vα = L−nO, Vβ = O
+We next consider a more complicated case that Vα is a general n-level descendant associated with the
+Virasoro generator L−n, and Vβ is still a primary. The two-point function of Vα and Vβ reads [64]
+⟨L−nO(w1, ¯w1)O(w2, ¯w2)⟩ = (n + 1)∆
+wn
+21
+|w12|−4∆.
+(26)
+We then compute the four-point function on Σ2. Under the conformal transformation, the level n
+descendant transforms as
+L−nO(wi, ¯wi) = (w′
+i)−(∆+n)( ¯w′
+i)−∆L−nO(zi, ¯zi) + ...
+(27)
+The ellipsis stands for operators with lower conformal dimensions, contributing to lower-order singu-
+larity in the correlation functions. Then at a late time
+⟨O(−n)(w1, ¯w1)O†(w2, ¯w2)O(−n)(w3, ¯w3)O†(w4, ¯w4)⟩Σ2
+∼
+�
+4
+�
+i=1
+|w′
+i|−2∆�
+(w′
+1)−n(w′
+3)−n⟨O(−n)(1)O†(2)O(−n)(3)O†(4)⟩Σ1.
+(28)
+We can pick out the most singular terms of the four-point function on the z-plane in (28). According to
+(12) and (60) in appendix A, the leading contribution at late time in ⟨O(−n)(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+should be
+(n − 1)∆
+zn
+41
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1 + −∂z4
+zn−1
+41
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+=
+�(n − 1)∆
+zn
+41
+− ∂z4
+zn−1
+41
+��(n − 1)∆
+zn
+23
+− ∂z2
+zn−1
+23
+�
+⟨O(1)O†(2)O(3)O†(4)⟩Σ1 + ...
+=|z13z24|−4∆d−1
+O (1 − η)−2∆¯η−2∆
+×
+�(n − 1)2∆2
+zn
+41zn
+23
+− (n − 1)∆
+zn
+41zn−1
+23
+· 2∆∂z2η
+1 − η
+− (n − 1)∆
+zn−1
+41 zn
+23
+· 2∆∂z4η
+1 − η
++
+1
+zn−1
+41 zn−1
+23
+·
+�2∆(2∆ + 1)∂z2η · ∂z4η
+(1 − η)2
++ 2∆∂z2∂z4η
+1 − η
+��
++ ...
+(29)
+Combine (26), (29) with t → ∞, the leading-order behavior of exp{−∆S(2)(T 1|2
+A (t))} is
+lim
+t→∞ e−∆S(2)�
+T 1|2
+A
+(t)
+�
+∼
+w2n
+12
+(n + 1)2∆2 ×
+1
+4nzn
+1 zn
+3
+d−1
+O
+�(n − 1)2∆2
+zn
+41zn
+23
+− (n − 1)∆
+zn
+41zn−1
+23
+· 2∆∂z2η
+1 − η
+− (n − 1)∆
+zn−1
+41 zn
+23
+· 2∆∂z4η
+1 − η
++
+1
+zn−1
+41 zn−1
+23
+·
+�2∆(2∆ + 1)∂z2η · ∂z4η
+(1 − η)2
++ 2∆∂z2∂z4η
+1 − η
+��
++ ...
+∼ 1
+dO
++ ...
+(30)
+7
+
+Again, the ellipsis denotes terms with sub-leading contributions. Hence the late-time limit of the 2nd
+pseudo-R´enyi entropy of the transition matrix constructed by a primary O and its n-level descendant
+L−nO is still log dO.
+3.3
+∆S(2)
+A
+for Vα = L−nO, Vβ = L−mO
+In this subsection, we use the conformal block and operator product expansion (OPE) to show the
+phenomenon discovered in previous subsections is true for a general case: Vα = L−nO, Vβ = L−mO.
+In terms of [64], the two-point function of Vα and Vβ reads6
+⟨L−nO(w1, ¯w1)L−mO(w2, ¯w2)⟩Σ1
+= 1
+12(−1)n(w1 − w2)−m−n
+1
+|w12|4∆
+�
+Γ(m + n)
+�
+cm
+�
+m2 − 1
+�
+n
+�
+n2 − 1
+�
++ 24∆(m + n)(m + n + 1)(mn − 1)
+�
+Γ(m + 2)Γ(n + 2)
++ 12∆(∆(m + 1)(n + 1) + 2)
+�
+.
+(31)
+The late-time behavior of the four point function on Σ2 of (6) can be derived according to(27)
+⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ2
+∼
+�
+4
+�
+i=1
+|w′
+i|−2∆�
+(w′
+1)−n(w′
+2)−m(w′
+3)−n(w′
+4)−m⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1.
+(32)
+We can next pick out the most singular terms of the four-point function on the z-plane in (32).
+According to (12) , the leading contribution at late time in ⟨O(−n)(1)O†(2)O(−n)(3)O†(4)⟩Σ1 comes
+6Here the following equation to simplify the result has been used
+m−1
+�
+k=1
+(k + m)(−k + m + 1)(k + n − 1)!
+(k + 1)!(n − 2)!
+= 2
+�
+m2n + m
+�
+2n2 − 1
+�
+− n(n + 1)
+�
+Γ(m + n)
+Γ(m + 1)Γ(n + 2)
+− m(m + 1)n + 2.
+8
+
+from the OPE of O(−n)(1)O†(4) and O†(2)O(−n)(3) .
+⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ2
+∼
+�
+4
+�
+i=1
+|w′
+i|−2∆�
+(w1 − w2)−m−n(w3 − w4)−m−nG(η, ¯η)(−1)m+n
+�
+−
+1
+12Γ(m + 2)2Γ(n + 2)2 ∆eiπn(−1)−m−n((m(m + 1)n − 2)Γ(m + 2)Γ(n + 2)
+− 2(m + 1)(m2n + m(2n2 − 1) − n(n + 1))Γ(m + n))((−(−1)2)−m−nΓ(m + n)
+(m(m2 − 1)eiπnn(−1)m+n(c(n2 − 1) + 24∆) + 24∆eiπm(n + 1)1m+n(m(2mn − m + n2 − 1) − n))
++ 12∆(−1)−nΓ(m + 2)Γ(n + 2)(1−m(n + 1)(∆ + m(∆ + n)) + (−1)−meiπm(2 − mn(n + 1))))
++ 1
+12∆(m + 1)(∆ + m)1−m−2n(−1)−2m−n(
+1
+n(n + 1)Γ(m + 2)Γ(n − 1)Γ(m + n)
+(m(m2 − 1)(−1)nn(−1)m+n(c(n2 − 1) + 24∆) + 24∆(−1)m(n + 1)1m+n(m(m(2n − 1) + n2 − 1) − n))
++ 12∆1n(n − 1)((−1)m(n + 1)(∆ + m(∆ + n)) + (−1)m+11m(mn(n + 1) − 2)))
+1
+12∆(2∆ + m)1−m−2n(−1)−2m−n(
+1
+mΓ(m)Γ(n + 2)Γ(m + n)
+(m(m2 − 1)(−1)nn(−1)m+n(c(n2 − 1) + 24∆) + 24∆(−1)m(n + 1)1m+n(m(m(2n − 1) + n2 − 1) − n))
++ 12∆(m + 1)1n((−1)m(n + 1)(∆ + m(∆ + n)) + (−1)m+11m(mn(n + 1) − 2)))
++
+1
+144(m + 1)Γ(m − 1)Γ(m)Γ(m + 2)2Γ(n − 1)Γ(n + 2)
+((−1)m+1m1−2(m+n)(−1)−m−n(c(m2 − 1) + 24∆)Γ(m + n)((m + 1)(−1)n+1n(−1)m+nΓ(m)Γ(m + 2)
+(c(n2 − 1) + 24∆)Γ(m + n) + 12∆1nΓ(m − 1)((m + 1)Γ(m)((−1)m(−n − 1)Γ(m + 2)Γ(n + 2)
+(∆ + m(∆ + n)) + (−1)m1m((mn(n + 1) − 2)Γ(m + 2)Γ(n + 2) + 2n(n + 1)Γ(m + n)))
+− 2(−1)m1m(n + 1)(2mn − m + n2 − 1)Γ(m + 2)Γ(m + n))))
+�
++ . . .
+(33)
+The complete derivation detail of Eq.(33) is shown in appendix B.
+Combine (31) and (33) and take the limit t → ∞, the leading-order behavior of exp{−∆S(2)(T 1|2
+A (t))}
+is
+lim
+t→∞ e−∆S(2)�
+T 1|2
+A
+(t)
+�
+∼ 1
+dO
++ ...
+(34)
+The ellipsis denotes terms with sub-leading contributions. The late-time limit of the 2nd pseudo-R´enyi
+entropy of the transition matrix constructed by a m-level descendant operator L−mO and a n-level
+descendant operator L−nO is log dO.
+4
+k-th pseudo-R´enyi entropy for generic descendent states
+In the previous section, the 2nd pseudo-R´enyi entropy corresponding to L−nO and L−mO is the same
+as the 2nd pseudo-R´enyi entropy of the corresponding primary operator O at a late time, and they
+all equal to the logarithm of the quantum dimension of the primary operator. However, to derive
+the pseudo-entropy, it is reasonable to consider the k-th pseudo-R´enyi entropy and take k analytic
+9
+
+continuation to 1. In this section, the late-time behavior of k-th pseudo-R´enyi entropy with two linear
+combination descendent operators will be checked whether it is still log dO.
+4.1
+∆S(k)
+A
+for Vα = L−nO, Vβ = L−mO
+We begin with calculating the general case discussed above: Vα = L−nO, Vβ = L−mO. Since the
+anti-holomorphic part of these operators is still primary, we only focus on the holomorphic part here.
+The late-time behavior of the 2k-point function on Σk can be derived according to (27)
+⟨O(−n)(w1)O(−m)†(w2) . . . O(−n)(w2k−1)O(−m)†(w2k)⟩Σk,
+∼F(w1, w2, . . . , w2k, m, n, ∆)⟨O(−n)(1)O(−m)†(2) . . . O(−n)(2k − 1)O(−m)†(2k)⟩Σ1 + . . . ,
+(35)
+where
+F(w1, w2, . . . , w2k, m, n, ∆) =
+� 2k
+�
+i=1
+|w′
+i|−2∆�
+(w′
+1)−n(w′
+2)−m . . . (w′
+2k−1)−n(w′
+2k)−m
+(36)
+is the leading factor coming from the conformal transformation between correlation functions on Σk
+and correlation functions on Σ1.
+According to (14), at the late-time limit, we can find the following relations
+lim
+t→∞(z2i+4 − z2i+1) ≃ w2 − w1
+kt
+e2πi i+1
+k t
+1
+k ≃ 0,
+lim
+t→∞(¯z2i+2 − ¯z2i+1) ≃ w1 − w2
+kt
+e−2πi
+i+ 1
+2
+k t
+1
+k ≃ 0.
+(37)
+Hence the leading term of 2k-point correlation function on Σ1 comes from the OPE of
+O(−n)(2i + 1)O(−m)†(2i + 4), i.e.,
+⟨O(−n)(1)O(−m)†(2) . . . O(−n)(2k − 1)O(−m)†(2k)⟩Σ1
+∼D1,4D3,6 . . . D2k−3,2k⟨O(1)O†(2) . . . O(2k − 1)O†(2k)⟩Σ1,
+(38)
+where D2i+1,2i+4 is a derivative operator that only contains constants related to the information of two
+descendant operators and derivatives coming from the most singular part of the OPE of O(−n)(2i +
+1)O(−m)†(2i + 4),
+D2i+1,2i+4 = D(∂2i+1, ∂2i+4; m, n, c, ∆).
+(39)
+See appendix B for a concrete example of the D-operator. We need to pick up the proper channel
+to expand the 2k-point function into the holomorphic and the anti-holomorphic part, as graphically
+shown in figure 1. In each channel, only the identity operator contributes to the final result. Hence,
+the 2k-point function breaks up into k two-point functions for the holomorphic part(and k for the
+anti-holomorphic part).
+⟨O(−n)(1)O(−m)†(2) . . . O(−n)(2k − 1)O(−m)†(2k)⟩Σ1
+∼(F00[O])k−1D1,4D3,6 . . . D2k−3,2k⟨O(1)O†(4)⟩Σ1⟨O(3)O†(6)⟩Σ1 . . . ⟨O(2k − 3)O†(2k)⟩Σ1
+∼(F00[O])k−1⟨O(−n)(1)O(−m)†(4)⟩Σ1⟨O(−n)(3)O(−m)†(6)⟩Σ1 . . . ⟨O(−n)(2k − 3)O(−m)†(2k)⟩Σ1.
+(40)
+10
+
+…
+1
+2
+3
+4
+6
+5
+2𝑘-3
+2𝑘-2
+2𝑘-1
+2𝑘
+…
+1
+2
+4
+6
+3
+2𝑘-2
+2𝑘-5
+2𝑘-1
+2𝑘
+(𝐹00 𝒪 )𝑘−2
+2𝑘-3
+…
+1
+2
+3
+4
+6
+5
+2𝑘-3
+2𝑘-2
+2𝑘-1
+2𝑘
+𝐹00 𝒪
++ …
++ …
+…
+1
+4
+3
+6
+2𝑘
+2𝑘-3
+2𝑘-1
+2
+(𝐹00 𝒪 )𝑘−1
++ …
+Figure 1: k − 1 fusion transformations to obtain ∆S(k)
+A
+In the last line, the fact that D2i+1,2i+3 is a linear operator, and coordinates zi and zj are independent
+for i ̸= j has been applied.
+Changing back into the w-coordinate, with the leading divergent term being transformed homo-
+geneously and keeping the most divergent term, we can find
+⟨O(−n)(w1)O(−m)†(w2) . . . O(−n)(w2k−1)O(−m)†(w2k)⟩Σk
+∼(F00[O])k−1F(w1, w2, . . . , w2k, m, n, ∆)
+⟨O(−n)(1)O(−m)†(4)⟩Σ1⟨O(−n)(3)O(−m)†(6)⟩Σ1 . . . ⟨O(−n)(2k − 3)O(−m)†(2k)⟩Σ1 + . . .
+∼(F00[O])k−1⟨O(−n)(w1)O(−m)†(w4)⟩Σk⟨O(−n)(w3)O(−m)†(w6)⟩Σk . . . ⟨O(−n)(w2k−3)O(−m)†(w2k)⟩Σk
++ . . .
+(41)
+The two-point function of descendent operators on Σk and that on Σ1 are similar at late time
+⟨O(−n)(w2i+1)O(−m)†(w2i+4)⟩Σk
+∼(w′
+2i+1)−∆−n(w′
+2i+4)−∆−m⟨O(−n)(z2i+1)O(−m)†(z2i+4)⟩Σ1
+∼(kzk−1
+2i+1e2πi i
+k )−∆−n(kzk−1
+2i+4e2πi i+1
+k )−∆−m
+C0(m, n)
+((z2i+1 − z2i+4)e2πi i+1
+k )2∆+m+n
+∼e(−2πi 1
+k )(−∆−n)(kt
+k−1
+k )−∆−n(kt
+k−1
+k )−∆−m
+C0(m, n)
+(w2i+1 − w2i+4)2∆+m+n (kt
+k−1
+k )2∆+m+n
+∼e(−2πi 1
+k )(−∆−n)⟨O(−n)(w1)O(−m)†(w2)⟩Σ1.
+(42)
+Therefore, at a late time, for two descendent operators with a single Virasoro generator, we still have
+lim
+t→∞ ∆S(k) = log dO, and its pseudo-entropy is log dO.
+4.2
+∆S(k)
+A
+for Linear combination of descendent operators
+Let us consider two linear combination operators constructed by operators in O’s conformal family.
