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expert
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What effect does reducing the corrugation period have?
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reduces peak surface fields, heating & HOMs
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Summary
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ where the bunch length $l$ is $$ l = \\frac { \\frac { \\pi } { 2 } + \\sqrt { \\mathcal { R } ^ { 2 } - 1 } - 1 } { k _ { n } } . $$ Evaluating the form factor at $k = k _ { n }$ produces $$ | F ( k _ { n } ) | = \\frac { 2 \\mathcal { R } } { \\mathcal { R } ^ { 2 } + \\pi - 2 } . $$ This result leads to the accelerating gradient $E _ { \\mathrm { a c c } }$ scaling with the inverse of the transformer ratio $\\mathcal { R }$ . [1] A. Zholents, S. Baturin, D. Doran, W. Jansma, M. Kasa, A. Nassiri, P. Piot, J. Power, A. Siy, S. Sorsher, K. Suthar, W. Tan, E. Trakhtenberg, G. Waldschmidt, and J. Xu, A compact high repetition rate free-electron laser based on the Advanced Wakefield Accelerator Technology, in Proceedings of the 11th International Particle Accelerator Conference, IPAC-2020, CAEN, France (2020), https:// ipac2020.vrws.de/html/author.htm. [2] A. Zholents et al., A conceptual design of a compact wakefield accelerator for a high repetition rate multi user X-ray Free-Electron Laser Facility, in Proceedings of the 9th International Particle Accelerator Conference, IPAC’18, Vancouver, BC, Canada (JACoW Publishing, Geneva, Switzerland, 2018), pp. 1266–1268, 10.18429/ JACoW-IPAC2018-TUPMF010. [3] G. Voss and T. Weiland, The wake field acceleration mechanism, DESY Technical Report No. DESY-82-074, 1982. [4] R. J. Briggs, T. J. Fessenden, and V. K. Neil, Electron autoacceleration, in Proceedings of the 9th International Conference on the High-Energy Accelerators, Stanford, CA, 1974 (A.E.C., Washington, DC, 1975), p. 278 [5] M. Friedman, Autoacceleration of an Intense Relativistic Electron Beam, Phys. Rev. Lett. 31, 1107 (1973). [6] E. A. Perevedentsev and A. N. Skrinsky, On the use of the intense beams of large proton accelerators to excite the accelerating structure of a linear accelerator, in Proceedings of 6th All-Union Conference Charged Particle Accelerators, Dubna (Institute of Nuclear Physics, Novosibirsk, USSR, 1978), Vol. 2, p. 272; English version is available in Proceedings of the 2nd ICFA Workshop on Possibilities and Limitations of Accelerators and Detectors, Les Diablerets, Switzerland, 1979 (CERN, Geneva, Switzerland,1980), p. 61
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1
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
With the minor radius and frequency selected, the corrugation profile is chosen to maximize the accelerating gradient as well as provide a high repetition rate. The $1 \\mathrm { - m m }$ minor radius of the CWG results in corrugation dimensions in the hundreds of $\\mu \\mathrm { m }$ which presents unique manufacturing challenges. Several fabrication methods have been investigated for constructing the CWG, with electroforming copper on an aluminum mandrel producing the most promising results [14]. Electroforming at these scales requires that neither the corrugation tooth width nor the vacuum gap is made excessively small since this would result in either a flimsy mandrel or a flimsy final structure. A sensible choice is to make the tooth width similar to the vacuum gap, resulting in $\\xi \\approx 0$ , while using the shortest practical corrugation period. The maximum radii and unequal radii geometries have similar characteristics when $\\xi \\approx 0$ and we have selected the maximum radii design for A-STAR. The final corrugation dimensions are shown in Table II and the electromagnetic characteristics of the $\\mathrm { T M } _ { 0 1 }$ and $\\mathbf { H E M } _ { 1 1 }$ modes are given in Table III.
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2
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
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reduces peak surface fields, heating & HOMs
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Summary
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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APPENDIX A: SCALING AND NORMALIZATION Here, we derive the scaling laws for the loss factor $\\kappa$ , group velocity $\\beta _ { g } ,$ and attenuation constant $\\alpha$ . We will assume that $\\sigma$ satisfies the conditions of a good conductor so that the field solutions are independent of conductivity. The time harmonic eigenmode solutions $E$ and $\\pmb { H }$ produced by CST are normalized such that the stored energy $U$ in the unit cell is 1 J and the frequency is $\\omega$ . Uniformly scaling the geometry by a constant $\\hat { \\boldsymbol a }$ while holding the stored energy fixed results in the scaled eigenmode solutions: $$ \\begin{array} { r l } & { E ^ { \\prime } ( x , y , z ) e ^ { j \\omega ^ { \\prime } t } = \\hat { a } ^ { - 3 / 2 } E ( x ^ { \\prime } , y ^ { \\prime } , z ^ { \\prime } ) e ^ { j \\omega _ { \\hat { a } } ^ { t } } } \\\\ & { H ^ { \\prime } ( x , y , z ) e ^ { j \\omega ^ { \\prime } t } = \\hat { a } ^ { - 3 / 2 } H ( x ^ { \\prime } , y ^ { \\prime } , z ^ { \\prime } ) e ^ { j \\omega _ { \\hat { a } } ^ { t } } , } \\end{array}
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1
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ We will now consider the effect of the bunch charge density $q ( s )$ on the accelerating field $E _ { z } ( s )$ in order to understand how $E _ { \\mathrm { a c c } }$ and the peak surface fields depend on $q ( s )$ . To begin, we write $E _ { z , n }$ due to a single mode as a convolution $$ E _ { z , n } ( s ) = \\int _ { - \\infty } ^ { \\infty } q ( s - s ^ { \\prime } ) 2 \\kappa _ { n } \\cos ( k _ { n } s ^ { \\prime } ) \\theta ( s ^ { \\prime } ) d s ^ { \\prime } . $$ Since $q ( s )$ is a real function, $$ E _ { z , n } ( s ) = 2 \\kappa _ { n } \\mathrm { R e } \\Bigg \\{ \\int _ { 0 } ^ { \\infty } q ( s - s ^ { \\prime } ) e ^ { j k _ { n } s ^ { \\prime } } d s ^ { \\prime } \\Bigg \\} .
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augmentation
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ q ( s ) = N \\times { \\left\\{ \\begin{array} { l l } { 1 } & { 0 < s < \\pi / ( 2 k _ { n } ) } \\\\ { k _ { n } s + ( 1 - \\pi / 2 ) } & { \\pi / ( 2 k _ { n } ) < s < l } \\\\ { 0 } & { { \\mathrm { e l s e } } } \\end{array} \\right. } $$ where $s$ is the longitudinal displacement from the head of the bunch, $k _ { n } = \\omega _ { n } / c$ is the wave number of the $\\mathrm { T M } _ { 0 1 }$ mode, $l = ( \\sqrt { \\mathcal { R } ^ { 2 } - 1 } + \\pi / 2 - 1 ) / k _ { n }$ is the bunch length, and $N = 2 k _ { n } q _ { 0 } / ( \\mathcal { R } ^ { 2 } + \\pi - 2 )$ is a normalization constant such that $\\textstyle \\int q ( s ) d s = q _ { 0 }$ is the total charge of the bunch. The accelerating wakefield behind the drive bunch is given by the convolution of the charge density $q ( s )$ with the Green’s function of the structure $h ( s )$ and can be calculated from Eqs. (26) and (27), and Eq. (B3), resulting in the accelerating field shown in Fig. 16 for the A-STAR design.
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augmentation
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ The integrals in $t$ and $t ^ { \\prime }$ produce Dirac delta functions leaving $$ \\begin{array} { l } { \\displaystyle P _ { w } = \\frac { c } { 2 \\pi } \\mathrm { R e } \\Bigg \\{ \\int _ { - \\infty } ^ { \\infty } d \\omega \\int _ { - \\infty } ^ { \\infty } d \\omega _ { 2 } \\int _ { - \\infty } ^ { \\infty } d \\omega _ { 1 } } \\\\ { \\displaystyle \\times I ( \\omega _ { 2 } ) I ( \\omega _ { 1 } ) Z _ { | | } ( \\omega ) \\delta ( \\omega _ { 1 } + \\omega _ { 2 } ) \\delta ( \\omega - \\omega _ { 1 } ) \\Bigg \\} . } \\end{array} $$ Using the sifting property of the delta function to evaluate the integral Eq. (B10) becomes $$ P _ { \\ w } = \\frac { c } { 2 \\pi } \\mathrm { R e } \\Bigg \\{ \\int _ { - \\infty } ^ { \\infty } I ( - \\omega ) I ( \\omega ) Z _ { | | } ( \\omega ) d \\omega \\Bigg \\} .
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augmentation
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Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ Q = \\frac { \\omega U } { P _ { d } } , $$ where $U$ is stored energy and $P _ { d }$ is the power dissipated in the cavity walls. The power dissipation density per unit area is $$ \\frac { d P _ { d } } { d A } = \\frac { 1 } { 2 } \\sqrt { \\frac { \\omega \\mu } { 2 \\sigma } } | { \\cal H } | ^ { 2 } . $$ In the scaled structure, the power dissipation and resulting quality factor become $$ P _ { d } ^ { \\prime } = \\hat { a } ^ { - 3 / 2 } P _ { d } , \\qquad Q ^ { \\prime } = \\hat { a } ^ { 1 / 2 } Q , $$ leading to the scaled attenuation constant from Eq. (9) $$ \\alpha ^ { \\prime } = \\hat { a } ^ { - 3 / 2 } \\alpha . $$ Scaling of the attenuation constant $\\alpha$ with conductivity is accomplished by multiplying $\\alpha$ by $\\sqrt { \\sigma / \\sigma ^ { \\prime } }$ where $\\sigma$ is the conductivity of the unscaled structure and $\\sigma ^ { \\prime }$ is the conductivity of the scaled structure.
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augmentation
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Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ Q _ { \\mathrm { d i s s } } = \\frac { E _ { \\mathrm { a c c } } ^ { 2 } } { 8 \\alpha \\kappa } ( e ^ { - 2 \\alpha L } + 2 \\alpha L - 1 ) . $$ According to Eq. (14), the amount of energy deposited on the CWG wall per unit length reaches a maximum after the electron bunch propagates a distance $z \\gg 1 / \\alpha$ . It is further convenient to approximate the CWG as a smooth cylinder of radius $a$ and elementary area $d S = 2 \\pi a d z$ , leading to the energy dissipation density on the cylinder wall: $$ \\frac { d Q _ { \\mathrm { d i s s } } ( z \\infty ) } { d S } = \\frac { E _ { \\mathrm { a c c } } ^ { 2 } } { 8 \\pi a \\kappa } . $$ Since the undulating wall of the CWG has a larger surface area per unit length than the smooth cylinder, Equation (16) is an upper bound on the average energy dissipation density in the CWG wall. From Eq. (16), we define the upper bound of the average thermal power dissipation density as
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augmentation
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ Here we notice the linear scaling of the energy dissipation with the minor radius, $a$ , which helps smaller diameter structures achieve less heating per pulse and thus higher bunch repetition rates. At a gradient of $E _ { \\mathrm { a c c } } = 9 0 ~ \\mathrm { M V } \\mathrm { m } ^ { - 1 }$ , a minor radius of $a = 1 ~ \\mathrm { m m }$ , and a repetition rate of $f _ { r } = 2 0 ~ \\mathrm { k H z }$ , the minimum theoretical thermal power dissipation density on the wall of the corrugated waveguide is roughly $3 6 ~ \\mathrm { W / c m ^ { 2 } }$ . This is well within the cooling capability of single phase cooling systems using water as a working fluid, see for example [27]. In addition to the steady-state thermal load, the transient heating of the corrugation plays an important role in limiting the attainable accelerating gradient. The transient temperature rise due to pulse heating causes degradation of the surface which eventually leads to nucleation sites where electric breakdown may occur [20]. Acceptable transient temperature rise is cited in the literature as $4 0 \\mathrm { K }$ [20], above which the structure begins to incur damage. The transient $\\Delta T$ at the surface is calculated from a Green’s function solution of the thermal diffusion equation in one dimension with Neumann boundary conditions as [20]:
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augmentation
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Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ E _ { z , n } ( s \\to \\infty ) = 2 \\kappa _ { n } q _ { 0 } \\mathrm { R e } \\{ e ^ { j k _ { n } s } F ( k _ { n } ) \\} $$ Expanding the real part $$ \\begin{array} { r } { E _ { z , n } ( s \\infty ) = 2 \\kappa _ { n } q _ { 0 } [ \\cos ( k _ { n } s ) \\mathrm { R e } \\{ F ( k _ { n } ) \\} } \\\\ { - \\sin ( k _ { n } s ) \\mathrm { I m } \\{ F ( k _ { n } ) \\} ] . \\qquad } \\end{array} $$ Since we are interested in the maximum value of the longitudinal accelerating field, we define $E _ { \\mathrm { a c c } }$ as the amplitude of $E _ { z , n } \\big ( s \\infty \\big )$ : $$ E _ { \\mathrm { a c c } } = 2 \\kappa q _ { 0 } \\sqrt { \\mathrm { R e } \\{ F ( k _ { n } ) \\} ^ { 2 } + \\mathrm { I m } \\{ F ( k _ { n } ) \\} ^ { 2 } }
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augmentation
|
Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ Since the current density $i ( t )$ is a purely real function, $I ( - \\omega ) = I ^ { * } ( \\omega )$ where $*$ denotes complex conjugation, leading to $$ P _ { \\nu } = \\frac { c } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } | I ( \\omega ) | ^ { 2 } \\operatorname { R e } \\{ Z _ { | | } ( \\omega ) \\} d \\omega . $$ Equation (B12) represents the power being converted from kinetic energy to electromagnetic energy in the frequency domain. Considering a single mode denoted by the subscript $n$ , the wake impedance is $$ Z _ { n | | } ( \\omega ) = \\int _ { - \\infty } ^ { \\infty } 2 \\kappa _ { n } \\cos ( \\omega _ { n } t ) \\theta ( t ) e ^ { - j \\omega t } d t . $$ Using the Fourier transform property $$ \\mathcal { F } \\{ f ( t ) \\cos ( a t ) \\} = \\frac { F ( \\omega - a ) + F ( \\omega + a ) } { 2 } ,
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augmentation
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Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
After determining the minor radius, $a$ , of $1 \\ \\mathrm { m m }$ , the frequency and corresponding aperture ratio of the synchronous $\\mathrm { T M } _ { 0 1 }$ accelerating mode must be chosen. We have shown in Figs. 10 and 12 that the peak surface fields and associated pulse heating increase with aperture ratio while the total power dissipation decreases, as shown by its dependence on the loss factor $\\kappa$ in Eq. (16) and Fig. 7. In addition to these considerations, the frequency must be compatible with the electromagnetic output couplers used to extract rf energy from the structure. An important feature of A-STAR is its ability to measure the trajectory of the bunch in the CWA using the $\\mathrm { H E M } _ { 1 1 }$ mode which is excited when the beam propagates off-axis. Due to mode conversion, the design of the coupler that extracts the $\\mathrm { H E M } _ { 1 1 }$ mode becomes increasingly challenging as the $\\mathrm { H E M } _ { 1 1 }$ wavelength shrinks with respect to the fixed aperture of the waveguide. For the $1 \\mathrm { - m m }$ minor radius cylindrical waveguide, the limiting factor in the $\\mathbf { H E M } _ { 1 1 }$ coupler design was converted to the $\\mathrm { T E } _ { 3 1 }$ mode which has a cutoff frequency of $2 0 0 \\ : \\mathrm { G H z }$ . To address this, the synchronous $\\mathbf { H E M } _ { 1 1 }$ mode was chosen to be $1 0 \\mathrm { G H z }$ below the $\\mathrm { T E } _ { 3 1 }$ cutoff frequency, resulting in a 190-GHz $\\mathrm { H E M } _ { 1 1 }$ mode and 180-GHz $\\mathrm { T M } _ { 0 1 }$ mode with an aperture ratio of $a / \\lambda = 0 . 6 0$ .
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augmentation
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Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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DOI: 10.1103/PhysRevAccelBeams.25.121601 I. INTRODUCTION A sub-terahertz accelerator (A-STAR) is being developed at Argonne National Laboratory to reduce the cost and footprint of a future hard x-ray free-electron laser (XFEL) facility [1,2]. A-STAR is a collinear wakefield accelerator (CWA) that uses a cylindrical corrugated waveguide (CWG) as a slow-wave structure, analogous to other CWA configurations [3–8] and drive beam decelerator in CLIC [9]. In operation, a high-charge drive electron bunch passing through the CWA generates an electromagnetic field, known as the wakefield, which accelerates a low charge witness electron bunch following close behind the drive bunch. The ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch is known as the transformer ratio $\\mathcal { R }$ and is limited to 2 for symmetric drive bunches [10]. The A-STAR design uses a 10-nC asymmetrical drive bunch [10,11] to achieve a transformer ratio of 5 and an accelerating gradient of $9 0 \\ \\mathrm { M V } \\mathrm { m } ^ { - 1 }$ , where the accelerating field is a $1 8 0 – \\mathrm { G H z }$ $\\mathrm { T M } _ { 0 1 }$ mode propagating with a group velocity of $0 . 5 7 c$ , where $c$ is the speed of light. The accelerator ends when the drive bunch exhausts almost all of its energy at which point the witness bunch reaches a maximum energy approaching $\\mathcal { E } _ { 0 } ( 1 + \\mathcal { R } )$ , where $\\mathcal { E } _ { 0 }$ is the initial energy of the beam. The entire CWA is composed of many $0 . 5 \\mathrm { - m }$ long modules connected in series, as shown in Fig. 1.
