$~500~µJ is t$_\\textnormal{n}~>$~2ms, whereas for the 105~µm fiber, the nucleation time is only half. Therefore there is increased heat dissipation for the 50~µm fiber, and thus a smaller efficiency. Therefore, the growth rate no longer increases linearly with the delivered energy, and larger growth rates require much more energy for the 50~µm fiber. \n\nInterestingly, extrapolating the linear fit of the 50~µm fiber in Figure~\\ref{fig: CW growth rates fibers}B shows a positive growth rate for zero energy, which is physically not possible. However, due to thermal dissipation, it is questionable whether the growth rate for the CW-laser would actually be linear with the energy, as thermal dissipation may reduce the energy efficiency. Therefore, even though the initial behaviour for both fibers seems linear, extrapolating this to find an energy threshold does not seem to have a physical meaning.\n\n\n\\begin{figure}[t!]\n \\centering\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/CW_growth_rate_incl_200-eps-converted-to.pdf}\n\n\\end{minipage}\n\\vline\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/CW_growth_rate_excl_200-eps-converted-to.pdf}\n\n\n\\end{minipage}\n \\caption{Average volumetric bubble growth rate for the CW laser. Figure (A) includes all three fiber sizes. As the energy and growth rates for the 200~µm fiber are much larger compared to the 50 and 105~µm fiber, figure (B) shows a close-up of the left bottom of (A), indicated by the dotted lines. Figure (B) includes the best linear fit for the 50 and 105~µm fiber.}\n \\label{fig: CW growth rates fibers}\n\\end{figure} \n\n\\subsubsection*{Secondary bubbles}\nFor small bubbles generated by the CW laser (R~$<$~150~µm), we observe the formation of a secondary bubble at the vapor-liquid interface, see~Figure~\\ref{fig: Secondary bubble}. These events are different from twin cavitation bubbles reported in Ref.~\\cite{Zhang2022}, where the initial bubble first partially collapses before forming a secondary bubble. In our case, the secondary bubble typically forms when the initial bubble reaches its maximum. Furthermore, in our case the volume of the secondary bubble is smaller compared to the first one, for which reason its appearance does not affect the parabolic shape of the bubble volume versus time.\nThese secondary bubbles can be explained by the fact that the laser remains irradiating during the bubble formation and, therefore, further heats the liquid during the bubble growth. In the case of large laser powers and short nucleation times, this heating during the bubble lifetime could create a secondary bubble. In the case of Figure~\\ref{fig: Secondary bubble}, the growth time of the bubble takes 40~µs, whereas the initial nucleation time was 135~µs, which means that after nucleation, another 30\\% of the initial energy is still delivered, which results in secondary nucleation at the vapor-liquid interface of the bubble. \n\nThe occurrence of these secondary bubbles depends on the laser power and nucleation time of the initial bubble. The secondary bubble forms during the lifetime of the bubble (typically 50-200~µs), during which enough additional energy has to be delivered to form a secondary bubble. We find that these secondary bubbles only appear when the intensities are sufficiently larger such that the initial nucleation time is smaller than approximately 450~µs, both for the 50 and 105~µm fiber. For the 200~µm fiber the intensities are smaller and thus the nucleation times are much larger we did not observe any secondary bubbles. However, we hypothesize that the use of a more powerful laser can also form these secondary bubbles with the 200~µm fiber. However, as mentioned, these secondary bubbles only happen for small nucleation times and large powers, which result in small bubbles overall. Therefore, we conclude that the larger and faster-growing primary bubbles are of more interest. \n\n\n\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}{.45\\textwidth}\n \\centering\n \\hspace{-1cm}\n \n \\includegraphics[width=1.05\\linewidth]{Figures/Secondary_Bubble_50_5_3-eps-converted-to.pdf}\n\n\\end{minipage}\n\\vline\n\\begin{minipage}{.45\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/Secondary_bubble_volume_curve_50_5_3-eps-converted-to.pdf}\n \n\\end{minipage}\n \\caption{\\textbf{Left:} 16 consecutive frames showing the typical dynamics of a small bubble generated by the CW laser. As the laser is still running during the bubble growth, more liquid will be heated, which results in a secondary (frame 7-12) and even a third bubble (frame 13-14). Snapshots are consecutive, from top to bottom and left to right, with an inter-frame time of 5~ms. The initial nucleation time (top left frame) was 135~µs, and the bubble reached its maximum size (frame 8) at 175~µs. Scalebar on the right bottom measures 100~µm. \\textbf{Right:} Calculated bubble volume for each frame, showing that the initial maximum is reached at frame 6, but due to secondary bubble formation, another maximum is reached at frame 8, although the difference is small.}\n \\label{fig: Secondary bubble}\n\\end{figure}\n\n\\subsection{Comparison between pulsed and CW}\n\nA direct comparison between the bubbles generated by the pulsed and CW laser requires a matching absorption coefficient such that the volumes over which the energy is absorbed are identical. Therefore, we compare the results of the 10~mM ARAC solution, as its absorption coefficient at 532~nm is nearly identical compared to the absorption coefficient of water at 1950~nm. In this subsection, we compare the bubbles generated by the 50 and 105~µm fiber, as they show similar results in the growth rate (1~-~$7*10^{-7} \\textnormal{m}^3/s$). The 200~µm fiber shows very different results for the CW laser, as the delivered energies are much larger (see Figure \\ref{fig: CW growth rates fibers}). \n\n\\subsubsection*{Energy efficiency}\nBudgeting of energy has practical implications on the right choice of laser source.\nThe growth rates for the CW and pulsed laser are shown in Figure~\\ref{fig: GR_pulsed_CW_50_105}, for the 50 and 105~µm fiber. For the same delivered optical energy, the pulsed laser results in faster growth rates, although the required energies are in the same order of magnitude. Typically, the CW laser requires two to three times more optical energy than the pulsed laser to generate a bubble with the same growth rate. The reduced efficiency is explained by three reasons. First of all, for the bubbles generated by the CW laser heat diffusion cannot be neglected, as the nucleation time is 0.1~-~5~ms, which is in the same order as the thermal diffusion timescale ($\\sim$~4~ms, see Equation~\\ref{eq: thermal diffusion}). Most of this heat will be lost, as it will heat up the liquid around the irradiated volume. Though, this heat increase in the proximate area would be limited, and most of it does not result in the phase transition. Second, the absorption coefficient of water around 1950~nm decreases with increasing temperature, and the absorption coefficient at 100$\\degree$C is only half the initial value at room temperature~\\cite{Jansen1994,Lange2002}. Therefore, for the heating phase with the CW laser, the average absorption length is much larger compared to the absorption length of the pulsed laser experiments.\nThird, for the pulsed laser, there might be non-linear absorption due to the high peak power. The measured absorption coefficient of the 10~mM ARAC is the same as the absorption coefficient of water for low optical intensities, but the absorption coefficient for the high-power pulses may be larger due to additional non-linear absorption by the liquid. This increased absorption would result in a smaller irradiated volume and thus a higher energy density, which will increase the bubble growth rate.\n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{Figures/Growth_rate_pulsed_CW_50_105-eps-converted-to.pdf}\n \\captionsetup{width=0.5\\linewidth}\n \\caption{Growth rate of bubbles generated by the pulsed (red) and CW laser (blue), for fibers with core diameters of 50 (dark colors) and 105~µm (light colors). The absorption coefficients for both laser are the same ($\\alpha$~$\\approx$~12000~m$^{-1}$).}\n \\label{fig: GR_pulsed_CW_50_105}\n\\end{figure} \n\n\\subsubsection*{Bubble dynamics}\nBubble dynamics govern the subsequent microfluidic jet parameters.\nAs shown above, the typical growth rates are similar for both lasers. However, identical average growth rates may not result in the exact same bubble characteristics. An increase in growth time, results in a larger bubble, assuming the average growth rate is the same. The maximum volume against the average growth rate is shown in Figure~\\ref{fig: GR_MV_pulsed_CW} for the bubbles generated by the pulsed (red) and CW (blue) laser. For identical average growth rates, the bubbles created by CW laser grow to a $\\sim 15\\%$ larger volume (due to the increased growth time). Alternatively, for bubbles with the same maximum volume, the bubbles created by the CW laser take longer to reach that volume. This reduced growth rate could be explained by the larger volume over which the delivered energy is dissipated, resulting in a less explosive phase transition. However, this also means that for the same average growth rate, the bubbles generated by the CW laser will push out the remaining liquid over a longer length and time. This increased time of energy transfer will mostly affect the jet tail, which is typically slower and more dispersed~\\cite{Krizek2020,AndrewUnpublished}. We hypothesize that this longer growth time results in a faster and more reproducible jet tail.