+Vα(w, ¯w) =
+�
+i
+CiVi(w, ¯w),
+Vi(w, ¯w) = L−{Ki} ¯L−{ ¯
+Ki}O(w, ¯w),
+(43)
+Vβ(w, ¯w) =
+�
+i
+C′
+iV ′
+i (w, ¯w),
+V ′
+i (w, ¯w) = L−{K′
+i} ¯L−{ ¯
+K′
+i}O†(w, ¯w),
+(44)
+where L−{Ki} ≡ L−ki1L−ki2...L−kini, (0 ≤ ki1 ≤ ki2 ≤ ... ≤ kini), and L−{ ¯
+Ki} ≡ L−¯ki1L−¯ki2...L−¯ki¯ni,
+(0 ≤ ¯ki1 ≤ ¯ki2 ≤ ... ≤ ¯ki¯ni). Likewise for L−{K′
+i} and L−{ ¯
+K′
+i}. If the combination coefficients Ci
+11
+
+(C′
+i) are required to be dimensionless, all Vi(w, ¯w)
+�
+V ′
+i (w, ¯w)
+�
+should have the same mass dimension,
+denoted as N (N′). This indicates that {Ki} and {K′
+i} satisfy
+|Ki| + | ¯Ki| = N,
+|K′
+i| + | ¯K′
+i| = N′,
+�
+|Ki| ≡
+ni
+�
+j=1
+kij, | ¯Ki| ≡
+¯ni
+�
+j=1
+¯kij
+�
+.
+(45)
+The two point function of Vα and Vβ is
+⟨Vα(w1, ��w1)Vβ(w2, ¯w2)⟩Σ1
+=
+�
+i
+�
+j
+CiC′
+j⟨L−{Ki}O(w1)L−{K′
+j}O†(w2)⟩Σ1⟨¯L−{ ¯
+Ki}O( ¯w1)¯L−{ ¯
+K′
+j}O†( ¯w2)⟩Σ1
+=
+�
+i
+�
+j
+CiC′
+j
+c0({Ki}, {K′
+j})
+(w1 − w2)2∆+|Ki|+|K′
+j|
+c0({ ¯Ki}, { ¯K′
+j})
+( ¯w1 − ¯w2)2∆+| ¯
+Ki|+| ¯
+K′
+j| ,
+(46)
+where the coefficient c0 depends on the decomposition of generic Virasoro generators. At the late
+time, the 2k-point function reads
+⟨Vα(w1, ¯w1)Vβ(w2, ¯w2) . . . Vα(w2k−1, ¯wi2k−1)Vβ(wj2k, ¯w2k)⟩Σk
+=
+�
+i1
+�
+j2
+· · ·
+�
+i2k−1
+�
+j2k
+Ci1C′
+j2 . . . Ci2k−1C′
+j2k
+⟨L−{Ki1}O(w1)L−{K′
+j2}O†(w2) . . . L−{Ki2k−1}O(w2k−1)L−{K′
+j2k}O†(w2k)⟩Σk
+⟨¯L−{ ¯
+Ki1}O( ¯w1)¯L−{ ¯
+K′
+j2}O†( ¯w2) . . . ¯L−{ ¯
+Ki2k−1}O( ¯w2k−1)¯L−{ ¯
+K′
+j2k}O†( ¯w2k)⟩Σk
+∼d−(k−1)
+O
+�
+i1
+�
+j2
+· · ·
+�
+i2k−1
+�
+j2k
+Ci1C′
+j2 . . . Ci2k−1C′
+j2k
+⟨L−{Ki1}O(w1)L−{K′
+j4}O†(w4)⟩Σk . . . ⟨L−{Ki2k−3}O(w2k−3)L−{K′
+j2k}O†(w2k)⟩Σk
+⟨¯L−{ ¯
+Ki1}O( ¯w1)¯L−{ ¯
+K′
+j2}O†( ¯w2)⟩Σk . . . ⟨¯L−{ ¯
+Ki2k−1}O( ¯w2k−1)¯L−{ ¯
+K′
+j2k}O†( ¯w2k)⟩Σk
+∼d−(k−1)
+O
+�
+i1
+�
+j2
+· · ·
+�
+i2k−1
+�
+j2k
+Ci1C′
+j2 . . . Ci2k−1C′
+j2k
+c0({Ki1}, {K′
+j4})
+(w1 − w4)2∆+|Ki1|+|K′
+j4| . . .
+c0({Ki2k−3}, {K′
+j2k})
+(w2k−3 − w2k)2∆+|Ki2k−3|+|K′
+j2k|
+c0({ ¯Ki1}, { ¯K′
+j2})
+( ¯w1 − ¯w2)2∆+| ¯
+Ki1|+| ¯
+K′
+j2| . . .
+c0({ ¯Ki2k−1}, { ¯K′
+j2k})
+( ¯w2k−1 − ¯w2k)2∆+| ¯
+K2k−1|+| ¯
+K′
+2k| .
+(47)
+The first formula transforms the correlation function on Σk into combinations of its holomorphic and
+anti-holomorphic parts. In the second formula, we have extracted its leading term on Σ1, separated
+it into k two-point functions using the fusion rule, and then changed the correlation function on Σ1
+back to Σk. The third formula has used the property of the two-point part on Σk (42).
+Combine (46) and (47), the k-th pseudo-R´enyi entropy for linear combination of descendent oper-
+12
+
+ators is
+∆S(k) =
+1
+1 − k log ⟨Vα(w1, ¯w1)Vβ(w2, ¯w2) . . . Vα(w2k−1, ¯wi2k−1)Vβ(wj2k, ¯w2k)⟩Σk
+(⟨Vα(w1, ¯w1)Vβ(w2, ¯w2)⟩Σ1)k
+∼
+1
+1 − k log
+�
+d−(k−1)
+O
+×
+�
+i1
+· · · �
+j2k
+Ci1 . . . C′
+j2kc0({Ki1}, {K′
+j4}) . . . c0({Ki2k−3}, {K′
+j2k})c0({ ¯Ki1}, { ¯K′
+j2}) . . . c0({ ¯Ki2k−1}, { ¯K′
+j2k})
+(�
+i
+�
+j
+CiC′
+jc0({Ki}, {K′
+j})c0({ ¯Ki}, { ¯K′
+j}))k
+�
+.
+(48)
+For the last formula, we have applied the restrictive condition (48) and the fact that when ϵ → 0, all
+zi and ¯zi are real.
+There are two types of contributions to the pseudo-entropy of two linear combination operators.
+The first one takes a universal form, depending on the quantum dimension of the corresponding
+primary operator. There may also have an extra contribution to the pseudo-entropy. To see this,
+consider the 2nd pseudo-R´enyi entropy,
+∆S(2) ∼ log dO
+− log
+�
+�
+�
+�
+i1,i3
+�
+j2,j4
+Ci1Ci3C′
+j2C′
+j4c0({Ki1}, {K′
+j4})c0({Ki3}, {K′
+j2})c0({ ¯Ki1}, { ¯K′
+j2})c0({ ¯Ki3}, { ¯K′
+j4})
+(�
+i
+�
+j
+CiC′
+jc0({Ki}, {K′
+j})c0({ ¯Ki}, { ¯K′
+j}))2
+�
+�
+� .
+(49)
+In general, the numerator is not equal to the denominator in the last line of (49). So the k-th pseudo-
+Renyi entropy may acquire additional correction. It is zero when correlation functions containing Vα
+and Vβ can be divided into the product of the holomorphic part and antiholomorphic part. How-
+ever, the extra correction may be nonzero in general. The following two examples can illustrate the
+phenomenon.
+• Example 1 with Vα(w1, ¯w1) = (L−1 + ¯L−1)O(w1, ¯w1) Vβ(w2, ¯w2) = (L−1 + ¯L−1)O(w2, ¯w2)
+The two-point function is
+⟨Vα(w1, ¯w1)Vβ(w2, ¯w2)⟩Σ1 = −4∆(4∆ + 1)
+(x1 − x2)4∆+2 .
+(50)
+Formula (50) is easy to check. Here, we replace x1 + t and x2 + t into w1 and w2 in the final
+result. The four-point function is
+⟨Vα(w1, ¯w1)Vβ(w2, ¯w2)Vα(w3, ¯w3)Vβ(w4, ¯w4)⟩Σ2 ∼ 8∆2(16∆ + 1 + 8∆)
+(x1 − x2)8∆+4
+.
+(51)
+We explicitly show how to compute the four-point function on Σ2 without showing the derivation
+of (51) in detail. Again we keep the final result replaced by x1 and x2.
+From (50) and (51), we can have
+∆S(2) ∼ log dO + log 2.
+(52)
+13
+
+In this case, the correlation function of Vα and Vβ can not be divided into the product of the
+holomorphic part and antiholomorphic part, and ∆S(2) contains an extra correction log 2 besides
+log dO.
+• Example 2 with Vα(w1, ¯w1) = L−{K1} ¯L−{ ¯
+K1}O((w1, ¯w1), Vβ(w2, ¯w2) = L−{K′
+2} ¯L−{ ¯
+K′
+2}O(w2, ¯w2)
+According to (46), the two-point function on Σ1 reads
+⟨Vα(w1, ¯w1)Vβ(w2, ¯w2)⟩Σ1
+=⟨L−{K1}O(w1)L−{K′
+2}O†(w2)⟩Σ1⟨¯L−{ ¯
+K1}O( ¯w1)¯L−{ ¯
+K′
+2}O†( ¯w2)⟩Σ1
+=
+c0({K1}, {K′
+2})
+(w1 − w2)2∆+|K1|+|K′
+2|
+c0({ ¯K1}, { ¯K′
+2})
+( ¯w1 − ¯w2)2∆+| ¯
+K1|+| ¯
+K′
+2| .
+(53)
+The 2k-point correlation function is
+⟨Vα(w1, ¯w1)Vβ(w2, ¯w2) . . . Vα(w2k−1, ¯wi2k−1)Vβ(wj2k, ¯w2k)⟩Σk
+=⟨L−{K1}O(w1)L−{K′
+2}O†(w2) . . . L−{K1}O(w2k−1)L−{K′
+2}O†(w2k)⟩Σk
+⟨¯L−{ ¯
+K1}O( ¯w1)¯L−{ ¯
+K′
+2}O†( ¯w2) . . . ¯L−{ ¯
+K1}O( ¯w2k−1)¯L−{ ¯
+K′
+2}O†( ¯w2k)⟩Σk
+∼d−(k−1)
+O
+⟨L−{K1}O(w1)L−{K′
+2}O†(w4)⟩Σk . . . ⟨L−{K1}O(w2k−3)L−{K′
+2}O†(w2k)⟩Σk
+⟨¯L−{ ¯
+K1}O( ¯w1)¯L−{ ¯
+K′
+2}O†( ¯w2)⟩Σk . . . ⟨¯L−{ ¯
+K1}O( ¯w2k−1)¯L−{ ¯
+K′
+2}O†( ¯w2k)⟩Σk
+∼d−(k−1)
+O
+c0({K1}, {K′
+2})
+(w1 − w4)2∆+|K1|+|K′
+2| . . .
+c0({K1}, {K′
+2})
+(w2k−3 − w2k)2∆+|K1|+|K′
+2|
+c0({ ¯K1}, { ¯K′
+2})
+( ¯w1 − ¯w2)2∆+| ¯
+K1|+| ¯
+K′
+2| . . .
+c0({ ¯K1}, { ¯K′
+2})
+( ¯w2k−1 − ¯w2k)2∆+| ¯
+K1|+| ¯
+K′
+2| .
+(54)
+In this case the correlation function of Vα and Vβ can be divided, and lim
+t→∞ ∆S(k) = log dO, i.e.
+∆S(k) has no extra correction.
+5
+Conclusion and prospect
+In this paper, we investigate the pseudo-R´enyi entropy of local descendent operators in RCFTs, ex-
+tending the previous studies in [30] [39] [45]. In [30] [45], it has been found that the late-time excess
+of the pseudo-R´enyi entropy of two primary states and the R´enyi entropy of a descendent state equal
+to the logarithmic quantum dimension of the primary operator in RCFTs. It is a natural question to
+consider the pseudo-R´enyi entropy of the descendent states.
+Firstly, we show that in some special cases: Vα = L−1O, Vβ = O and Vα = L−nO, Vβ = O
+with O being primary, the late-time excess of the 2nd pseudo-R´enyi entropy (6) is still logarithmic
+of the quantum dimension of the primary operator. Using the conformal block and operator product
+expansion, we compute the 2nd pseudo-R´enyi entropy constructed by two descendent operators with
+different Virasoro generators.
+We show that their 2nd pseudo-R´enyi entropy is the same as their
+14
+
+primaries for such states. Although the calculation looks quite complicated, the leading divergent
+terms in the late time limit are simple, behaving as the one for primary operators.
+Further, we compute k-th pseudo-R´enyi entropy with two descendent operators L−nO and L−mO.
+We extract the most divergent term of the 2k-point function on Σk with an overall factor F (35), and
+then associate the 2k-point function of descendent operators with the 2k-point function of primary
+operators (38) with some derivative operators of the form
+D2i+1,2i+4 = D(∂2i+1, ∂2i+4; m, n, c, ∆).
+(55)
+We find the 2k-point function breaks up into k two-point functions for the holomorphic part(and k for
+the anti-holomorphic part). The two-point function only depends on the conformal weight and some
+constant (42). As a result, in this case, the pseudo-entropy of the descendent operators is the same as
+primaries.
+Finally, we discuss the most generic descendent operators, which are two linear combination oper-
+ators constructed by operators in O’s conformal family
+Vα(w1, ¯w1) =
+�
+i
+CiVi(w1, ¯w1),
+Vβ(w2, ¯w2) =
+�
+j
+C′
+jVj(w2, ¯w2).
+(56)
+Unsurprisingly, we find that the pseudo-R´enyi entropy of these operators is generally different from
+that of the primary operator O. The entropies are the same as the ones of the primary when the
+correlation function of Vα and Vβ can be divided into the product of the holomorphic part and the
+anti-holomorphic part. A typical example is
+Vα(w1, ¯w1) = L−{K1} ¯L−{ ¯
+K1}O(w1, ¯w1),
+Vβ(w2, ¯w2) = L−{K′
+2} ¯L−{ ¯
+K′
+2}O(w2, ¯w2).
+(57)
+Otherwise, there is an extra contribution. A typical example of extra contribution is
+Vα(w1, ¯w1) = (L−1 + ¯L−1)O(w1, ¯w1),
+Vβ(w2, ¯w2) = (L−1 + ¯L−1)O(w2, ¯w2).
+(58)
+In general, the k-th pseudo-R´enyi entropy for two linear combination operators only depends on the
+coefficients of the two-point functions and the combination coefficients,
+∆S(k) ∼
+1
+1 − k log
+�
+d−(k−1)
+O
+×
+�
+i1
+· · · �
+j2k
+Ci1 . . . C′
+j2kc0({Ki1}, {K′
+j4}) . . . c0({Ki2k−3}, {K′
+j2k})c0({ ¯Ki1}, { ¯K′
+j2}) . . . c0({ ¯Ki2k−1}, { ¯K′
+j2k})
+(�
+i
+�
+j
+CiC′
+jc0({Ki}, {K′
+j})c0({ ¯Ki}, { ¯K′
+j}))k
+�
+.
+(59)
+Noticing the current results in RCFTs, one can directly calculate the pseudo-entropy of generic local
+operators in Liouville CFT, holographic CFTs, non-diagonal CFTs, etc. Since the spectra in such
+theories have different structures, the associated pseudo-entropy will be highly different from those
+in RCFTs.
+In particular, since holomorphic and antiholomorphic conformal blocks have different
+structures in non-diagonal CFTs, the late-time behavior of the entanglement entropy and pseudo-
+entropy associated with locally excited states will not be the same as the ones demonstrated in the
+current paper. We would like to leave them to future work.
+15
+
+Acknowledgements
+We thank Wu-zhong Guo, Linlin Huang, Pak Hang Chris Lau, Yang Liu, Yuan Sun, and Long Zhao
+for the valuable discussions. S.H. would appreciate the financial support from Jilin University, Max
+Planck Partner group, and Natural Science Foundation of China Grants (No.12075101, No.12047569).