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augmentation
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ \\frac { E _ { \\mathrm { m a x } } ^ { 3 0 } t _ { p } ^ { 5 } } { \\mathrm { B D R } } = \\mathrm { c o n s t . } $$ From a design perspective, reducing the BDR is achieved by reducing the peak surface fields and the pulse length. Calculation of the absolute threshold value of the fields that induce breakdown in sub-THz structures is an active area of research [19,22,23] and reliable models have not yet been developed. The modified Poynting vector introduced in [21] has been used to predict rf breakdown in structures operating up to $3 0 ~ \\mathrm { G H z }$ , but there are limited data for its applicability at higher frequencies. For this reason, the BDR and maximum gradient of the CWA must ultimately be determined experimentally. For the purpose of optimization, we choose the peak surface fields $E _ { \\mathrm { m a x } }$ and $H _ { \\mathrm { m a x } }$ as figures of merit which should be minimized to increase the attainable accelerating gradient. Since some evidence suggests that pulse heating is of fundamental importance to the initiation of rf breakdown in high frequency accelerating structures [24], we give additional weight to the minimization of the peak magnetic surface field. This choice leads to a higher overall thermal efficiency which will be discussed further in Sec. VI.
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augmentation
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Yes
| 0
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expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ \\begin{array} { l } { { \\displaystyle E _ { z } ( s ) = \\int _ { - \\infty } ^ { \\infty } q ( s - s ^ { \\prime } ) h ( s ^ { \\prime } ) d s ^ { \\prime } } } \\\\ { { \\displaystyle h ( s ) = \\sum _ { n = 0 } ^ { \\infty } 2 \\kappa _ { n } \\cos { ( k _ { n } s ) } \\theta ( s ) . } } \\end{array} $$ The wakefield falls off exponentially behind the drive bunch due to ohmic loss in the wall material, leading to an rf pulse power envelope at the end of a half-meter section of A-STAR resembling that in Fig. 17. In the copper structure, the trailing edge of the power envelope is attenuated by $8 5 \\%$ after a half meter. For structures longer than ${ \\sim } 1 / \\alpha$ , the pulse length becomes saturated and is determined by the conductivity of the wall material rather than the length of the CWG, where lower conductivity produces a shorter pulse. Because the witness bunch follows close behind the drive bunch, the loss in accelerating gradient for a lossy wall CWG can be small while the overall thermal load remains unchanged according to Eq. (16). This feature of the CWA can potentially be exploited to allow fabrication from lossy materials or use of surface coatings to improve performance. The reduction in pulse length may be used as a way to increase the rf breakdown threshold of the structure.
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augmentation
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Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
VI. THERMAL LOADING Thermal loading of the corrugated waveguide places a limit on the maximum repetition rate $f _ { r }$ of the accelerator, where $f _ { r }$ is the number of bunches injected into the structure per second. The thermal loading depends on the electromagnetic properties of the $\\mathrm { T M } _ { 0 1 }$ mode as well as the length of the corrugated waveguide and the conductivity of the wall material. Achieving a high repetition rate requires active cooling of the structure as well as an optimally designed corrugation profile. Here we focus on designing a corrugation that minimizes the steady-state thermal load and transient pulse heating. The thermally induced stresses due to temperature gradients in the wall pose additional design considerations which are discussed further in [26]. Because the group velocity $v _ { g }$ of the electromagnetic wave is less than the electron bunch velocity, the length of the rf pulse behind the bunch grows as it traverses the structure. This causes the thermal energy density deposited in the CWG wall to increase along the direction of propagation. At a distance $z$ from the beginning of the CWG, the field strength of the rf pulse induced by the electron bunch entering at time $t = 0$ is
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augmentation
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Yes
| 0
|
expert
|
What effect does reducing the corrugation period have?
|
reduces peak surface fields, heating & HOMs
|
Summary
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
Maintaining the fundamental $\\mathrm { T M } _ { 0 1 }$ and $\\mathrm { H E } _ { 1 1 }$ frequencies within a $\\pm 5$ GHz-bandwidth specified by the design of the output couplers requires dimensional tolerances of roughly $\\pm 1 0 ~ { \\mu \\mathrm { m } } ,$ as shown by Fig. 5. The most sensitive dimension to manufacturing error is the corrugation depth, which must be carefully controlled to produce the desired frequency. Mode conversion due to the straightness of the CWG is not expected to change the acceleration properties over the short length scale between the drive and witness bunch. However, such effects may become relevant in the operation of the output couplers and are a subject of future analysis. Table: Caption: TABLE II. A-STAR key operating parameters. Body: <html><body><table><tr><td colspan="2">Parameter</td></tr><tr><td>a</td><td>1 mm Corrugation minor radius</td></tr><tr><td>d 264 μm</td><td>Corrugation depth</td></tr><tr><td>g 180 μm</td><td>Corrugation vacuum gap</td></tr><tr><td>t 160 μm</td><td>Corrugation tooth width</td></tr><tr><td>80 μm rt.g</td><td>Corrugation corner radius</td></tr><tr><td>P 340 μm</td><td>Corrugation period</td></tr><tr><td>0.06</td><td>Spacing parameter</td></tr><tr><td>L</td><td>50 cm Waveguide module length</td></tr><tr><td>R 5</td><td>Transformer ratio</td></tr><tr><td>|F| 0.382</td><td>Bunch form factor</td></tr><tr><td>q0 10 nC</td><td>Bunch charge</td></tr><tr><td>90 MVm-1 Eacc</td><td>Accelerating gradient</td></tr><tr><td>325 MV m-1 Emax</td><td>Peak surface E field</td></tr><tr><td>610 kA m-1 Hmax</td><td>Peak surface H field</td></tr><tr><td>74°</td><td>Phase advance</td></tr><tr><td>fr 20 kHz</td><td>Repetition rate</td></tr><tr><td>Pdiss 1050 W</td><td>Power dissipation per module</td></tr><tr><td>W 55 W/cm²</td><td>Power density upper bound</td></tr><tr><td>△T 9.5K</td><td>Pulse heating</td></tr></table></body></html>
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augmentation
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Yes
| 0
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Expert
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What electron beam energy was used in the experimental demonstration?
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30 keV
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Fact
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haeusler-et-al-2022-boosting-the-efficiency-of-smith-purcell-radiators-using-nanophotonic-inverse-design.pdf
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allows to design the spectrum $( \\omega )$ , spatial distribution $\\mathbf { \\Pi } ( \\mathbf { r } )$ , and polarization (e) of radiation by favoring one kind $| \\mathbf { e } { \\cdot } \\mathbf { E } ( \\mathbf { r } , \\omega ) |$ and penalizing others, $- | \\mathbf { e } ^ { \\prime } { \\boldsymbol { \\cdot } } \\mathbf { E } ( \\mathbf { r } ^ { \\prime } , \\omega ^ { \\prime } ) |$ , with possibly orthogonal polarization $\\mathbf { e ^ { \\prime } }$ . Lifting the periodicity constraint opens the space to complex metasurfaces, which would for example enable designs for focusing or holograms.18,22‚àí27,40 Future efforts could also target the electron dynamics to achieve (self-)bunching and, hence, coherent enhancement of radiation. In that case, the objective function would aim at the field inside the electron channel rather than the far-field emission. This would favor higher quality factors at the cost of lower out-coupling efficiencies. However, direct inclusion of the electron dynamics through an external multiphysics package proves challenging as our inverse design implementation requires differentiability of the objective function with respect to the design parameters. Instead, one may choose to use an analytical expression for the desired electron trajectory or an approximate form for the desired field pattern.
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1
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Yes
| 0
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Expert
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What electron beam energy was used in the experimental demonstration?
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30 keV
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Fact
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haeusler-et-al-2022-boosting-the-efficiency-of-smith-purcell-radiators-using-nanophotonic-inverse-design.pdf
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$$ { \\bf J } ( { \\bf r } , \\omega ) = \\frac { q } { 2 \\pi } { ( 2 \\pi \\sigma _ { x } ^ { 2 } ) } ^ { - 1 / 2 } \\mathrm { e } ^ { - x ^ { 2 } / 2 \\sigma _ { x } ^ { 2 } } \\mathrm { e } ^ { - i k _ { y } y } \\widehat { { \\bf y } } $$ with $k _ { y } = \\omega / \\nu$ . Using this expression, the electromagnetic field was calculated via Maxwell’s equations for linear, nonmagnetic materials. As this is a 2D problem, the transverse-electric mode $E _ { z }$ decouples from the transverse-magnetic mode $H _ { z } ,$ where only the latter is relevant here. A typical 2D-FDFD simulation took 1 s on a common laptop, and the algorithm needed about 500 iterations to converge to a stable maximum. Simulated Radiation Power. From the simulated electromagnetic field, we calculate the total energy $W$ radiated by a single electron per period $a$ of the grating. In the time domain, this would correspond to integrating the energy flux $\\mathbf { \\boldsymbol { s } } ( \\mathbf { \\boldsymbol { r } } , \\ t )$ through the area surrounding the grating over the time it takes for the particle to pass over one period of the grating. In the frequency domain, one needs to integrate $\\mathbf { \\Delta } \\mathbf { S } ( \\mathbf { r } , \\omega )$ through the area around one period over all positive frequencies, that is,36
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Finally, we note that the inverse design was operated with the beam at the center of the $2 6 0 ~ \\mathrm { n m }$ wide channel, whereas the rectangular grating worked at minimal beam-structure distance for maximal efficiency. The simulations assumed the distance $d = 7 0 \\ \\mathrm { n m } .$ , whereas Figure S3 suggests that the actual distance in the experiment was $5 8 ~ \\mathrm { n m }$ . Would the rectangular grating have been operated at $d = 1 3 0 \\ \\mathrm { n m } _ { \\cdot }$ , the simulations predict that the inverse design could improve peak spectral radiation density and overall radiation power by factors of 96 and 42, respectively. It is interesting to relate the quantum efficiency of our inverse-designed structure to that of other silicon gratings reported elsewhere (Table 1). Roques-Carmes et al.6 state a quantum efficiency of $0 . 1 3 \\%$ in a similar experiment at $\\lambda =$ $1 4 0 0 \\ \\mathrm { n m }$ . Although their interaction length was $1 3 \\times$ longer, and the distance to the grating was with $\\bar { d } = 2 3 ~ \\mathrm { n m }$ just a fifth of ours, the inverse-designed structure surpasses it with its quantum efficiency of $0 . 2 \\dot { 2 } \\%$ .
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File Name:haeusler-et-al-2022-boosting-the-efficiency-of-smith-purcell-radiators-using-nanophotonic-inverse-design.pdf Boosting the Efficiency of Smith−Purcell Radiators Using Nanophotonic Inverse Design Urs Haeusler,\\*,∥ Michael Seidling,\\*,∥ Peyman Yousefi, and Peter Hommelhoff\\* Cite This: ACS Photonics 2022, 9, 664−671 ACCESS 山 Metrics & More 国 Article Recommendations Supporting Information ABSTRACT: The generation of radiation from free electrons passing a grating, known as Smith−Purcell radiation, finds various applications, including nondestructive beam diagnostics and tunable light sources, ranging from terahertz toward X-rays. So far, the gratings used for this purpose have been designed manually, based on human intuition and simple geometric shapes. Here we apply the computer-based technique of nanophotonic inverse design to build a $1 4 0 0 ~ \\mathrm { n m }$ Smith−Purcell radiator for subrelativistic $3 0 \\mathrm { \\ k e V }$ electrons. We demonstrate that the resulting silicon nanostructure radiates with a $3 \\times$ higher efficiency and $2 . 2 \\times$ higher overall power than previously used rectangular gratings. With better fabrication accuracy and for the same electron−structure distance, simulations suggest a superiority by a factor of 96 in peak efficiency. While increasing the efficiency is a key step needed for practical applications of free-electron radiators, inverse design also allows to shape the spectral and spatial emission in ways inaccessible with the human mind.
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3D Simulations. 3D finite-element-method (FEM) frequency-domain simulations were performed in COMSOL to analyze effects originating from the finite height of the structure and beam. The structures were assumed to be $1 . 5 \\mu \\mathrm { m }$ high on a flat silicon substrate (Figure 1b). The spectral current density had a Gaussian beam profile of width $\\sigma = 2 0$ nm: $$ \\mathbf { J } ( \\mathbf { r } , \\omega ) = \\frac { - e } { 2 \\pi } \\big ( 2 \\pi \\sigma ^ { 2 } \\big ) ^ { - 1 } \\mathrm { e } ^ { - \\big ( x ^ { 2 } + z ^ { 2 } \\big ) / 2 \\sigma ^ { 2 } } \\mathrm { e } ^ { - i k _ { y } y } \\widehat { \\mathbf { y } } $$ Experimental Setup. The experiment was performed within an FEI/Philips XL30 SEM providing an $1 1 \\mathrm { \\ n A }$ electron beam with $3 0 \\mathrm { \\ k e V }$ mean electron energy. The structure was mounted to an electron optical bench with full translational and rotational control. The generated photons were collected with a microfocus objective SchaÃàfter+Kirchhoff 5M-A4.0-00-STi with a numerical aperture of 0.58 and a working distance of $1 . 6 ~ \\mathrm { m m }$ . The objective can be moved relative to the structure with five piezoelectric motors for the three translation axes and the two rotation axes transverse to the collection direction. The front lens of the objective was shielded with a fine metal grid to avoid charging with secondary electrons in the SEM, which would otherwise deflect the electron beam, reducing its quality. The collected photons were focused with a collimator into a ${ 3 0 0 } { - } \\mu \\mathrm { m }$ -core multimode fiber guiding the photons outside the SEM, where they were detected with a NIRQuest $^ { \\cdot + }$ spectrometer.
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A 200-period-long version of the inverse-designed structure was fabricated by electron beam lithography $\\bar { ( } 1 0 0 \\ \\mathrm { k V } )$ and cryogenic reactive-ion etching of $1 - 5 \\Omega \\cdot \\mathrm { c m }$ phosphorus-doped silicon to a depth of $1 . 3 \\big ( 1 \\big ) \\mathsf { \\bar { \\mu } m }$ .35 The surrounding substrate was etched away to form a $5 0 ~ \\mu \\mathrm { m }$ high mesa (Figure 2a). We note that unlike in most previous works the etching direction is here perpendicular to the radiation emission, enabling the realization of complex 2D geometries. The radiation generation experiment was performed inside a scanning electron microscope (SEM) with an 11 nA beam of $3 0 \\mathrm { \\ k e V }$ electrons. The generated photons were collected with an objective (NA 0.58), guided out of the vacuum chamber via a $3 0 0 ~ \\mu \\mathrm { m }$ core multimode fiber and detected with a spectrometer (Figure 2b and Methods). RESULTS We compare the emission characteristics of the inversedesigned structure to two other designs: First, a rectangular 1D grating with groove width and depth of half the periodicity $a _ { \\mathrm { { ; } } }$ , similar to the one used in refs 6 and 36. And second, a dual pillar structure with two rows of pillars, $\\pi$ -phase shifted with respect to each other, and with a DBR on the back. This design was successfully used in dielectric laser acceleration, the inverse effect of SPR. $\\mathbf { \\lambda } ^ { 3 0 , 3 2 , 3 3 , 3 5 , 3 7 - } 3 9$ It further represents the manmade design closest to our result of a computer-based optimization.
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KEYWORDS: light‚àímatter interaction, free-electron light sources, Smith‚àíPurcell radiation, inverse design, nanophotonics T ehle Smith‚àíPurcell effect describes the emission of ctromagnetic radiation from a charged particle propagating freely near a periodic structure. The wavelength $\\lambda$ of the far-field radiation follows1 $$ \\lambda = \\frac { a } { m } ( \\beta ^ { - 1 } - \\cos { \\theta } ) $$ where $a$ is the periodicity of the structure, $\\beta = \\nu / c$ is the velocity of the particle, $\\theta$ is the angle of emission with respect to the particle propagation direction, and $m$ is the integer diffraction order. The absence of a lower bound on the electron velocity in eq 1 makes Smith‚àíPurcell radiation (SPR) an interesting candidate for an integrated, tunable free-electron light source in the low-energy regime.2‚àí8 While the power efficiency of this process is still several orders of magnitude smaller than conventional light sources, it can be enhanced by super-radiant emission from coherent electrons.9 For this, prebunching of the electrons is a possible avenue ,10‚àí13 but also self-bunching due to the interaction with the excited nearfield of the grating is observed above a certain current threshold.14‚àí16 The use of coherent electrons is particularly interesting in combination with resonant structures, such as near bound states in the continuum.5,17
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haeusler-et-al-2022-boosting-the-efficiency-of-smith-purcell-radiators-using-nanophotonic-inverse-design.pdf
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DESIGN The inverse design optimization was carried out via an opensource Python package34 based on a 2D frequency-domain (FD) simulation. At the center of the optimization process is the objective function $G$ , which formulates the desired performance of the design, defined by the design variable $\\phi$ (Methods). Here, we aimed for maximum radiation in negative $x$ -direction at the design angular frequency $\\omega$ corresponding to $\\lambda = 1 . 4 \\mu \\mathrm { m }$ (Figure 1). To this end, the Poynting vector S was numerically measured in the far field of the structure and integrated over one period $a$ , giving the objective function $$ G ( \\phi ) = - \\int _ { 0 } ^ { a } \\mathrm { d } y \\ S _ { x } ( x _ { \\mathrm { f a r f i e l d } } , y ) $$ The resulting design is depicted in Figure 1a and reveals two gratings on each side of the vacuum channel, which are similar in shape but $\\pi$ -phase shifted with respect to each other. The back of the double-sided grating results in a structure that resembles a distributed Bragg reflector (DBR). This way, the radiation to the left is $4 6 9 \\times$ higher than to the right.