\n\nA comparison between the growth and collapse can be made by normalizing the volume and time, as shown in Figure~\\ref{fig: Volume_time_normalized}. For each individual bubble, the volume is normalized by its maximum volume and the time by the growth time. The lines show an average value over all bubbles, and the shaded region is the standard deviation. This figure shows that the initial normalized growth rate of the bubbles generated by the pulsed laser is larger compared to bubbles generated by the CW laser, as indicated by the steeper curve for small times. \nFurthermore, it shows that for the pulsed laser, the collapse is on average $\\sim 20\\%$ slower compared to the growth, whereas for the CW they take equal time. Both of these findings are in agreement with previous studies on the bubble dynamics for pulsed laser~\\cite{Zwaan2007,Sun2009} and CW~\\cite{OyarteGalvez2020}. However, this difference is mainly caused by the $\\sim 15\\%$ smaller growth time for the pulsed laser, see Figure~\\ref{fig: GR_MV_pulsed_CW}. If both curves were normalized by the same time, the time of collapse would only vary by $<5\\%$.\n\n\\begin{figure}\n\\centering\n\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures/Growth_rate_Max_size_shade-eps-converted-to.pdf}\n \\captionsetup{width=0.9\\linewidth}\n \\caption{Maximum bubble volume vs average growth rate for individual bubbles generated by the pulsed laser (red) and the CW laser (blue). The shaded area encapsulates all data.}\n \\label{fig: GR_MV_pulsed_CW}\n\\end{minipage}\n\\vline\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures/Growth_collapse_norm_pulsed_CW-eps-converted-to.pdf}\n \\captionsetup{width=0.9\\linewidth}\n \\caption{Normalized bubble size vs normalized time, for CW (blue) and pulsed (red). Lines indicate averages over all bubbles, shaded regions indicate standard deviation. For each individual bubble, the volume is normalized by the maximum volume, and the time is normalized by the growth time.}\n \\label{fig: Volume_time_normalized}\n\\end{minipage}\n\\end{figure}\n\n\\subsubsection*{Reproducibility}\nReproducibility is an important factor considering application in health care. The error bars in Figure~\\ref{fig: GR_pulsed_CW_50_105} show that the deviation in the bubble growth rate is larger for the CW laser (blue colors) compared to the pulsed laser (red colors). This indicates that even with identical initial conditions for the CW laser, the bubble dynamics can vary each time slightly. As discussed in section~\\ref{sec: CW laser}, the energy delivered by the CW laser is not controlled directly but depends on the laser power and nucleation time. As nucleation is a stochastic event, there is a variance in the nucleation time, resulting in less or more delivered energy for early or late nucleation, respectively. \n\nFigure~\\ref{fig: CW growth individual} shows the growth rate of all individual bubbles generated by the CW laser together with the mean and standard deviation for a group of 6 or more with identical initial conditions (laser power). Each color indicates a different laser power, which influences the delivered energy. However, even for measurements with identical laser power, the delivered energy can vary up to $\\pm~50$~µJ. For each of these laser powers, the individual results are typically in the bottom-left or top-right quadrant of the error bars. This indicates that there is a correlation between the additional energy for delayed nucleation and the bubble growth rate. Furthermore, per laser power, the individual data points are fitted with a linear fit, which is shown by the colored lines. The slope of these colored lines is very similar to the slope of the black line, which is the best fit when comprising all data. Therefore, the variance in nucleation time directly affects the bubble growth rate, which means that delayed nucleation results in more energy and a faster-growing bubble, and early nucleation results in less energy and a slower-growing bubble. However, as nucleation is a stochastic process, this results in a random deviation and reduces reproducibility.\n\nFor the pulsed laser, the standard deviation in delivered energy is only affected by the laser specifications. On average, it is much smaller with 2\\% of the delivered energy, compared to 8\\% for CW. As this deviation also directly affects the bubble growth rate, the bubble growth rate for the pulsed laser is thus more reproducible. \n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=1\\textwidth]{Figures/CW_growth_individual-eps-converted-to.pdf}\n \\captionsetup{width=1\\linewidth}\n \\caption{Growth rate for individual bubble events for the 105~µm fiber with the CW laser. Each color indicates a different laser power. The delivered energy mainly depends on the laser power but also on the moment of nucleation (e.g. late nucleation results in larger delivered energy). The circles are individual measurements, and the error bars indicate the mean and standard deviation of those individual measurements with the same laser power (at least 6). The colored lines indicate the best linear fit per laser power. The black line indicates the best linear fit for all data points. The slope of the colored lines is similar to the black line, indicating a correlation between the time of nucleation and bubble growth rate.}\n \\label{fig: CW growth individual}\n\\end{figure} \n\n\\subsection{Practical considerations of laser choice for applications}\nEngineering a medical device based on laser-induced cavitation is limited by the available offer of laser sources on the market. Yet, choosing the right laser emitter is vital for the success of any potential product or prototype, where not only the best physical and optical parameters shall be considered. Other aspects like cost, the robustness of operation, size or compliance with safety hazards play a key role. Intuitively, the cheaper, smaller and safer source is favourable, but these are usually contradictory parameters. The results of this study bring a better understanding of what kind of laser parameters are critical to generate fast traveling liquid droplets of relevance for its use in eventual devices using laser-induced cavitation, e.g. needle-free jet injectors. \nAlthough an in-depth techno-economical discussion on the selection of laser sources for specific applications is beyond the scope of this study, we will briefly discuss the impact of the different laser and device parameters here. The main differences are shown in Table \\ref{Table: CW vs Pulsed}.\n\nFor applications where reproducibility and reliability of the bubble dynamics are most important, the pulsed laser is preferred over the CW laser. Due to the reduced control over the delivered energy by the CW laser, there is a larger deviation in individual bubble growth rates. In the case of a jet injection device, this affects the jet velocity and thus injection depth. However, the exact influence of the deviation in bubble dynamics on the injection depth should be investigated in a further study.\n\n\\begin{table}[t]\n\\caption{Overview of the difference of the laser and bubble characteristics of the CW and pulsed laser.}\n\\label{Table: CW vs Pulsed}\n\\begin{tabular}{|l|l|l|}\n\\hline\n & \\textbf{Pulsed laser} & \\textbf{CW laser} \\\\ \\hline\n\\textbf{Timescale} & 5~ns & 1~ms \\\\ \\hline\n\\textbf{Peak power} & 10$^4$~W & 1~W \\\\ \\hline\n\\textbf{Typical price$^*$} & $\\sim$~1000~-~50000~\\$ & $\\sim$~100~-~1000~\\$ \\\\ \\hline\n\\textbf{Typical size} & Table top device & Handheld device \\\\ \\hline\n\\textbf{Control over delivered energy} & Direct (input of laser) & Indirect (depends on P \\& t$_\\textnormal{n}$) \\\\ \\hline\n\\textbf{Reproducibility (variation)} & 2\\% & 8\\% \\\\ \\hline\n\\textbf{Ratio growth to collapse time} & 1:1.2 & 1:1 \\\\ \\hline\n\\textbf{\\begin{tabular}[c]{@{}l@{}}Influence of increase in fiber \\\\ size (laser input parameters \\\\ constant)\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Reduction in energy density\\\\ Increase in energy threshold\\\\ Smaller/slower growing bubbles\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Delayed nucleation\\\\ Increase in delivered energy\\\\ Larger/faster growing bubbles\\end{tabular} \\\\ \\hline\n\\textbf{\\begin{tabular}[c]{@{}l@{}}Influence of increase in \\\\ absorption coefficient/\\\\ dye concentration\\end{tabular}} & \\begin{tabular}[c]{@{}l@{}}Larger energy density\\\\ Decrease in energy threshold\\\\ Increase of bubble size\\end{tabular} & Not examined in this study$^{**}$. \\\\ \\hline\n\\end{tabular}\n$^{*}$Laser prices are highly dependent on wavelength and exact characteristics. Given prices are just an indication for typical consumer price of a single device. \\\\\n$^{**}$Refs. \\cite{AfanadorDelgado2019,Zhang2022} show that an increase in dye concentration results in shorter nucleation times and smaller bubbles.