+A
+Reduction of ⟨O(−n)(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+With following the standard way [64], we can compute the four-point function ⟨O(−n)(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+in this section.
+⟨O(−n)(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+= −
+1
+2πi
+4
+�
+i=2
+�
+C(zi)
+dz(z − z1)−n+1⟨T(z)O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+= −1
+2πi
+�
+C(z2)
+dz
+(z − z1)n−1
+�∆⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+(z − z2)2
++ ∂z2⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+z − z2
++ reg(z − z2)
+�
++ −1
+2πi
+�
+C(z3)
+dz
+(z − z1)n−1
+�n(n2 − 1)c/12 + 2n∆
+(z − z3)n+2
+⟨O(1)O†(2)O(3)O†(4)⟩Σ1
++
+n−1
+�
+k=1
+(n + k)⟨O(1)O†(2)O(−(n−k))(3)O†(4)⟩Σ1
+(z − z3)k+2
++ (∆ + n)⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+(z − z3)2
++ ∂z3⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+z − z3
++ reg(z − z3)
+�
++ −1
+2πi
+�
+C(z4)
+dz
+(z − z1)n−1
+�∆⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+(z − z4)2
++ ∂z4⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+z − z4
++ reg(z − z4)
+�
+=(n − 1)∆
+zn
+21
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1 + −∂z2
+zn−1
+21
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
++ (n − 1)∆
+zn
+41
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1 + −∂z4
+zn−1
+41
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
++ (−1)n
+�
+n(n2 − 1)c/12 + 2n∆
+�
+(2n − 1)!
+(n + 1)!(n − 2)!
+⟨O(1)O†(2)O(3)O†(4)⟩Σ1
+z2n
+13
++ (−1)n
+n−1
+�
+k=1
+(n + k)!
+(k + 1)!(n − 2)!
+⟨O(1)O†(2)O(−(n−k))(3)O†(4)⟩Σ1
+zn+k
+13
++ (n − 1)(∆ + n)
+zn
+31
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1 + −∂z3
+zn−1
+31
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1
+(60)
+B
+Reduction of⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1
+In terms of (12), the most divergent term of ⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1 should only
+contain z14 and z23, as any terms containing z13,z24,z12 and z34 are subleading. So we can firstly
+16
+
+expand O(1)’s Virasoro generator,
+⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1
+∼ −
+1
+2πi
+�
+C(z4)
+dz
+(z − z1)n−1 ⟨T(z)O(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1
+∼ −
+1
+2πi
+�
+C(z4)
+dz
+(z − z1)n−1 ⟨O(1)O(−m)†(2)O(−n)(3)(m(m2 − 1)c/12 + 2m∆
+(z − z4)m+2
+O†(4)
++
+m−1
+�
+k=1
+(m + k)
+(z − z4)k+2 O−(m−k)†(4) + (∆ + m)
+(z − z4)2 O(−m)† + ∂3O(−m)†(4)
+z − z4
+)⟩Σ1
+∼(−1)m
+(n + m − 1)!
+(m + 1)!(n − 2)!
+m(m2 − 1)c/12 + 2m∆
+zn+m
+41
+⟨O(1)O(−m)†(2)O(−n)(3)O†(4)⟩Σ1
++ (−1)n
+m−1
+�
+k=1
+(n + k − 1)!
+(k + 1)!(n − 2)!
+(m + k)
+zn+k
+14
+⟨O(1)O(−m)†(2)O(−n)(3)O(−(m−k))†(4)⟩Σ1
++ (n − 1)(∆ + m)
+zn
+41
+⟨O(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1
+−
+∂4
+zn−1
+41
+⟨O(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1.
+(61)
+The correlation function with four Virasoro generators is deformed into correlation functions containing
+no more than 3 Virasoro generators. We can then expand O(4)’s Virasoro generator,
+⟨O(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1
+∼ −
+1
+2πi
+�
+c(z1)
+dz
+(z − z4)m−1 ⟨T(z)O(1)O(−m)†(2)O(−n)(3)O†(4)⟩Σ1
+∼ −
+�
+c(z1)
+dz
+(z − z4)m−1 ⟨(
+∆
+(z − z1)2 O(1) + ∂1O(1)
+Z − Z1
+)O(−m)†(2)O(−n)(3)O†(4)⟩Σ1
+∼(m − 1)∆
+zm
+14
+⟨(O(1)O(−m)†(2)O(−n)(3)O†(4)⟩Σ1 −
+∂1
+zm−1
+14
+⟨(O(1)O(−m)†(2)O(−n)(3)O†(4)⟩Σ1.
+(62)
+Form (61) and (62), we can read the exact form of D1,4 introduced in (38)
+D1,4 =(−1)m
+(n + m − 1)!
+(m + 1)!(n − 2)!
+m(m2 − 1)c/12 + 2m∆
+zn+m
+41
++ (−1)n
+m−1
+�
+k=1
+(n + k − 1)!
+(k + 1)!(n − 2)!
+(m + k)
+zn+k
+14
+((m − k − 1)∆
+zm−k
+14
+−
+∂1
+zm−k−1
+14
+)
++ (n − 1)(∆ + m)
+zn
+41
+((m − 1)∆
+zm
+14
+−
+∂1
+zm−1
+14
+)
+−
+∂4
+zn−1
+41
+((m − 1)∆
+zm
+14
+−
+∂1
+zm−1
+14
+).
+(63)
+17
+
+We can expand O(2)’s Virasoro generator and O(3)’s Virasoro generator in a similar way,
+⟨O(1)O(−m)†(2)O(−n)(3)O†(4)⟩Σ1
+∼ −
+�
+c(z3)
+dz
+(z − z2)m−1 ⟨O(1)T(z)O†(2)O(−n)(3)O†(4)⟩Σ1
+∼ −
+�
+c(z3)
+dz
+(z − z2)m−1 ⟨O(1)O†(2)(n(n2 − 1)c/12 + 2n∆
+(z − z3)n+2
+O(3)
++
+n−1
+�
+l=1
+(n + l)
+(z − z3)l+2 O−(n−l)(3) + (∆ + n)
+(z − z3)2 O−n(3) +
+∂3
+z − z3
+O(−n)(3))O†(4)⟩Σ1
+∼(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+⟨O(1)O†(2)O(3)O†(4)⟩Σ1
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+⟨O(1)O†(2)O(−(n−l)(3)O†(4)⟩Σ1
++ (m − 1)(∆ + n)
+zm
+32
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1 −
+∂3
+zm−1
+32
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1.
+(64)
+Finally, we can have
+⟨O(1)O†(2)O(−n)(3)O†(4)⟩Σ1 ∼ (n − 1)∆
+zn
+23
+G −
+∂2
+zn−1
+23
+G,
+(65)
+where G is ⟨O(1)O†(2)O(3)O†(4)⟩Σ1, and in late-time limit, it’s d−1
+O (1 − η)−2∆¯η−2∆.
+18
+
+Combining (61),(62),(64) and (65), we have
+⟨O(−n)(1)O(−m)†(2)O(−n)(3)O(−m)†(4)⟩Σ1
+∼(−1)m
+(n + m − 1)!
+(m + 1)!(n − 2)!
+m(m2 − 1)c/12 + 2m∆
+zn+m
+41
+{(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+G −
+∂2
+zn−l−1
+23
+G)
++ (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+G −
+∂2
+zn−1
+23
+G) −
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+G + (n − 1)∆∂3
+zn
+23
+G − (n − 1)∂2
+zn
+23
+G − ∂3∂2
+zn−1
+23
+G)}
++ (−1)n
+m−1
+�
+k=1
+(n + k − 1)!
+(k + 1)!(n − 2)!
+(m + k)
+zn+k
+14
+{(m − k − 1)∆
+zm−k
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+G −
+∂2
+zn−l−1
+23
+G)
++ (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+G −
+∂2
+zn−1
+23
+G) −
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+G + (n − 1)∆∂3
+zn
+23
+G − (n − 1)∂2
+zn
+23
+G − ∂3∂2
+zn−1
+23
+G)]
+−
+1
+zm−k−1
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+∂1G+
+(−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+∂1G − ∂1∂2
+zn−l−1
+23
+G) + (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+∂1G − ∂1∂2
+zn−1
+23
+G)
+−
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+∂1G + (n − 1)∆∂1∂3
+zn
+23
+G − (n − 1)∂1∂2
+zn
+23
+G − ∂1∂3∂2
+zn−1
+23
+G)]}
++ (n − 1)(∆ + m)
+zn
+41
+{(m − 1)∆
+zm
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+G −
+∂2
+zn−l−1
+23
+G)
++ (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+G −
+∂2
+zn−1
+23
+G) −
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+G + (n − 1)∆∂3
+zn
+23
+G − (n − 1)∂2
+zn
+23
+G − ∂3∂2
+zn−1
+23
+G)]
+−
+1
+zm−1
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+∂1G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+∂1G − ∂1∂2
+zn−l−1
+23
+G) + (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+∂1G − ∂1∂2
+zn−1
+23
+G)
+−
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+∂1G + (n − 1)∆∂1∂3
+zn
+23
+G − (n − 1)∂1∂2
+zn
+23
+G − ∂1∂3∂2
+zn−1
+23
+G)]}
+−
+1
+zn−1
+41
+{m(m − 1)∆
+zm+1
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+G −
+∂2
+zn−l−1
+23
+G) + (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+G −
+∂2
+zn−1
+23
+G)
+−
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+G + (n − 1)∆∂3
+zn
+23
+G − (n − 1)∂2
+zn
+23
+G − ∂3∂2
+zn−1
+23
+G)]
+19
+
++ (m − 1)∆
+zm
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+∂4G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+∂4G − ∂4∂2
+zn−l−1
+23
+G)
++ (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+∂4G − ∂4∂2
+zn−1
+23
+G)
+−
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+∂4G + (n − 1)∆∂4∂3
+zn
+23
+G − (n − 1)∂4∂2
+zn
+23
+G − ∂4∂3∂2
+zn−1
+23
+G)]
+− (m − 1)
+zm
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+∂1G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+∂1G − ∂1∂2
+zn−l−1
+23
+G) + (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+∂1G − ∂1∂2
+zn−1
+23
+G)
+−
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+∂1G + (n − 1)∆∂1∂3
+zn
+23
+G − (n − 1)∂1∂2
+zn
+23
+G − ∂1∂3∂2
+zn−1
+23
+G)]
+−
+1
+zm−1
+14
+[(−1)n
+(n + m − 1)!
+(n + 1)!(m − 2)!
+n(n2 − 1)c/12 + 2n∆
+zm+n
+32
+∂4∂1G
++ (−1)m
+n−1
+�
+l=1
+(m + l − 1)!
+(l + 1)!(m − 2)!
+(n + l)
+zm+l
+23
+((n − l − 1)∆
+zn−l
+23
+∂4∂1G − ∂4∂1∂2
+zn−l−1
+23
+G)
++ (m − 1)(∆ + n)
+zm
+32
+((n − 1)∆
+zn
+23
+∂4∂1G − ∂4∂1∂2
+zn−1
+23
+G)
+−
+1
+zm−1
+32
+(n(n − 1)∆
+zn+1
+23
+∂4∂1G + (n − 1)∆∂4∂1∂3
+zn
+23
+G − (n − 1)∂4∂1∂2
+zn
+23
+G − ∂4∂1∂3∂2
+zn−1
+23
+G)]}.
+(66)
+The correlation function of four descendent operators becomes the correlation functions of their cor-
+responding primary operators with some constants and derivatives.
+For i ̸= j ̸= k ̸= l, we can have
+∂iG = 2∆∂iη
+1 − η G,
+∂j∂iG = 2∆∂j∂iη
+1 − η
+G + 2∆(2∆ + 1)∂jη∂iη
+(1 − η)2
+G,
+∂k∂j∂iG = 2∆∂k∂j∂iη
+1 − η
+G + 2∆(2∆ + 1)[∂j∂iη∂kη + ∂j∂kη∂iη + ∂k∂iη∂jη
+(1 − η)2
+G + (2∆ + 2)∂jη∂iη∂kη
+(1 − η)3
+G]
+∼ 2∆(2∆ + 1)[∂j∂iη∂kη + ∂j∂kη∂iη + ∂k∂iη∂jη
+(1 − η)2
+G + (2∆ + 2)∂jη∂iη∂kη
+(1 − η)3
+G],
+∂l∂k∂j∂iG = 2∆∂l∂k∂j∂iη
+1 − η
+G + (2∆(2∆ + 1))
+(1 − η)2
+(∂k∂j∂iη∂lη + ∂k∂j∂lη∂iη + ∂l∂j∂iη∂kη + ∂k∂l∂iη∂jη
++ ∂i∂jη∂l∂kη + ∂i∂kη∂j∂lη + ∂i∂lη∂k∂jη)G + 2∆(2∆ + 1)(2∆ + 2)
+(1 − η)3
+(∂j∂iη∂kη∂lη + ∂j∂kη∂iη∂lη
++ ∂k∂iη∂jη∂lη + ∂l∂iη∂kη∂jη + ∂j∂lη∂kη∂iη + ∂l∂kη∂iη∂jη)G
++ 2∆(2∆ + 1)(2∆ + 1)(2∆ + 3)
+(1 − η)4
+∂iη∂jη∂kη∂lηG
+∼ (2∆(2∆ + 1))
+(1 − η)2
+(∂i∂jη∂l∂kη + ∂i∂kη∂j∂lη + ∂i∂lη∂k∂jη)G + 2∆(2∆ + 1)(2∆ + 1)
+(1 − η)3
+(∂j∂iη∂kη∂lη + ∂j∂kη∂iη∂lη
++ ∂k∂iη∂jη∂lη + ∂l∂iη∂kη∂jη + ∂j∂lη∂kη∂iη + ∂l∂kη∂iη∂jη)G + 2∆(2∆ + 1)(2∆ + 2)(2∆ + 3)
+(1 − η)4
+∂iη∂jη∂kη∂lηG.
+(67)
+20
+
+References
+[1] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv.
+Theor. Math. Phys. 2 (1998) 231–252, [hep-th/9711200].
+[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncritical
+string theory, Phys. Lett. B 428 (1998) 105–114, [hep-th/9802109].
+[3] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291,
+[hep-th/9802150].
+[4] H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600
+(2004) 142–150, [hep-th/0405111].
+[5] P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.
+0406 (2004) P06002, [hep-th/0405152].
+[6] A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404,
+[hep-th/0510092].
+[7] H. Casini, I. Salazar Landea and G. Torroba, The g-theorem and quantum information theory,
+JHEP 10 (2016) 140, [1607.00390].
+[8] T. Nishioka, Entanglement entropy: holography and renormalization group, Rev. Mod. Phys. 90
+(2018) 035007, [1801.10352].
+[9] E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on
+entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003,
+[1803.04993].
+[10] H. Casini and M. Huerta, Lectures on entanglement in quantum field theory, 2201.13310.
+[11] M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42
+(2010) 2323–2329, [1005.3035].
+[12] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013)
+781–811, [1306.0533].
+[13] M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, vol. 931. Springer, 2017,
+10.1007/978-3-319-52573-0.
+[14] S. W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976)
+2460–2473.
+[15] S. D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26
+(2009) 224001, [0909.1038].
+21
+
+[16] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?,
+JHEP 02 (2013) 062, [1207.3123].
+[17] G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09
+(2020) 002, [1905.08255].
+[18] A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and
+the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063, [1905.08762].
+[19] Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka and Z. Wei, New holographic generalization of
+entanglement entropy, Phys. Rev. D 103 (2021) 026005, [2005.13801].
+[20] Y. Aharonov and L. Vaidman, The two-state vector formalism: an updated review, Time in
+quantum mechanics (2008) 399–447.
+[21] Y. Aharonov, P. G. Bergmann and J. L. Lebowitz, Time symmetry in the quantum process of
+measurement, Phys. Rev. 134 (Jun, 1964) B1410–B1416.