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Figure 2c shows the photon count rate as a function of the electron beam position. Maximum photon count rate is observed when focusing the beam into the channel of the inverse design structure at medium height. The spatial confinement in vertical direction points at the presence of a confined mode, as found in cavities. By contrast, Figure S1 reveals a nonresonating nature of the other two structures, with only slight dependence on the beam height. The efficiency of a design is quantified by comparing three different figures of merit: the peak spectral radiation density $\\mathrm { { ( p W / n m ) } }$ , the total radiation $\\left( \\mathrm { p W } \\right)$ , and the quantum efficiency $( \\% )$ , defined as the number of photons generated per electron. All three quantities are determined in the experimentally accessible window, which is limited by the numerical aperture of the fiber. Its angular acceptance window acts as an effective spectral filter with a Gaussian shape centered around $1 4 0 0 ~ \\mathrm { n m }$ and a full width at half maximum of $1 7 5 \\ \\mathrm { n m }$ (Figure 3 and Methods).
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The measurements in Figure 3a show overall a similar performance of the inverse-designed structure and the dual pillar structure. In terms of overall power, the inverse design is with $2 1 . 9 ~ \\mathrm { p W }$ around $12 \\%$ weaker than the dual pillars. This can be understood by the larger channel width of $2 6 0 \\ \\mathrm { n m }$ compared to the $1 8 0 ~ \\mathrm { n m }$ of the dual pillars (Figure 1a). By contrast, the rectangular grating is single sided, and the beam was steered as closely as possible to the grating to yield maximum radiation. Even then, the inverse design radiated 2.2(1)-times as strong as the rectangular grating. The superiority becomes even more pronounced when looking at the peak spectral radiation density. The inverse design reaches $0 . 1 6 \\mathrm { \\ p W / n m }$ at $1 3 8 5 \\mathrm { n m }$ , which is 3.0(1)-times as high as that of the rectangular grating. It also surpasses marginally the dual pillar peak efficiency. This is a first indicator for the narrowband emission of the inverse design, in contrast to the broadband emission of the other two designs (Figure 1c).
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For a further study of the different structures, we performed 2D time-domain and 3D frequency-domain simulations. While both time and frequency domain are in principal legitimate ways to calculate the radiation spectrum from single electrons, they differ in computational complexity and precession. The time-domain simulation (Figure 3b and Videos S1, S2, and S3) can capture the instantaneous response to a structure of finite length. This is computationally expensive because the field of the entire grating needs to be calculated at each point in time. The frequency-domain simulation (Figure 3c), on the other hand, calculates the radiation density at each frequency of the spectrum. This is computationally less complex because it is sufficient to consider a single unit cell with periodic boundaries, which allowed us to perform 3D simulations. It can therefore take into account the limited height of the electron beam and the structure, which is on the order of the wavelength. This is particularly relevant here, because the inverse design yielded a double-sided grating that forms a resonator. The mirrors of the resonator are plane parallel and therefore do not form a stable resonator. Both the 2D time-domain and 3D frequency-domain simulations show similar results. For the inverse design, they predict a total radiation of 108(14) pW, a quantum efficiency of $1 . 1 ( 2 ) \\% ,$ and a peak spectral radiation density of 1.8(2) $\\mathrm { { \\ p W / n m } }$ . In terms of total power, this corresponds to an increase by $8 0 \\%$ compared to the dual pillar design and a colossal boost of $9 8 0 \\%$ with respect to the rectangular grating. The contrast in terms of peak efficiency within the experimentally accessible range from 1200 to $1 6 0 0 ~ \\mathrm { { n m } }$ is even more drastic. It reaches an increase by $2 9 0 \\%$ compared to the dual pillars and $1 6 5 0 \\%$ relative to the rectangular grating.
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DISCUSSION Comparing the measured emission spectrum of the inverse design to its simulated profile shows that the observed emission was not as powerful and spectrally broader. We identify two causes: First, the electron beam current deteriorates as the beam diverges, where electrons hit the boundaries of the channel and are lost. By measuring the current after the structure, we determined an effective current $I _ { \\mathrm { e f f } }$ for each design (Figure 3e and Figure S2). The effective current is smallest for the dual pillar design, which has the narrowest channel, and largest for the single-sided rectangular grating. Another factor that reduces the efficiency of the inversedesigned structure are the deviations of the fabricated structure from its design. Figure 4 shows that the structure was not perfectly vertically etched but has slightly conical features. This leads to a reduction of the quality factor of the inversedesigned structure, which is reflected in a less powerful $( - 6 7 \\% )$ and more broadband emission of radiation. By contrast, the efficiencies of the dual pillar structure and the rectangular grating are expected to be less affected by conical features due to their lack of pronounced resonance.
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What electron beam energy was used in the experimental demonstration?
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METHODS Inverse Design. The inverse design optimization was carried out via an open-source Python package34 based on a 2D finite-difference frequency-domain (FDFD) simulation at the design angular frequency $\\omega$ corresponding to $\\lambda = 1 . 4 \\mu \\mathrm { m }$ . The simulation cell used for this purpose is presented in Figure 5. The design $\\varepsilon _ { \\mathrm { r } } ( \\phi )$ was parametrized with the variable $\\phi ( \\mathbf { r } )$ . Sharp features $\\left( < 1 0 0 \\ \\mathrm { \\ n m } \\right)$ in the design were avoided by convolving $\\phi ( \\mathbf { r } )$ with a 2D circular kernel of uniform weight. Afterward the convolved design $\\tilde { \\phi }$ was projected onto a sigmoid function of the form tanh $( \\gamma \\tilde { \\phi } )$ . This results in a closeto-binary design where the relative permittivity $\\varepsilon _ { \\mathrm { { r } } } ( { \\bf { r } } )$ only takes the values of silicon $\\left( \\varepsilon _ { \\mathrm { r } } = 1 2 . 2 \\right) ^ { 4 \\mathrm { f } }$ or vacuum $\\left( \\varepsilon _ { \\mathrm { r } } \\ = \\ 1 \\right)$ ). We observed good results by starting the optimization with small values $\\gamma = 2 0$ and slowly increasing $\\gamma$ to 1000.
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To improve the convergence of the algorithm, we enforced mirror symmetry in $x$ -direction onto the design. This reflects the symmetry of SPR under $\\theta = 9 0 ^ { \\circ }$ and reduces the parameter space by a factor of 2. Furthermore, we observed improved convergence when starting with a large grid spacing $\\left( 1 0 \\ \\mathrm { n m } \\right)$ , which is then slowly reduced to $3 \\ \\mathrm { n m }$ as the optimization progresses. 2D Simulations. The source term of our 2D simulations is given by the current density of a line charge with density $q =$ $- e / \\Delta z$ traversing the structure with velocity $\\nu$ along $\\hat { \\mathbf { y } }$ . The choice of the length $\\Delta z$ is crucial to obtain meaningful intensities from a 2D simulation.36 By choosing $\\Delta z = 1 \\mathrm { \\ ' } \\mu \\mathrm { m }$ throughout, we obtained 2D results that were on average only $1 4 \\%$ off the 3D values. In the transverse direction, we assumed a Gaussian charge distribution of width $\\sigma _ { x } = 2 0 ~ \\mathrm { n m }$ such that the spectral current density reads
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$$ W = \\int _ { S } \\mathrm { d } \\mathbf { A \\cdot } \\int _ { 0 } ^ { \\infty } \\mathrm { d } \\omega ~ \\mathbf { S } ( \\mathbf { r } , \\omega ) , $$ $$ { \\bf S } ( { \\bf r } , \\omega ) = 4 { \\cdot } 2 \\pi \\mathrm { R e } \\left\\{ \\frac { 1 } { 2 } { \\bf E } ( { \\bf r } , \\omega \\omega ) \\times { \\bf H } ^ { * } ( { \\bf r } , \\omega \\omega ) \\right\\} $$ where we chose a surface $\\begin{array} { r } { \\int _ { \\mathrm { s } } \\mathrm { d } \\mathbf { A } \\ = \\ - a \\hat { \\mathbf { x } } \\int \\mathrm { d } z } \\end{array}$ parallel to the grating as we were only interested in the radiation in the negative $x$ -direction. For 2D simulations, the area $A = a { \\cdot } \\Delta z$ is determined by the assumed length $\\Delta z$ of the line charge density $q = - e / \\Delta z$ corresponding to one electron.
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What electron beam energy was used in the experimental demonstration?
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Numerical Instabilities. We observed that the optimization for a single frequency is very sensitive to numerical instabilities, which is why we optimized our design for multiple frequencies $\\omega _ { i } ( i = 1 , . . . , \\dot { N } )$ simultaneously. A suitable objective function could be the sum over all $G ( \\phi , \\dot { \\omega } _ { i } )$ , but we found that the min-function $$ f _ { \\mathrm { o b j } } ( \\phi ) = \\operatorname* { m i n } _ { i } G ( \\phi , \\omega _ { i } ) $$ was even more robust against numerical instabilities. Our design was optimized for the three $\\omega$ ’s corresponding to $\\lambda _ { 1 - 3 } =$ 1350, 1400, and $1 4 5 0 ~ \\mathrm { n m }$ . Dual Pillar Design. The dual pillar design is inspired from ref 35. Pillar radii and DBR thicknesses were optimized using the same gradient-based algorithm as for inverse design. Pillars that are $\\pi$ -phase shifted with respect to each other are preferred over symmetric rows of pillars because they yield a stronger phase difference in $E _ { y }$ and therefore stronger coupling to the far field.
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Collection Range. The measured Gaussian spectrum from Figure 3a can be explained by the limited numerical aperture of the collection fiber. Smith‚àíPurcell radiation that is emitted in the nonperpendicular direction is offset from the optical axis for collection. This leads to a loss in collection efficiency, which we modeled with the function $\\exp \\{ - 2 r ^ { 2 } / ( { f } \\mathrm { { \\cdot } N A } ) ^ { 2 } \\} ,$ where $r$ is the offset measured at the collimator, $f = 1 2 ~ \\mathrm { m m }$ is the focal length of the collimator, and NA is the numerical aperture of the fiber. We found good agreement with the experimental data for $\\mathrm { N A } = 0 . 1 1$ , which is below the 0.22 stated by the manufacturer and might have been a result of misalignment. ASSOCIATED CONTENT $\\bullet$ Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphotonics.1c01687. Dependence of radiation on electron beam height within the structure; Determination of effective current; Dependence on beam-grating distance (PDF) 2D time-domain simulation of the inverse design structure (MP4) 2D time-domain simulation of the dual pillar structure with DBR (MP4)
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What electron beam energy was used in the experimental demonstration?
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30 keV
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2D time-domain simulation of the rectangular grating (MP4) AUTHOR INFORMATION Corresponding Authors Urs Haeusler − Department Physik, Friedrich-AlexanderUniversität Erlangen-Nürnberg (FAU), Erlangen 91058, Germany; Present Address: Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, U.K; $\\circledcirc$ orcid.org/0000-0002-6818-0576; Email: uph20@cam.ac.uk Michael Seidling − Department Physik, Friedrich-AlexanderUniversität Erlangen-Nürnberg (FAU), Erlangen 91058, Germany; $\\circledcirc$ orcid.org/0000-0002-9261-9040; Email: michael.seidling@fau.de Peter Hommelhoff − Department Physik, FriedrichAlexander-Universität Erlangen-Nürnberg (FAU), Erlangen 91058, Germany; Email: peter.hommelhoff@fau.de Author Peyman Yousefi − Department Physik, Friedrich-AlexanderUniversität Erlangen-Nürnberg (FAU), Erlangen 91058, Germany; Present Address: Fraunhofer-Institut für Keramische Technologien und Systeme IKTS, Ä ussere Nürnberger Strasse 62, Forchheim 91301, Germany. Complete contact information is available at: https://pubs.acs.org/10.1021/acsphotonics.1c01687 Author Contributions ∥ These authors contributed equally to this work. U.H., M.S., and P.H. conceived the project and prepared the manuscript. U.H. designed and P.Y. fabricated the structures. M.S. and U.H. acquired and analyzed data and performed simulations. Funding This project has received funding from the Gordon and Betty Moore Foundation Grants 4744 (ACHIP) and 5733 (QEMII), as well as ERC Advanced Grant 884 217 (AccelOnChip). Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS We thank Ian A. D. Williamson for helpful support with the inverse design software. We thank R. Joel England and Andrzej Szczepkowicz for fruitful discussions about the design and
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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g ( \\mathbf { k } ) \\triangleq \\int f ( \\mathbf { r } ) \\mathrm { e } ^ { - i \\mathbf { k } \\cdot \\mathbf { r } } \\mathrm { d } \\mathbf { r } , g ( \\mathbf { r } ) \\triangleq \\frac { 1 } { \\left( 2 \\pi \\right) ^ { 3 } } \\int g ( \\mathbf { k } ) \\mathrm { e } ^ { i \\mathbf { k } \\cdot \\mathbf { r } } \\mathrm { d } \\mathbf { k } $$ Numerical methods. The photonic band structure in Fig. 4b is calculated via the eigenfrequency calculation in COMSOL Multiphysics. Numerical radiation intensities (Figs. 1d,e, 2b,c, 3d and 4d–f) are obtained via the frequency-domain calculation in the radiofrequency module in COMSOL Multiphysics. A surface (for 3D problems) or line (for 2D problems) integral on the Poynting vector is calculated to extract the radiation intensity at each frequency. Experimental set-up and sample fabrication. Our experimental set-up comprises a conventional SEM with the sample mounted perpendicular to the stage. A microscope objective was placed on the SEM stage to collect and image the light emission from the surface. The collected light was then sent through a series of
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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A surprising feature of the limits in equations (4), (5a) and (5b) is their prediction for optimal electron velocities. As shown in Fig. 1c, when electrons are in the far field of the structure $( \\kappa _ { \\rho } d \\gg 1 )$ , stronger photon emission and energy loss are achieved by faster electrons—a well-known result. On the contrary, if electrons are in the near field $( \\kappa _ { \\rho } d \\ll 1 )$ , slower electrons are optimal. This contrasting behaviour is evident in the asymptotics of equation (5b), where the $1 / \\beta ^ { 2 }$ or $\\mathrm { e } ^ { - 2 \\kappa _ { \\rho } d }$ dependence is dominant at short or large separations. Physically, the optimal velocities are determined by the incident-field properties (equation (2)): slow electrons generate stronger near-field amplitudes although they are more evanescent (Supplementary Section 2). There has been recent interest in using low-energy electrons for Cherenkov10 and Smith–Purcell31 radiation; our prediction that they can be optimal at subwavelength interaction distances underscores the substantial technological potential of non-relativistic free-electron radiation sources. The tightness of the limit (equations (4), (5a) and (5b)) is demonstrated by comparison with full-wave numerical calculations (see Methods) in Fig. 1d,e. Two scenarios are considered: in Fig. 1d, an electron traverses the centre of an annular Au bowtie antenna and undergoes antenna-enabled transition radiation $( \\eta \\approx 0 . 0 7 \\% )$ , while, in Fig. 1e, an electron traverses a Au grating, undergoing Smith–Purcell radiation $( \\eta \\approx 0 . 9 \\% )$ . In both cases, the numerical results closely trail the upper limit at the considered wavelengths, showing that the limits can be approached or even attained with modest effort.
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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free-space optical elements, enabling simultaneous measurement of the spectrum and of the spatial radiation pattern. The SEM used for the experiment was a JEOL JSM-6010LA. Its energy spread at the gun exit was in the range 1.5 to $2 . 5 \\mathrm { e V }$ for the range of acceleration voltages considered in this paper. The SEM was operated in spot mode, which we controlled precisely to align the beam so that it passes tangentially to the surface near the desired area of the sample. A Nikon TU Plan Fluor 10x objective with a numerical aperture of 0.30 was used to collect light from the area of interest. The monochrome image of the radiation was taken using a Hamamatsu CCD (chargecoupled device). The spectrometer used was an Action SP-2360-2300i with a lownoise Princeton Instruments Pixis 400 CCD. A 1D grating (Au-covered single-crystalline Si: periodicity, ${ 1 4 0 } \\mathrm { n m } ;$ filling factor, $5 0 \\%$ ; patterned Si thickness, $5 3 \\pm 1 . 5 \\mathrm { { n m } }$ ; Au thickness $4 4 \\pm 1 . 5 \\mathrm { n m }$ was used as the sample in our experiment. The original nanopatterned linear silicon stamp was obtained from LightSmyth Technologies and coated using an electron beam evaporator with a $2 \\mathrm { n m T i }$ adhesion layer and $4 0 \\mathrm { n m }$ of Au at $1 0 ^ { - 7 }$ torr. The sample was mounted inside the SEM chamber to enable the alignment of free electrons to pass in close proximity to the stamps. The emitted light was coupled out of the SEM chamber to a spectrometer, while a camera was used to image the surface of the sample.
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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The Smith–Purcell effect epitomizes the potential of free-electron radiation. Consider an electron at velocity $\\beta = \\nu / c$ traversing a structure with periodicity $a$ ; it generates far-field radiation at wavelength $\\lambda$ and polar angle $\\theta$ , dictated by2 $$ \\lambda = \\frac { a } { m } \\left( \\frac { 1 } { \\beta } - \\mathrm { c o s } \\theta \\right) $$ where $m$ is the integer diffraction order. The absence of a minimum velocity in equation (1) offers prospects for threshold-free and spectrally tunable light sources, spanning from microwave and terahertz14–16, across visible17–19, and towards $\\bar { \\mathrm { X - r a y } } ^ { 2 0 }$ frequencies. In stark contrast to the simple momentum-conservation determination of wavelength and angle, there is no unified yet simple analytical equation for the radiation intensity. Previous theories offer explicit solutions only under strong assumptions (for example, assuming perfect conductors or employing effective medium descriptions) or for simple, symmetric geometries21–23. Consequently, heavily numerical strategies are often an unavoidable resort24,25. In general, the inherent complexity of the interactions between electrons and photonic media have prevented a more general understanding of how pronounced spontaneous electron radiation can ultimately be for arbitrary structures, and consequently, how to design the maximum enhancement for free-electron light-emitting devices.