\n\\end{table}\n\nOn the other hand, if the price and/or size of the device is of more importance, a CW laser would be preferred over a pulsed laser. Due to their lower power, they are smaller and more affordable. Although, portable pulsed lasers with moderate pulse energies, as the ones used in our studies, have become more widely available over the past decade. \n\nFurthermore, the use of optical fiber allows for a small handheld device, even in the case of a large laser source. The laser source, including electronics, can then be placed in a bench-top part. \nThe choice of optical fiber has a large influence on bubble dynamics. For the pulsed laser, a smaller fiber reduces the required pulse energy. However, the smallest fiber (50~µm diameter) is limited in the range of bubble growth rate due to laser-induced damage of the fiber tip. For the CW laser, an increase in fiber size results in faster-growing bubbles, although it requires a more powerful laser (P~$>$~2W) to create a larger range of growth rates. \n\nChanging the dye concentration has a large influence on the bubble dynamics. For the pulsed laser, increase of dye concentration and thus absorption coefficient results reduces the required pulse energies, as discussion in Section~\\ref{sec: pulsed laser}. This allows for the use of smaller and more affordable pulsed lasers. Simultaneously, a single device could create a large range of bubbles by changing the absorption coefficient of the liquid. The influence of the absorption coefficient on the bubbles generated by the CW laser have not been investigated in this study. However, previous studies showed an influence on the nucleation time and bubble size~\\cite{AfanadorDelgado2019,Zhang2022}. Moreover, adding the dye into drug formulation might bear further toxicity and regulatory issues. Future studies should focus on the exact influence of absorption coefficient on bubble dynamics.\n\n\\section{Conclusion}\\label{sec: conclusion}\n\nWe compared the dynamics of bubbles generated by two lasers with different timescales (ns and ms) inside an open-ended capillary. Our comparative study is the first to show the resulting bubble dynamics in the same fluidic confinement with two different laser types. We have shown that these lasers create bubbles of comparable growth rates proportional to the delivered energy. This linear increase is in agreement with previously found linear increase between the energy and the jet velocity. \n\nFor the pulsed laser set-up, we found that the energy threshold for nucleation is proportional to the irradiated volume. This volume depends on the fiber core radius, beam divergence and absorption coefficient. Therefore, a decrease in irradiated volume, either by changing the fiber or increasing the absorption coefficient, resulting in a reduction of the energy threshold. This allows for the use of more affordable and/or less powerful pulsed laser sources while creating the same bubble. \n\nFor the continuous-wave laser, we found an efficiency increase and better reproducibility compared to previous experiments with free-space optics. This is explained by a reduction in the number of interfaces and easier alignment. Furthermore, we show that the delivered energy can be controlled by the laser power and the fiber size. For each fiber, a decrease in laser power results in an increase in delivered energy, allowing the creation of faster-growing bubbles. Furthermore, a larger fiber results in an increase in delivered energy for fixed laser power due to a longer nucleation time. Therefore, the 200~µm fiber results in much larger and faster-growing bubbles than the 50 or 105~µm fiber, when the same laser power is used. \n\nA comparison between the two laser sources shows that the pulsed laser requires slightly less optical energy to create the same bubble growth rate, which we attribute to heat dissipation and a reduction in absorption coefficient during the CW laser heating. However, for the same average growth rate, bubbles generated by the CW laser grow for a longer time and are larger. \n\nFinally, we made a comparison between both methods in terms of practical usage. Since the delivered energy by the CW laser cannot be controlled directly, there is a larger deviation in the delivered energy compared to the pulsed lasers (8\\% and 2\\% respectively), which also results in a larger deviation in bubble growth rates. This would be unfavourable for laser-based jet injection, as it decreases the control over jet velocity and, therefore, injection depth. However, if this variation is within the allowable error industry standards, then the CW laser could be advantageous due to its smaller size and lower price compared to pulsed lasers. \n\n\n\\section*{Acknowledgements}\nJ.J.S. would like to thank dr. Dani\\\"el J\\'auregui-V\\'azquez and dr. Jose Alvarez Chavez for their help with the optical set-up of the CW laser. J.J.S and D.F.R. acknowledge the funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 851630), and NWO Take-off phase 1 program funded by the Ministry of Education, Culture and Science of the Government of the Netherlands (No. 18844). J.K. and Ch.M. acknowledge the Innosuisse BRIDGE Proof of Concept grant funding. The authors are thankful for the insightful discussions with Dr. M. A. Quetzeri Santiago, D.L. van der Ven, K. Mohan and D. de Boer.\n\n\\section*{Competing interest}\nThe authors declare that they have no known competing financial\ninterests or personal relationships that could have appeared to influence the work reported in this paper.\n\n\\section*{CRediT authorship contribution statement}\n\\textbf{Jelle Schoppink:} Conceptualization, Methodology, Formal analysis, Investigation, Data Curation, Writing - Original Draft, Visualization\n\\textbf{Jan Krizek:} Conceptualization, Methodology, Writing - Review \\& Editing\n\\textbf{Christophe Moser:} Conceptualization, Funding acquisition, Writing - Review \\& Editing\n\\textbf{David Fernandez Rivas:} Conceptualization, Supervision, Project administration, Funding acquisition, Writing - Review \\& Editing.\n\n\\printbibliography \n\n\\begin{figure}[b!]\n \\centering\n \\includegraphics[width=0.7\\textwidth]{Figures/TOC_Only_Graphical_abstract.pdf}\n \\captionsetup{width=1\\linewidth}\n \\caption{For Table of Contents Only}\n \\label{fig: Graphical abstract}\n\\end{figure} \n\n\\end{document}\n\\section*{Appendix A: Allura Red AC absorption coefficient}\\label{sec: Appendix}\nThe measured absorption coefficients for the Allura Red AC dye (ARAC, 80\\%, Sigma Aldrich) are shown in Figure~\\ref{fig: alpha_vs_concentration}. The values are calculated from the transmission of the 532~nm laser through a rectangular capillary with a 100~µm path length. The values are compared with reported values in literature~\\cite{Bevziuk2017,Garcia2017}, which closely matches our measured values for concentrations up to 5~mM. For larger concentrations, we note that the absorption coefficient does no longer increase linearly with the dye concentration.\n\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=.7\\linewidth]{Figures/Absorption_coefficient_concentration-eps-converted-to.pdf}\n \\caption{Measured absorption coefficients (blue symbols) for Allura Red AC (ARAC). The values are compared with an extrapolated value from literature (red curve)~\\cite{Bevziuk2017,Garcia2017}. It can clearly be observed that the absorption coefficient does not behave linearly at these concentrations.}\n \\label{fig: alpha_vs_concentration}\n\\end{figure}\n\\newpage\n\\section*{Appendix B: Calculation of irradiated volume}\\label{sec: Appendix B}\nAs the fiber core radii R$_\\textnormal{f}$ ($25\\textnormal{~µm}<\\textnormal{R}_\\textnormal{f}<100\\textnormal{~µm}$) are of similar size of the absorption length d ($52\\textnormal{~µm}<\\textnormal{d}<131\\textnormal{~µm}$), the beam divergence $\\beta$ cannot be neglected when calculating the irradiated volume. The beam divergence is calculated by the NA of the fiber and the refractive index of the liquid $n$ (see Figure \\ref{fig: BeamRadDiv})\n\\begin{equation*}\n \\beta = \\arcsin{\\frac{\\textnormal{NA}}{n}} = \\arcsin{\\frac{\\textnormal{0.22}}{1.33}} \\approx 9.5\\degree.\n\\end{equation*}\nThen, the beam radius R along the propagation axis $x$ is calculated as\n\\begin{equation*}\n \\textnormal{R}(x) = \\textnormal{R}_\\textnormal{f}+x\\tan{\\beta}.\n\\end{equation*}\n\nThe irradiated volume $V$ can be calculated by integrating the beam area ($\\pi$R$(x)^2$) from the fiber tip ($x=0$) to the absorption length ($x=\\textnormal{d}$)\n\n\\begin{align*}\n V = \\pi\\int_{0}^{\\textnormal{d}}\\textnormal{R}^2 dx = \\pi\\int_{0}^{\\textnormal{d}}(\\textnormal{R}_\\textnormal{f}+x\\tan{\\beta})^2 dx = \\pi\\int_{0}^{\\textnormal{d}}(\\textnormal{R}_\\textnormal{f}^2+2\\textnormal{R}_\\textnormal{f}x\\tan{\\beta}+x^2\\tan^2{\\beta})dx \\\\\n = \\pi |\\textnormal{R}_\\textnormal{f}^2x + \\textnormal{R}_\\textnormal{f}x^2\\tan{\\beta}+\\frac{1}{3}x^3\\tan^2{\\beta}|_0^\\textnormal{d} = \\pi (\\textnormal{R}_\\textnormal{f}^2\\textnormal{d} + \\textnormal{R}_\\textnormal{f}\\textnormal{d}^2\\tan{\\beta}+\\frac{1}{3}\\textnormal{d}^3\\tan^2{\\beta)}\n\\end{align*}\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width = \\linewidth]{Figures/BeamRadiusDivergence-eps-converted-to.pdf}\n \\caption{Schematic indicating the irradiated volume (green) of the liquid (red) near the fiber tip. Beam radius increases along the propagation axis due to the divergence. At the fiber tip, the radius is equal to the core radius. The irradiated volume is calculated by integrating the beam area from the fiber tip to the absorption length \\textnormal{d}.}\n \\label{fig: BeamRadDiv}\n\\end{figure}\n\\newpage\n\\section*{Appendix C: CW bubble growth}\\label{sec: Appendix C}\nFigure~\\ref{Appendix fig: CW 105 and 200} shows a few snapshots of the bubbles generated by the CW laser at the 105 and 200~µm fiber. For the 200~µm fiber (right snapshots), the bubble has a much larger energy, due to the longer nucleation time. Therefore, the bubble grows more explosively and always grows beyond the capillary edge. Therefore, no maximum bubble size or average growth rate could be obtained for the 200~µm fiber.