+[22] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan and R. W. Boyd, Colloquium: Understanding
+quantum weak values: Basics and applications, Rev. Mod. Phys. 86 (Mar, 2014) 307–316.
+[23] I. Akal, T. Kawamoto, S.-M. Ruan, T. Takayanagi and Z. Wei, Page curve under final state
+projection, Phys. Rev. D 105 (2022) 126026, [2112.08433].
+[24] A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka and Z. Wei, Pseudo Entropy in Free
+Quantum Field Theories, Phys. Rev. Lett. 126 (2021) 081601, [2011.09648].
+[25] G. Camilo and A. Prudenziati, Twist operators and pseudo entropies in two-dimensional
+momentum space, 2101.02093.
+[26] A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka and Z. Wei, Aspects of pseudoentropy in
+field theories, Phys. Rev. Res. 3 (2021) 033254, [2106.03118].
+[27] T. Nishioka, T. Takayanagi and Y. Taki, Topological pseudo entropy, JHEP 09 (2021) 015,
+[2107.01797].
+[28] K. Goto, M. Nozaki and K. Tamaoka, Subregion spectrum form factor via pseudoentropy, Phys.
+Rev. D 104 (2021) L121902, [2109.00372].
+[29] J. Mukherjee, Pseudo Entropy in U(1) gauge theory, JHEP 10 (2022) 016, [2205.08179].
+[30] W.-z. Guo, S. He and Y.-X. Zhang, On the real-time evolution of pseudo-entropy in 2d CFTs,
+JHEP 09 (2022) 094, [2206.11818].
+[31] M. Miyaji, Island for gravitationally prepared state and pseudo entanglement wedge, JHEP 12
+(2021) 013, [2109.03830].
+22
+
+[32] Y. Ishiyama, R. Kojima, S. Matsui and K. Tamaoka, Notes on pseudo entropy amplification,
+PTEP 2022 (2022) 093B10, [2206.14551].
+[33] A. Bhattacharya, A. Bhattacharyya and S. Maulik, Pseudocomplexity of purification for free
+scalar field theories, Phys. Rev. D 106 (2022) 086010, [2209.00049].
+[34] W.-z. Guo, S. He and Y.-X. Zhang, Constructible reality condition of pseudo entropy via
+pseudo-Hermiticity, 2209.07308.
+[35] K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki, Pseudo Entropy in dS/CFT and
+Time-like Entanglement Entropy, 2210.09457.
+[36] Z. Li, Z.-Q. Xiao and R.-Q. Yang, On holographic time-like entanglement entropy, 2211.14883.
+[37] F. C. Alcaraz, M. I. Berganza and G. Sierra, Entanglement of low-energy excitations in
+Conformal Field Theory, Phys. Rev. Lett. 106 (2011) 201601, [1101.2881].
+[38] M. Nozaki, T. Numasawa and T. Takayanagi, Quantum Entanglement of Local Operators in
+Conformal Field Theories, Phys. Rev. Lett. 112 (2014) 111602, [1401.0539].
+[39] S. He, T. Numasawa, T. Takayanagi and K. Watanabe, Quantum dimension as entanglement
+entropy in two dimensional conformal field theories, Phys. Rev. D 90 (2014) 041701,
+[1403.0702].
+[40] M. Nozaki, Notes on Quantum Entanglement of Local Operators, JHEP 10 (2014) 147,
+[1405.5875].
+[41] P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal
+field theories, PTEP 2014 (2014) 093B06, [1405.5946].
+[42] P. Caputa, J. Sim´on, A. ˇStikonas and T. Takayanagi, Quantum Entanglement of Localized
+Excited States at Finite Temperature, JHEP 01 (2015) 102, [1410.2287].
+[43] W.-Z. Guo and S. He, R´enyi entropy of locally excited states with thermal and boundary effect in
+2D CFTs, JHEP 04 (2015) 099, [1501.00757].
+[44] P. Caputa and A. Veliz-Osorio, Entanglement constant for conformal families, Phys. Rev. D 92
+(2015) 065010, [1507.00582].
+[45] B. Chen, W.-Z. Guo, S. He and J.-q. Wu, Entanglement Entropy for Descendent Local
+Operators in 2D CFTs, JHEP 10 (2015) 173, [1507.01157].
+[46] P. Caputa, T. Numasawa and A. Veliz-Osorio, Out-of-time-ordered correlators and purity in
+rational conformal field theories, PTEP 2016 (2016) 113B06, [1602.06542].
+23
+
+[47] T. Numasawa, Scattering effect on entanglement propagation in RCFTs, JHEP 12 (2016) 061,
+[1610.06181].
+[48] S. He, Conformal bootstrap to R´enyi entropy in 2D Liouville and super-Liouville CFTs, Phys.
+Rev. D 99 (2019) 026005, [1711.00624].
+[49] W.-Z. Guo, S. He and Z.-X. Luo, Entanglement entropy in (1+1)D CFTs with multiple local
+excitations, JHEP 05 (2018) 154, [1802.08815].
+[50] L. Apolo, S. He, W. Song, J. Xu and J. Zheng, Entanglement and chaos in warped conformal
+field theories, JHEP 04 (2019) 009, [1812.10456].
+[51] P. Caputa, T. Numasawa, T. Shimaji, T. Takayanagi and Z. Wei, Double Local Quenches in 2D
+CFTs and Gravitational Force, JHEP 09 (2019) 018, [1905.08265].
+[52] L. Bianchi, S. De Angelis and M. Meineri, Radiation, entanglement and islands from a boundary
+local quench, 2203.10103.
+[53] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between
+Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602, [1507.07555].
+[54] M. Miyaji, Butterflies from Information Metric, JHEP 09 (2016) 002, [1607.01467].
+[55] J. Zhang and P. Calabrese, Subsystem distance after a local operator quench, JHEP 02 (2020)
+056, [1911.04797].
+[56] X. Wen, P.-Y. Chang and S. Ryu, Entanglement negativity after a local quantum quench in
+conformal field theories, Phys. Rev. B 92 (2015) 075109, [1501.00568].
+[57] J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross
+sections in holographic theories, Phys. Rev. D 99 (2019) 106014, [1808.00446].
+[58] J. Kudler-Flam, I. MacCormack and S. Ryu, Holographic entanglement contour, bit threads, and
+the entanglement tsunami, J. Phys. A 52 (2019) 325401, [1902.04654].
+[59] J. Kudler-Flam, Y. Kusuki and S. Ryu, Correlation measures and the entanglement wedge
+cross-section after quantum quenches in two-dimensional conformal field theories, JHEP 04
+(2020) 074, [2001.05501].
+[60] J. Kudler-Flam, M. Nozaki, S. Ryu and M. T. Tan, Entanglement of local operators and the
+butterfly effect, Phys. Rev. Res. 3 (2021) 033182, [2005.14243].
+[61] J. Kudler-Flam, Y. Kusuki and S. Ryu, The quasi-particle picture and its breakdown after local
+quenches: mutual information, negativity, and reflected entropy, JHEP 03 (2021) 146,
+[2008.11266].
+24
+
+[62] G. W. Moore and N. Seiberg, Naturality in Conformal Field Theory, Nucl. Phys. B 313 (1989)
+16–40.
+[63] P. Calabrese and J. L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J.
+Stat. Mech. 0504 (2005) P04010, [cond-mat/0503393].
+[64] P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory. Graduate Texts in
+Contemporary Physics. Springer-Verlag, New York, 1997, 10.1007/978-1-4612-2256-9.
+25
+

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+1
+Recent advances of transition radiation: fundamentals and
+applications
+Ruoxi Chen1,2, Zheng Gong1,2, Jialin Chen1,2,3, Xinyan Zhang1,2, Xingjian Zhu4, Hongsheng
+Chen1,2,5,6,*, and Xiao Lin1,2,*
+1Interdisciplinary Center for Quantum Information, State Key Laboratory of Extreme Photonics and
+Instrumentation, ZJU-Hangzhou Global Scientific and Technological Innovation Center, College of Information
+Science & Electronic Engineering, Zhejiang University, Hangzhou 310027, China.
+2International Joint Innovation Center, the Electromagnetics Academy at Zhejiang University, Zhejiang University,
+Haining 314400, China.
+3Department of Electrical and Computer Engineering, Technion-Israel Institute of Technology, Haifa 32000,
+Israel.
+4School of Physics, Zhejiang University, Hangzhou 310027, China.
+5Key Laboratory of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, Jinhua Institute of
+Zhejiang University, Zhejiang University, Jinhua 321099, China.
+6Shaoxing Institute of Zhejiang University, Zhejiang University, Shaoxing 312000, China.
+*Corresponding authors: xiaolinzju@zju.edu.cn (X. Lin); hansomchen@zju.edu.cn (H. Chen)
+Transition radiation is a fundamental process of light emission and occurs whenever a charged
+particle moves across an inhomogeneous region. One feature of transition radiation is that it
+can create light emission at arbitrary frequency under any particle velocity. Therefore,
+transition radiation is of significant importance to both fundamental science and practical
+applications. In this paper, we provide a brief historical review of transition radiation and its
+recent development. Moreover, we pay special attention to four typical applications of
+transition radiation, namely the detection of high-energy particles, coherent radiation sources,
+beam diagnosis, and excitation of surface waves. Finally, we give an outlook for the research
+tendency
+of
+transition
+radiation,
+especially
+its
+flexible
+manipulation
+by
+exploiting
+artificially-engineered materials and nanostructures, such as gain materials, metamaterials,
+spatial-temporal materials, meta-boundaries, and layered structures with a periodic or
+non-periodic stacking.
+
+2
+
+3
+Free-electron radiation originates from the particle-matter interaction and is a fundamental
+process of light emission [1-8]. Since free-electron radiation is able to create light emission at
+arbitrary frequency, it is of paramount importance to numerous applications [9-15], including
+high-energy
+particles
+detector,
+particle-beam
+diagnosis,
+free-electron
+lasers,
+high-power
+microwave/terahertz sources, electron microscopy, biomedical imaging, medical therapy, optical
+communications, and security.
+Due to the exotic particle-matter interactions, the free-electron radiation could occur under
+different scenarios. Correspondingly, there are various types of free-electron radiation, including
+Cherenkov
+radiation
+[16-19],
+transition
+radiation
+[20-23],
+Smith-Purcell
+radiation
+[24-28],
+bremsstrahlung radiation [29-34], and synchrotron radiation [35-40], as shown in Fig. 1. Cherenkov
+radiation is the most famous type of free-electron radiation, and it occurs in a homogeneous matter
+only when a charged particle moves with a velocity larger than the Cherenkov threshold, namely the
+phase velocity of light in that matter [41-45]. Different from Cherenkov radiation, the occurrence of
+transition radiation has no specific requirement on the particle velocity, and it could happen
+whenever a charged particle moves across an inhomogeneous region [46-50], such as an optical
+interface. Smith-Purcell radiation emerges when a charged particle moves parallel and close to the
+surface of an optical diffraction grating [24-28]. Bremsstrahlung radiation, also known as the braking
+radiation, would appear if the charge particle decelerates or accelerates [29-34]. Synchrotron
+radiation originates from the circular motion of charged particles [35-40]. Due to the complexity of
+particle-matter interactions, the free-electron radiation, such as transition radiation, is a subject of
+extensive researches over the last several decades and is still a hot topic [51-55].
+
+4
+Below, we focus on the discussion of transition radiation. A brief review of the development of
+transition radiation is provided, including its interesting history and recent advances. To highlight the
+importance of transition radiation in science and technology, several typical applications of transition
+radiation are discussed in depth, ranging from high-energy particle detectors [56-65], coherent
+radiation source [66-71], beam diagnostics [72-76], to excitation of surface plasmon [77-84]. Due to
+the recent rapid development in material science and nanotechnology, an outlook on how to flexibly
+control the behavior of transition radiation is given by exploiting exotic artificially-engineered
+materials
+and
+nanostructures
+[85-90],
+including
+gain
+materials,
+spatial-temporal
+materials,
+meta-boundaries, and layered structures with a random stacking.
+Brief development history of transition radiation
+We begin with a brief review of the development history of transition radiation, as summarized
+in Fig. 2. The simplest case of transition radiation was first proposed by Ginzburg and Frank in 1946
+[20] by exploring the bombardment of an electron at the interface between vacuum and a metal. The
+theory of transition radiation was experimentally confirmed in the visible range by Goldsmith and
+Jelley in 1959 [91], after the analysis of the collected radiation fields, including their polarization,
+excitation function and absolute yield. In their setup, a Van de Graaff generator was adopted to
+produce a 5 mega-electron-volt (MeV) beam bunch, which was then injected towards the
+vacuum-metal interface. Soon after the discovery of transition radiation, the quantum effect of
+transition radiation was also investigated by Garibian in 1960 [92].
+The development of transition radiation is closely related to the interesting history of Ferrell
+radiation [93], which is featured with a peak near the plasma frequency in the radiation spectrum. In
+1958, Ferrell proposed an approximate theory for the radiation of plasma oscillation and claimed that
+
+5
+under suitable circumstances, the plasma oscillations would give off electromagnetic radiation [93].
+Since Ferrell radiation could facilitate the plasma-frequency measurement of metals [93], Ferrell
+radiation was soon experimentally observed in 1960 [94,95]. In 1961, Silin and Fetisov pointed out
+that Ferrell radiation is merely the transition radiation [96]. In 1962, Stern argued that Ferrell’s
+method and Ginzburg and Frank’s theory of transition radiation are two distinct ways to consider the
+same phenomenon [97]. Stern highlighted that the physical mechanism of Ferrell radiation was
+misinterpreted by Silin and Fetisov, and it is “a surface effect” [97], namely “the contribution of
+radiative surface plasma oscillation (SPO)” [98], instead of the bulk longitudinal plasma oscillation
+in their study. In 1969, Economou emphasized that “there are no radiative SPO in the present
+geometry” [98]. Actually, he preferred the explanation of transition radiation, after the mathematical
+analysis of the denominator in the expression of transition radiation near the peak of Ferrell radiation
+[98]. Due to the recent advances in plasmonics [99-105], it is now acknowledged that the radiative
+SPO is essentially a leaky mode, denoted as the Ferrell mode [84,97,98,106]. In 2022, the Ferrell
+radiation was re-investigated in the time domain and was found able to occur far beyond the
+formation time, since it is supported by a long tail of bulk plasmons following the electron’s
+trajectory deep into the plasmonic medium [107]. This way, the plasmonic tail is capable to mix
+surface and bulk effects and provides a sustained channel for electron-interface interaction. This
+time-domain finding may settle the historical debate in Ferrell radiation, regarding whether it is a
+surface or bulk effect, from transition radiation or plasmonic oscillation. Moreover, with the aid of
+Ferrell modes in uniaxial epsilon-near-zero materials, such as a thin hexagonal boron nitride (hBN)
+slab, an exotic phenomenon of low-velocity-favored transition radiation was recently proposed [108].
+When the low-velocity-favored transition radiation occurs, the light emission from ultralow-energy
+
+6
+particles with extremely-low velocities could exhibit comparable intensity as that from high-energy
+particles [108].
+Due to the recent advances in nanofabrication, artificially-engineered materials or nanostructures,
+such as metamaterials [109-119], 2D materials [120-128], and photonic crystals [129-135], begin to
+play an important role in the flexible manipulation of transition radiation.
+In 2009, Ref. [136] analyzed the case that a charged particle crosses an interface between a
+positive-index medium and a negative-index medium. Under this scenario, the transition radiation
+from the interface and the reversed Cherenkov radiation from the negative-index material would
+interfere and make the light emission complex. In 2012, Ref. [137] further studied the transition
+radiation
+from
+an
+average
+zero-index
+metamaterial,
+comprised
+of
+periodically
+alternating
+negative-index and positive-index layers. A strong radiation enhancement up to three orders of
+magnitude was predicted, due to the gigantic increase in the density of states
+at the
+positive-index/negative-index interface.