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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We begin our analysis by considering an electron (charge $- e$ ) of constant velocity $\\nu \\hat { \\mathbf { x } }$ traversing a generic scatterer (plasmonic or dielectric, finite or extended) of arbitrary size and material composition, as in Fig. 1a. The free current density of the electron, ${ \\bf \\dot { J } } ( { \\bf r } , t ) = - { \\hat { \\bf x } } e \\nu \\delta ( y ) { \\bf \\ddot { \\delta } } ( z ) \\delta ( x - \\nu t )$ , generates a frequency-dependent $( { \\bf e } ^ { - i \\omega t }$ convention) incident field26 $$ \\mathbf { E } _ { \\mathrm { i n c } } ( \\mathbf { r } , \\omega ) = \\frac { e \\kappa _ { \\rho } \\mathrm { e } ^ { i k _ { \\nu } x } } { 2 \\pi \\omega \\varepsilon _ { 0 } } [ \\hat { \\mathbf { x } } i \\kappa _ { \\rho } K _ { 0 } ( \\kappa _ { \\rho } \\rho ) - \\hat { \\mathbf { \\rho } } \\hat { \\mathbf { e } } k _ { \\nu } K _ { 1 } ( \\kappa _ { \\rho } \\rho ) ]
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$$ written in cylindrical coordinates $( x , \\rho , \\psi )$ ; here, $K _ { n }$ is the modified Bessel function of the second kind, $k _ { \\nu } = \\omega / \\nu$ and $k _ { \\rho } = \\sqrt { k _ { \\nu } ^ { 2 } - k ^ { 2 } } =$ k/βγ $\\scriptstyle ( k = \\omega / c$ , free-space wavevector; $\\gamma = 1 / \\sqrt { 1 - \\beta ^ { 2 } }$ , Lorentz factor). Hence, the photon emission and energy loss of free electrons can be treated as a scattering problem: the electromagnetic fields $\\mathbf { F } _ { \\mathrm { i n c } } { = } ( \\mathbf { E } _ { \\mathrm { i n c } } , \\ Z _ { 0 } \\mathbf { H } _ { \\mathrm { i n c } } ) ^ { \\mathrm { T } }$ (for free-space impedance $Z _ { 0 } ^ { \\mathrm { ~ \\tiny ~ { ~ \\chi ~ } ~ } }$ are incident on a photonic medium with material susceptibility $\\overline { { \\overline { { \\chi } } } }$ (a $6 { \\times } 6$ tensor for a general medium), causing both absorption and far-field scattering—that is, photon emission—that together comprise electron energy loss (Fig. 1a).
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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As recently shown in refs 27–29, for a generic electromagnetic scattering problem, passivity—the condition that polarization currents do no net work—constrains the maximum optical response from a given incident field. Consider three power quantities derived from $\\mathbf { F } _ { \\mathrm { i n c } }$ and the total field F within the scatterer volume $V !$ the total power lost by the electron, $P _ { \\mathrm { l o s s } } = - ( 1 / 2 ) \\mathrm { R e } \\int _ { \\mathrm { V } } \\mathbf { J } ^ { * } \\cdot \\mathbf { E d } V = ( \\epsilon _ { 0 } \\omega / 2 ) \\mathrm { I m } \\hat { \\int _ { V } } \\mathbf { F } _ { \\mathrm { ~ i n c } } ^ { \\dagger } \\overline { { \\chi } } \\mathbf { F d } V ,$ the power absorbed by the medium, $P _ { \\mathrm { { a b s } } } \\mathrm { { = } } \\left( \\epsilon _ { 0 } \\omega / 2 \\right) \\mathrm { I m } \\stackrel { \\cdot } { \\int } _ { V } \\mathbf { F } ^ { \\dagger } \\overline { { \\chi } } \\mathbf { F } \\mathrm { { d } } V ,$ and their difference, the power radiated to the far field, $P _ { \\mathrm { r a d } } { = } P _ { \\mathrm { l o s s } } { - } P _ { \\mathrm { a b s } }$ . Treating $\\mathbf { F }$ as an independent variable, the total loss $P _ { \\mathrm { l o s s } }$ is a linear function of $\\mathbf { F }$ , whereas the fraction that is dissipated is a quadratic function of F. Passivity requires non-negative radiated power, represented by the inequality $P _ { \\mathrm { a b s } } { < } P _ { \\mathrm { l o s s } } ,$ which in this framework is therefore a convex constraint on any response function. Constrained maximization (see Supplementary Section 1) of the energy-loss and photon-emission power quantities, $P _ { \\mathrm { l o s s } }$ and $P _ { \\mathrm { r a d } } ,$ directly yields the limits
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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$$ P _ { \\tau } ( \\omega ) \\leq \\frac { \\varepsilon _ { 0 } \\omega \\xi _ { \\tau } } { 2 } \\int _ { V } \\mathbf { F } _ { \\mathrm { i n } } ^ { \\dagger } \\overline { { \\overline { { \\chi } } } } ^ { \\dagger } ( \\mathrm { I m } \\overline { { \\overline { { \\chi } } } } ) ^ { - 1 } \\overline { { \\overline { { \\chi } } } } \\mathbf { F } _ { \\mathrm { i n c } } \\mathrm { d } V $$ where $\\tau \\in \\{ \\mathrm { r a d } , \\log \\}$ and $\\xi _ { \\tau }$ accounts for a variable radiative efficiency $\\eta$ (defined as the ratio of radiative to total energy loss): $\\xi _ { \\mathrm { l o s s } } = 1$ and $\\xi _ { \\mathrm { r a d } } = \\eta ( 1 - \\eta ) \\le 1 / 4$ . Hereafter, we consider isotropic and non-magnetic materials (and thus a scalar susceptibility $\\chi$ ), but the generalizations to anisotropic and/or magnetic media are straightforward.
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Combining equations (2) and (3) yields a general limit on the loss or emission spectral probabilities $\\bar { T _ { \\tau } } ( \\omega ) = \\bar { P _ { \\tau } } ( \\omega ) / \\hbar \\omega$ : $$ \\Gamma _ { \\tau } ( \\omega ) \\leq \\frac { \\alpha \\xi _ { \\tau } c } { 2 \\pi \\omega ^ { 2 } } \\int _ { V } \\frac { \\left. \\chi \\right. ^ { 2 } } { \\mathrm { I m } \\chi } [ \\kappa _ { \\rho } ^ { 4 } K _ { 0 } ^ { 2 } \\left( \\kappa _ { \\rho } \\rho \\right) + \\kappa _ { \\rho } ^ { 2 } k _ { \\nu } ^ { 2 } K _ { 1 } ^ { 2 } \\left( \\kappa _ { \\rho } \\rho \\right) ] \\mathrm { d } V $$ where $\\alpha$ is the fine-structure constant. Equation (4) imposes, without solving Maxwell’s equations, a maximum rate of photon generation based on the electron velocity $\\beta$ (through $k _ { \\nu }$ and $\\kappa _ { \\rho } \\mathrm { , }$ , the material composition $\\chi ( \\mathbf { r } )$ and the volume $V .$ .
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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The limit in equation (4) can be further simplified by removing the shape dependence of $V$ , since the integrand is positive and is thus bounded above by the same integral for any enclosing structure. A scatterer separated from the electron by a minimum distance $d$ can be enclosed within a larger concentric hollow cylinder sector of inner radius $d$ and outer radius $\\infty$ ‚Äã. For such a sector (height $L$ and opening azimuthal angle $\\scriptstyle \\psi \\in ( 0 , 2 \\pi ] )$ , equation (4) can be further simplified, leading to a general closed-form shape-independent limit (see Supplementary Section 2) that highlights the pivotal role of the impact parameter $\\kappa _ { \\rho } d$ : $$ T _ { \\tau } ( \\omega ) \\leq \\frac { \\alpha \\xi _ { \\tau } } { 2 \\pi c } \\frac { | \\chi | ^ { 2 } } { \\mathrm { I m } \\chi } \\frac { L \\psi } { \\beta ^ { 2 } } [ ( \\kappa _ { \\rho } d ) K _ { 0 } ( \\kappa _ { \\rho } d ) K _ { 1 } ( \\kappa _ { \\rho } d ) ]
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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$$ $$ \\propto \\frac { 1 } { \\beta ^ { 2 } } \\Bigg \\{ \\ln ( 1 / \\kappa _ { \\rho } d ) \\mathrm { f o r } \\kappa _ { \\rho } d \\ll 1 , $$ The limits of equations (4), (5a) and (5b) are completely general; they set the maximum photon emission and energy loss of an electron beam coupled to an arbitrary photonic environment in either the non-retarded or retarded regimes, given only the beam properties and material composition. The key factors that determine maximal radiation are identified: intrinsic material loss (represented by $\\mathrm { I m } \\chi \\dot { } ,$ ), electron velocity $\\beta$ and impact parameter $\\kappa _ { \\rho } d$ . The metric $| \\chi | ^ { 2 } / \\mathrm { I m } \\chi$ reflects the influence of the material choice, which depends sensitively on the radiation wavelength (Fig. 1b). The electron velocity $\\beta$ also appears implicitly in the impact parameter $\\kappa _ { \\rho } d = k d / \\beta \\gamma ,$ showing that the relevant length scale is set by the relativistic velocity of the electron. The impact parameter $\\kappa _ { \\rho } d$ reflects the influence of the Lorentz contraction $d / \\gamma ;$ a well-known feature of both electron radiation and acceleration20,26,30.
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Next, we specialize in the canonical Smith–Purcell set-up illustrated in Fig. 1e inset. This set-up warrants a particularly close study, given its prominent historical and practical role in free-electron radiation. Aside from the shape-independent limit (equations (5a) and (5b)), we can find a sharper limit (in per unit length for periodic structure) specifically for Smith–Purcell radiation using rectangular gratings of filling factor $\\varLambda$ (see Supplementary Section 3) $$ \\frac { \\mathrm { d } { \\varGamma } _ { \\tau } ( \\omega ) } { \\mathrm { d } x } \\leq \\frac { \\alpha \\xi _ { \\tau } } { 2 \\pi c } \\frac { | \\boldsymbol \\chi | ^ { 2 } } { \\mathrm { I m } \\chi } \\varLambda \\mathcal { G } ( \\beta , k d ) $$ The function ${ \\mathcal { G } } ( { \\boldsymbol { \\beta } } , k d )$ is an azimuthal integral (see Supplementary Section 3) over the Meijer G-function $G _ { 1 , 3 } ^ { 3 , 0 }$ (ref. $^ { 3 2 } )$ that arises in the radial integration of the modified Bessel functions $K _ { n }$ . We emphasize that equation (6) is a specific case of equation (4) for grating structures without any approximations and thus can be readily generalized to multi-material scenarios (see Supplementary equation (37)).
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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The grating limit (equation (6)) exhibits the same asymptotics as equations (5a) and (5b), thereby reinforcing the optimal-velocity predictions of Fig. 1c. The $( \\beta , k \\dot { d } )$ dependence of $\\mathcal { G }$ (see Fig. 2a) shows that slow (fast) electrons maximize Smith–Purcell radiation in the small (large) separation regime. We verify the limit predictions by comparison with numerical simulations: at small separations (Fig. 2b), radiation and energy loss peak at velocity $\\beta \\approx 0 . 1 5$ , consistent with the limit maximum; at large separations (Fig. 2c), both the limit and the numerical results grow monotonically with $\\beta$ . The derived upper limit also applies to Cherenkov and transition radiation, as well as bulk loss in electron energy-loss spectroscopy. For these scenarios where electrons enter material bulk, a subtlety arises for the field divergence along the electron’s trajectory $( \\rho = 0$ in equation (2)) within a potentially lossy medium. This divergence, however, can be regularized by introducing natural, system-specific momentum cutoffs26, which then directly permits the application of our theory (see Supplementary Section 6). Meanwhile, there exist additional competing interaction processes (for example, electrons colliding with individual atoms). However, they typically occur at much smaller length scales.
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Finally, we turn our attention to an ostensible peculiarity of the limits: equation (4) evidently diverges for lossless materials $( \\mathrm { I m } \\chi \\to 0 ) \\dot { { \\frac { . } { . } } }$ ), seemingly providing little insight. On the contrary, this divergence suggests the existence of a mechanism capable of strongly enhancing Smith–Purcell radiation. Indeed, by exploiting high-Q resonances near bound states in the continuum (BICs)13 in photonic crystal slabs, we find that Smith–Purcell radiation can be enhanced by orders of magnitude, when specific frequency-, phaseand polarization-matching conditions are met. A 1D silicon $\\left( \\chi = 1 1 . 2 5 \\right)$ -on-insulator $\\mathrm { \\ S i O } _ { 2 } ,$ $\\chi = 1 . 0 7 \\$ ) grating interacting with a sheet electron beam illustrates the core conceptual idea most clearly. The transverse electric (TE) $( E _ { x } , H _ { y } , E _ { z } )$ band structure (lowest two bands labelled $\\mathrm { T E } _ { 0 }$ and $\\mathrm { T E } _ { 1 . } ^ { \\cdot }$ ), matched polarization for a sheet electron beam (supplementary equation S41b)), is depicted in Fig. 4b along the $\\Gamma { \\mathrm { - } } \\mathrm { X }$ direction. Folded electron wavevectors, $k _ { \\nu } = \\omega / \\nu ,$ are overlaid for two distinct velocities (blue and green). Strong electron–photon interactions are possible when the electron and photon dispersions intersect: for instance, $k _ { \\nu }$ and the $\\mathrm { T E } _ { 0 }$ band intersect (grey circles) below the air light cone (light yellow shading). However, these intersections are largely impractical: the $\\mathrm { T E } _ { 0 }$ band is evanescent in the air region, precluding free-space radiation. Still, analogous ideas, employing similar partially guided modes, such as spoof plasmons33, have been explored for generating electron-enabled guided waves34,35.
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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To overcome this deficiency, we theoretically propose a new mechanism for enhanced Smith–Purcell radiation: coupling of electrons with $\\mathrm { B I C } s ^ { 1 3 }$ . The latter have the extreme quality factors of guided modes but are, crucially, embedded in the radiation continuum, guaranteeing any resulting Smith–Purcell radiation into the far field. By choosing appropriate velocities $\\beta = a / m \\lambda$ ( $m$ being any integer; $\\lambda$ being the BIC wavelength) such that the electron line (blue or green) intersects the $\\mathrm { T E } _ { 1 }$ mode at the BIC (red square in Fig. 4b), the strong enhancements of a guided mode can be achieved in tandem with the radiative coupling of a continuum resonance. In Fig. 4c, the incident fields of electrons and the field profile of the BIC indicate their large modal overlaps. The BIC field profile shows complete confinement without radiation, unlike conventional multipolar radiation modes (see Supplementary Fig. 9). The Q values of the resonances are also provided near a symmetry-protected $\\mathrm { B I C ^ { 1 3 } }$ at the $\\Gamma$ point. Figure 4d,e demonstrates the velocity tunability of BIC-enhanced radiation—as the phase matching approaches the BIC, a divergent radiation rate is achieved.
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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The BIC-enhancement mechanism is entirely accordant with our upper limits. Practically, silicon has non-zero loss across the visible and near-infrared wavelengths. For example, for a period of $a = 6 7 6 \\mathrm { n m }$ , the optimally enhanced radiation wavelength is $\\approx 1 { , } 0 5 0 \\mathrm { n m }$ , at which $\\chi _ { \\mathrm { s i } } \\approx 1 1 . 2 5 + 0 . 0 0 1 \\mathrm { i }$ (ref. 36). For an electron– structure separation of $3 0 0 \\mathrm { n m }$ , we theoretically show in Fig. 4f the strong radiation enhancements ${ > } 3$ orders of magnitude) attainable by BIC-enhanced coupling. The upper limit (shaded region; 2D analogue of equation (4); see Supplementary Section 10) attains extremely large values due to the minute material loss $( | \\chi | ^ { 2 } / \\mathrm { I m } \\chi \\approx 1 0 ^ { 5 } )$ ; nevertheless, BIC-enhanced coupling enables the radiation intensity to closely approach this limit at several resonant velocities. In the presence of an absorptive channel, the maximum enhancement occurs at a small offset from the BIC where the $Q$ -matching condition (see Supplementary Section 11) is satisfied (that is, equal absorptive and radiative rates of the resonances).
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The upper limit demonstrated here is in the spontaneous emission regime for constant-velocity electrons, and can be extended to the stimulated regime by suitable reformulation. Stronger electron– photon interactions can change electron velocity by a non-negligible amount that alters the radiation. If necessary, this correction can be perturbatively incorporated. In the case of external optical pumping37, the upper limit can be revised by redefining the incident field as the summation of the electron incident field and the external optical field. From a quantum mechanical perspective, this treatment corresponds to stimulated emission from free electrons, which multiplies the limit by the number of photons in that radiation mode. This treatment could also potentially translate our limit into a fundamental limit for particle acceleration38,39, which is the time-reversal of free-electron energy loss and which typically incorporates intense laser pumping. In the multi-electron scenario, the radiation upper limit will be obtained in the case of perfect bunching, where all electrons radiate in phase. In this case, our singleelectron limit should be multiplied by the number of electrons to correct for the superradiant nature of such coherent radiation. Methods Methods, including statements of data availability and any associated accession codes and references, are available at https://doi. org/10.1038/s41567-018-0180-2
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Author contributions Y.Y., O.D.M., I.K. and M.S. conceived the project. Y.Y. developed the analytical models and numerical calculations. A.M. prepared the sample under study. Y.Y., A.M., C.R.-C., S.E.K. and I.K. performed the experiment. Y.Y., T.C. and O.D.M. analysed the asymptotics and bulk loss of the limit. S.G.J., J.D.J., O.D.M., I.K. and M.S. supervised the project. Y.Y. wrote the manuscript with input from all authors. Competing interests The authors declare no competing interests. Additional information Supplementary information is available for this paper at https://doi.org/10.1038/ s41567-018-0180-2. Reprints and permissions information is available at www.nature.com/reprints. Correspondence and requests for materials should be addressed to Y.Y. or O.D.M. or I.K. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Methods Fourier transform convention. Throughout the paper, we adopt the following Fourier transform conventions $$ f ( \\omega ) \\triangleq \\int f ( t ) \\mathrm { e } ^ { i \\omega t } \\mathrm { d } t , f ( t ) \\triangleq \\frac { 1 } { 2 \\pi } \\int f ( \\omega ) \\mathrm { e } ^ { - i \\omega t } \\mathrm { d } \\omega $$ $$
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What experimental structure was used to validate the theoretical upper limit with Smith-Purcell measurements?