\n\n\\begin{figure}[ht]\n \\centering\n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/Appendix_Bubble105-eps-converted-to.pdf}\n\n\\end{minipage} \n\\begin{minipage}{.49\\textwidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{Figures/Appendix_Bubble200-eps-converted-to.pdf}\n\n\n\\end{minipage}\n \\caption{Snapshots of the bubble generated at a 105~µm (left) and 200~µm fiber (right) by the CW laser with a power of approximately 0.7~W. The bubbles generated at the 200~µm fiber have a much larger energy and grow beyond the capillary edge and expel all the liquid. Therefore, there is no maximum bubble size or average growth rate observed.}\n \\label{Appendix fig: CW 105 and 200}\n\\end{figure} \n\n\\printbibliography \n\\end{document}"}}},{"rowIdx":83598,"cells":{"uuid":{"kind":"number","value":1116691499413,"string":"1,116,691,499,413"},"dataset":{"kind":"string","value":"arxiv"},"text":{"kind":"string","value":"\\section{Introduction} \\label{sec:Introduction}\n\nOnly the most violent objects in the universe are capable of producing \nhigh-energy particles in non-thermal acceleration processes and emit \nphotons with energies in the X-ray band and above. Spectral and \nmorphological studies in the X-ray and gamma-ray bands have become \nestablished tools to study the non-thermal emission processes of various \nastrophysical sources \\cite{Frederick2010}. However, many of the regions \nof interest (black hole vicinities, formation zones of relativistic \njets, etc.) are too small to be spatially resolved with current and \nfuture instruments. Spectro-polarimetric X-ray observations are capable \nof providing additional information~-- namely (i) the energy-resolved \nfraction of linear polarization (e.g. what fraction of emission is \npolarized), and (ii) the projected orientation of the polarization plane \n(defined by the electric field vector of the photon) with respect to the \nemitting source. Various emission mechanisms of compact objects lead to \ncomparable spectral signatures, but would differ in the polarization \ncharacteristics. The measurements of polarization properties would \ntherefore help to constrain the geometry of the inner regions of \nrelativistic plasma jets, mass-accreting black holes (BHs) and neutron \nstars \\cite{Lei1997, Krawcz2011}.\n\nSo far, only a few space-borne missions have successfully measured \npolarization in the X-ray regime. The Crab nebula is the only source for \nwhich X-ray polarization has been established with a high level of \nconfidence. In measurements with the OSO-8 satellite, the Crab exhibits \na polarization fraction of $20 \\%$ at energies of $2.6 - 5.2 \\, \n\\rm{keV}$ and a direction angle of $30^{\\circ}$ with respect to the \nX-ray jet observed in the nebula \\cite{Weisskopf1978}. At energies above \n$100 \\, \\rm{keV}$, measurements resulted in a polarization fraction of \n$46 \\% \\pm 10 \\%$ with the direction aligned with the jet \n\\cite{Dean2008, Forot2008}. A second astrophysical source emitting \npolarized X-rays was identified recently. {\\it INTEGRAL}\\xspace observations of the \nX-ray binary Cygnus\\,X-1 indicate a high fraction of polarization of \n$(67 \\pm 30)\\%$ in the $400 \\, \\rm{keV} - 2 \\, \\rm{MeV}$ band, whereas a \n$20\\%$ upper limit was derived for the $250-400 \\, \\rm{keV}$ band \n\\cite{Laurent2011}. Various authors have reported tentative evidence for \npolarized hard X-ray/soft gamma-ray emission from different gamma-ray \nbursts \\cite{Coburn2003, Kalemci2007, Yonetoku2011, Kostelecky2013}. \nHowever, all of these detections have somewhat marginal significance, \npossibly being impacted by unknown systematic effects in their \nrespective instrumentation.\n\nModel predictions of polarized emission for various source types lie \nslightly below the sensitivity of the past OSO-8 mission, making future, \nmore sensitive polarimetry missions particularly interesting. However, \nthere are currently no dedicated missions in orbit that are capable of \nmeasuring X-ray polarization fractions in the $<10\\%$ regime, a \nrequirement to study the corresponding emission mechanisms for a variety \nof astrophysical source classes (see Sec.~\\ref{sec:Science}). More \nrecently, various experiments have been proposed that could change the \nsituation as they combine broadband sensitivity with a high detection \nefficiency. The proposed polarimeters use focusing mirrors to collect \nphotons from the source. Photoelectric effect polarimeters, like the \n{\\it Gravity and Extreme Magnetism SMEX} (GEMS) mission \\cite{GEMS} and \nXIPE \\cite{XIPE}, track the direction of photo electrons ejected in \nphotoelectric effect interactions of the X-rays. The {\\it ASTRO-H} \nmission (to be launched in 2015) will carry the {\\it Soft Gamma-Ray \nImager}. The detector combines an X-ray and gamma-ray collimator with a \nSi scatterer and CdTe absorber. The mission will be able to do \nscattering polarimetry at $E > 50 \\, \\rm{keV}$ energies \n\\cite{Tajima2010}. Our group is working on a proposal of the space borne \nscattering polarimeter PolSTAR which uses a lower-Z LiH stick as a \nscatterer to enable the detection of $3 - 80 \\, \\rm{keV}$ X-rays. The \nPolSTAR concept will be described in a forthcoming paper. Missions like \nGEMS, PolSTAR and XIPE aim at obtaining $1 \\%$ minimum detectable \npolarization fractions, see Eq.~(\\ref{eqn:MDP}), for mCrab sources. \nThese missions would allow very high-signal-to-noise detections of \nbright galactic sources (e.g. Cyg\\,X-1, GRS\\,1915+105, Her\\,X-1) and \ncould detect $\\sim1 \\%$ polarization fractions for extra-galactic \nsources (e.g. NGC\\,4151).\n\nScattering polarimeters detect the direction into which photons scatter \nwhen interacting in the detector. Although different scattering \nprocesses dominate at different energies (coherent scattering below a \nfew keV, Thomson scattering at intermediate energies, and Compton \nscattering at energies $> 20 \\, \\rm{keV}$), all scattering processes \nshare the property that the photons scatter preferentially perpendicular \nto the polarization plane (electric field vector) of the incoming \nphoton. For example, the angular dependence of Compton scattering \nprocesses is given by the Klein-Nishina cross section \\cite{Evans1955}:\n\n\\begin{equation}\n\\label{eq:KleinNishinaCrossSection} \n\\frac{\\rm{d}\\sigma}{\\rm{d}\\Omega} = \\frac{r_{0}^{2}}{2} \n\\frac{k_{1}^{2}}{k_{0}^{2}} \\left[ \\frac{k_{0}}{k_{1}} + \n\\frac{k_{1}}{k_{0}} -2 \\sin^{2} \\theta \\cos^{2} \\eta \\right], \n\\end{equation}\nwhere $\\eta$ is the angle between the electric vector of the incident \nphoton and the scattering plane, $r_{0}$ is the classical electron \nradius, $\\bf{k}_{0}$ and $\\bf{k}_{1}$ are the wave-vectors before and \nafter scattering, and $\\theta$ is the scattering angle. The azimuthal \ndistribution of scattered events shows a sinusoidal modulation with a \n$180^{\\circ}$ periodicity and a maximum at $\\pm 90^{\\circ}$ to the \npreferred electric field direction of a polarized X-ray signal.\n\nIn this paper we describe the design and performance of a scattering \npolarimeter, X-Calibur. The polarimeter utilizes a plastic scintillator \nas scatterer which can scatter $E > 20 \\, \\rm{keV}$ X-rays efficiently. \nThis is ideal for the operation on a balloon where the energy threshold \nis well matched to the low energy cutoff of the transmissivity caused by \nthe residual atmosphere above the balloon altitude of 125,000 feet. An \nassembly of Cadmium Zinc Telluride (CZT) detectors surrounds the \nscintillator in order to record the azimuthal distribution of the \nscattered photons, allowing one to reconstruct the polarization \nproperties of the incoming X-ray beam. The background of charged \nparticles and high-energy photons is suppressed by an active CsI shield.\n\nThe paper is structured as follows. Section~\\ref{sec:Science} gives a \nbrief overview over the scientific potential of hard X-ray polarimetry. \nThe design of the X-Calibur polarimeter is described in \nSec.~\\ref{sec:XCLB}. The data analysis methods are described in \nSec.~\\ref{sec:Analysis}, followed by an outline of the simulation \nprocedure in Sec.~\\ref{sec:Simulations}. \nSection~\\ref{sec:DetectorCharacterization} describes the calibration and \ncharacterization studies of the CZT detectors. Measurements with the \nassembled X-Calibur polarimeter to characterize the scattering \nscintillator and the shield are described in \nSec.~\\ref{sec:XCLB_Characterization}. Polarization measurements with \nX-Calibur are described in Sec.~\\ref{sec:XCLB_PolarimetryMeasurements}. \nThe paper ends with a summary and outlook in Sec.~\\ref{sec:Conclusion}. \nThe X-Calibur data presented in this paper were taken (i) in a \nlaboratory environment at Washington University, (ii) at the Cornell \nHigh Energy Synchrotron Source (CHESS), and (iii) in Ft.~Sumner, NM, \nduring a preparation campaign for an upcoming balloon flight.