+In 2012, Ref. [138] used graphene to tailor the transition radiation. Although the graphene is
+atomically thin, the free electron can efficiently excite graphene plasmons with probabilities in the
+order of one per electron [139]. In addition to the monolayer graphene [78,84,140-142], Ref. [142]
+also investigated the photonic and plasmonic transition radiation from multilayer graphene.
+Due to the resonance effect, the analysis of transition radiation from periodic structures [143-147]
+is rather complicated. The related research could be dated back to the transition radiation in a
+periodically stratified plasma [148]. In 2003, Ref. [149] revealed that the behavior of Cherenkov
+radiation inside a two-dimensional photonic crystal is intrinsically coupled with the transition
+radiation and would appear without the Cherenkov threshold. In 2018, Ref. [150] further revealed the
+
+7
+connection between Cherenkov radiation and transition radiation. Actually, Ref. [150] proposed a
+new mechanism – by exploiting the resonance transition radiation from one-dimensional photonic
+crystals – to create the effective Cherenkov radiation. This mechanism is capable to control the
+effective Cherenkov angles with high sensitivity, under any desired range of particle momentum, and
+is thus promising for the design of novel particle detectors.
+High-energy particle detectors based on transition radiation
+Particle detectors provide a powerful tool to detect, track and identify high-energy particles
+[151-155]. In high-energy physics, the charged particle could have a Lorentz factor
+��� = 1/
+1 − ���2/���2 up to 104, where ��� is the particle velocity and ��� is the speed of light in free space.
+For such high-energy particles, the sensitivity of common particle detectors (e.g. Cherenkov
+detectors [156-159]) is generally very low, and thus their detection is full of challenges. To address
+this issue, Garibian expanded the theory of transition radiation into the X-ray regime in 1959 and
+found the radiation energy is linearly proportional to the Lorentz factor ���, along with the radiation
+peak emerging at ���max = ���−1 , if ��� ≫ 1 [160,161]. Therefore, the unique relation between the
+radiation energy and the Lorentz factor offers a new route to detect ultra-relativistic particles, even
+when their kinetic energy is up to tera-eV (TeV). Correspondingly, the particle detector based on the
+X-ray transition radiation is now known as the transition radiation detector [56-65].
+The transition radiation detector, along with other particle detectors, has made a significant
+contribution to many famous experiments and the finding of new particles (e.g. W and Z bosons
+[162], the Higgs boson [163]). Actually, the transition radiation detector is widely used in many
+particle-physics laboratories (e.g. CERN, Femi Lab), as exemplified in Fig. 3. For example, in the
+H1 experiment, electron and positron were identified at the HERA electron proton collider [56]. In
+
+8
+the experiment E799, the Fermi lab designed a transition radiation detector with a large aperture to
+provide π/e rejection [57]. In the NOMAD experiment, transition radiation detector discriminated
+electron and pion and could search for the ������ → ������ oscillation [58]. In the ALICE experiment, the
+transition radiation detector is served as a trigger on high ������ ���+ ���− pairs to reduce the collision rate
+to the readout event rate [59]. In the super proton synchrotron (SPS) of CERN, an inorganic
+scintillator-based Compton-scatter transition radiation detector in Fig. 3a is designed to detect
+high-energy electrons with an ultra-high Lorentz factor [60]. In the ATLANS experiment, the
+transition radiation tracker is a fast detector with thin detector layers realized with straw tubes and
+could provide both tracking information and particle identification; see its cutaway view in Fig. 3b
+[61]. In the compressed baryonic matter (CBM) experiment, the transition radiation detector is used
+to measure common hadrons, such as low-mass dileptons, charmed hadrons, and multistrange
+baryons [62].
+The performance of transition radiation detectors is mainly determined by the properties of the
+associated radiator and detector. For experiments with relatively-low particle multiplicity, transition
+radiation detectors generally use multi-wire proportional chambers (MWPC) or straw tubes, filled
+with a Xenon based gas mixture to efficiently absorb the emitted photons [63]. For experiments with
+relatively-high particle multiplicity, the efficiency of traditional transition radiation detectors would
+decrease significantly due to the channel occupancy. Ref. [64] showed that the performance of
+transition radiation detectors could be improved by replacing MWPC or straw tubes with a
+high-granularity-micro-pattern gas detector, such as gas electron multipliers as shown in Fig. 3c. In
+addition, Ref. [150] found that the one-dimensional photonic crystal could create the resonance
+transition radiation with ultrahigh directivity in Fig. 3d and acts as a new type of radiator.
+
+9
+Correspondingly, the photonic crystal offers a promising versatile platform well suited for the
+identification of particles at high energy with enhanced sensitivity.
+Coherent light sources based on transition radiation
+If the electron beam has a length much shorter than the working wavelength, all electrons can be
+considered to emit in phase. As a result, the related transition radiation is coherent. The phenomenon
+of coherent transition radiation was first observed in 1991 [164]. One feature of the coherent
+transition radiation is that its energy could be enhanced by a factor of ��� than that of the incoherent
+transition radiation, where ��� is the electron number.
+For any type of free-electron radiation, it takes a finite space domain instead of a point for
+photons to be emitted. This finite space domain is now known as the formation zone [165], which is
+a useful concept for the coherent transition radiation. In the context of free-electron radiation, this
+concept was first presented by Ter-Mikaelian in Landau’s seminar in 1952 [166], and later developed
+by Landau himself [167,168], with its experimental confirmation in the 1990s. Later, Ginzburg
+extended this concept into transition radiation [169]. The concept of formation zone provides
+valuable guidance for practical applications based on the transition radiation. Particularly, the
+influence of formation zone is generally avoided in the design of transition radiation detectors [170].
+According to Ginzburg’s work [169], the length of the formation zone for the transition radiation,
+namely the formation length ���f, is defined as the distance that the charge field ������ and the radiation
+field ������ separate from each other. In other words, the contribution of the interference term ������ ⋅ ������
+to the total field energy (proportional to
+������ + ������ 2 ) is very small outside the formation zone.
+According to the definition, we have
+
+10
+���f =
+2���
+��� ���±��� ��� 1−
+���⊥2���2
+���2
+(1)
+where ��� is the angular frequency, ���⊥ is the component of wavevector perpendicular to the electron
+velocity, and ±
+correspond to the forward and backward radiation, respectively.
+Ginzburg’s
+estimation on the formation zone is actually only applicable to the photon emission during the
+process of transition radiation. In 2017, along the line of Ginzburg’s thought, the concept of
+formation zone was extended to the emitted surface plasmons during the process of transition
+radiation [78].
+With the knowledge of formation zone, the generation of coherent transition radiation mainly
+relies on two ways. One way is to use an electron beam to directly bombard the optical interface. For
+example, Ref. [66] shows that the coherent terahertz (THz) radiation could be generated by an
+electron bunch with a duration in the order of tens of femtoseconds to picoseconds penetrating
+through the plasma-vacuum interface; see Fig. 4a. Such electron bunches are produced by a
+laser-plasma accelerator [66]. Similarly, the THz coherent transition radiation with energies in the
+order of sub-mJ/pulse could also be induced by letting the laser-driven electron beams cross a
+dielectric-vacuum interface [67] in Fig. 4b. When an electron beam has a femtosecond duration and
+hundreds of kiloampere peak current and penetrates through the plasma-vacuum interface, the
+terawatt ultraviolet coherent transition radiation could happen [68]; see the associated ring intensity
+distribution in Fig. 4c. The other way is to use an ultrahigh-power laser to irradiate on the target
+material, which further induces the photoelectric effect and makes electrons escape from the
+rear-side surface of the target material. Ref. [69] shows that the terahertz radiation with a field
+
+11
+strength up to 100 GV/m in Fig. 4d is produced by two-color, ultrashort optical pulses interacting
+with under-dense helium gases at ultrahigh intensities.
+Beam diagnostics based on transition radiation
+In other to facilitate the implementation of high-gain free-electron laser or high-power electron
+beam, the shape of electron beams should be monitored. The coherent transition radiation offers a
+feasible method to diagnose the electron beam in high-energy equipment. This method mainly relies
+on the measurement of angular spectral energy density [169]. To be specific, the angular spectral
+energy density
+���2���total
+������������
+of the coherent transition radiation [169] is
+���2���total
+������������ ≅ ���2��� ���
+���2���single
+������������
+(2)
+where
+���2���single
+������������
+is the angular spectral energy density for the transition radiation from a single
+electron, and ��� ���
+is a function of longitudinal and transverse distribution parameters for the
+electron beam after the Fourier transformation. With the knowledge of
+���2���single
+������������
+and
+���2���total
+������������ , the
+information of electron beams, namely ��� ��� , can be straightforwardly obtained in the experiment.
+The beam diagnosis based on transition radiation was proposed in 1975 by Ref. [171], which
+uses two parallel foils. The phase and energy information of electron bunches can be inferred from
+the angular distribution of the interference pattern of transition radiation. In the following decades,
+more optical components (e.g. wire grid, Michelson interferometer) are adopted to provide not only
+the longitudinal and transverse distributions of electron beams but also their divergent angle with the
+help of coherent transition radiation [72-76].
+Figure 5 shows a variety of setups for beam diagnosis, which are widely used in high-energy
+experiments and facilities, such as free-electron laser, wake-field acceleration, and large particle
+collider. When the electron beam of 42 MeV passes through an aluminum foil, the coherent transition
+
+12
+radiation with a millimeter or submillimeter wavelength is observed in Fig. 5a [72]. The length of
+electron beams is measured by using a 45°-tilted foil and a polarizing Michelson interferometer in
+Fig. 5b, because the spectral power of light emission is dependent on the degree of coherency, which
+strongly relates to the beam size [73]. More precise measurement for the electron beam is proposed
+in AWAKE experiments through the usage of seeded self-modulation as shown in Fig. 5c [74]. At the
+CLARA facility, the coherent transition radiation is used to diagnose the longitudinal beam profile
+for the dielectric wakefield accelerator and coherent Cherenkov diffraction radiation [75]; see the
+setup in Fig. 5d.
+Excitation of surface wave by using transition radiation
+While surface waves can mold the flow of light at the subwavelength scale, their excitation is
+generally difficult, due to their momentum mismatch with propagating waves. In addition to the
+conventional schemes like gratings or prism matching, the transition radiation provides a powerful
+scheme to excite surface waves [77-84], especially for those with extremely-high spatial confinement
+such as graphene plasmons [78,172-175].
+In 1957, Ritchie theoretically proposed a mechanism to excite surface waves by using transition
+radiation [176]. This prediction was observed in experiments in 2006 [77] by using an electron beam
+of 50 keV injected onto a flat gold surface as shown in Fig. 6a. In 2017, Ref. [78] theoretically
+revealed a splashing transient of graphene plasmons launched by swift electrons. During the process
+of transition radiation, a jet-like rise of excessive charge concentration emerges and is analogous to
+the hydrodynamic Rayleigh jet in a splashing phenomenon before the launching of ripples. When the
+free electrons interact with a mesoscopic structure composed of an array of nanoscale holes in a gold
+film, the surface waves are firstly excited and soon transformed into propagating waves [79]. As a
+
+13
+result, ultrashort chirped electromagnetic wave packets could be created under the irradiation of
+30-200 keV electron beams [79]. In 2019, the interaction between free electrons and photonic
+topological crystals is experimentally studied in Ref. [80]. The robust edge states are excited and
+observed, when a charged particle passes from one photonic crystal into another one.
+Outlook
+Despite the numerous applications enabled by the transition radiation, the transition radiation
+itself still suffers from low intensity and low directionality, especially if the charged particles have
+ultra-low energy. How to strengthen the particle-interface interaction and control the transition
+radiation in the desired way remains a long-standing challenge that is highly sought after. While
+metamaterials, 2D materials, and photonic crystals have been applied to tackle this challenge, there is
+still plenty of room to tailor the transition radiation by exploiting artificially-engineered materials
+and nanostructures. For example, we show in Fig. 7 that the gain materials, tilted hyperbolic
+materials, meta-boundaries, spatial-temporal materials, and layered structures with a periodic or
+random stacking might offer an enticing platform to enhance the particle-interface interaction and to
+achieve exotic features of transition radiation.
+The gain material in Fig. 7a in practice can be implemented, for example, by using
+negative-resistance components [177-180] (e.g. microwave tunnel diodes) and optically pumped dye
+molecules [181,182] (e.g. Rhodamine 800 dye molecules). While gain materials could provide a
+universal way to amplify the light emission, the free-electron radiation from gain systems, including
+the transition radiation, has been rarely discussed. Particularly, the influence of optical gain and the
+slab thickness on the directionality of transition radiation remains elusive, while a larger optical gain
+is generally thought to have a larger enhancement of the intensity of transition radiation. More rich
+
+14
+physics of free-electron radiation could be expected in systems simultaneously with optical gain and
+optical loss, such as those with parity-time symmetry [183-187]. However, the influence of the
+interplay between optical gain and optical loss on the transition radiation has never been explored
+before.
+The hyperbolic material [188-196] is a uniaxial material that has a hyperbolic iso-frequency
+contour and is featured with an extraordinarily-large photonic density state. Due to these unique
+features, hyperbolic materials can greatly enhance both the particle-matter interaction and the
+particle-interface interaction. However, once these high-k hyperbolic modes are excited, they cannot
+be safely coupled into free space, partly due to the existence of total internal reflection and material
+losses. How to overcome this issue becomes crucial for the development of novel on-chip light
+sources based on ultralow-energy electrons. To mitigate this issue, the photonic hyper-crystals are
+experimentally studied in Ref. [197]. The photonic hyper-crystals have combined virtues of strong
+light
+outcoupling
+in
+photonic
+crystals
+and
+large
+photonic
+density-of-states
+in
+hyperbolic
+metamaterials. Moreover, the broadband enhancement of on-chip photon extraction is achievable by
+using tilted hyperbolic metamaterials in Fig. 7b [198], since their eigenmodes now become
+momentum-matched with propagating waves in free space. Due to the exotic features of photonic
+hyper-crystals [197,199,200] and tilted hyperbolic metamaterials [198], the transition radiation from
+these exotic materials deserves more in-depth exploration.
+The judicious design of electromagnetic boundary could provide a key route to control the
+free-electron
+radiation,
+including
+the
+transition
+radiation.
+Due
+to
+the
+recent
+advent
+of
+two-dimensional materials (e.g. graphene [201-208], hBN [209-217], twisted photonic structures
+[218-227]) and metasurfaces [89, 228-235], the boundary is uniquely featured with a surface
+
+15
+conductivity, which can be rather complex but provide an extra degree of freedom to regulate the
+free-electron radiation. Without loss of generality, the boundary with a non-zero surface conductivity
+is termed as the meta-boundary in Fig. 7c [236]. According to the electromagnetic boundary
+conditions, the meta-boundary could be categorized into four types, including isotropic, anisotropic,
+biisotropic and bianisotropic meta-boundaries [236]. While the transition radiation from the simple
+isotropic meta-boundary (e.g. graphene) has been extensively studied [78,84,140-142], the transition
+radiation from more complex meta-boundaries remains largely unexplored and awaits more
+systematic investigation. Due to the powerfulness of metamaterial and meta-boundaries, the
+manipulation of free-electron radiation (e.g. transition radiation) at will might be enabled by
+combining meta-boundaries and metamaterials.
+Spatial-temporal materials in Fig. 7d, such as photonic-time crystal and temporal metamaterials,
+have their optical response dependent both on space and time [237-239]. The emergence of
+spatial-temporal materials gets rid of fundamental limitations presented in space-engineered media
+and enable many counterintuitive phenomena, such as magnet-free nonreciprocity [240] and
+Cherenkov radiation in the vacuum [241,242]. Therefore, the spatial-temporal material in principle
+can provide a versatile platform to tailor the transition radiation. For example, Ref. [243] showed that
+a swift electron moving in a photonic-time crystal spontaneously emits radiation. When associated
+with momentum-gap modes, the process of free-electron radiation is exponentially amplified by the
+modulation of the refractive index [243]. Actually, the exploration of free-electron radiation from
+spatial-temporal material is still in its infancy.