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A one-dimensional, 50%-filling-factor, Au-covered single-crystalline silicon grating.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Data availability. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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File Name:Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf Maximal spontaneous photon emission and energy loss from free electrons Yi Yang $\\textcircled { 1 0 } 1 \\star$ , Aviram Massuda1, Charles Roques-Carmes $\\oplus 1$ , Steven E. Kooi $\\oplus 2$ , Thomas Christensen1, Steven G. Johnson1, John D. Joannopoulos1,2, Owen D. Miller $\\textcircled { 1 0 } 3 \\star$ , Ido Kaminer $\\textcircled { 1 0 } 1 , 4 \\star$ and Marin Soljačić1 Free-electron radiation such as Cerenkov1, Smith–Purcell2 and transition radiation3,4 can be greatly affected by structured optical environments, as has been demonstrated in a variety of polaritonic5,6, photonic-crystal7 and metamaterial8–10 systems. However, the amount of radiation that can ultimately be extracted from free electrons near an arbitrary material structure has remained elusive. Here we derive a fundamental upper limit to the spontaneous photon emission and energy loss of free electrons, regardless of geometry, which illuminates the effects of material properties and electron velocities. We obtain experimental evidence for our theory with quantitative measurements of Smith–Purcell radiation. Our framework allows us to make two predictions. One is a new regime of radiation operation—at subwavelength separations, slower (non-relativistic) electrons can achieve stronger radiation than fast (relativistic) electrons. The other is a divergence of the emission probability in the limit of lossless materials. We further reveal that such divergences can be approached by coupling free electrons to photonic bound states in the continuum11–13. Our findings suggest that compact and efficient free-electron radiation sources from microwaves to the soft X-ray regime may be achievable without requiring ultrahigh accelerating voltages.
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The upper limit demonstrated here is in the spontaneous emission regime for constant-velocity electrons, and can be extended to the stimulated regime by suitable reformulation. Stronger electron– photon interactions can change electron velocity by a non-negligible amount that alters the radiation. If necessary, this correction can be perturbatively incorporated. In the case of external optical pumping37, the upper limit can be revised by redefining the incident field as the summation of the electron incident field and the external optical field. From a quantum mechanical perspective, this treatment corresponds to stimulated emission from free electrons, which multiplies the limit by the number of photons in that radiation mode. This treatment could also potentially translate our limit into a fundamental limit for particle acceleration38,39, which is the time-reversal of free-electron energy loss and which typically incorporates intense laser pumping. In the multi-electron scenario, the radiation upper limit will be obtained in the case of perfect bunching, where all electrons radiate in phase. In this case, our singleelectron limit should be multiplied by the number of electrons to correct for the superradiant nature of such coherent radiation. Methods Methods, including statements of data availability and any associated accession codes and references, are available at https://doi. org/10.1038/s41567-018-0180-2
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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In closing, we have theoretically derived and experimentally probed a universal upper limit to the energy loss and photon emission from free electrons. The limit depends crucially on the impact parameter $\\kappa _ { \\rho } d$ , but not on any other detail of the geometry. Hence, our limit applies even to the most complex metamaterials and metasurfaces, given only their constituents. Surprisingly, in the near field, slow electrons promise stronger radiation than relativistic ones. The limit predicts a divergent radiation rate as the material loss rate goes to zero, and we show that BIC resonances enable such staggering enhancements. This is relevant for the generation of coherent Smith–Purcell radiation14,34,35. The long lifetime, spectral selectivity and large field enhancement near a BIC can strongly bunch electrons, allowing them to radiate coherently at the same desired frequency, potentially enabling low-threshold Smith–Purcell freeelectron lasers. The combination of this mechanism and the optimal velocity prediction reveals prospects of low-voltage yet high-power free-electron radiation sources. In addition, our findings demonstrate a simple guiding principle to maximize the signal-to-noise ratio for electron energy-loss spectroscopy through an optimal choice of electron velocity, enabling improved spectral resolution. The predicted slow-electron-efficient regime still calls for direct experimental validation. We suggest that field-emitter-integrated free-electron devices (for example, ref. 10) are ideal to confirm the prediction due to the achievable small electron–structure separation and high electron beam quality at relatively large currents. Alternatively, the microwave or terahertz frequencies could be suitable spectral ranges for verifying the slow-electron-efficient regime, where the subwavelength separation requirement is more achievable.
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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The Smith–Purcell effect epitomizes the potential of free-electron radiation. Consider an electron at velocity $\\beta = \\nu / c$ traversing a structure with periodicity $a$ ; it generates far-field radiation at wavelength $\\lambda$ and polar angle $\\theta$ , dictated by2 $$ \\lambda = \\frac { a } { m } \\left( \\frac { 1 } { \\beta } - \\mathrm { c o s } \\theta \\right) $$ where $m$ is the integer diffraction order. The absence of a minimum velocity in equation (1) offers prospects for threshold-free and spectrally tunable light sources, spanning from microwave and terahertz14–16, across visible17–19, and towards $\\bar { \\mathrm { X - r a y } } ^ { 2 0 }$ frequencies. In stark contrast to the simple momentum-conservation determination of wavelength and angle, there is no unified yet simple analytical equation for the radiation intensity. Previous theories offer explicit solutions only under strong assumptions (for example, assuming perfect conductors or employing effective medium descriptions) or for simple, symmetric geometries21–23. Consequently, heavily numerical strategies are often an unavoidable resort24,25. In general, the inherent complexity of the interactions between electrons and photonic media have prevented a more general understanding of how pronounced spontaneous electron radiation can ultimately be for arbitrary structures, and consequently, how to design the maximum enhancement for free-electron light-emitting devices.
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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We begin our analysis by considering an electron (charge $- e$ ) of constant velocity $\\nu \\hat { \\mathbf { x } }$ traversing a generic scatterer (plasmonic or dielectric, finite or extended) of arbitrary size and material composition, as in Fig. 1a. The free current density of the electron, ${ \\bf \\dot { J } } ( { \\bf r } , t ) = - { \\hat { \\bf x } } e \\nu \\delta ( y ) { \\bf \\ddot { \\delta } } ( z ) \\delta ( x - \\nu t )$ , generates a frequency-dependent $( { \\bf e } ^ { - i \\omega t }$ convention) incident field26 $$ \\mathbf { E } _ { \\mathrm { i n c } } ( \\mathbf { r } , \\omega ) = \\frac { e \\kappa _ { \\rho } \\mathrm { e } ^ { i k _ { \\nu } x } } { 2 \\pi \\omega \\varepsilon _ { 0 } } [ \\hat { \\mathbf { x } } i \\kappa _ { \\rho } K _ { 0 } ( \\kappa _ { \\rho } \\rho ) - \\hat { \\mathbf { \\rho } } \\hat { \\mathbf { e } } k _ { \\nu } K _ { 1 } ( \\kappa _ { \\rho } \\rho ) ]
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ written in cylindrical coordinates $( x , \\rho , \\psi )$ ; here, $K _ { n }$ is the modified Bessel function of the second kind, $k _ { \\nu } = \\omega / \\nu$ and $k _ { \\rho } = \\sqrt { k _ { \\nu } ^ { 2 } - k ^ { 2 } } =$ k/βγ $\\scriptstyle ( k = \\omega / c$ , free-space wavevector; $\\gamma = 1 / \\sqrt { 1 - \\beta ^ { 2 } }$ , Lorentz factor). Hence, the photon emission and energy loss of free electrons can be treated as a scattering problem: the electromagnetic fields $\\mathbf { F } _ { \\mathrm { i n c } } { = } ( \\mathbf { E } _ { \\mathrm { i n c } } , \\ Z _ { 0 } \\mathbf { H } _ { \\mathrm { i n c } } ) ^ { \\mathrm { T } }$ (for free-space impedance $Z _ { 0 } ^ { \\mathrm { ~ \\tiny ~ { ~ \\chi ~ } ~ } }$ are incident on a photonic medium with material susceptibility $\\overline { { \\overline { { \\chi } } } }$ (a $6 { \\times } 6$ tensor for a general medium), causing both absorption and far-field scattering—that is, photon emission—that together comprise electron energy loss (Fig. 1a).
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The grating limit (equation (6)) exhibits the same asymptotics as equations (5a) and (5b), thereby reinforcing the optimal-velocity predictions of Fig. 1c. The $( \\beta , k \\dot { d } )$ dependence of $\\mathcal { G }$ (see Fig. 2a) shows that slow (fast) electrons maximize Smith–Purcell radiation in the small (large) separation regime. We verify the limit predictions by comparison with numerical simulations: at small separations (Fig. 2b), radiation and energy loss peak at velocity $\\beta \\approx 0 . 1 5$ , consistent with the limit maximum; at large separations (Fig. 2c), both the limit and the numerical results grow monotonically with $\\beta$ . The derived upper limit also applies to Cherenkov and transition radiation, as well as bulk loss in electron energy-loss spectroscopy. For these scenarios where electrons enter material bulk, a subtlety arises for the field divergence along the electron’s trajectory $( \\rho = 0$ in equation (2)) within a potentially lossy medium. This divergence, however, can be regularized by introducing natural, system-specific momentum cutoffs26, which then directly permits the application of our theory (see Supplementary Section 6). Meanwhile, there exist additional competing interaction processes (for example, electrons colliding with individual atoms). However, they typically occur at much smaller length scales.
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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To overcome this deficiency, we theoretically propose a new mechanism for enhanced Smith–Purcell radiation: coupling of electrons with $\\mathrm { B I C } s ^ { 1 3 }$ . The latter have the extreme quality factors of guided modes but are, crucially, embedded in the radiation continuum, guaranteeing any resulting Smith–Purcell radiation into the far field. By choosing appropriate velocities $\\beta = a / m \\lambda$ ( $m$ being any integer; $\\lambda$ being the BIC wavelength) such that the electron line (blue or green) intersects the $\\mathrm { T E } _ { 1 }$ mode at the BIC (red square in Fig. 4b), the strong enhancements of a guided mode can be achieved in tandem with the radiative coupling of a continuum resonance. In Fig. 4c, the incident fields of electrons and the field profile of the BIC indicate their large modal overlaps. The BIC field profile shows complete confinement without radiation, unlike conventional multipolar radiation modes (see Supplementary Fig. 9). The Q values of the resonances are also provided near a symmetry-protected $\\mathrm { B I C ^ { 1 3 } }$ at the $\\Gamma$ point. Figure 4d,e demonstrates the velocity tunability of BIC-enhanced radiation—as the phase matching approaches the BIC, a divergent radiation rate is achieved.
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Author contributions Y.Y., O.D.M., I.K. and M.S. conceived the project. Y.Y. developed the analytical models and numerical calculations. A.M. prepared the sample under study. Y.Y., A.M., C.R.-C., S.E.K. and I.K. performed the experiment. Y.Y., T.C. and O.D.M. analysed the asymptotics and bulk loss of the limit. S.G.J., J.D.J., O.D.M., I.K. and M.S. supervised the project. Y.Y. wrote the manuscript with input from all authors. Competing interests The authors declare no competing interests. Additional information Supplementary information is available for this paper at https://doi.org/10.1038/ s41567-018-0180-2. Reprints and permissions information is available at www.nature.com/reprints. Correspondence and requests for materials should be addressed to Y.Y. or O.D.M. or I.K. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Methods Fourier transform convention. Throughout the paper, we adopt the following Fourier transform conventions $$ f ( \\omega ) \\triangleq \\int f ( t ) \\mathrm { e } ^ { i \\omega t } \\mathrm { d } t , f ( t ) \\triangleq \\frac { 1 } { 2 \\pi } \\int f ( \\omega ) \\mathrm { e } ^ { - i \\omega t } \\mathrm { d } \\omega $$ $$
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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g ( \\mathbf { k } ) \\triangleq \\int f ( \\mathbf { r } ) \\mathrm { e } ^ { - i \\mathbf { k } \\cdot \\mathbf { r } } \\mathrm { d } \\mathbf { r } , g ( \\mathbf { r } ) \\triangleq \\frac { 1 } { \\left( 2 \\pi \\right) ^ { 3 } } \\int g ( \\mathbf { k } ) \\mathrm { e } ^ { i \\mathbf { k } \\cdot \\mathbf { r } } \\mathrm { d } \\mathbf { k } $$ Numerical methods. The photonic band structure in Fig. 4b is calculated via the eigenfrequency calculation in COMSOL Multiphysics. Numerical radiation intensities (Figs. 1d,e, 2b,c, 3d and 4d–f) are obtained via the frequency-domain calculation in the radiofrequency module in COMSOL Multiphysics. A surface (for 3D problems) or line (for 2D problems) integral on the Poynting vector is calculated to extract the radiation intensity at each frequency. Experimental set-up and sample fabrication. Our experimental set-up comprises a conventional SEM with the sample mounted perpendicular to the stage. A microscope objective was placed on the SEM stage to collect and image the light emission from the surface. The collected light was then sent through a series of
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What general limit does Equation (4) impose on free-electron radiation?
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It establishes a shape-independent upper limit on the spectral probabilities of energy loss and photon emission by free electrons in any photonic environment.
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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free-space optical elements, enabling simultaneous measurement of the spectrum and of the spatial radiation pattern. The SEM used for the experiment was a JEOL JSM-6010LA. Its energy spread at the gun exit was in the range 1.5 to $2 . 5 \\mathrm { e V }$ for the range of acceleration voltages considered in this paper. The SEM was operated in spot mode, which we controlled precisely to align the beam so that it passes tangentially to the surface near the desired area of the sample. A Nikon TU Plan Fluor 10x objective with a numerical aperture of 0.30 was used to collect light from the area of interest. The monochrome image of the radiation was taken using a Hamamatsu CCD (chargecoupled device). The spectrometer used was an Action SP-2360-2300i with a lownoise Princeton Instruments Pixis 400 CCD. A 1D grating (Au-covered single-crystalline Si: periodicity, ${ 1 4 0 } \\mathrm { n m } ;$ filling factor, $5 0 \\%$ ; patterned Si thickness, $5 3 \\pm 1 . 5 \\mathrm { { n m } }$ ; Au thickness $4 4 \\pm 1 . 5 \\mathrm { n m }$ was used as the sample in our experiment. The original nanopatterned linear silicon stamp was obtained from LightSmyth Technologies and coated using an electron beam evaporator with a $2 \\mathrm { n m T i }$ adhesion layer and $4 0 \\mathrm { n m }$ of Au at $1 0 ^ { - 7 }$ torr. The sample was mounted inside the SEM chamber to enable the alignment of free electrons to pass in close proximity to the stamps. The emitted light was coupled out of the SEM chamber to a spectrometer, while a camera was used to image the surface of the sample.
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Andersson_2008.pdf
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INTRODUCTION Synchrotron radiation (SR) sources based on electron storage rings are among the primary tools in materials research, physics, chemistry, and biology to study the structure of matter on the atomic scale [1]. However, phase transitions, chemical reactions as well as changes of molecular conformation, electronic or magnetic structure take place on the sub-picosecond scale which cannot be resolved by conventional synchrotron radiation pulses which are constrained to tens of picoseconds by the longitudinal beam dynamics in a storage ring. The femtosecond regime has been accessed by lasers at near-visible wavelengths and with high-harmonic generation, and more recently by high-gain free-electron lasers (FELs) in the extreme ultraviolet and X-ray regime [2]. While X-ray FELs serve one user at a time with the repetition rate of a linear accelerator and their number is worldwide still below ten, there are about 50 SR sources supplying multiple beamlines simultaneously with laser modulator radiator CHG chicane WW EEHG -1 0 z/ z/2 laser modulator laser modulator radiator chicane chicane 8 ： 0.5 before after 0.5 凯 E 正0.5 -0.5 -0.5 modulation 0 z/2L 0 z/2 0 stable and tunable radiation at a rate of up to ${ 5 0 0 } \\mathrm { M H z }$ . It is therefore worthwhile to consider possibilities of extending SR sources towards shorter pulse duration.
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Andersson_2008.pdf
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Workshop on Beam Diagnostics and Dynamics in Ultra-Low Emittance Rings In the future, synchrotron radiation sources and $\\mathrm { e + / e - }$ colliders will require high-quality beams with ultra-low emittance. To assess the beam quality and stability, technological breakthroughs in beam diagnostics are necessary to evaluate the beam quality achieved at a level that previous accelerators have never reached. In order to direct the development of the beam diagnostic system in the optimal direction for ultra-low emittance beams, it is essential to enhance the understanding of beam dynamics and to facilitate the exchange of information and knowledge between the beam diagnostics side and the beam dynamic side. Based on these ideas, the workshop on beam diagnostics and dynamics in ultra-low emittance rings was organized [2]. The workshop period was from April 25 to 29, 2022, during the international pandemic. During this period, it was challenging to proceed with workshop activities. However, the workshop was conducted online by the Karlsruhe Institute of Technology (KIT) with participants from around the globe. Due to differing time zones, the scientific sessions were held only in the afternoon Central European Time. The workshop included sessions on beam diagnostics and beam dynamics, with the objective of exchanging information and knowledge between the two sides of beam diagnostics and beam dynamics. Topical sessions included discussions of beam injection and collective beam instabilities from both the beam dynamics and diagnostics perspectives. Other topics included longterm beam stability, such as mechanical vibrations occurring in experimental halls and slow drift motions due to ambient temperature, as well as the application of machine learning to beam diagnostic systems. The workshop’s contents have also been summarized briefly and published as the I.FAST project report [3].