\n\n\n\\section{Scientific Potential} \\label{sec:Science}\n\nThis section discusses the scientific potential for a scattering \npolarimeter such as X-Calibur from a balloon platform. X-rays from \ncosmic sources can be polarized owing to the anisotropy in the source \ngeometry and/or the emission characteristics of various processes \n\\cite{Lei1997}. Non-thermal emission, like synchrotron radiation, \nresults in a large polarization fraction $r$. Synchrotron emission will \nresult in linearly polarized photons with their electric fields oriented \nperpendicular to the magnetic field lines (projected); the observed \npolarization map can therefore be used to trace the magnetic field \nstructure of the source, a common practice in radio and optical \npolarimetry. An electron population with a spectral energy distribution \nof $\\rm{d}N/\\rm{d}E \\propto E^{-p}$ emitting in a uniform magnetic field \nwill lead to an observable fraction of polarization of \n\\cite{Korchakov1962}:\n\n\\begin{equation}\nr_{\\rm{sync}} = \\frac{p+1}{p+7/3}, \\quad \\rm{with} \\, (p + 3) = \n\\frac{\\alpha + 1} {\\alpha + 5/3}.\n\\end{equation}\nHere, $\\alpha$ is the index of the X-ray power law spectrum. An observed \npolarization fraction close to this limit can therefore be interpreted \nas an indication of a highly ordered magnetic field since \nnon-uniformities in the magnetic field will reduce $r_{\\rm{sync}}$. The \npolarized synchrotron photons can in turn be inverse-Compton scattered \nby relativistic electrons~-- weakening the fraction of polarization (but \nnot erasing it) and imprinting a scattering angle dependence to the \nobserved fraction of polarization \\cite{Krawczynski2012b}. Such \ninverse-Compton signals will usually (but not always) appear in hard \ngamma-rays, where polarimetry is difficult, due to multiple scattering \nin pair production detectors. Another important mechanism for polarizing \nphotons is Thomson scattering which creates a polarization perpendicular \nto the scattering plane \\cite{Rybicki1991}. Curvature radiation is \npolarized, as well. The scientific potentials of spectro-polarimetric \nobservations over the broadest possible energy range are summarized \nbelow; more detailed discussions can be found in \\citet{Krawcz2011} and \n\\citet{Lei1997}.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Schnittman_Fig3}\n\\end{center}\n\n\\caption{Ray-traced image of direct radiation from a thermal disk around \na BH including returning radiation (observer located at an inclination \nangle of $75^{\\circ}$, gas on the left side of the disk moving toward \nthe observer causing the characteristic increase in intensity due to \nrelativistic beaming). The image is adopted from \\citet{Schnittman2009}. \nThe observed intensity is color-coded on a logarithmic scale and the \nenergy-integrated polarization vectors are projected onto the image \nplane with lengths proportional to the fraction of polarization.}\n\n\\label{fig:BH_Polarization}\n\\end{figure}\n\n{\\it Binary black hole systems.} Particle scattering in a Newtonian \naccretion disk surrounding a BH will lead to the emission of polarized \nX-rays. Relativistic aberration and beaming, gravitational lensing, and \ngravito-magnetic frame-dragging will result in an energy-dependent \nfraction of polarization since photons with higher energies originate \ncloser to the BH than the lower-energy photons \\cite{Connors1977}. \n\\citet{Schnittman2009} calculate the expected polarization signature \nincluding (i) the effects of deflection of photons emitted in the disk \nby the strong gravitational forces in the regions surrounding the BH and \n(ii) re-scattering these photons by the accretion disk \n\\cite{Schnittman2009, Schnittman2010}. The resulting effect is a swing \nin the polarization direction from being horizontal at low energies to \nvertical at high energies, i.e., parallel to the spin axis of the BH. \nSpectro-polarimetric observations can therefore be used to constrain the \nmass and spin of the BH \\cite{Schnittman2009}, as well as the \ninclination of the inner accretion disk and the shape of the corona \n\\cite{Schnittman2010}, see Fig.~\\ref{fig:BH_Polarization}. In principle, \nX-ray polarization can also be used to test General Relativity in the \nstrong gravity regime \\cite{Krawczynski2012c}.\n\n{\\it Pulsars.} High-energy particles in pulsar magneto-spheres are \nexpected to emit synchrotron and/or curvature radiation which are \ndifficult to distinguish from one another, solely based on the observed \nphoton energy spectrum. However, since the orbital planes for \naccelerating charges that govern these two radiation processes are \northogonal to each other, their polarized emission will exhibit \ndifferent behavior in position angle and polarization fraction as \nfunctions of energy and the rotation phase of the pulsar \n\\cite{Dean2008}. An illustration of the models of phase dependence of \nthe X-ray/gamma-ray polarization signatures in pulsars can be found in \n\\citet{Dyks2004}. In magnetars, the highly-magnetized cousins of \npulsars, polarization-dependent resonant Compton up-scattering is a \nleading candidate for generating the observed hard X-ray tails \n\\cite{BaringHarding2007}. In both these classes, phase-dependent \nspectro-polarimetry can probe the emission mechanism, and provide \ninsights into the magnetospheric locale of the emission region.\n\nCyclotron lines arising from transitions between Landau levels in \nintense magnetic fields that occur in the polar regions of neutron stars \nand magnetars are polarized. Also, the absorption scattering \ncross-sections from the Landau levels are dependent on the polarization \nstate of the X-rays, so that the radiative transfer through such plasma \nwill lead to a polarization of the emergent radiation. The first such \ncyclotron feature was observed by \\citet{Truemper1978} in Her\\,X-1 and \nis interpreted as due to an absorption line around $40 \\, \\rm{keV}$ \n\\cite{Staubert2007}. Such features allow an estimate of the magnetic \nfield strength \\cite{Coburn2002}. Observations of polarized X-rays will \nstrongly confirm that these features are indeed due to cyclotron lines \nin magnetic fields of $3-5 \\times 10^{12} \\, \\rm{G}$. A detailed \ndiscussion of the theoretical aspects of cyclotron radiation is given in \n\\citet{Leahy2010}.\n\n{\\it Pulsar wind nebulae.} The compact object (e.g. pulsar) resulting \nfrom a previous supernova explosion can be surrounded by a synchrotron \nemitting nebula. The nebula extends far beyond the magneto-sphere of the \ncentral pulsar, but its emission is believed to be driven by the pulsar \nwhich injects relativistic electrons/positrons that are further shock \naccelerated in the nebula. Spectro-polarimetric observations can be used \nto constrain the magnetic field and particle populations in such pulsar \nwind nebulae~-- such as the Crab nebula, the leading driver for this \nfield of X-ray polarimetry. Given the more compact emission regions at \nhigh energies, these objects potentially show a higher polarization \nfraction at hard X-rays as compared to soft X-rays, reflecting the \ncontrast between jet and more diffuse nebular contributions \n\\cite{Forot2008}.\n\n{\\it Supernova remnants.} Supernova remnants (SNRs) present an \nopportunity to perform X-ray polarimetry, as well. The remnants possess \ntangled magnetic fields on large scales in their interiors, as is \nevidenced in the classic radio polarization map of the Crab nebula \n\\cite{Velusamy1985}. Both, radio and X-ray signals, are believed to be \ndue to synchrotron emission, and so it is reasonably presumed that X-ray \nemission from SNRs should be significantly polarized. The X-ray spectra \nof such remnants are typically steeper than spectra in the radio, which \nleads to the expectation of higher polarization fractions in the X-ray \nband. Yet, since the electrons generating X-rays will diffuse on larger \nspatial scales than their radio-emitting counterparts do, the X-ray \nsignals should capture the field morphology on larger scales. It may or \nmay not be more coherent than the field structure on smaller (radio) \nscales. Due to instrumental limitations in angular resolution at X-ray \nenergies, however, it will not be possible to resolve individual \nregions~-- depending on the angular size of the remnant. The \nobservational challenge will therefore be to overcome the competition \nbetween compact regions causing highly polarized emission on one hand, \nand on the other hand an averaging effect of different emission regions \nwith different orientations of their magnetic fields. An example of \nexpectations for the SNR synchrotron polarization properties can be \nfound in \\citet{Bykov2009}.\n\n{\\it Relativistic jets in active galactic nuclei.} Relativistic \nelectrons in jets of active galactic nuclei (AGN) emit polarized \nsynchrotron radiation at radio/optical wavelengths. The same electron \npopulation is believed to produce hard X-rays by inverse-Compton \nscattering off a photon field. Simultaneous measurements of the \npolarization angle and the fraction of polarization in the radio to hard \nX-ray band could help to disentangle the following scenarios: (i) If the \nelectrons mainly up-scatter the co-spatial synchrotron photon field \n(synchrotron self Compton), the polarization of the hard X-rays is \nexpected to track the polarization at radio/optical wavelengths \n\\cite{Poutanen1994}. The fraction of X-ray polarization could be close \nto the fraction of polarization of the synchrotron emission measured in \nthe radio/optical bands and the polarization directions between radio, \noptical and X-rays should be identical. (ii) If the electrons dominantly \nup-scatter an external photon field (external Compton, e.g. photons of \nthe cosmic microwave background or from the accretion disk surrounding \nthe super-massive black hole) the hard X-rays will have a relatively \nsmall ($<$10\\%) fraction of polarization \\cite{McNamara2009}. Hadronic \njet emission models for low-synchrotron-peaked AGN, on the other hand, \npredict an even higher fraction of polarization at high energies, \ncompared to the leptonic SSC models \\cite{Zhang2013}.