+Due to the appearance of multiple interfaces in layered structures, the transition radiation from
+each interface of the layered structure may interfere constructively or destructively. This way,
+
+16
+through the judicious structural design, the layered structure may provide an extra degree of freedom
+to tailor the particle-interface interaction. For example, if the layered structure has a periodic
+stacking in Fig. 7e, the constructive interface of transition radiation from each interface can be
+regarded as the excitation of high-k Bloch modes inside the photonic crystals. As advantageous to the
+high-k modes in hyperbolic materials, the excited high-k Block modes in layered structures with a
+periodic stacking might be safely coupled into free space, due to the umklapp scattering. Therefore,
+the
+particle-interface
+interaction
+is
+promising
+to
+efficiently
+extract
+the
+information
+from
+ultralow-energy electron, which however remains further studies. Moreover, the dispersion is
+unavoidable in layered structures with a periodic stacking, either from the material-induced
+dispersion or the structural-periodicity-induced dispersion. How to reduce or even eliminate the
+dispersion in layered structures so that the dispersionless resonance transition radiation can be
+created is still a challenge in science and technology.
+If the layered structure has a non-periodic stacking in Fig. 7f, the interference of transition
+radiation from each interface becomes more random but offers more degrees of freedom to tailor the
+particle-interface interaction. Due to the disappearance of eigenmodes or Bloch modes in the
+randomly-stacked layered structures, we can no longer predict the behavior of transition radiation
+simply from the optical response of the randomly-stacked layered structures. To enable the
+theoretical prediction, the rigorously analytical solution of transition radiation from the layered
+structures, although being quite tedious, is mandatory. As a result, whether we can achieve the
+constructive interference of transition radiation from each interface at a specific radiation angle
+remains elusive. From the perspective of applications, the associated resonance conditions for
+transition radiation from the randomly-stacked layered structures are highly wanted. Moreover, due
+
+17
+to the non-period nature of the randomly-stacked layered structures, whether they can be adopted to
+achieve broadband dispersionless resonance transition radiation is also worth further investigation.
+
+18
+Competing interests
+The authors declare no competing interests.
+Acknowledgement
+X.L. acknowledges the support partly from the National Natural Science Fund for Excellent Young
+Scientists Fund Program (Overseas) of China, the National Natural Science Foundation of China
+(NSFC) under Grant No. 62175212, Zhejiang Provincial Natural Science Fund Key Project under
+Grant
+No.
+LZ23F050003,
+the
+Fundamental
+Research
+Funds
+for
+the
+Central
+Universities
+(2021FZZX001-19), and Zhejiang University Global Partnership Fund. H.C. acknowledges the
+support from the Key Research and Development Program of the Ministry of Science and
+Technology under Grants No. 2022YFA1404704, 2022YFA1404902, and 2022YFA1405200, the
+National Natural Science Foundation of China (NNSFC) under Grants No.11961141010 and No.
+61975176. J.C. acknowledges the support from the Chinese Scholarship Council (CSC No.
+202206320287).
+Reference
+[1] A. Konečná, F. Iyikanat, & F. J. G. de Abajo, Entangling free electrons and optical excitations,
+Sci. Adv. 8 (2022) eabo7853.
+[2] A. Fisher, et al., Single-pass high-efficiency terahertz free-electron laser, Nat. Photon. 16 (2022)
+441-447.
+[3] J. S. Hummelt, et al., Coherent Cherenkov-cyclotron radiation excited by an electron beam in a
+metamaterial waveguide, Phys. Rev. Lett. 117 (2016) 237701.
+[5] Y. Yang, et al., Maximal spontaneous photon emission and energy loss from free electrons, Nat.
+Phys. 14 (2018) 894-899.
+[4] L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, & M. Soljacic, Towards graphene
+plasmon-based free-electron infrared to X-ray sources, Nat. Photon. 10 (2016) 46-52.
+[6] G. Adamo, et al., Light well: A tunable free-electron light source on a chip, Phys. Rev, Lett. 103
+(2009) 113901.
+[7] J. Gardelle, J. Labrouche, & J. L. Rullier, Direct observation of beam bunching produced by a
+high-power microwave free-electron laser, Phys. Rev. Lett. 76 (1996) 4532-4535.
+[8] Genevet, P. et al., Controlled steering of Cherenkov surface plasmon wakes with a
+one-dimensional metamaterial, Nat. Nanotech. 10 (2015) 804-809.
+[9] W. Wang, et al., Free-electron lasing at 27 nanometres based on a laser wakefield accelerator,
+Nature 595 (2021) 516-520.
+[10] W. Decking, et al., A MHz-repetition-rate hard X-ray free-electron laser driven by a
+superconducting linear accelerator, Nat. Photon. 14 (2020) 391-397.
+[11] B. McNeil, & N. Thompson, X-ray free-electron lasers, Nat. Photon. 4 (2010) 814-821.
+[12] E. Prat, et al., A compact and cost-effective hard X-ray free-electron laser driven by a
+high-brightness and low-energy electron beam, Nat. Photon. 14 (2020) 748-754.
+[13]X. Lin, et al., A Brewster route to Cherenkov detectors, Nat. Commun. 12 (2021) 5554.
+[14] I. Nozawa, et al., Measurement of <20 fs bunch length using coherent transition radiation, Phys.
+Rev. ST Accel. Beams 17 (2014) 072803.
+[15] T. Shaffer, E. Pratt, & J. Grimm, Utilizing the power of Cerenkov light with nanotechnology,
+
+19
+Nat. Nanotech. 12 (2017) 106-117.
+[16] P. A. Cherenkov, Visible glow under exposure of gamma radiation, Dokl. Akad. Nauk SSSR 2
+(1934) 451-454.
+[17] H. Hu, et al., Surface Dyakonov-Cherenkov radiation, eLight 2 (2022) 2.
+[18] H. Hu, et al., Broadband enhancement of Cherenkov radiation using dispersionless plasmons,
+Adv. Sci. 9 (2022) 2200538.
+[19] F. Liu, et al., Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,
+Nat. Photon. 11 (2017) 289-292.
+[20] V. L. Ginzburg, et al., Radiation of a uniformly moving electron due to its transition from one
+medium into another, Zh. Eksp. Teor. Fiz. 16 (1946) 15-28.
+[21] V. L. Ginzburg, & V. N. Tsytovich, Several problems of the theory of transition radiation and
+transition scattering. Physics Reports 49 (1979) 1-89.
+[22] H. Lihn, et al, Observation of stimulated transition radiation, Phys. Rev. Lett. 76 (1996) 4163.
+[23] Karataev, P. et al., First observation of the point spread function of optical transition radiation,
+Phys. Rev. Lett. 107 (2011) 174801.
+[24] S. J. Smith, & E. M. Purcell, Visible light from localized surface charges moving across a
+grating, Phys. Rev. 92 (1953) 1069.
+[25] L. Jing, et al., Polarization shaping of free-electron radiation by gradient bianisotropic
+metasurfaces, Laser & Photon. Rev. 15 4 (2021) 2000426.
+[26] I. Kaminer, et al., Spectrally and spatially resolved Smith-Purcell radiation in plasmonic crystals
+with short-range disorder, Phys. Rev. X 7 (2017) 011003.
+[27] Z. Wang, K. Yao, M. Chen, H. Chen, & Y. Liu, Manipulating Smith-Purcell emission with
+Babinet metasurfaces, Phys. Rev. Lett. 117 (2016) 157401.
+[28] K. Mizuno, J. Pae, T. Nozokido, et al., Experimental evidence of the inverse Smith-Purcell
+effect, Nature 328 (1987) 45-47.
+[29] Y. S. Tsai, Pair production and bremsstrahlung of charged leptons, Rev. Mod. Phys. 46 (1974)
+815.
+[30] L. I. Schiff, Energy-angle distribution of thin target bremsstrahlung, Phys. Rev. 83 (1951) 252.
+[31] F. A. Berends, R. Kleiss, P. de Causmaecker, R. Gastmans, & T. T. Wu, Single bremsstrahlung
+processes in gauge theories, Phys. Rev. B 103 (1981) 124-128.
+[32] F. V. Bunkin, & M. V. Fedorov, Bremsstrahlung in a strong radiation field, Sov. Phys. JETP 22
+(1966) 844-847.
+[33] E. Aprile, et al., Search for light dark matter interactions enhanced by the Migdal effect or
+Bremsstrahlung in XENON1T, Phys. Rev. Lett. 123 (2019) 241803.
+[34] M. Vassholz, & T. Salditt, Observation of electron-induced characteristic X-ray and
+bremsstrahlung radiation from a waveguide cavity, Sci. Adv. 7 (2021) eabd5677.
+[35] F. R. Elder, A. M. Gurewitsch, R. V. Langmuir, & H. C. Pollock, Radiation from electrons in a
+synchrotron, Phys. Rev. 71 (1947) 829-830.
+[36] A. A. Sokolov, & Ternov, I. M. Synchrotron radiation, Akademia Nauk SSSR (1966).
+[37] Y. Hikosaka, et al., Coherent control in the extreme ultraviolet and attosecond regime by
+synchrotron radiation, Nat. Commun. 10(2019) 4988.
+[38] F. Xie, et al., Vela pulsar wind nebula X-rays are polarized to near the synchrotron
+limit, Nature 612 (2022) 658-660.
+[39] J. Hastings, D. Siddons, U. van Bürck, R. Hollatz, & U. Bergmann, Mössbauer spectroscopy
+using synchrotron radiation, Phys. Rev. Lett. 66 (1991) 770.
+[40] A. Zholents, & M. J. Zolotorev, Femtosecond x-ray pulses of synchrotron radiation, Phys. Rev.
+Lett. 76 (1996) 912.
+[41]P. Cherenkov, Radiation from high-speed particles, Science. 131 (1960) 136-142.
+
+20
+[42] I. M. Frank, & I. Tamm, Coherent visible radiation of fast electrons passing through matter,
+Dokl. Akad. Nauk SSSR 14 (1937) 109-114.
+[43]I. M. Frank, Optics of light sources moving in refractive media, Science 131 (1960) 702-712.
+[44] H. Hu, et al., Nonlocality induced Cherenkov threshold, Laser & Photon. Rev. 14 (2020)
+2000149.
+[45] Liu, S. et al., Surface polariton Cherenkov light radiation source, Phys. Rev. Lett. 109 (2012)
+153902.
+[46] V. L. Ginzburg, & V. N. Tsytovich, Transition Radiation and Transition Scattering, CRC Press,
+1990.
+[47] V. L. Ginzburg, Transition radiation and transition scattering, Physica Scripta 1982 (1982) 182.
+[48] H. Boersch, C. Radeloff, & G. Sauerbrey, Experimental detection of transition radiation, Phys.
+Rev. Lett. 7 (1961) 52.
+[49] P. Karataev, et al., First observation of the point spread function of optical transition radiation,
+Phys. Rev. Lett. 107 (2011) 174801.
+[50]
+P.
+B.
+Glek,
+&
+A.
+M.
+Zheltikov,
+Enhanced
+coherent
+transition
+radiation
+from
+midinfrared-laser-driven microplasmas, Sci. Rep. 12 (2022) 7660.
+[51] I. Kaminer, et al., Quantum Čerenkov radiation: Spectral cutoffs and the role of spin and orbital
+angular momentum, Phys. Rev. X 6 (2016) 011006.
+[52] V. Ginis, J. Danckaert, I. Veretennicoff, & P. Tassin, Controlling Cherenkov radiation with
+transformation-optical metamaterials, Phys. Rev. Lett. 113 (2014) 167402.
+[53] Z. Su, et al., Manipulating Cherenkov radiation and Smith-Purcell radiation by artificial
+structures, Adv. Opt. Mater. 7 (2019) 1801666.
+[54] V. V. Vorobev, & A. V. Tyukhtin, Nondivergent Cherenkov radiation in a wire metamaterial,
+Phys. Rev. Lett. 108 (2012) 184801.
+[55] H. Ren, X. Deng, Y. Zheng, N. An, & X. Chen, Nonlinear Cherenkov radiation in an anomalous
+dispersive medium, Phys. Rev. Lett. 108 (2012) 223901.
+[56] G. A. Beck, et al., e± identification using the drift chambers and transition radiators of the H1
+forward track detector, Nucl. Instrum. Meth. A 367 (1995) 228-232.
+[57] G. E. Graham, et al., Design and test results of a transition radiation detector for a Fermilab
+fixed target rare kaon decay experiment, Nucl. Instrum. Meth. A 367 (1995) 224-227.
+[58] G. Bassompierre, et al., Performance of the NOMAD transition radiation detector, Nucl. Instrum.
+Meth. A 411 (1998) 63-74.
+[59] T. Mahmoud, The ALICE transition radiation detector, Nucl. Instrum. Meth. A 502 (2003)
+127-132.
+[60] G. L. Case, et al., Measurements of Compton scattered transition radiation at high Lorentz
+factors, Nucl. Instrum. Meth. A 524 (2004) 257-263.
+[61] A. Andronic, & J. P. Wessels, Transition radiation detectors, Nucl. Instrum. Meth. A 666 (2012)
+130-147.
+[62] V. Friese, The CBM experiment at GSI/FAIR, Nuclear Physics A 774 (2006) 377-386.
+[63] F. Barbosa, et al., A new Transition radiation detector based on GEM technology, Nucl. Instrum.
+Meth. A 942 (2019) 162356.
+[64] E. Barbarito, et al., A large area transition radiation detector to measure the energy of muons in
+the Gran Sasso underground laboratory, Nucl. Instrum. Meth. A 365 (1995) 214-223.
+[65] K. D. de Vries, & S. Prohira, Coherent transition radiation from the geomagnetically induced
+current in cosmic-ray air showers: Implications for the anomalous events observed by ANITA, Phys.
+
+21
+Rev. Lett. 123 (2019) 091102.
+[66] W. P. Leemans, et al., Observation of terahertz emission from a laser-plasma accelerated
+electron bunch crossing a plasma-vacuum boundary, Phys. Rev. Lett. 91 (2003) 074802.
+[67] G. Q. Liao, et al., Demonstration of coherent terahertz transition radiation from relativistic
+laser-solid interactions, Phys. Rev. Lett. 116 (2016) 205003.
+[68] X. Xu, et al., Generation of terawatt attosecond pulses from relativistic transition radiation, Phys.
+Rev. Lett. 126 (2021) 094801.
+[69] J. Déchard, A. Debayle, X. Davoine, L. Gremillet, & L. Bergé, Terahertz pulse generation in
+underdense relativistic plasmas: From photoionization-induced radiation to coherent transition
+radiation, Phys. Rev. Lett. 120 (2018) 144801.
+[70] G. Adamo, et al., Electron-beam-driven collective-mode metamaterial light source, Phys. Rev.
+Lett. 109 (2012) 217401.
+[71] L. Yi, & T. Fülöp, Coherent diffraction radiation of relativistic terahertz pulses from a
+laser-driven microplasma waveguide, Phys. Rev. Lett. 123 (2019) 094801.
+[72] Y. Shibata, et al., Observation of coherent transition radiation at millimeter and submillimeter
+wavelengths, Phys. Rev. A 45 (1992) R8340.
+[73] A. Murokh, et al., Bunch length measurement of picosecond electron beams from a
+photoinjector using coherent transition radiation, Nucl. Instrum. Meth. A 410 (1998) 452-460.
+[74] F. Braunmueller, M. Martyanov, S. Alberti, & P. Muggli, Novel diagnostic for precise
+measurement of the modulation frequency of seeded self-modulation via coherent transition radiation
+in AWAKE, Nucl. Instrum. Meth. A 909 (2018) 76-79.
+[75] K. Fedorov, et al., Development of longitudinal beam profile monitor based on coherent
+transition radiation effect for CLARA accelerator, Journal of Instrumentation 15(2020) C06008.