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Andersson_2008.pdf
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The assembly consists of a cylindrical vacuum tank housing the instrument with a motor driving a protruding hollow shaft onto which are mounted two titanium forks which support a carbon fibre ‘wire’ of $3 4 \\mu \\mathrm { m }$ diameter (see Fig. 1). The wire is rotated 270 degrees at high speed, crossing the beam at upto $2 0 \\mathrm { m s ^ { - 1 } }$ , scattering some particles which are detected downstream by a scintillator. A precise wire position measurement and particle loss flux are combined to produce a transverse beam intensity profile. The wires are electrically insulated from the forks and connected at each end to copper cables which exit the vacuum via a feedthrough. This allows additional wire properties such as resistance and current flow to be measured. The instruments performed well in all machines, with more than $7 0 ^ { \\circ } 0 0 0$ scans made in the first year [3]. However, early in 2023 operations an incident occurred with the SPS scanners. There are 4 wire scanners in the SPS ring, 2 mounted sequentially in the horizontal (H) plane (operational and ‘hot spare’ scanner) in the ‘BA5’ straight section and 2 sequentially in the vertical (V) plane, several hundred meters away in ‘BA4’.
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Andersson_2008.pdf
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File Name:EXPERIMENTAL_DESIGNS_OF_COHERENT_SYNCHROTRON.pdf EXPERIMENTAL DESIGNS OF COHERENT SYNCHROTRONRADIATION IN COMPLEX BEAMS O. H. Ramachandran1,2‚àó , G. Ha1,2, C.-K. Huang3, X. Lu1,2, J. Power2, and Ji Qiang4 1Northern Illinois University, DeKalb, IL 60115, USA 2Argonne National Laboratory, Lemont, IL 60439, USA 3Los Alamos National Laboratory, Los Alamos, NM 87545, USA 4Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Abstract Coherent synchrotron radiation (CSR) is one critical beam collective effect in high-energy accelerators, which impedes the generation of high-brightness beams. The Argonne Wakefield Accelerator (AWA) facility is unique in the experimental investigation of CSR effects in complex beams, offering a large parameter space for the bunch charge and size, various bunch profiles (round and flat beams), and the capability of generating shaped bunches through both laser shaping and the emittance exchange approach. This presentation will outline planned experiments at AWA and their designs, including a CSR shielding study using a dipole chamber with a variable gap size, and the effect of CSR on the beam phase space in a laser-shaped short electron bunch. This work is part of a comprehensive study involving self-consistent CSR code development and experimental investigation. The experimental component aims to provide benchmarking with the advanced codes under development, explore the boundaries of 1/2/3D CSR effects on beam dynamics, evaluate CSR effects in complex beams, and eventually propose CSR mitigation strategies.
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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File Name:OPERATION_OF_THE_ESRF_-EBS_LIGHT_SOURCE.pdf OPERATION OF THE ESRF-EBS LIGHT SOURCE J. L. Revol, C. Benabderrahmane, P. Borowiec, E Buratin, N. Carmignani, L. Carver, A. D’Elia, M.Dubrulle, F. Ewald, A Franchi, G. Gautier, L. Hardy, L. Hoummi, J. Jacob, G. Le Bec, L. Jolly, I Leconte, S. M. Liuzzo, T. Perron, Q. Qin, B. Roche, K. B. Scheidt, V. Serrière, R. Versteegen, S. White, European Synchrotron Radiation Facility, Grenoble, France Abstract The European Synchrotron Radiation Facility - Extremely Brilliant Source (ESRF-EBS) is a facility upgrade allowing its scientific users to take advantage of the first high-energy $4 ^ { \\mathrm { t h } }$ generation storage ring light source. In December 2018, after 30 years of operation, the beam stopped for a 12-month shutdown to dismantle the old storage ring and to install the new $\\mathrm { \\Delta X }$ -ray source. On 25th August 2020, the user programme restarted with beam parameters very close to nominal values. Since then beam is back for the users at full operation performance and with an excellent reliability. This paper reports on the present operation performance of the source, highlighting the ongoing and planned developments. INTRODUCTION The ESRF, located in Grenoble France, is a facility supported and shared by 22 partner nations. This light source, in operation since 1994 [1, 2, 3], has been delivering 5500 hours of beam time per year on up to 42 beam-lines. The chain of accelerators consists of a $2 0 0 \\mathrm { M e V }$ linac, a $4 ~ \\mathrm { H z }$ full-energy booster synchrotron and a $6 \\mathrm { G e V }$ storage ring (SR) $8 4 4 ~ \\mathrm { m }$ in circumference. A large variety of insertion devices (in-air, in-vacuum and cryo-in-vacuum undulators, as well as wigglers) [4] are installed along the 28 available straight sections. Bending-magnet radiation, now produced by short bends and wigglers, is used by 12 beamlines.
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Table: Caption: Table 1 Nominal (no IDs) and measured parameter values at the observation point, together with derived emittances and emittance ratio Maximum error margins are linearly added when deducing the maximum emittance and emittance ratio errors. Body: <html><body><table><tr><td>Parameter</td><td>Nominal value</td><td>Measured value</td><td>Max.error margin</td></tr><tr><td>σs (%)</td><td>0.086</td><td>1</td><td>+0.009/-0.000</td></tr><tr><td>βx (m)</td><td>0.452</td><td>0.431</td><td>±0.009</td></tr><tr><td>nx (mm)</td><td>29</td><td>27.3</td><td>±1.0</td></tr><tr><td>σex (μm)</td><td>56</td><td>57.3</td><td>±1.5</td></tr><tr><td>εx (nmrad)</td><td>5.6</td><td>6.3</td><td>+0.7/-0.9</td></tr><tr><td>βy (m)</td><td>14.3</td><td>13.55</td><td>±0.14</td></tr><tr><td>ny (mm)</td><td>0</td><td>2.3</td><td>±0.55</td></tr><tr><td>δeyo (μm)</td><td>1</td><td>6.8</td><td>±0.5</td></tr><tr><td>εy (pmrad)</td><td>1</td><td>3.2</td><td>±0.7</td></tr><tr><td>g (%)</td><td>1</td><td>0.05</td><td>±0.02</td></tr></table></body></html> The diagnostic beamline comprising the two optics schemes is described in Section 2. The pinhole camera scheme is still under development. Preliminary results have been presented elsewhere [4]. Here we place emphasis on the $\\pi$ –polarization method. The model for the SR emission and focusing is described in Section 3. In Section 4 we present measured data at SLS and compare it to the SRW predictions of a finite emittance beam. In Section 5 we perform the emittance determinations while estimating different error contributions. Finally in Section 6, we discuss whether the vertical emittance minimization is of local or global nature. 2. The diagnostic beamline The source point of the beamline is the centre of the middle-bending magnet in the SLS triple bend achromat lattice (see Table 1 for machine parameters). Fig. 1 shows a schematic top view of the beamline. The angular separation of the vis–UV branch and the $\\mathrm { \\Delta X }$ -ray branch is 5 mrad, corresponding to an arc length of $3 0 \\mathrm { m m }$ .
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Andersson_2008.pdf
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4. Beam size measurements All measurements presented are performed with $\\pi$ -polarized vis–UV range SR in $3 5 0 / 4 0 0 \\mathrm { m A }$ multi-bunch top-up operation mode corresponding to $0 . 8 6 / 0 . 9 8 \\mathrm { n C }$ per bunch (390 out of 480 buckets populated). Most measurements were performed during user operation with IDs at arbitrary positions/excitations (see Table 1 for machine parameters). Fig. 4 shows the beam size measurement application displaying an acquired image by the $\\pi$ -polarization method. An IEEE-1394 Firewire camera [34], using a SonyTM $1 / 3 ^ { \\prime \\prime }$ CCD chip of $1 0 2 4 \\times 7 6 8$ pixels and 8 bit resolution writes image data directly to the EPICS control system. A region of interest selected from the application defines an EPICS sub-array record containing relevant data that can be retrieved at a faster rate than the complete image. In this way images can be updated and evaluated at rates of up to $1 0 \\mathrm { H z }$ . Filter configurations allowing exposure times of $0 . 5 \\mathrm { m s }$ are mostly chosen. Vibrations of the beam or the experimental setup with frequencies of less than approx. $2 0 0 \\mathrm { H z }$ thus will not enter into the result of the beam size measurement. However, higher frequency vibrations will make the beam size appear larger than it is. Special care is taken to adjust the noise level and to check the linearity of the camera, since these properties can change slightly over time. The rms vertical beam size, $\\sigma _ { \\mathrm { e y } }$ , is derived from the summation of the pixel intensities within the vertical narrow corridor. A pre-SRW-calculated table is then used to convert the valley-to-peak intensity ratio to a value for $\\sigma _ { \\mathrm { e y } }$ . The rms horizontal beam size, $\\sigma _ { \\mathrm { e x } }$ , is derived from integrating over the pixel intensities within the horizontal broad corridor, encapsulating the whole image.
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Andersson_2008.pdf
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A one-dimensional profile of the intensity distribution through the two maxima, $I ( x _ { \\mathrm { m a x } } , y )$ , gives a distribution of the vertically polarized focused light that displays a dual peak separated by a zero minimum at the centre, $I ( x _ { \\mathrm { m a x } } , 0 ) = 0$ . A vertical beam size may be determined even for the smallest of finite vertical beam sizes where the minimum of the acquired image significantly remains nonzero. While results presented in Section 4 demonstrate support for the Chubar model, it is worth noting that, with the present set-up at SLS, results to an accuracy of within $10 \\%$ may already be achieved through use of the approximate model [36], which uses the square of Eq. (1) as the FBSF. For high current measurements a vertically thin ‘‘finger’’ absorber is inserted to block the intense mid-part of SR. It is incorporated into the model in Section 4. The vertical acceptance angle of 9.0 mrad, being slightly smaller than the total SR opening angles at the observed wavelengths, is also included in the model. However, these modifications only marginally affect the FBSF.
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Andersson_2008.pdf
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The FWHM value is used for converting to $\\sigma _ { \\mathrm { e x } }$ . During measurements small beam ellipse rotations, originating either from betatron coupling or from a local vertical dispersion, are sometimes present. In this case, the vertically measured quantity is sey0 such that sey0osey, since only the vertical size of the central beam region is observed. This quantity will be termed the central rms vertical beam size. Horizontally the result is unaffected by a beam rotation, since the full image is used to obtain $\\sigma _ { \\mathrm { e x } }$ . 4.1. Horizontal measurements While it is more challenging to determine the beam size in the vertical plane, it is also of interest to verify that the model predictions agree with measurements in the horizontal plane, where the beam size is typically much larger. Fig. 5 shows a profile of the horizontal image in its entirety. The lines are profile predictions calculated from the SRW model by convoluting the FBSF of the vertically polarized light with a Gaussian electron beam and then integrating over the entire image, as is done with the on-line monitor. In this particular example $\\sigma _ { \\mathrm { e x } } = ( 5 7 . 0 \\pm 1 . 5 ) \\mu \\mathrm { m }$ is measured at the monitor. The error margins represent the maximum systematic errors. The statistical rms error is considerably smaller, $0 . 3 \\mu \\mathrm { m }$ for a single sample. Of particular note is the good agreement in the tails, which deviate from a Gaussian shape (dashed line). This is as anticipated since the predicted profile is a convolution of a Gaussian and a function of approximate form $\\operatorname { s i n c } ^ { 2 } ( x )$ .
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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$$ where $\\gamma = E / m _ { \\mathrm { e } } c ^ { 2 }$ , $E$ is the electron energy, $\\lambda$ is the observed radiation wavelength, $\\lambda _ { \\mathrm { c } } = 4 \\pi R / 3 \\gamma ^ { 3 }$ is the critical wavelength, $R$ is the radius of the electron trajectory, $p$ and $p ^ { \\prime }$ are the distances from the source point to the lens and from the lens to the image plane, respectively, and $E _ { \\pi 0 }$ is a constant. Squaring Eq. (1) gives an intensity distribution in the image plane as shown in Fig. 2, where we have used numbers resembling our actual imaging scheme at SLS. This two-dimensional distribution function is of the form $f ( x ) g ( y )$ , where $f ( x ) = \\mathrm { s i n c } ^ { 2 } ( x )$ and $g ( y )$ is given by the square of the integral expression in Eq. (1). The model used to describe our $\\pi$ -polarization scheme was outlined by Chubar [37,38]. It is based on a near-field SR calculation at the first optical element, using the Fourier transform of the retarded scalar- and vector potentials [21], preserving all phase information as the electron moves along its trajectory. The integral theorem of Helmholtz and Kirchoff [23] is now applied to this Fourier transform (rather than the more usual spherical wave) at different apertures in the beamline. One benefit of this approach is that the model now includes, in a natural way, the so-called depth-of-field effect appearing in the image plane. Using Fourier optical methods, the SRW code, based on the described model, calculates the intensity distribution, $I ( x , y )$ , in the image plane. This distribution, resulting from a single relativistic electron, is termed the ‘‘filament-beam-spread function’’ (FBSF). It is equivalent to point-spread functions for optical systems in the case of virtual point sources. The intensity distribution is shown in Fig. 3, for the same SLS case as above, and is seen to no longer maintain the simple $f ( x ) g ( y )$ form. This is a consequence of the fact that the wavefront produced by the relativistic electron is more complicated than that of a point source [39].
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Andersson_2008.pdf
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File Name:DESIGN_OF_A_NON-INVASIVE_BUNCH_LENGTH_MONITOR_USING.pdf DESIGN OF A NON-INVASIVE BUNCH LENGTH MONITOR USING COHERENT SYNCHROTRON RADIATION SIMULATIONS C. Swain1,2,‚àó, J. Wolfenden1,2, L. Eley1,2, C. P. Welsch1,2 1Department of Physics, University of Liverpool, UK 2Cockcroft Institute, Warrington, UK Abstract Synchrotron radiation (SR) is a phenomena found in most accelerator facilities. Whilst many look to reduce the amount of SR produced to minimise beam losses, its existence allows for several types of novel non-invasive beam instrumentation. The aim of this study is to use SR in the development of a non-invasive, high resolution, longitudinal bunch length monitor. The monitor will be capable of sub 100 fs bunch measurements, which are becoming more common in novel acceleration and free electron laser facilities. This contribution details the simulation work carried out in Synchrotron Radiation Workshop (SRW), which allows for complex studies into the production and features of coherent synchrotron radiation (CSR). The design of the monitor has also been discussed, alongside simulations of the planned optical setup performed in Zemax OpticStudio (ZOS). INTRODUCTION As accelerator upgrades and novel acceleration have lead to multi-GeV beams and fs scale bunch lengths, new diagnostic options are needed to provide the resolution necessary to properly study them. One option under consideration across the beam instrumentation community is the utilisation of coherent synchrotron radiation (CSR). Synchrotron radiation is is produced in any facility where the beam passes through a magnetic field causing it to bend. Whilst SR can prove problematic for some operations, its availability as a possible non-invasive diagnostic can be very useful. Some sections of the radiation emitted are coherent, where the wavelength is equal to or greater than the bunch length $( \\lambda \\geq \\sigma )$ ). CSR is of specific interest for bunch length diagnostic applications, as the spectral content is directly affected by the charge distribution of the bunch.
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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$$ since the dispersive contributions to the particle distribution are of course correlated horizontally and vertically. Since the vertical beam size, $\\sigma _ { \\mathrm { e y 0 } }$ , is obtained from integration over a narrow corridor (width $\\leqslant \\sigma _ { \\mathrm { e x } }$ , see Fig. 4) the correction factor in Eq. (3) has to be applied to correctly de-convolute the dispersive contribution from the emittance contribution. In our case, however, the dispersive contribution is rather small, resulting in a rotation angle of only $\\vartheta = 1 4$ mrad of the beam ellipse, which is barely detectable with our experimental set-up (the corresponding vertical rms beam size is $\\sigma _ { \\mathrm { e y } } = 6 . 8 5 \\mu \\mathrm { m } \\approx \\sigma _ { \\mathrm { e y 0 } } )$ . In the horizontal, the full image is used to obtain $\\sigma _ { \\mathrm { e x } }$ and the simple de-convolution of Eq. (2) can be applied. In conclusion, we state that the vertical rms emittance at the observation point is $\\varepsilon _ { y } = ( 3 . 2 \\pm 0 . 7 )$ pmrad, where the error margins represent linearly added maximum systematic errors of measured quantities. Correspondingly, the emittance ratio, $g$ , is determined to be $g = ( 0 . 0 5 { \\pm } 0 . 0 2 ) \\%$ . With no skew quadrupoles excited, the vertical rms emittance is larger by a factor of 2.
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Expert
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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Table 1 summarizes the emittance determinations. Measured values for the machine functions and for the rms beam sizes are presented. From the estimated maximum error in the measurement we give maximum error margins for the different quantities. The beam relative energy spread, $\\sigma _ { \\delta }$ , is the only quantity not to be measured. However, an energy spread deviation from the natural one can be probed, and most likely excluded, since while modulating the RF levels in the Higher Harmonic Cavities [40], no effect was seen on the horizontal beam size. Still, we estimate a maximum value of $10 \\%$ increase due to possible RF noise. The dispersion values can be measured, assuming a known momentum compaction, by the camera with a $0 . 2 5 \\mathrm { m m }$ precision error. We use the $2 \\sigma$ values as an estimate of the maximum deviations. The influence on the dispersion of the momentum compaction uncertainty (maximum $2 \\%$ ) is $0 . 5 \\mathrm { m m }$ horizontally and $0 . 0 5 \\mathrm { m m }$ vertically. In comparison to the measured $\\eta _ { y } = 2 . 3 \\ : \\mathrm { m m }$ , spurious vertical dispersion is also measured at all BPMs, resulting in an rms value of $3 . 0 \\mathrm { m m }$ . The beta function values cannot be measured at the source point but only in the adjacent quadrupoles. We perform an entire measurement of the (average) beta functions in all 177 quadrupoles and use this to fit the model beta functions. From this we get the values at the observation point. The precision of the horizontal and vertical beta function value measurements are $1 \\%$ and $0 . 5 \\%$ , respectively. The $2 \\sigma$ values are used as an estimate of the maximum model deviation from the actual value at the observation point. The maximum deviations in the beam size values are estimated from systematic profile fitting errors and errors due to possible optics wavefront distortions.