\n\nPolarization also allows one to test the structure of the magnetic field \nof the jet. Particles accelerated in a helical field which are moving \nthrough a standing shock can cause an X-ray synchrotron flare with a \ncontinuous (in time) swing in polarization direction. Such an event was \nobserved from BL\\,Lacertae at optical wavelengths \\cite{Marscher2008}.\n\n{\\it Gamma-ray bursts.} Gamma-ray bursts are believed to be connected to \nhyper-nova explosions and the formation/launch of relativistic jets \n\\cite{Woosley1993}. As in the case of the jets in AGN, the structure of \ntheir jets and the particle distribution responsible for gamma-ray \nbursts can be revealed by X-ray polarization measurements \n\\cite{Kostelecky2013}. The X-ray emission of a gamma-ray burst, however, \nusually lasts for only a few minutes at most, so that rapid follow-up \nobservations in the X-ray band below $30 \\, \\rm{keV}$ would be the main \nchallenge for studying their polarization properties.\n\n\nOn a one-day balloon flight, we would achieve $5-15 \\%$ MDPs, see \nEq.~(\\ref{eqn:MDP}), for between 1 and 4 sources. For the Crab, the \nphase resolved polarimetry would allow us to decide between emission \nmodels. For Cyg\\,X-1 and GRS\\,1915+105 we could test corona models, and \nfor Her\\,X-1, we could get a first estimate of the polarization fraction \nof the X-ray emission.\n\n\n\\section{Design of X-Calibur} \\label{sec:XCLB}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{X-Calibur_InFocus}\n\\end{center}\n\n\\caption{Schematic view of the functionality of X-Calibur (not to \nscale). X-rays from an astrophysical source are focused with a grazing \nincidence mirror onto the scattering rod of the polarimeter. The \nscattered X-ray is recorded in one of the surrounding CZT detectors. The \npolarimeter is embedded by a shield to suppress background of particles \nnot originating from the mirror.}\n\n\\label{fig:XCLB_Functionality}\n\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[height=0.38\\textheight]{X-Calibur_Schematic}\n\\hfill\n\\includegraphics[height=0.38\\textheight]{P1080730_LabVersion4}\n\\hfill\n\\includegraphics[height=0.38\\textheight]{P1090499_X-Calibur}\n\\end{center}\n\n\\caption{Design of the X-Calibur polarimeter. {\\bf Left:} Schematic \nillustration. Side view: incoming X-rays Compton-scatter in the \nscintillator rod (aligned with the optical axis, read out by a PMT) and \nare subsequently photo-absorbed in one of the surrounding CZT detectors \n($8\\times8$ pixels each, see Fig.~\\ref{fig:DetectorRing}). A group of \nfour detectors surrounding the scintillator at a given depth are \nreferred to as ring ({\\it R1} to {\\it R8}). Each ring can be further \ndivided into top and bottom sub rings, as illustrated for ring {\\it R8}. \nTop view: Detector ring viewed along the optical axis, looking into the \nX-ray beam. The polarization signature is imprinted in the azimuthal \nscattering distribution (see Fig.~\\ref{fig:X-Calibur_Configurations}, \ntop, for more details). The four sides of detectors are referred to as \nboards ({\\it Bd0} to {\\it Bd3}), each board comprising all eight \ndetectors per side. 3D view (`exploded'): four sides of detector columns \n({\\it Bd0} to {\\it Bd3}) surround the central scintillator rod~-- \ncovering the whole range of azimuthal scattering angles. {\\bf Middle:} \nPartly assembled polarimeter with two detector sides removed for better \nvisibility. The red wires provide the high voltage to the detectors. The \nread-out electronics is stacked at the backside of the detectors. {\\bf \nRight:} Fully assembled polarimeter (flipped upside-down: X-ray entering \nfrom the bottom).}\n\n\\label{fig:Design}\n\n\\end{figure*}\n\n\\begin{table}[t!]\n\n\\begin{tabular}{rrrrr}\n\nID & Serial-No. & Date & $C_{\\rm{ch}}$ & $C_{\\rm{wu/ft}}$ \\\\\n\\hline \\hline\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Endicott 5\\,mm}} \\\\\n\\hline\n\nEN$_{5}$1 & 672992-01 & 01/11 & 0/5 & 0/5 \\\\\nEN$_{5}$2 & 672992-02 & 01/11 & 1/5 & 1/5 \\\\\nEN$_{5}$3 & 672992-03 & 01/11 & 2/5 & 2/5 \\\\\nEN$_{5}$4 & 672992-04 & 01/11 & 3/5 & 3/5 \\\\\nEN$_{5}$5 & 672994-04 & 02/11 & 0/4 & 0/4 \\\\\nEN$_{5}$6 & 672994-03 & 02/11 & 1/4 & 1/4 \\\\\nEN$_{5}$7 & 672994-02 & 02/11 & 2/4 & 2/4 \\\\\nEN$_{5}$8 & 672994-01 & 02/11 & 3/4 & --- \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Creative Electron 5\\,mm}} \\\\\n\\hline\n\nCE$_{5}$1 & 721613 & 03/11 \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf QuikPak 5\\,mm}} \\\\\n\\hline\n\nQP$_{5}$1 & 3627 & 03/11 & --- & 3/4 \\\\\nQP$_{5}$3 & 3611 & 03/11 & --- & --- \\\\\nQP$_{5}$4 & 3834 & 03/11 & 3/1 & 3/1 \\\\\nQP$_{5}$6 & 6292 & 03/11 & 0/1 & 0/1 \\\\\nQP$_{5}$7 & 6345 & 03/11 & 2/1 & 2/1 \\\\\nQP$_{5}$8 & 721c602 & 03/11 & 1/1 & 1/1 \\\\\n\n\nQP$_{5}$13 & 6886 & 04/12 & 0/3 & 0/3 \\\\\nQP$_{5}$14 & 6977 & 04/12 & 2/3 & 2/3 \\\\\nQP$_{5}$16 & 10814 & 04/12 & 2/2 & 2/2 \\\\\nQP$_{5}$17 & 10819 & 04/12 & 0/2 & 0/2 \\\\\nQP$_{5}$18 & 10829 & 04/12 & 1/2 & 1/2 \\\\\nQP$_{5}$19 & 10847 & 04/12 & 3/3 & 3/3 \\\\\nQP$_{5}$20 & 10848 & 04/12 & 3/2 & 3/2 \\\\\nQP$_{5}$21 & 10860 & 04/12 & 1/3 & 1/3 \\\\\n\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Endicott 2\\,mm}} \\\\\n\\hline\n\nEN$_{2}$1 & 674326-01 & 02/11 & 0/6 & 0/8 \\\\\nEN$_{2}$2 & 674327-01 & 02/11 & 2/8 & 2/8 \\\\\nEN$_{2}$3 & 674328-01 & 02/11 & 2/6 & 2/6 \\\\\nEN$_{2}$4 & 674328-02 & 02/11 & 0/8 & 0/6 \\\\\nEN$_{2}$5 & 674329-01 & 02/11 & --- & --- \\\\\nEN$_{2}$6 & 674330-01 & 02/11 & 1/6 & 1/6 \\\\\nEN$_{2}$7 & 674331-01 & 02/11 & 3/6 & --- \\\\\nEN$_{2}$8 & 674332-01 & 02/11 & --- & 3/8 \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{5}{l}{{\\bf Creative Electron 2\\,mm}} \\\\\n\\hline\n\nCE$_{2}$1 & 2180 & 03/11 & 0/7 & 0/7 \\\\\nCE$_{2}$2 & 720612 & 03/12 & 3/8 & 3/6 \\\\\nCE$_{2}$3 & 720511 & 03/12 & 1/8 & 1/8 \\\\\nCE$_{2}$4 & 721541i & 03/12 & 1/7 & 1/7 \\\\\nCE$_{2}$5 & 06172A & 03/12 & 2/7 & 2/7 \\\\\nCE$_{2}$6 & 726712 & 03/12 & 3/7 & 3/7 \\\\\n\n\\end{tabular}\n\n\\caption{CZT detector sample. ID as in this paper (two letters for the \ncompany, index for the detector thickness $[\\rm{mm}]$, and a number), \nserial number, date of delivery [MM/YY], and position in polarimeter \n[Bd/ring] for the configurations $C_{\\rm{ch}}$ and $C_{\\rm{wu/ft}}$ (see \nFig.~\\ref{fig:X-Calibur_Configurations}).}\n\n\\label{tab:Detectors}\n\n\\end{table}\n\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[height=0.255\\textheight]{DetectorGold_Rotated}\n\\hfill\n\\includegraphics[height=0.255\\textheight]{Detector_BondedAndASICs}\n\\hfill\n\\includegraphics[height=0.255\\textheight]{p1060223_zoom}\n\\end{center}\n\n\\caption{Definition of a detector `ring'. {\\bf Left:} $2 \\times 2 \\times \n0.2 \\, \\rm{cm}^{3}$ CZT detector with 64 pixels (anode side). {\\bf \nMiddle:} $2 \\times 2 \\times 0.2 \\, \\rm{cm}^{3}$ CZT detector bonded to a \nceramic chip carrier which is plugged into 2 ASIC readout boards. The \nhigh-voltage cable is glued to the detector cathode (red wire). {\\bf \nRight:} Four CZT detectors surrounding the scintillator (blueish glow) \nform a unit referred to as `ring'~-- covering the whole $360^{\\circ}$ \nazimuthal scattering range with $4 \\times 8 = 32$ pixels. The whole \npolarimeter is equipped with eight rings.}\n\n\\label{fig:DetectorRing}\n\\end{figure*}\n\nThe balloon-borne version of X-Calibur is a low-Z Compton scattering \npolarimeter that will allow one to measure polarization fractions in the \n$20 - 80 \\, \\rm{keV}$ band down to the percentage level. X-Calibur will \nbe used in the focal plane of the X-ray mirror of the {\\it InFOC$\\mu$S}\\xspace telescope \nwith a field-of-view of $10 \\, \\rm{arc min}$; X-Calibur does not provide \nimaging capabilities. Owing to the fact that a grazing incidence mirror \nreflects only under very shallow angles, it changes the polarization \nproperties of X-rays by less than $1\\%$ \\cite{Katsuta2009}. The \nadvantages of the X-Calibur design can be described as follows. (i) A \nhigh detection efficiency is achieved, using roughly $80 \\%$ of photons \nimpinging on the polarimeter. (ii) The use of a focusing optics instead \nof a large detector volume results in a compact instrument design that \ncan be shielded efficiently~-- strongly reducing the background level. \n(iii) The continuous rotation of the polarimeter strongly reduces \npossible systematic effects that can hamper non-rotating polarimeters \ndue to asymmetric azimuthal detector responses. These characteristics, \nas outlined in the following sections, make it a well-suited experiment \nto study several sources mentioned in Sec.~\\ref{sec:Science} in a \none-day balloon flight. The energy-dependent detection efficiency of the \npolarimeter depends on (i) the effective area and point-spread function \nof the X-ray mirror, (ii) the fraction of scatterings compared to \ncompeting interactions such as photo-absorption, and (iii) the \ngeometrical detector coverage to record a high fraction of scattered \nX-rays (minimization of possible escape paths) \\cite{Guo2010}. This \nsection describes the overall design of the polarimeter, as well as the \ncharacteristics of its individual components.