+[76] A. H. Lumpkin, et al., First observation of z-dependent electron-beam microbunching using
+coherent transition radiation, Phys. Rev. Lett. 86 (2001) 79.
+[77] M. V. Bashevoy, et al., Generation of traveling surface plasmon waves by free-electron impact,
+Nano letters 6 (2006) 1113-1115.
+[78] X. Lin, et al., Splashing transients of 2D plasmons launched by swift electrons, Sci. Adv. 3
+(2017), e1601192.
+[79] N. Talebi, S. Meuret, S. Guo, et al., Merging transformation optics with electron-driven photon
+sources, Nat. Commun. 10 (2019) 599.
+[80] Y. Yu, et al., Transition radiation in photonic topological crystals: quasiresonant excitation of
+robust edge states by a moving charge, Phys. Rev. Lett. 123 (2019) 057402.
+[81] A. Yurtsever, M. Couillard, & D. A. Muller, Formation of guided Cherenkov radiation in
+silicon-based nanocomposites, Phys. Rev. Lett. 100 (2008) 217402.
+[82] F. J. G. de Abajo, Optical excitations in electron microscopy, Rev. Mod. Phys. 82 (2010) 209.
+[83] C. H. Chen, & J. Silcox, Detection of optical surface guided modes in thin graphite films by
+high-energy electron scattering, Phys. Rev. Lett. 35 (1975), 389.
+[84] J. Chen, H. Chen, & X. Lin, Photonic and plasmonic transition radiation from graphene, Journal
+of Optics 23 (2021) 034001.
+[85] Y. Zhong, et al., Toggling near-field directionality via polarization control of surface waves.
+Laser & Photon. Rev. 15 (2021) 2000388.
+[86] Y. Jiang, et al., Group-velocity-controlled and gate-tunable directional excitation of polaritons in
+graphene-boron nitride heterostructures. Laser & Photon. Rev. 12 (2018) 1800049.
+
+22
+[87] M. Kadic, G. W. Milton, M. van Hecke, & M. Wegener, 3D metamaterials, Nat. Rev. Phys. 1
+(2019) 198-210.
+[88] A. M. Urbas, et al., Roadmap on optical metamaterials, Journal of Optics 18 (2016) 093005.
+[89] O. Quevedo Teruel, et al., Roadmap on metasurfaces, Journal of Optics 21(2019) 073002.
+[90] K. S. Novoselov, L. Colombo, P. R. Gellert, M. G. Schwab, & K. Kim, A roadmap for graphene,
+Nature 490 (2012) 192-200.
+[91] P. Goldsmith & J. V. Jelley, Optical transition radiation from protons entering metal surfaces,
+Philosophical Magazine, 4 (1959) 836-844.
+[92] G. M. Garibyan, Phenomenological quantum electrodynamics in the case of two media, Zhur.
+Eksptl'. i Teoret. Fiz. 39 (1960).
+[93] R. A. Ferrell, Predicted radiation of plasma oscillations in metal films, Phys. Rev. 111 (1958)
+1214-1222.
+[94] W. Steinmann, Experimental verification of radiation of plasma oscillations in thin silver films,
+Phys. Rev. Lett. 5 (1960) 470-472.
+[95] R. W. Brown, P. Wessel, & E. P. Trounson, Plasmon reradiation from silver films, Phys. Rev.
+Lett. 5 (1960) 472-473.
+[96] V. P. Silin, & E. P. Fetisov, Interpretation of the electromagnetic radiation from electron passage
+through metal films, Phys. Rev. Lett. 7 (1961) 374-377.
+[97] E. A. Stern, Transition radiation from metal films, Phys. Rev. Lett. 8 (1962) 7-10.
+[98] E. N. Economou, Surface plasmons in thin films, Phys. Rev. 182 (1969) 539-554.
+[99] X. Lin, et al., All-angle negative refraction of highly squeezed plasmon and phonon polaritons in
+graphene-boron nitride heterostructures, Proceedings of the National Academy of Sciences USA 114
+(2017) 6717-6721.
+[100] X. Zhang, et al., Confined transverse-electric graphene plasmons in negative refractive-index
+systems, npj 2D Materials and Applications 4 (2020) 25.
+[101] M. Chen, et al., Configurable phonon polaritons in twisted α-MoO3, Nat. Mater. 19(2020)
+1307-1311 (2020).
+[102] N. Wu, et al., Tunable high-Q plasmonic metasurface with multiple surface lattice resonances,
+Progress in Electromagnetics Research 172 (2021) 23-32.
+[103] L. Liu, & Z. Li, Spoof surface plasmons arising from corrugated metal surface to structural
+dispersion waveguide, Progress in Electromagnetics Research 173 (2022) 93-127.
+[104] T. Low, et al., Polaritons in layered two-dimensional materials, Nat. Mater. 16 (2017) 182-194.
+[105] D. N. Basov, M. M. Fogler, & G. de Abajo, Polaritons in van der Waals materials, Science 354
+(2016) aag1992.
+[106] N. Yamamoto, K. Araya, A. Toda, & H. Sugiyama, Light emission from surfaces, thin films
+and particles induced by high-energy electron beam, Surf. Interface Anal. 31 (2001) 79-86.
+[107] F. Tay, et al., Anomalous free-electron radiation beyond the conventional formation time,
+arXiv (2022) 2211.14377.
+[108] J. Chen, el. al., Low-velocity-favored transition radiation, arXiv (2022) 2212.13066.
+[109] S. Xu, et al., Broadband surface-wave transformation cloak, Proceedings of the National
+Academy of Sciences 112 (2015) 7635-7638.
+[110] Yao, D. et al. Miniaturized photonic and microwave integrated circuits based on surface
+plasmon polaritons. Progress in Electromagnetics Research 175, 105-125 (2022).
+[111] Z. Guo, C. Lu, X. Lin, & X. Ni, Optical hyperbolic metamaterials, Frontiers in Materials 9
+
+23
+(2022) 1115744.
+[112] Z. Chen, el. al., Wide-angle giant photonic spin Hall effect, Phys. Rev. B 106 (2022) 075409.
+[113] X. Lin and B. Zhang, Normal Doppler frequency shift in negative refractive-index systems,
+Laser & Photon. Rev. 13 (2019) 1900081.
+[114] W. J. Padilla, & R. D. Averitt, Imaging with metamaterials, Nat. Rev. Phys. 4 (2022) 85-100.
+[115] C. Wang, et al., Enhancing directivity of terahertz photoconductive antennas using spoof
+surface plasmon structure, New Journal of Physics 24.7 (2022) 073046.
+[116] L. Zhang, et al., Acoustic non-Hermitian skin effect from twisted winding topology, Nat.
+Commun. 12.1 (2021) 6297.
+[117] H. Chen, et al., Left-handed materials composed of only S-shaped resonators, Phys. Rev. E 70
+(2004) 057605.
+[118] Z. Fan, et al., Homeostatic neuro-metasurfaces for dynamic wireless channel management, Sci.
+Adv. 8 (2022) eabn7905.
+[119] H. Huang, et al., Millimeter-wave wideband high effeciency circular airy OAM multibeams
+with multiplexing OAM modes based on transmission metasurfaces, Progress in Electromagnetics
+Research, 173 (2022), 151-159.
+[120] Y. Jiang et al., Directional polaritonic excitation of circular, Huygens and Janus dipoles in
+graphene-hexagonal boron nitride heterostructures, Progress in Electromagnetics Research 170 (2021)
+169-176.
+[121] S. Dai, et al. Tunable phonon polaritons in atomically thin van der Waals crystals of boron
+nitride, Science 343 (2014) 1125-1129.
+[122] A. J. Giles, et al. Ultralow-loss polaritons in isotopically pure boron nitride, Nat. Mater. 17
+(2018) 134-139.
+[123] H. Hu, X. Lin, and Y. Luo, Free-electron radiation engineering via structured environments,
+Progress in Electromagnetics Research, 171 (2021) 75-88.
+[124] C. Wang, et al., Superscattering of light in refractive-index near-zero environments, Progress in
+Electromagnetics Research 168 (2020) 15-23.
+[125] S. Huang, et al., From anomalous to normal: temperature dependence of the band gap in
+two-dimensional black phosphorus, Phys. Rev. Lett. 125 (2020) 156802.
+[126] S. Huang, et al., Layer-dependent pressure effect on the electronic structure of 2D black
+phosphorus, Phys. Rev. Lett. 127 (2021) 186401.
+[127] Wang, F., Wang, C., Chaves, A. et al. Prediction of hyperbolic exciton-polaritons in monolayer
+black phosphorus, Nat Commun. 12, 5628 (2021).
+[128] X. Lin, et al., Ab initio study of electronic and optical behavior of two-dimensional silicon
+carbide, Journal of Materials Chemistry C 1 (2013) 2131-2135.
+[129] J. Cao, et al., Tamm states and gap topological numbers in photonic crystals, Progress in
+Electromagnetics Research 173 (2022) 141-149.
+[130]
+Y.
+Akahane,
+et
+al.,
+High-Q
+photonic
+nanocavity
+in
+a
+two-dimensional
+photonic
+crystal, Nature 425 (2003) 944-947.
+[131] J. D. Joannopoulos, et al., Molding the flow of light, Princeton Univ. Press, Princeton, NJ
+(2008).
+[132] B. Xie, et al. Higher-order quantum spin Hall effect in a photonic crystal, Nat. Commun. 11
+(2020) 3768.
+
+24
+[133] X. D. Chen, et al. Direct observation of corner states in second-order topological photonic
+crystal slabs, Phys. Rev. Lett. 122 (2019) 233902.
+[134] S. S. Sunku, et al. Photonic crystals for nano-light in moiré graphene superlattices, Science 362
+(2018) 1153-1156.
+[135] K. Chen, et al. Graphene photonic crystal fibre with strong and tunable light-matter interaction.
+Nat. Photon. 13 (2019) 754-759.
+[136]
+S.
+N.
+Galyamin,
+A.
+V.
+Tyukhtin,
+A.
+Kanareykin,
+&
+P.
+Schoessow,
+Reversed
+Cherenkov-transition radiation by a charge crossing a left-handed medium boundary, Phys. Rev. Lett.
+103 (2009) 194802.
+[137] A. Yanai, & U. Levy, Radiation of a uniformly moving line charge in a zero-index
+metamaterial and other periodic media, Optics Express 20 (2012) 18515-18524.
+[138] V. M. Sukharev, M. N. Strikhanov, & A. A. Tishchenko, Transition radiation from graphene, J.
+Phys.: Conf. Ser. 357 (2012) 012015.
+[139] G. de Abajo, F. J. Multiple excitation of confined graphene plasmons by single free electrons,
+ACS Nano 7 (2013) 11409-11419.
+[140] K. C. Zhang, et al., Transition radiation from graphene plasmons by a bunch beam in the
+terahertz regime, Optics Express 25 (2017) 20477-20485.
+[141] K. Akbari, S. Segui, Z. L. Mišković, J. L. Gervasoni, & N. R. Arista, Energy losses and
+transition radiation in graphene traversed by a fast charged particle under oblique incidence, Phys.
+Rev. B 98 (2018) 195410.
+[142] K. Akbari, Z. L. Miskovic, S. Segui, J. L. Gervasoni, & N. R. Arista, Energy losses and
+transition radiation in multilayer graphene traversed by a fast charged particle, ACS Photon. 4 (2017)
+1980-1992.
+[143] A. E. Kaplan, C. T. Law, & P. L. Shkolnikov, X-ray narrow-line transition radiation source
+based on low-energy electron beams traversing a multilayer nanostructure, Phys. Rev. E 52 (1995)
+6795.
+[144] B. Pardo, & J. M. Andre, Transition radiation from periodic stratified structures, Phys. Rev. A
+40 (1989) 1918.
+[145] B. Lastdrager, A. Tip, & J. Verhoeven, Theory of Čerenkov and transition radiation from
+layered structures, Phys. Rev. E 61(2000) 5767.
+[146] B. Pardo, & J. M. André, Classical theory of resonant transition radiation in multilayer
+structures, Phys. Rev. E 63 (2000) 016613.
+[147] K. Yamada, T. Hosokawa, & H. Takenaka, Observation of soft x rays of single-mode resonant
+transition radiation from a multilayer target with a submicrometer period, Phys. Rev. A 59 (1999)
+3673.
+[148] K. F. Casey, & C. Yeh, Transition radiation in a periodically stratified plasma, Phys. Rev. A 2
+(1970) 810.
+[149] C. Luo, M. Ibanescu, S. G. Johnson, & J. D. Joannopoulos, Cerenkov radiation in photonic
+crystals, Science 299 (2003) 368-371.
+[150] X. Lin, et al., Controlling Cherenkov angles with resonance transition radiation, Nat. Phys. 14
+(2018) 816.
+[151] W. Galbraith, & J. V. Jelley, Light pulses from the night sky associated with cosmic rays,
+Nature 171, 349-350 (1953).
+[152] T. Ypsilantis, & J. Seguinot, Theory of ring imaging Cherenkov counters, Nucl. Instrum. Meth.
+
+25
+A 343 (1994) 30-51.
+[153] E. Nappi, Aerogel and its applications to RICH detectors, Nuclear Physics B (Proc. Suppl.) 6
+(1998) 270-276.
+[154] A. Abashian, et al., The Belle detector, Nucl. Instrum, Meth. A 478 (2002) 117-232.
+[155] M. Adinolfi, et al., Performance of the LHCb RICH at the LHC, Eur. Phys. J. C. 73 (2013),
+2431.
+[156] O. Chamberlain, E. Segrè, C. Wiegand, & T. Ypsilantis, Observation of antiprotons, Phys. Rev.
+100 (1955) 947-950.
+[157] J. J. Aubert, et al., Experimental observation of a heavy particle J, Phys. Rev. Lett. 33 (1974)
+1404-1406.
+[158] J. E. Augustin et al., Discovery of a narrow resonance in e+e- annihilation, Phys. Rev. Lett. 33
+(1974) 1406.
+[159] IceCube Collaboration, Evidence for high-energy extraterrestrial neutrinos at the IceCube
+detector, Science 342 (2013) 1242856.
+[160] G. M. Garibyan, Contribution to the theory of transition radiation, JETP (USSR) 33 (1957)
+1403.
+[161] G. M. Garibyan, Transition radiation effects in particle energy losses, JETP (USSR) 37 (1959)
+527-533.
+[162] D. Haidt, The discovery of the weak neutral currents, CERN Courier, 44 (2004) 8, 21.
+[163] J. Heilprin, Higgs Boson discovery confirmed after physicists review Large Hadron Collider
+data at CERN, Huffington Post 14 (2013) 03-13.
+[164] U. Happek, A. J. Sievers, and E. B. Blum, Observation of coherent transition radiation, Phys.
+Rev. Lett. 67 (1991) 2962.
+[165] L. C. L. Yuan, C. L. Wang, H. Uto, & S. Prünster, Formation-zone effect in transition radiation
+due to ultrarelativistic particles, Phys. Rev. Lett. 25 (1970) 1513.
+[166] M. L. Ter Mikaelian, High energy electromagnetic processes in condensed media, Wiley,
+1972.
+[167] L. D. Landau, & I. Pomeranchuk, The limits of applicability of the theory of bremsstrahlung by
+electrons and of the creation of pairs at large energies, Dokl. Akad. Nauk SSSR 92 (1953) 535.
+[168] L. D. Landau, & I. Pomeranchuk, Electron cascade process at very high-energies, Dokl. Akad.
+Nauk Ser. Fiz 92 (1953) 735-738.
+[169] V. L. Ginzburg, & V. N. Tsytovich, Transition scattering, Sov. Phys. JETP 38 (1974) 909.
+[170] B. Dolgoshein, Transition radiation detectors, Nucl. Instrum. Methods. Phys. Res. A 326 (1993)
+434-469.
+[171] L. Wartski, S. Roland, J. Lasalle, M. Bolore, & G. Filippi, Interference phenomenon in optical
+transition radiation and its application to particle beam diagnostics and multiple‐scattering
+measurements, Journal of Applied Physics 46 (1975) 3644-3653.