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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The branch used for the $\\pi$ -polarization method has a maximum clearance of $7 \\mathrm { { m r a d } _ { H } \\times 9 \\mathrm { { m r a d } _ { V } } }$ . The vis–UV light is twice directed through $9 0 ^ { \\circ }$ angles due to space constraints. This arrangement is also of benefit for optical reasons as a partial polarization of the light is obtained from the substantial attenuation of the (horizontal) $\\sigma$ -polarization component. The first mirror is made of SiC, a material, which has an advantageous ratio of thermal conductivity to expansion (a factor 6 better than $\\mathrm { { C u ) } }$ . This allows for low current measurements $\\mathrm { \\Gamma } ( < 1 0 \\mathrm { m A } )$ , when the vertical gradient of the surface power density on the mirror is still moderate. At higher currents, however, the mirror must be protected from an excessive heat load that would otherwise result in its deformation. To achieve this, a horizontal ‘‘finger’’ absorber, of $4 \\mathrm { m m }$ height, has been inserted immediately before the mirror, while obstructing only the mid $\\pm 0 . 4 5$ mradV of the SR. This prevents $98 \\%$ of the $2 3 0 \\mathrm { W }$ power, at $4 0 0 \\mathrm { m A }$ current, reaching the mirror, while blocking only $1 \\%$ of the $\\pi$ -polarized spectral flux in the vis–UV range used for measurements. Cooling of the SiC mirror was not a viable option since mirror deformation would inevitably result from the large vertical temperature gradient. The second mirror is an angular movable aluminized fused silica (FS) mirror. Both mirrors have a surface flatness better than $3 0 \\mathrm { n m }$ peak to valley. A FS symmetric spherical lens is positioned $5 . 0 7 8 \\mathrm { m }$ from the source point, between the two mirrors. It has a surface accuracy of $4 0 \\mathrm { n m }$ peak to valley. The vis–UV light is guided out from within the vacuum region through a FS window at the end of the beamline, approx. $9 \\mathrm { m }$ from the source point. This serves to minimize the light footprint on the vacuum window, which, in the absence of optical window specifications, was deemed necessary in order to minimize the risk of wavefront distortions. Blades that determine the acceptance angle of the light are positioned in the locality of the lens. External to the vacuum, grey filters, narrow bandpass (BP) filters and a Glan–Taylor polarizer are all placed on remotely controlled rotational movers. These, together with a neighbouring Pointgrey FleaTM [34] CCD camera (pixel size $4 . 6 5 \\mu \\mathrm { m }$ ), can also be remotely moved longitudinally in order to adjust for a new image plane when different BP filters are used. The camera roll error is less than 10 mrad. Since the filters and polarizer are situated close to the image plane, the specifications for their surface accuracies need not be the most stringent, and can duly be purchased from off-the-shelf optical component vendors. The magnifications in the vis–UV branch for the chosen wavelengths are $0 . 8 5 4 / 0 . 8 4 1 / 0 . 8 2 0$ at $\\lambda = 4 0 3 / 3 6 4 / 3 2 5 \\mathrm { n m }$ , respectively.
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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To explore further the model predictions we measured horizontal image profiles — retaining the setup for vertically polarized light — for different horizontal apertures set by the blocking blades at the lens position. Fig. 6 shows the results, where we have plotted measured and predicted FWHM/2.355 of the images, against the inverse accepted horizontal SR opening angle. The solid line is the prediction from SRW, which is a convolution of an $\\sigma _ { \\mathrm { e x } } = 5 5 . 6 \\mu \\mathrm { m }$ Gaussian electron distribution and the calculated FBSF for the different opening angles. We have also indicated (dashed line) the result given from a convolution of the assumed electron beam distribution and a $\\mathrm { s i n c } ^ { 2 } ( x )$ distribution resulting from the simplified assumption of treating the filament beam as a far away point source (Fraunhofer diffraction case). For small acceptance angles the Fraunhofer approximation is correct, while for larger acceptance angles there is a clear discrepancy originating from the more complicated phase relations of the SR electric field emission over the arc, compared to a virtual point source. Even though it is a small effect at 5.9 mrad acceptance angle, we could still verify this experimentally with visible light $( 4 0 3 \\mathrm { n m } )$ . The effect is even more pronounced at shorter wavelengths. We also show the measured two-dimensional image (Fig. 7) at $\\lambda = 3 6 4 \\mathrm { n m }$ , and 5.9 mrad horizontal acceptance angle. Here we begin to see the predicted horizontal asymmetry (compare FBSF in Fig. 3; horizontal directions are reversed by the camera), even though it is heavily masked by the relatively large horizontal beam size.
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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6. Discussion So far we have tacitly assumed that the concept of horizontal and vertical emittance of a particle beam is valid. In the case of uncoupled motion, $\\mathit { \\varepsilon } _ { \\mathit { \\varepsilon } _ { x } }$ and $\\varepsilon _ { y }$ are defined as the two transverse rms phase space areas divided by $\\pi$ . In reality, there is always a coupled motion between the transverse phase space planes due to magnet misalignments. In this case, let $\\varepsilon _ { x }$ and $\\varepsilon _ { y }$ be the projections of the four-dimensional rms phase space volume onto the twodimensional horizontal and vertical phase spaces [41]. With this definition $\\varepsilon _ { x }$ and $\\varepsilon _ { y }$ are synonymous to the measured quantities. However, they are not invariants, but vary with the longitudinal coordinate, $s$ . Three equilibrium invariants, the normal mode emittances [42], $\\varepsilon _ { \\mathrm { I } } , \\varepsilon _ { \\mathrm { I I } }$ and $\\varepsilon _ { \\mathrm { I I I } }$ , may be calculated in the fully coupled case for an electron storage ring. Assuming only transverse coupling, $\\varepsilon _ { x } ( \\mathrm { s } )$ and $\\varepsilon _ { y } ( \\mathrm { s } )$ will approach $\\varepsilon _ { \\mathrm { I } }$ and $\\varepsilon _ { \\mathrm { I I } }$ , respectively, as we move towards a perfectly aligned machine. In the limit the (invariant) vertical emittance still has a finite value, set by the fundamental quantum nature of SR emission, which for SLS is $\\varepsilon _ { y } = 0 . 5 5$ pmrad.
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Expert
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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The emittances are deduced according to $$ \\varepsilon _ { x } = ( \\sigma _ { \\mathrm { e x } } ^ { 2 } - ( \\sigma _ { \\delta } \\eta _ { x } ) ^ { 2 } ) / \\beta _ { x } $$ and $$ \\varepsilon _ { y } = ( \\sigma _ { \\mathrm { e y 0 } } ^ { 2 } - ( \\sigma _ { \\delta } \\eta _ { y } ) ^ { 2 } ( 1 - ( \\sigma _ { \\delta } \\eta _ { x } / \\sigma _ { \\mathrm { e x } } ) ^ { 2 } ) ) / \\beta _ { y } . $$ In the presence of horizontal and vertical dispersion, $\\eta _ { x }$ and $\\eta _ { y }$ , the beam ellipse in the $( x , y )$ plane is not only widened but also rotated by an angle, W, given by $$ 1 2 \\vartheta = 2 \\eta _ { x } \\eta _ { y } \\sigma _ { \\delta } ^ { 2 } \\Big / ( \\sigma _ { \\mathrm { e x } } ^ { 2 } - \\sigma _ { \\mathrm { e y } } ^ { 2 } ) ,
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Expert
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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The sensitivity of the valley-to-peak intensity ratio at small vertical beam sizes can be slightly increased by either blocking a large part of the central SR or detecting at shorter wavelengths. Both methods bring the peaks of the FBSF closer together. We prefer the latter method, since it preserves the possibility to cross-check the optics quality from behaviour of the tails. In the former case, one is essentially moving towards a pure interferometer [15] method, where tails and possible beam rotations are being obscured by a growing fringe pattern. Hence, to further verify the small beam size measurements, additional measurements at two different wavelengths were undertaken. BP and grey filters were exchanged and the image plane was re-adjusted. Fig. 9 shows the predicted valley-topeak intensity ratios as functions of central vertical rms beam size at wavelengths of 325, 364 and $4 0 3 \\mathrm { n m }$ . For four different skew quadrupole settings the measured ratio at each wavelength has been superimposed onto the predicted curve. Error margins represent estimated maximum systematic errors from fitting and wavefront distortions. A $\\sigma _ { \\mathrm { e y 0 } }$ value and its error margins are given by a reading on the abscissa. There is agreement within the error margins, except for the $4 0 3 \\mathrm { n m }$ values at small vertical beam sizes. The explanation is that our $4 0 3 \\mathrm { n m }$ band pass filter had a broader transmission band (approx. $3 \\mathrm { n m }$ instead of $1 . 5 \\mathrm { n m }$ at FWHM). This illustrates the need for very narrow band pass filters if small vertical beam sizes are to be determined. The fact that the results agree well at 325 and $3 6 4 \\mathrm { n m }$ for the smallest vertical beam size, supports the conclusion that the optics quality is not limiting the resolution.
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Expert
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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The horizontal acceptance angle of the X-ray branch is 0.8 mrad. The water cooled pinhole array, fabricated from a $1 5 0 \\mu \\mathrm { m }$ thick tungsten sheet interspersed with $1 5 \\mu \\mathrm { m }$ diameter holes, is located $4 . 0 2 0 \\mathrm { m }$ from the source point. The light escaping these holes carries low power and can be released through a $2 5 0 \\mu \\mathrm { m }$ thick non-cooled aluminium window. Monochromating molybdenum filters and phosphor $( 6 \\mu \\mathrm { m }$ thick P43) are placed on a common optical table at the end of the beamline. The same type of camera as in the vis–UV branch is used to observe the phosphor via a zoom and focus adjustable lens system [35]. The magnification in the $\\mathrm { \\Delta X }$ -ray branch, to the phosphor screen, is 1.276. 3. SR imaging model The ideal goal would be to capture an exact image of the electron distribution in the transverse plane. However, certain features inherent to SR, such as a narrow vertical opening angle and radiation generation along the longitudinal electron trajectory, make this impossible. A more realistic scenario would be to form an image, which although affected by the afore-mentioned SR features, is nevertheless free from optical component aberrations. The transverse electron distribution could then be derived from a model that describes the image of a single electron, or ‘‘filament’’ beam. The acquired image is, to a good approximation, given by the convolution of the single electron image and the transverse electron distribution. Conversely, the transverse electron distribution is a deconvolution of the acquired image with the ‘‘filament’’ beam image. For stable beam conditions, the transverse electron distribution can be assumed to be a twodimensional Gaussian of unknown widths, which simplifies the de-convolution.
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What is SRW (Synchrotron Radiation Workshop)?
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SRW (Synchrotron Radiation Workshop) is a wave optics simulation code used to model synchrotron radiation emission and propagation through optical systems.
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Fact
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Andersson_2008.pdf
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A larger number of skew quads in the storage ring, would increase the risk of building up local coupling bumps when applying this straightforward, empirical method of minimization. Other methods, such as measurements of the fully coupled response matrix and the SVDbased minimization of the off-diagonal elements using all skew quads, would therefore have to be applied to assure a global minimization. 7. Conclusions We have described a method using vertically polarized SR in the visible to ultra-violet range, capable of determining vertical rms beam sizes below $7 \\mu \\mathrm { m }$ . The SLS can operate in standard user $4 0 0 \\mathrm { m A }$ multi-bunch $( 0 . 9 8 { \\mathrm { n C } } /$ bunch) top-up mode over several days providing a vertical rms emittance of $\\varepsilon _ { y } = ( 3 . 2 \\pm 0 . 7 )$ pmrad, or an emittance ratio of $g = ( 0 . 0 5 \\pm 0 . 0 2 ) \\%$ . It was also shown that the achieved vertical emittance is not due to a local minimization, but rather a global one. Acknowledgements We would like to thank Dr. Oleg Chubar and Prof. Leonid Rivkin for encouraging discussions and Martin Rohrer for the mechanical design of the beamline.
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Carilli_2024.pdf
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We can apply this reasoning to a particle in our system. We can compute integrals on the expression of the separatrices to evaluate the area of each region based on the value of $\\eta$ , and trace the evolution of the actions of the particles. We see that the process is di!erent if $| \\alpha | \\le 1$ or if $| { \\boldsymbol { \\alpha } } | > 1$ . In the first case (the process for $\\alpha = 0$ is outlined in Fig. 2) it is possible to prove that, if the initial value of the rescaled action is $J _ { 0 }$ , it Figure 1: Transition of phase-space topologies as a function of the main parameters of the Hamiltonian of Eq. (2). The behaviour for $\\alpha < 0$ can be retrieved using the transformation $\\alpha \\to - \\alpha$ , $\\eta - \\eta$ (adapted from Ref. [10]). Table: Caption: Table 1: Analysis of the stability type of the fixed points of the Hamiltonian of Eq. (2) (from Ref. [10]). Body: <html><body><table><tr><td>Solution</td><td>Stability type stable</td><td>unstable</td></tr><tr><td>Φ=0,π</td><td>α≤1 and -1<η<1-2α</td><td>α≥1 and 1-2α<η<-1</td></tr><tr><td>π3π 2'2 Φ=</td><td>α≥-1 and -1-2α<η<1</td><td>α≤-1 and 1<η<-1-2α</td></tr><tr><td>J=1 J=0</td><td>never ln|>1</td><td>always lnl<1</td></tr></table></body></html> will ultimately need to become $1 - J _ { 0 }$ . This means that, going back to the original coordinates, the final $I _ { x }$ corresponds to the initial $I _ { y }$ , and vice versa; therefore, for a given particle distribution, the two emittances are exchanged. For example, following the scheme shown in Fig. 2, a particle starting at action $J _ { 0 }$ will be trapped inside one lobe enclosed by a green separatrix or one of the blue. In either case, the new action $J _ { 1 }$ will be given by the area of the region at the crossing instant: we have $J _ { 1 } = J _ { 0 } / 2$ for the eight-shaped region or $J _ { 1 } = ( 1 - J _ { 0 } ) / 2$ for the lateral lobe. Later, as the separatrices shrink, the particle is released in the circle, assuming the area $J _ { 2 } = 1 - 2 J _ { 1 } = 1 - J _ { 0 }$ in the first case or $J _ { 2 } = 2 J _ { 1 } = 1 - J _ { 0 }$ in the second. This is not always true if $| { \\boldsymbol { \\alpha } } | > 1$ : the presence of more complex phase space structures (as in the central scheme of the first row of Fig. 1) results in the fact that only a fraction of particles will have a final action of $1 - J _ { 0 }$ , thus contributing to the emittance exchange. All the mathematical details of these area and probability calculations are given in Ref. [10].
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Carilli_2024.pdf
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For each optics, we mainly look for the optimal Working Point (WP), i.e. the combination of horizontal and vertical tunes $( Q _ { x } , Q _ { y } )$ that yields the highest DA. It is however paramount to consider that, when doing a tune scan, a split — the difference between the fractional part of the horizontal and vertical tunes — of at least $5 \\times 1 0 ^ { - 3 }$ is required to avoid loss of transverse Landau damping, hence beam instabilities from impedance, due to $x - y$ coupling [12]. The upper tune split diagonal is represented as a dashed blue line in all relevant plots. In addition, we choose a target of $6 ~ \\sigma$ , represented as a green contour in all relevant plots. Beam Separation Collapse For beam stability, the most critical phase of the cycle is the collapse of the beam separation bumps. During this phase, beams go from being fully separated in all IPs, to fully HO in IPs 1 and 5, partially HO in $\\mathrm { I P } 8$ to reach a target luminosity of $2 \\times 1 0 ^ { 3 3 } \\mathrm { c m } ^ { - 2 } / \\mathrm { s }$ , and partially HO in $\\mathrm { I P } 2$ to keep $5 \\sigma$ of separation.
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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ION INSTABILITIES AT DIFFERENT STAGES WITH DIFFERENT FILLING PATTERNS Di!erent filling patterns have been studied and the chosen stable filling patterns are listed in Table. 2. The first two stages are considered early commissioning stages when vacuum conditions are not optimal and transverse multi-bunch feedback (TMBF) and harmonic cavity (HC) are unavailable. For these stages, higher chromaticity and more ion clearing gaps are needed to maintain beam stability [3]. At stage 3, normal operation is expected and it has been verified that the standard filling pattern is stable at nominal chromaticity with the aid of TMBF and HC. Table: Caption: Table 2: Stable Filling Patterns at Each Vacuum Conditioning Stage Body: <html><body><table><tr><td>Stable Filling Patterns</td></tr><tr><td>Stage 1 7-bucket gap,20 trains 50 mA, chromaticity 5 7-bucket gap,30 trains</td></tr><tr><td>w/o TMBF,HC Stage 2 7-bucket gap,20 trains 100 mA,chromaticity 5</td></tr><tr><td>7-bucket gap,30 trains w/o TMBF,HC</td></tr><tr><td>Stage 3 Standard filling pattern 300 mA,chromaticity 2 (7-bucket gap,5 trains)</td></tr></table></body></html> The beam vertical oscillation amplitude in the presence of ions at di!erent stages are illustrated in Fig. 2. In stages 1 and 2, the beam is tracked for 5,000 turns using two distinct filling patterns. It has been found that increasing the number of gaps is slightly beneficial to mitigate the ion instability in the early stages. To align with the standard filling pattern, which has 7-bucket gaps, filling patterns with 7-bucket gaps and 20/30 trains are deemed suitable for the early stages. In stage 3, the beam with the standard filling pattern is tracked for 10,000 turns. In all cases, the vertical emittance remains stable at approximately $8 \\mathrm { p m }$ . The maximum oscillation amplitude is less than $5 \\%$ of the nominal beam size for stages 1 and 2. With the aid of TMBF, the maximum oscillation amplitude in stage 3 is even less than $2 . 5 \\%$ of the nominal beam size.