\n\n\n\n\\subsection{Design}\n\nThe conceptual design of the X-Calibur polarimeter is illustrated in \nFigures~\\ref{fig:XCLB_Functionality} and \\ref{fig:Design}. A low-Z \nscintillator rod aligned with the optical axis of the telescope is used \nas Compton-scatterer~-- leading to a polarization-dependent azimuthal \nscattering distribution that is recorded by the surrounding assembly of \nCZT detectors. The azimuthal distribution is resolved by $4\\times8 = 32$ \npixels for each of the $64$ depth bins along the optical axis. Detailed \ninformation on the simulations to optimize the X-Calibur design (as \npresented in this paper) can be found in \\citet{Krawcz2011} and \n\\citet{Guo2013}.\n\nThroughout the paper, we refer to a detector ring as a set of four CZT \ndetectors surrounding the scintillator rod on four sides at a given \ndepth along the optical axis (see Fig.~\\ref{fig:DetectorRing}, right). \nRing {\\it R1} is situated at the polarimeter entrance (top in \nFig.~\\ref{fig:Design}, left), and ring {\\it R8} is situated at its rear \nend. Each ring covers the whole $360^{\\circ}$ azimuthal scattering \nrange. A ring can further be subdivided into sub rings, eg. {\\it \nR8$_{\\rm{t}}$} and {\\it R8$_{\\rm{b}}$} for the top and bottom half, \nrespectively (see Fig.~\\ref{fig:Design}, left). The smallest possible \nsubdivision is a ring consisting of only one pixel row, referred to as \n{\\it single-pixel ring}\\xspace. All eight detectors situated on one of the four sides are \nreferred to as detector board {\\it Bd0} to {\\it Bd3} (see \nFig.~\\ref{fig:Design}, left).\n\nIn the scintillator, photoelectric effect interactions dominate below \n$\\sim$$15 \\, \\rm{keV}$. At $20 \\, \\rm{keV}$, however, the cross section \nof the photoelectric absorption already drops to $0.1 \\, \n\\rm{cm}^{2}/\\rm{g}$ and can be neglected as compared to the cross \nsection of Compton scattering for which X-Calibur is designed. This \nenergy regime therefore defines the detection threshold of the \npolarimeter. The mean free path for Compton scattering in the \nscintillator material is $\\approx$$4 \\, \\rm{cm}$, so that the length of \nthe scattering rod of $14 \\, \\rm{cm}$ covers $\\simeq 3.5$ path lengths. \nThis translates into a $\\simeq$$90 \\%$ probability for \nCompton-scattering in the energy regime of $20 - 80 \\, \\rm{keV}$. For \nsufficiently energetic photons, the Compton interaction produces enough \nscintillation light to trigger a photo-multiplier tube (PMT) attached to \nthe end of the scintillator. The trigger efficiency of the \nscintillator/PMT unit is discussed in \nSec.~\\ref{subsec:ScintillatorEfficiency}. The scattered X-rays are in \nturn photo-absorbed in the surrounding rings of high-Z CZT detectors. \nThis combination of scatterer/absorber leads to a high fraction of \nunambiguously detected Compton events~-- in contrast to background \nevents not entering the polarimeter along the optical axis for which the \nscintillator/CZT coincidences are strongly reduced (see \nSec.~\\ref{subsec:BG_Data}). Linearly polarized X-rays will preferably \nCompton-scatter perpendicular to their electric field vector, see \nEq.~(\\ref{eq:KleinNishinaCrossSection}). This will result in a \nmodulation of the measured azimuthal scattering distribution that is \nused to determine the polarization properties of the X-rays (see \nSec.~\\ref{subsec:AnalysisPolarization}).\n\n\n\\subsection{The CZT Detectors}\n\nX-ray photons that penetrate a semi-conductor X-ray detector deposit \ntheir energy in the detector volume, usually through photo-electric \neffect interactions. A charge cloud proportional to the X-ray energy is \ncreated and accelerated by a high-voltage electric field that is applied \nbetween the detector cathode and the pixels on the anode side. The \nmoving charge is measured and digitized and the grid position of the \ncorresponding pixel reflects the position of the interaction. CZT is the \nsemi-conductor material of choice for X-ray detectors operating in the \n$E>5 \\, \\rm{keV}$ to MeV energy band with a high probability for \nphoto-electric effect interactions, see for example \\citet{VarPitch}.\n\nThe CZT detectors used in X-Calibur were ordered from different \ncompanies\\footnote{Endicott Interconnect: http://www.evproducts.com, \nQuikpak/Redlen: http://redlen.ca, Creative Electron: \nhttp://creativeelectron.com}. The sample of detectors is listed in \nTab.~\\ref{tab:Detectors}. Each detector ($2\\times2 \\, \\rm{cm}^{2}$) is \ncontacted with a 64-pixel anode grid ($2.5 \\, \\rm{mm}$ pixel pitch) and \na monolithic cathode facing the scintillator rod. Two different detector \nthicknesses ($2 \\, \\rm{mm}$ and $5 \\, \\rm{mm}$) are used in the \npolarimeter (Fig.~\\ref{fig:Design}, left). Historically, our group had \nbeen working with detector thicknesses of $5 \\, \\rm{mm}$ and higher. \nTherefore, five of the polarimeter rings are equipped with $5 \\, \n\\rm{mm}$ detectors. The remaining three rings are equipped with $2 \\, \n\\rm{mm}$ detectors which are still sufficient to absorb more than $99\\%$ \nof X-rays in the energy regime relevant for X-Calibur, and at the same \ntime measure a lower level of background (which scales roughly with the \ndetector volume). The cathodes of the detectors are biased at \n$V_{\\rm{bi,5}} = -500 \\, \\rm{V}$ ($5 \\, \\rm{mm}$ detectors) and at \n$V_{\\rm{bi,2}} = -150 \\, \\rm{V}$ ($2 \\, \\rm{mm}$ detectors), \nrespectively.\n\nEach CZT detector is permanently bonded (anode side) to a ceramic chip \ncarrier which is plugged into the electronic readout board. \nFigure~\\ref{fig:DetectorRing} (left) shows a single CZT detector unit \nwith an $8 \\times 8$ pixel matrix on the anode side as well as the \nreadout electronics. Each CZT detector is read out by two digitizer \nboards, each consisting of a 32 channel ASIC and a 12-bit \nanalog-to-digital converter. The ASIC was developed by G.~De~Geronimo \n(BNL) and E.~Wulf (NRL) \\cite{Wulf2007}. The ASICs are operated at a \nmedium amplification (gain) of $28.5 \\, \\rm{mV/fC}$ and a signal peaking \ntime of $0.5 \\, \\mu \\rm{s}$. These settings are a result of a previous \noptimization to achieve an optimal energy resolution and low noise. Each \nASIC has a built-in capacitor that allows one to directly inject a \nprogrammable amount of charge into the individual readout channels for \ntesting purposes. The readout noise of the ASIC is as low as $2.5 \\, \n\\rm{keV}$ FWHM (see Fig.~\\ref{fig:TempStudiesNoise} in \nSec.~\\ref{subsec:TempStudies}). All 16 digitizer boards (reading eight \nCZT detectors) are read out by one harvester board ({\\it Bd0}-{\\it Bd3}, \nsee Fig.~\\ref{fig:Design}) transmitting the data to a PC-104 computer \nwith a rate of 6.25~Mbits/s. X-Calibur comprises 2048 data channels. The \ntime to read and process a triggered event is about $130 \\, \\mu\\rm{s}$ \n(ASIC dead time). However, only the ASIC involved in the triggered event \nwill be dead during the read-out. All other ASICs will still be \nsensitive and can store events that will be read out once the previous \nread-out cycle is completed.\n\n\\subsection{The Scintillator}\n\nA plastic scintillator rod is used as Compton-scatterer. The advantage, \ncompared to other scattering materials, is the scintillation light \nproduced in the scattering interaction. The light is read by a PMT and \ncan be used (optional) in the analysis. The EJ-200 scintillator \n(Hydrogen:Carbon ratio of $5.17:4.69$, $\\left< Z \\right> = 3.4$, $\\rho \n\\approx 1 \\, \\rm{g} / \\rm{cm}^{3}$, decay time $2.1 \\, \\rm{nsec}$) is \nused, read by a Hamamatsu R7600U-200 PMT with a high quantum efficiency \nsuper-bi-alkali photo cathode. To increase the optical yield, the \nscintillator is wrapped in white {\\it tyvek$^{\\tiny \\circledR}$} paper. \nThe PMT signal is amplified and digitized. A discriminator tests whether \nthe digitized PMT pulse exceeds a programmable trigger threshold and \nactivates a corresponding flag ($f_{\\rm{sci}}$). The flag is kept high \nfor $6 \\, \\mu\\rm{s}$ and is merged into the data stream of triggered CZT \ndetector events. The trigger efficiency of the scintillator is studied \nin Sec.~\\ref{subsec:ScintillatorEfficiency}. The $f_{\\rm{sci}}$ flag \nallows one to select scintillator/CZT events from the data, which \nrepresent likely Compton-scattering candidates~-- strongly suppressing \nother backgrounds (see Sec.~\\ref{subsec:BG_Data}). However, the \npolarimeter can be operated without the PMT trigger information, with \nthe scintillator acting only as a passive scatterer.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[height=0.31\\textheight]{beilicke_FRAWS_2013_02_fig03}\n\\includegraphics[height=0.31\\textheight]{P1090353_MotorAndBearing_Plus_TungstenCap}\n\\hfill\n\\includegraphics[height=0.31\\textheight]{P1090507_X-CaliburInVessel_Top}\n\\hfill\n\\includegraphics[height=0.31\\textheight]{InfocusTelescope}\n\\end{center}\n\n\\caption{Active/passive shield, rotation design, and {\\it InFOC$\\mu$S}\\xspace X-ray \ntelescope. {\\bf Left:} X-Calibur polarimeter, embedded by the CsI active \nshield, as well as the electronic readout and azimuthal rotation \nbearing. {\\bf 2$^{\\rm{nd}}$ from left, top:} Tungsten cap with \ncollimator, only allowing X-rays from the mirror to enter the shield. \n{\\bf 2$^{\\rm{nd}}$ from left, bottom:} Motor and ring bearing to rotate \npolarimeter/shield assembly. {\\bf 2$^{\\rm{nd}}$ from right:} Opened \npressure vessel with shield/X-Calibur installed (tungsten cap removed). \nThe X-ray beam enters the polarimeter through the hole in the top plate. \n{\\bf Right:} The {\\it InFOC$\\mu$S}\\xspace balloon gondola/truss with active pointing \ncontrol and the X-ray mirror installed. The pressure vessel is installed \nin the back.