+[172] X. Zhang et al., High-efficiency threshold-less Cherenkov radiation generation by a graphene
+hyperbolic grating in the terahertz band, Carbon 183 (2021) 225-231.
+[173] S. Gong, et al., Direction controllable inverse transition radiation from the spatial dispersion in
+a graphene-dielectric stack, Photon. Res. 7(2019) 1154-1160.
+[174] G. Dobrik, et al., Large-area nanoengineering of graphene corrugations for visible-frequency
+graphene plasmons, Nat. Nanotechnol. 17 (2022) 61-66.
+[175] L. Cui, J. Wang, & M. Sun, Graphene plasmon for optoelectronics, Reviews in Physics 6 (2021)
+
+26
+100054.
+[176] R. H. Ritchie. Plasma losses by fast electrons in thin films, Phys. Rev. 106 (1957) 874.
+[177] C. Qian, et al., Breaking the fundamental scattering limit with gain metasurfaces, Nat.
+Commun. 13 (2022) 4383.
+[178] D. Ye, K. Chang, L. Ran, & H. Xin, Microwave gain medium with negative refractive index,
+Nat. Commun. 5 (2014) 5841.
+[179] W. Xu, W. J. Padilla, and S. Sonkusale, Loss compensation in metamaterials through
+embedding of active transistor based negative differential resistance circuits, Opt. Express 20 (2012)
+22406-22411.
+[180] T. Jiang, K. Chang, L. M. Si, L. Ran, & H. Xin, Active microwave negative-index
+metamaterial transmission line with gain, Phys. Rev. Lett. 107 (2011) 205503.
+[181] I. de Leon, & P. Berini, Amplification of long-range surface plasmons by a dipolar gain
+medium, Nat. Photon. 4 (2010) 382-387.
+[182] M. Gather, et al., Net optical gain in a plasmonic waveguide embedded in a fluorescent
+polymer, Nat. Photon. 4 (2010) 457-461.
+[183] Y. Yang, et al., Radiative Anti-parity-time plasmonics, Nat. Commun. 13 (2022) 7678.
+[184] X. Lin, et al., Loss induced amplification of graphene plasmons, Optics Letters 41 (2016)
+681-684.
+[185] Ş. K. Özdemir, et al., Parity-time symmetry and exceptional points in photonics, Nat. Mater. 18
+(2019) 783-798.
+[186] Wu, Y. et al. Observation of parity-time symmetry breaking in a single-spin system, Science
+364 (2019) 878-880.
+[187] Cao, W. et al. Fully integrated parity-time-symmetric electronics, Nat. Nanotech. 17 (2022)
+262-268.
+[188] S. Dai, et al. Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial, Nat.
+Nanotech. 10 (2015) 682-686.
+[189] S. Dai, et al. Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic
+material, Nat. Commun. 6 (2015) 6963.
+[190] C. Qian, et al. Multifrequency superscattering from subwavelength hyperbolic structures, ACS
+Photon. 5 (2018) 1506-1511.
+[191] J. Jiang, X. Lin, & B. Zhang, Broadband negative refraction of highly squeezed hyperbolic
+polaritons in 2D materials, Research (2018) 2532819.
+[192] N. C. Passler, et al., Hyperbolic shear polaritons in low-symmetry crystals, Nature 602 (2022)
+595-600.
+[193] A. Poddubny, I. Iorsh, P. Belov, & Y. Kivshar, Hyperbolic metamaterials, Nat. Photon. 7
+(2013) 948-957.
+[194] A. Nemilentsau, T. Low, & G. Hanson, Anisotropic 2D materials for tunable hyperbolic
+plasmonics, Phys. Rev. Lett. 116 (2016) 066804.
+[195] E. E. Narimanov, & A. V. Kildishev, Naturally hyperbolic, Nat. Photon. 9 (2015) 214-216.
+[196] V. M. García Chocano, J. Christensen, & J. Sánchez-Dehesa, Negative refraction and energy
+funneling by hyperbolic materials: An experimental demonstration in acoustics, Phys. Rev. Lett. 112
+(2014) 144301.
+[197] T. Galfsky, J. Gua, E. E. Narimanov, and V. Menon. Photonic hypercrystals for control of
+light-matter interactions, Proc. Natl. Acad. Sci. 114 (2017) 5125-5129.
+[198] L. Shen, et al., Broadband enhancement of on-chip singlephoton extraction via tilted
+
+27
+hyperbolic metamaterials, Appl. Phys. Rev. 7 (2020) 021403.
+[199] J. Yang, et al., Near-field excited Archimedean-like tiling patterns in phonon-polaritonic
+crystals, ACS Nano 15 (2021) 9134-9142.
+[200] F. J. Alfaro-Mozaz, et al., Hyperspectral nanoimaging of van der Waals polaritonic crystals,
+Nano Lett. 21 (2021) 7109-7115.
+[201] X. Shi, et al., Superlight inverse Doppler effect, Nat. Phys. 14 (2018) 1001-1005.
+[202] Li, Y. et al., Nonlinear co-generation of graphene plasmons for optoelectronic logic operations,
+Nat. Commun. 13 (2022) 3138.
+[203] X. Lin, et al., Transverse-electric Brewster effect enabled by nonmagnetic two-dimensional
+materials, Phys. Rev. A 94 (2016) 023836.
+[204] X. Lin, et al., Tailoring the energy distribution and loss of 2D plasmons, New Journal of
+Physics 18 (2016) 105007.
+[205] J. Zhang, et al., The roadmap of graphene: from fundamental research to broad applications,
+Adv. Funct. Mater. 32 (2022) 2270232.
+[206] X. Guo, et al., Polaritons in van der Waals heterostructures, Adv. Mater. (2022) 2201856.
+[207] Q. Zhang, et al. Interface nano-optics with van der Waals polaritons, Nature 597 (2021)
+187-195.
+[208] X. R. Wang, et al., Optically transparent microwave shielding hybrid film composited by metal
+mesh and graphene, Progress in Electromagnetics Research 170 (2021) 187-197.
+[209] A. Gottscholl, et al., Spin defects in hBN as promising temperature, pressure and magnetic
+field quantum sensors, Nat. Commun. 12 (2021) 4480.
+[210] P. Huang, et al., Ultra-long carrier lifetime in neutral graphene-hBN van der Waals
+heterostructures under mid-infrared illumination, Nat. Commun. 11 (2020) 863.
+[211] M. Y. Musa, et al., Confined transverse electric phonon polaritons in hexagonal boron nitrides,
+2D Materials 5 (2018) 015018.
+[212] A. Bylinkin, et al., Real-space observation of vibrational strong coupling between propagating
+phonon polaritons and organic molecules, Nat. Photon. 15 (2021) 197-202.
+[213] A. J. Giles, et al., Imaging of anomalous internal reflections of hyperbolic phonon-polaritons in
+hexagonal boron nitride, Nano Letters 16 (2016) 3858-3865.
+[214] N. Li, et al., Direct observation of highly confined phonon polaritons in suspended monolayer
+hexagonal boron nitride, Nat. Mater. 20 (2021) 43-48.
+[215] S. Dai, et al., Phonon polaritons in monolayers of hexagonal boron nitride, Adv. Mater. 31
+(2019) 1806603.
+[216] I. H. Lee, et al., Image polaritons in boron nitride for extreme polariton confinement with low
+losses, Nat. Commun. 11 (2020) 3649.
+[217] S. Dai, et al., Tunable phonon polaritons in atomically thin van der Waals crystals of boron
+nitride. Science 343 (2014) 1125-1129.
+[218] X. Lin et al., Electronic structures of multilayer two-dimensional silicon carbide with oriented
+misalignment, Journal of Materials Chemistry C 3 (2015) 9057-9062.
+[219] M. Papaj, & C. Lewandowski, Plasmonic nonreciprocity driven by band hybridization in moiré
+materials, Phys. Rev. Lett. 125 (2020) 066801.
+[220] L. Zhang, et al., Van der Waals heterostructure polaritons with moiré-induced nonlinearity,
+Nature 591 (2021) 61-65.
+[221] X. Zhang, et al., Emerging chiral optics from chiral interfaces, Phys. Rev. B 103 (2021)
+
+28
+195405.
+[222] X. Lin, et al., Chiral plasmons with twisted atomic bilayers, Phys. Rev. Lett. 125 (2020)
+077401.
+[223] Sheng, L. et al. Exotic photonic spin Hall effect from a chiral interface, Laser & Photonics
+Reviews 16 (2022) 2200534.
+[224] J. Chen, et al., A perspective of twisted photonic structures, Appl. Phys. Lett. 119 (2021)
+240501.
+[225] L. Brey, T. Stauber, T. Slipchenko, & L. Martín Moreno, Plasmonic Dirac cone in twisted
+bilayer graphene, Phys. Rev. Lett. 125 (2020) 256804.
+[226] E. Y. Andrei, et al., The marvels of moiré materials, Nat. Rev. Mat. 6 (2021) 201-206.
+[227] N. C. H. Hesp, et al., Observation of interband collective excitations in twisted bilayer
+graphene, Nat. Phys. 17 (2021) 1162-1168.
+[228] C. Qian, et al., Experimental observation of superscattering, Phys. Rev. Lett. 122, 063901
+(2019).
+[229] T. Han, K. Wen, Z. Xie, & X. Yue, An ultra-thin wideband reflection reduction metasurface
+based on polarization conversion, Progress in Electromagnetics Research 173 (2022) 1-8.
+[230] A. Arbabi, et al., Planar metasurface retroreflector, Nat. Photon. 11 (2017) 415-420.
+[231] N. Yu, et al., Light propagation with phase discontinuities: generalized laws of reflection and
+refraction, Science 334 (2011) 333-337.
+[232] S. Sun, et al. Gradient-index meta-surfaces as a bridge linking propagating waves and surface
+waves, Nat. Mater. 11 (2012) 426-431.
+[222] N. Yu, & F. Capasso, Flat optics with designer metasurfaces, Nat. Mater. 13 (2014) 139-150.
+[234] C. Qian, et al., Performing optical logic operations by a diffractive neural network, Light Sci.
+Appl. 9 (2020) 59.
+[235] Y. Zhong, et al., Optical interface engineering with on-demand magnetic surface conductivities,
+Phys. Rev. B 106 (2022) 035304.
+[236] X. Zhang, et al., A perspective on meta-boundaries, arXiv (2022) 2211.00903.
+[237] X. Li, et al., A memristors‐based dendritic neuron for high‐efficiency spatial‐temporal
+information processing, Adv. Mater. (2022) 2203684.
+[238] E. K. W. Tan, et al., Density modulation of embedded nanoparticles via spatial, temporal, and
+chemical control elements, Adv. Mater. 31 (2019) 1901802.
+[239] B. Liao, et al., Spatial-temporal imaging of anisotropic photocarrier dynamics in black
+phosphorus, Nano Letters 17 (2017), 3675-3680.
+[240] Z. Yu, & S. Fan, Complete optical isolation created by indirect interband photonic transitions,
+Nat. Photon. 3 (2009) 91-94.
+[241] D. Oue, K. Ding, & J. B. Pendry, Čerenkov radiation in vacuum from a superluminal grating,
+Phys. Rev. Res. 4 (2022) 013064.
+[242] P. A. Huidobro, E. Galiffi, S. Guenneau, R. V. Craster, & J. B. Pendry, Fresnel drag in
+space-time-modulated metamaterials, Proceedings of the National Academy of Sciences 116 (2019)
+24943-24948.
+[243] A. Dikopoltsev, et al., Light emission by free electrons in photonic time-crystals, Proceedings
+of the National Academy of Sciences 119 (2022) e2119705119.
+
+29
+Fig. 1 Schematic of different types of free-electron radiation. a, Cherenkov radiation. b,
+Transition radiation. c, Smith-Purcell radiation. d, Bremsstrahlung radiation. e, Synchrotron
+radiation.
+
+electron30
+Fig. 2 Brief history of transition radiation, along with its typical applications.
+
+theory and applications of transtion radiation
+high-energy particledetector
+high-powerradiation source
+beamdiagnosis
+surface wave excitation31
+Fig. 3 High-energy particle detectors based on transition radiation. a, An inorganic
+scintillator-based transition radiation detector at the super proton synchrotron (SPS) of CERN [60]. b,
+Cutaway view of the ATLAS inner barrel detectors [61]. d, Transition radiation detector based on the
+gas electron multipliers (GEM) technology [64], which can improve the detector performance to
+identify multiple particles. d, Design of Cherenkov detectors by using the resonance transition
+radiation [150].
+
+NAlDetecrors
+Pb
+Beam
+S2
+S3
+S1
+RadiatorStacks
+photonic crystal
+5.2m
+2.1m
+Barrel semiconductortracker
+Pixeldetectors
+Barreltransitionradiationtracker
+air
+End-captransitionradiationtracker
+End-capsemiconductortracker
+β=0.532
+Fig. 4 High-power radiation source by exploiting the coherent transition radiation (CTR). a,
+High
+power
+terahertz emission
+from
+a laser-plasma
+accelerated
+electron
+bunch
+[66]. b,
+Demonstration of terahertz emissions with energy of sub-mJ/pulse [67], when the laser-produced
+electron beam passes through the rear dielectric-vacuum interface. c, Coherent transition radiation by
+the wakefield-accelerated electrons to yield a field strength of ~100 GV/m [68]. d, Terawatt
+attosecond pulses from ultraviolet transition radiation [69].
+
+IP@O
+Bolometer
+PD3@-60
+XPD2@45°
+CCD
+PH
+THz
+Si
+Phosphor
+beam
+PD4
+PD1
+Magnet
+Laser bean
+@-75'
+@75°
+ICT
+Electron
+IPstack
+beam
+Jet
+Foil
+54°
+OAP
+Mass-limited
+Metal-PE
+PE
+CTR
+PIR+Laser33
+Fig. 5 Beam diagnosis technology based on transition radiation. a, Schematic of the experimental
+setup to observe the coherent transition radiation at millimeter and submillimeter wavelengths [72]. b,
+Bunch length measurement of picosecond electron beams [73]. c, A waveguide-integrated heterodyne
+diagnostic at the AWAKE (Advanced WAKEfield Experiment) of CERN via the coherent transition
+radiation [74]. d, A longitudinal beam profile monitor based on the coherent transition radiation in
+CLARA (Compact Linear Accelerator for Research and Applications) [75].
+
+a
+ LINAC
+e-beam
+CTR foil
+P
+P5
+P4
+P3
+MI
+e
+collecting lens
+M3
+45°-polarizer
+1
+MI
+1
+2
+6
+detector
+reference
+detector
+90°-polarizer
+P2
+4
+C
+M21
+5
+SPECTROMETER
+translator
+retroreflectors
+c
+output window
+CTR
+LO
+TR target
+Mixer
+TRmirror
+IF
+IF
+Mixer
+e-beam
+concavemirror
+TR
+LO34
+Fig. 6 Excitation of surface waves by transition radiation. a, A free-electron beam is injected into
+a gold surface to create a highly-localized source of surface plasmons [77], which would be further
+scattered into free space after their interaction with the grating. b, Plasmonic splashing from
+transition radiation [78]. c, Femtosecond photon bunches from the coherent scattering of hyperbolic
+surface waves [79], which are excited by the transition radiation. d, Generation of edge waves when
+an electron beam passes through an interface between two different photonic crystals [80].
+
+b
+Light
+E-beam
+Grating
+Gold film
+electron
+Plasmon wave
+e
+X
+hw
+Au35
+Fig.
+7
+Manipulation
+of
+transition
+radiation
+via
+artificially-engineered
+materials
+or
+nanostructures. a, Gain material. b, Titled hyperbolic material. c, Meta-boundary, such as twisted
+bilayer graphene. d, Spatial-temporal material. e, Layered structure with a periodic stacking. F,
+Layered structure with a non-periodic (or random) stacking.
+
+electron
+:(r,t)
+μ(r,t)