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Carilli_2024.pdf
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$$ When $| p _ { \\theta } | < p _ { \\theta } ^ { \\mathrm { c r i t } }$ , the polynomial $\\mathcal { P } _ { 8 } ( r )$ has three positive roots: $$ 0 < r _ { * } ^ { \\mathrm { s t } } < r _ { * } ^ { \\mathrm { s e p } } < 1 < r _ { * } ^ { \\mathrm { u n } } . $$ Here, $r _ { * } ^ { \\mathrm { s e p } }$ corresponds to the unstable fixed point with a separatrix that isolates stable trajectories and $r _ { * } ^ { \\mathrm { u n } }$ is the second unstable fixed point. When the absolute value of the angular momentum exceeds the critical value, the two equilibria $r _ { * } ^ { \\mathrm { s t } }$ and $r _ { * } ^ { \\mathrm { s e p } }$ collide and annihilate in a saddle-node bifurcation. For $| a | > 2$ , there is only one unstable equilibrium $r _ { * } ^ { \\mathrm { u n s t } } > 1$ and the global dynamics is unstable, thus we will omit any further consideration.
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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Table: Caption: Table 1: Averaged Total Pressure at Di!erent Vacuum Condition Stages Body: <html><body><table><tr><td></td><td>Active NEG Coating</td><td>Saturated NEG Coating</td></tr><tr><td>Stage 1</td><td>0.9 nTorr</td><td>3.8 nTorr</td></tr><tr><td>Stage 2</td><td>0.5 nTorr</td><td>2.7 nTorr</td></tr><tr><td>Stage 3</td><td>0.5 nTorr</td><td>2.9 nTorr</td></tr></table></body></html> The normalised maximum vertical oscillation amplitude is shown in Fig. 1. The results indicate that the simplified model provides an acceptable level of agreement, slightly overestimating the ion e!ects compared to elementby-element tracking. As such, the ILMATRIX model can be used to study the upper bound of the ion instability, sacrificing accuracy for faster execution time. If a filling pattern is stable in the ILMATRIX model, it should be stable in element-by-element tracking. This simplified model has been used to scan filling patterns and settings in di!erent scenarios. The stable filling patterns will be verified with element-by-element tracking in future studies. VACUUM CONDITIONING STAGES As a first approximation of how vacuum conditioning might take place we have studied the following three stages: Stage 1: at $1 \\mathrm { A } \\cdot \\mathrm { h }$ vacuum condition, using $5 0 \\mathrm { m A }$ beam to run for 180 hours until $1 0 \\mathrm { A \\cdot h }$ . Stage 2: at $1 0 \\mathrm { A \\cdot h }$ vacuum condition, using $1 0 0 \\mathrm { m A }$ beam to run for 900 hours until $1 0 0 \\mathbf { A } \\cdot \\mathbf { h }$ .
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augmentation
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NO
| 0
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IPAC
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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The transverse splitting of the MTE only works with a beam of low momentum spread. Consequently the RF voltage must be small $( V _ { \\mathrm { R F } } \\simeq 1 2 \\mathrm { k V }$ from one single cavity) during the about $1 5 0 ~ \\mathrm { m s }$ of the process. These are very unfavourable conditions for the evolution of longitudinal coupled-bunch instabilities, and quadrupolar coupled-bunch bunch oscillations had been previously observed at intensities as low as $1 . 7 \\cdot 1 0 ^ { 1 3 }$ protons [8]. An important reduction of the RF cavity impedances by evolved feedback has been achieved with the upgrades in the framework of the LIU project [5, 9]. However, the synchronization of the barrierbucket with the kicker rise times and the circulating beam in the SPS requires the critical low-voltage phase to be performed at fixed RF frequency. This practically excludes the application of a beam phase loop, as well as conventional feedback to improve longitudinal stability. Front Porch To fit the time required for the barrier-bucket manipulation and synchronization within the constraint of the overall cycle duration (1.2 s), the low- and medium energy parts are compressed as much as possible (Fig. 1). Hence no controlled longitudinal emittance blow-up to modify the distribution for improved longitudinal stability can be applied before the arrival on the intermediate plateau (Fig. 1, grey shaded area). Strong dipole coupled-bunch oscillations evolve before the bunch-pair splitting, as illustrated in Fig. 2. The dipole coupled-bunch oscillations rising before the bunchpair splitting are probably seeded by the residual phase and energy errors of the bunches injected from the four different PSB rings. Increased controlled longitudinal emittance blow-up starting from the arrival at the plateau stabilizes them, but at the expense of a longitudinal emittance too large for transition crossing on harmonic $h = 1 6$ .
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augmentation
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NO
| 0
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IPAC
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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Table: Caption: Table 1: Stability Under Different P Values Body: <html><body><table><tr><td>Value of P</td><td>Critical ω</td><td>Stability (%)</td></tr><tr><td>0</td><td>3 kHz</td><td>-1.5-0.7</td></tr><tr><td>0.4</td><td>6 kHz</td><td>-0.8-0.5</td></tr><tr><td>0.8</td><td>10 kHz</td><td>-0.5-0.3</td></tr><tr><td>1.2-2</td><td>10 kHz</td><td>-0.45-0.3</td></tr><tr><td>>2</td><td>/</td><td>unstable</td></tr></table></body></html> The impact of the I term is also investigated. By varying the I value, the simulation results reveal that the I value is the primary determinant of control performance and convergence speed. Specifically, the proposed algorithm is compared with a P-type iteration-learning-control (ILC) algorithm, where the integral term is not applied in the feedforward table, but still exhibits good robustness and effectiveness [3]. The simulation results, presented in Fig. 6, clearly demonstrate that the convergence speed of the proposed controller is much faster than that of P-ILC. Study on the Non-causal Filter It can be seen from Fig. 5 there still remains a small oscillation in the early stage of the cycle, though it decreases with the increase of filter band-width. This phenomenon may be attributed to the phase lag of the FIR filter [4]. As a consequence, the filtered signal will experience a time delay in the time domain relative to the original signal. Alternatively, this could be mitigated by using a non-causal filter without any phase lag, such as a zero-phase FIR/IIR filter or a non-causal moving average filter. Fig. 7 compares the control performance with different filters. The results demonstrate that the oscillations have been nicely suppressed by either type of the non-causal filters. The peak-to-peak stability is improved to $\\pm 0 . 1 5 \\%$ .
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augmentation
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NO
| 0
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Expert
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
|
In radio interferometry, the voltages at each element are measured by phase coherent receivers and amplifiers, and visibilities are generated through subsequent cross correlation of these voltages using digital multipliers (Thomson, Moran, Swenson 2023; Taylor, Carilli, Perley 1999). In the case of optical aperture masking, interferometry is performed by focusing the light that passes through the mask ( $=$ the aperture plane element array), using reimaging optics (effectively putting the mask in the far-field, or Fraunhofer diffraction), and generating an interferogram on a CCD detector at the focus. The visibilities can then be generated via a Fourier transform of these interferograms or by sinusoidal fitting in the image plane. However, the measurements can be corrupted by distortions introduced by the propagation medium, or the relative illumination of the holes, or other effects in the optics, that can be described, in many instances, as a multiplicative element-based complex voltage gain factor, $G _ { a } ( \\nu )$ . Thus, the corrupted measurements are given by: $$ V _ { a b } ^ { \\prime } ( \\nu ) = G _ { a } ( \\nu ) V _ { a b } ( \\nu ) G _ { b } ^ { \\star } ( \\nu ) ,
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4
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NO
| 1
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Expert
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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These images show the expected behaviour, with the diffraction pattern covering more of the CCD for the $3 \\mathrm { m m }$ hole image vs. the 5mm hole. Note that the total counts in the field is very large (millions of photons), and hence the Airy disk is visible beyond the first null, right to the edge of the field. This extent may be relevant for the closure phase analysis below. Figure 4 shows the corresponding image for a 5-hole mask with 3 mm holes. The interference pattern is clearly more complex given the larger number of non-redundant baselines sampled $\\mathrm { \\Delta N _ { b a s e l i n e s } = ( N _ { h o l e s } * ( N _ { h o l e s } - 1 ) ) / 2 = 1 0 }$ for $\\mathrm { N } _ { \\mathrm { h o l e s } } = 5$ ). B. Fourier Domain Data are acquired as CCD two-dimensional arrays of size $1 2 9 6 \\times 9 6 6$ . We first remove the constant offset which is due to a combination of the bias and the dark current. We use a fixed estimate of this offset obtained by examination of the darkest areas of the CCD and the FFT of the image. We find a bias of 3.7 counts per pixel. Errors in this procedure accumulate in the central Fourier component, corresponding to the zero spacing, or total flux (u,v = 0,0), and contribute to the overall uncertainty of the beam reconstruction.
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4
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NO
| 1
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Expert
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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File Name:Carilli_2024.pdf Deriving the size and shape of the ALBA electron beam with optical synchrotron radiation interferometry using aperture masks: technical choices Christopher L. Carilli∗ National Radio Astronomy Observatory, P. O. Box 0, Socorro, NM 87801, US Laura Torino† and Ubaldo Iriso‡ ALBA - CELLS Synchrotron Radiation Facility Carrer de la Llum 2-26, 08290 Cerdanyola del Vall\\`es (Barcelona), Spain Bojan Nikolic§ Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK Nithyanandan Thyagarajan Commonwealth Scientific and Industrial Research Organisation (CSIRO), Space & Astronomy, P. O. Box 1130, Bentley, WA 6102, Australia (Dated: June 2024) We explore non-redundant aperture masking to derive the size and shape of the ALBA synchrotron light source at optical wavelengths using synchrotron radiation interferometry. We show that nonredundant masks are required due to phase fluctuations arising within the experimental set-up. We also show, using closure phase, that the phase fluctuations are factorizable into element-based errors. We employ multiple masks, including 2, 3, 5, and 6 hole configurations. We develop a process for self-calibration of the element-based amplitudes (square root of flux through the aperture), which corrects for non-uniform illumination over the mask, in order to derive visibility coherences and phases, from which the source size and shape can be derived. We explore the optimal procedures to obtain the most reliable results with the 5-hole mask, based on the temporal scatter in measured coherences and closure phases. We find that the closure phases are very stable, and close to zero (within $2 ^ { o }$ ). Through uv-modeling, we consider the noise properties of the experiment and conclude that our visibility measurements per frame are likely accurate to an rms scatter of $\\sim 1 \\%$ .
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4
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NO
| 1
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Expert
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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Figure 13 also shows the closure phase for the 3-hole data, which has only one triad (holes 0-1-2). The 3-hole closure phase for triad 0-1-2 has a mean of $- 1 . 1 6 ^ { o }$ with an RMS of the time series of $0 . 2 7 ^ { o }$ . For comparison, the values for the 5-hole data for this triad were $- 2 . 1 7 ^ { o }$ and $0 . 5 9 ^ { o }$ . The values ought to be the same, all else being equal. The difference could arise from: (i) the source changed (unlikely), (ii) the geometry of the mask changed (could only be a rotation): but the uv-sampling points found by photom are within 0.1 pixels, (iii) the centering of the diffraction pattern on the CCD is different, which leads to a different sampling of the outer Airy disk. We are investigating these phenomena. For now, we can conclude is that $\\sim 2 ^ { o }$ is the limit to a reliable closure phase measurement from experiment to experiment, for the current data. Table: Caption: Body: <html><body><table><tr><td>Triad</td><td>Mean Closure Phase RMS degrees degrees</td></tr><tr><td>0-1-2</td><td>-2.17 0.59</td></tr><tr><td>0-1-3</td><td>-1.19 0.42</td></tr><tr><td>0-1-4</td><td>0.09 0.36</td></tr><tr><td>0-2-3</td><td>1.44 0.38</td></tr><tr><td>0-2-4</td><td>2.11 0.39</td></tr><tr><td>0-3-4</td><td>0.74 0.30</td></tr><tr><td>1-2-3</td><td>0.46 0.49</td></tr><tr><td>1-2-4</td><td>-0.15 0.68</td></tr><tr><td>1-3-4</td><td>-0.54 0.40</td></tr><tr><td>2-3-4</td><td>0.07 0.37</td></tr></table></body></html>
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2
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NO
| 0
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IPAC
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
|
Carilli_2024.pdf
|
$$ \\begin{array} { c } { { J _ { x } ^ { 1 / 2 } = ( - 1 ) ^ { k + 1 } \\displaystyle \\frac { 3 | { \\cal G } | } { 4 \\alpha _ { x x } } ( 1 \\pm \\sqrt { 1 - \\displaystyle \\frac { 1 6 \\alpha _ { x x } \\delta } { 9 { \\cal G } ^ { 2 } } } ) , } } \\\\ { { \\phi _ { x } = \\displaystyle \\frac { k \\pi - \\phi _ { 0 } } { 3 } , } } \\end{array} $$ where $k = \\pm 1 , 3$ and $k = \\pm 2 , 0$ for either three SFPs or unstable fixed points (UFPs). Equation (3) implies $J _ { x } ^ { 1 / 2 }$ has four possible solutions but only two are physical because $J _ { x } ^ { 1 / 2 } > 0$ . Therefore, one solution is for SFPs and the other for UFPs. If the solution of the action of the SFPs $( J _ { x \\_ S F P } )$ is either too small $( \\sim 0 )$ or too large (exceed the physical aperture), no TRIBs will be observed. If two solutions of 𝑆𝐿1/2 requires di!erent 𝑏 (𝑏1 = ±1, 3 and 𝑏2 = ±2, 0), three
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1
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NO
| 0
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Expert
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
|
Carilli_2024.pdf
|
$$ V _ { a b } ( \\nu ) = \\int _ { \\mathrm { s o u r c e } } A _ { a b } ( \\hat { \\bf s } , \\nu ) I ( \\hat { \\bf s } , \\nu ) e ^ { - i 2 \\pi { \\bf u } _ { a b } \\cdot \\hat { \\bf s } } \\mathrm { d } \\Omega , $$ where, $a$ and $b$ denote a pair of array elements (eg. holes in a mask), $\\hat { \\pmb s }$ denotes a unit vector in the direction of any location in the image, $A _ { a b } ( \\hat { \\mathbf { s } } , \\nu )$ is the spatial response (the ‘power pattern’) of each element (in the case of circular holes in the mask, the power pattern is the Airy disk), ${ \\mathbf { u } } _ { a b } = { \\mathbf { x } } _ { a b } ( \\nu / c )$ is referred to as the “baseline” vector which is the vector spacing $\\left( { \\bf x } _ { a b } \\right)$ between the element pair expressed in units of wavelength, and $\\mathrm { d } \\Omega$ is the differential solid angle element on the image (focal) plane.
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augmentation
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NO
| 0
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Expert
|
What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
|
Carilli_2024.pdf
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Masks were made with hole diameters of 3mm and 5mm, to investigate decoherence caused by possible phase fluctuations across a given hole. Observations were made with integration times (frame times) of 1 ms and 3 ms, to investigate decoherence by phase variations in time. Thirty frames are taken, each separated by 1 sec. We estimate the pixel size in the CCD referenced to the source plane of 0.138 arcsec/pixel, using the known hole separations (baselines), and the measured fringe spacings, either in the image itself, or in the Fourier transformed u,v distribution. IV. STANDARD PROCESSING AND RESULTS A. Images Figure 3 shows two images made with the 3-hole mask, one with 3 mm holes and one with 5 mm holes. Any three hole image will show a characteristic regular grid diffraction pattern, modulated by the overall power pattern of the individual holes (Thyagarajan $\\&$ Carilli 2022). This power pattern envelope (the ’primary beam’ for the array elements), is set by the hole size and shape, which, for circular holes with uniform illumination, appears as an Airy disk. The diameter of the Airy disk is $\\propto \\lambda / D$ , where $\\lambda$ is the wavelength and $D$ is the diameter. Also shown in Figure 3 are the Fourier transforms of the images (see Section IV B). The point here is that the size of the uv-samples decreases with increasing beam size = decreasing hole size. The primary beam power pattern (Airy disk) multiples the image-plane, which then corresponds to a convolution in the uv-plane. So a smaller hole has a larger primary beam and hence a smaller convolution kernel in the Fourier domain.
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augmentation
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NO
| 0
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Expert
|
What is a stable closure phase?
|
A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
|
Fact
|
Carilli_2024.pdf
|
IX. SUMMARY AND FUTURE DIRECTIONS A. Summary We have described processing and Fourier analysis of multi-hole interferometric imaging at optical wavelengths at the ALBA synchrotron light source to derive the size and shape of the electron beam using non-redundant masks of 2, 3, and 5 holes, plus a 6-hole mask with some redundancy. The techniques employed parallel those used in astronomical interferometry, with the addition of gain amplitude self-calibration. Self-calibration is possible in the laboratory case due to the vastly higher number of photons available relative to the astronomical case. We have considered varying hole size and varying frame time. The main conclusions from this work are: • The size of the Airy disk behaves as expected for changing hole sizes. There are many photons (millions), such that the diffraction pattern is sampled beyond the first null of the Airy disk, to the edge of the CCD field. • We develop a technique of self-calibration assuming a Gaussian model to simultaneously solve for the source size and the relative illumination of the mask (the hole-based voltage gains). The gains are stable to within $1 \\%$ over 30 seconds, and relative illumination of different holes can differ by up to $3 0 \\%$ in voltage solutions. Hence, gain corrections are required to derive visibility coherences, and hence the source size. • We show visibility phases have a peak-to-peak variation over 30 seconds of $\\sim 5 0 ^ { o }$ . Further, coherences for 3 ms frame-times for the 5-hole data are systematically lower than those for $1 \\mathrm { m s }$ frame time by up to $1 0 \\%$ , and the 3 ms coherences are much noisier than $1 \\mathrm { m s }$ . We also find the phase fluctuations are correlated on two longer and similar baselines.
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augmentation
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NO
| 0
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