}\n\n\\label{fig:ActiveShieldAndTelescope}\n\n\\end{figure*}\n\n\\subsection{The Shield}\n\nDuring the balloon flight, the polarimeter will be hit by charged and \nneutral particle backgrounds with different spectral signatures and \nintensities. These backgrounds reduce the signal-to-noise ratio and, in \nthe case of non-isotropic fluxes, can even lead to a fake polarization \nsignature. In order to suppress these backgrounds, the polarimeter and \nthe front-end readout electronics are operated inside an active CsI(Na) \nanti-coincidence shield. The $2.7 \\, \\rm{cm}$ thick CsI crystal of the \nshield covers the sides and the bottom of the polarimeter and produces \nscintillation light when particles interact. The top is protected by a \npassive tungsten plate/collimator \n(Fig.~\\ref{fig:ActiveShieldAndTelescope}, left), blocking X-rays and \nparticles that do not come from the X-ray mirror. The (active) CsI \nscintillator of the shield is read out by four Hamamtsu PMTs R\\,6233 \nwhich are biased at $V_{\\rm{bi}} = +800 \\, \\rm{V}$. The analog signal of \nall four PMTs is merged and in turn digitized. A programmable, digital \ndiscriminator decides on whether a shield flag $f_{\\rm{shld}}$ is set on \nthe CZT readout board (kept up for $6 \\, \\mu\\rm{s}$) and is merged into \nthe data stream. The values of the discriminator and the width of the \nflag were optimized using a radio-active source to maximize the shield \nefficiency and minimize chance coincidences (see \nSec.~\\ref{subsec:BG_Data}).\n\n\nIn order to reduce the systematic uncertainties of the polarization \nmeasurements, the polarimeter and the active shield will be rotated \naround the optical axis with $\\sim 2 \\, \\rm{rpm}$ using a ring bearing \n(see Fig.~\\ref{fig:ActiveShieldAndTelescope}). The angle between the \npolarimeter/shield and the mounting fixture is read out by a code wheel \nwith the accuracy of $1^{\\circ}$. A counter-rotating mass can be used to \ncancel the net angular momentum of the rotating polarimeter assembly \nduring the balloon flight. The computer reading the PMT and CZT events \nis part of the rotating assembly, and referred to as polarimeter CPU.\n\n\n\\subsection{The {\\it InFOC$\\mu$S}\\xspace X-ray Telescope}\n\nThe X-Calibur polarimeter will be flown in a pressurized vessel located \nin the focal plane of the {\\it InFOC$\\mu$S}\\xspace X-ray telescope \n\\cite{InFocus_FirstFlight}. The telescope is shown in the right panel of \nFig.~\\ref{fig:ActiveShieldAndTelescope}. A Wolter grazing incidence \nmirror focuses the X-rays onto the polarimeter. The X-Calibur \nscintillator rod will be aligned with the optical axis of the {\\it InFOC$\\mu$S}\\xspace \nX-ray telescope (see Sec.~\\ref{subsec:FtSumnerData}). The focal length \nof the mirror is $8 \\, \\rm{m}$ and the field of view is $\\rm{FWHM} = 10 \n\\, \\rm{arc min}$. The telescope truss of {\\it InFOC$\\mu$S}\\xspace is only coupled to the \ngondola by a ball joint in a support cup with floating oil, allowing for \nfull inertial pointing of the telescope with an accuracy of $7 ''$ and \n$15 ''$ RMS in altitude and azimuth, respectively. To maintain the \ndecoupling between truss and gondola, any communication between the two \nsystems is done by wireless connections. In addition to the rotating \npolarimeter CPU (see above), a second CPU as part of the X-Calibur \nsubsystem (motor CPU) is installed in the pressure vessel (non-rotating) \nand controls the motors, and a temperature system. Power and data \ncommunication between the polarimeter CPU and the motor CPU is achieved \nby a {\\it Mercotac$^{\\tiny \\circledR}$} 830-SS rotating ring of mercury \nsliding contacts. Communication between the pressure vessel and the \ntelescope gondola will be done via a wireless network. The data will be \nstored on solid state drives and will in parallel be down-linked to the \nground.\n\n\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[width=0.44\\textwidth]{X-Calibur_ConfigurationsCZT} \\\\\n\\includegraphics[width=0.46\\textwidth]{X-Calibur_Configurations}\n\\end{center}\n\n\\caption{Different X-Calibur configurations referred to via \n$C_{\\rm{env}}^{\\rm{stp}}$. {\\bf Top row:} CZT detector geometries (see \nFig.~\\ref{fig:Design}, left) viewed along the optical axis looking into \nthe X-Ray beam. Each detector is tangentially translated by $\\Delta = 1 \n\\, \\rm{mm}$. $C^{1}$: CZT detector cathodes being $c = 16 \\, \\rm{mm}$ \naway from the optical axis (regular-sized sealed radioactive source fits \nin between). $C^{2}$: Reduced distance of $c =11 \\, \\rm{mm}$ (only \ncompact sources). $C^{3}$: same as $C^{2}$ but with the scintillator rod \ninstalled. The definitions of $\\Phi$ and $\\Delta \\Phi$ as used in \nSec.~\\ref{sec:XCLB_Characterization} are illustrated. {\\bf Bottom:} \nSchematic sketches (not to scale) of X-Calibur installed in the \ndifferent environments. $C_{\\rm{wu}}$: Washington University; detector \ncalibration in copper box ($C_{\\rm{wu1a}}$) or in passive CsI shield \n($C_{\\rm{wu1b}}$); performance measurements in the active shield with \ndifferent source positions ($C_{\\rm{wu2a}}$, $C_{\\rm{wu2b}}$, and \n$C_{\\rm{wu2c}}$). $C_{\\rm{ch}}$: X-Calibur/CHESS setup in hutch {\\it \nC1}. The beam intensity is reduced by absorption foils; platinum \n($C_{\\rm{ch1}}$), or platinum and lead ($C_{\\rm{ch3}}$). The X-Calibur \npolarimeter can be rotated around its optical axis (angle $\\alpha$) \nwhich is aligned with the CHESS beam. $C_{\\rm{ft}}$: X-Calibur installed \nin the {\\it InFOC$\\mu$S}\\xspace X-ray telescope during a flight preparation campaign in \nFt.~Sumner.}\n\n\\label{fig:X-Calibur_Configurations}\n\n\\end{figure}\n\n\n\\subsection{X-Calibur Configurations and Data Sets}\n\nIn order to characterize the different components and aspects of the \nX-Calibur polarimeter, different types of measurements were performed \nwith different geometrical configurations of the instrument (e.g. with \nand without the shield, measurements without the scattering rod to \ncalibrate the CZT detector response itself, illumination of the \ninstrument with different X-ray sources from different angles, etc.). \nThe measurements were performed at different facilities/locations (which \nwe refer to as environments). The energy calibration and \ncharacterization of the CZT detectors was performed in the laboratory at \nWashington University (Sec.~\\ref{sec:DetectorCharacterization}). \nMeasurements of a polarized X-ray beam to study the performance of the \npolarimeter were conducted at the CHESS synchrotron facility at Cornell \nUniversity (Sec.~\\ref{subsec:CHESS}). Data were also taken with the \nfully integrated X-Calibur/{\\it InFOC$\\mu$S}\\xspace telescope in a field campaign in \nFt.Sumner, NM (Sec.~\\ref{subsec:FtSumnerData}). Measurements of the \nbackground (Sec.~\\ref{subsec:BG_Data}) were performed at Washington \nUniversity, CHESS, and in Ft.Sumner. Throughout the paper, each \nconfiguration and location is referred to as $C_{\\rm{env}}^{\\rm{stp}}$ \n(`env' referring to the environment/location and `stp' referring to the \nexperimental setup). These different configurations are shown in \nFig.~\\ref{fig:X-Calibur_Configurations} and will be referred to in the \nsections to follow in which the corresponding results are presented.\n\nIt should be noted, that some of the data presented in this paper were \ntaken without the CsI shield (e.g. the data taken at the CHESS \nfacility), not allowing one to use the shield veto for background \nsuppression. Since separate background runs were taken and subtracted \nfrom the data, and the majority of measurements is signal-dominated, \nthis does not affect any result or conclusion presented in this paper.\n\n\n\\section{Reconstruction of Polarization Properties} \\label{sec:Analysis}\n\nThe events recorded by X-Calibur consist of the digitized pulse height \nof one (or more) detector pixel(s), a time stamp, and flags describing \nwhether the shield and/or the scintillator triggered. These raw events \nare first transformed into measured energies using the detector \ncalibration. The reconstructed events are further processes in order to \nderive energy spectra and the polarization properties of the measured \nX-ray beam. This section outlines the corresponding procedures.\n\n\n\\subsection{Definitions}\n\nThe modulation in the measured azimuthal scattering distribution is the \ndefining signature from which the polarization properties are derived. \nThe modulation factor $\\mu$ describes the polarimeter response to a \npolarized beam and is used to reconstruct the polarization properties. \nAssuming a $100 \\%$ linearly polarized X-ray beam, the minimum \n($C_{\\rm{min}}$) and maximum ($C_{\\rm{max}}$) number of counts of the \nazimuthal scattering distribution define:\n\n\\begin{equation}\n\\label{eq:ModulationFactor}\n\\mu = \\frac{C_{\\rm{max}} - C_{\\rm{min}}}{C_{\\rm{max}} + C_{\\rm{min}}}.\n\\end{equation}\nIt represents the modulation amplitude of a $100 \\%$ polarized beam and \ndepends on the polarimeter design and the physics of Compton-scattering. \n\nThe performance of a polarimeter can be characterized by the minimum \ndetectable polarization (MDP) as the minimum fraction of polarization \nthat can be detected at the $99 \\%$ confidence level for a given time of \nobservation $T$. Assuming a polarimeter that detects all \nCompton-scattered photons with an ideal angular resolution~-- in this \ncase $\\mu$ becomes the modulation amplitude averaged over all solid \nangles and the Klein-Nishina cross section~-- one can estimate the MDP \nby integrating the scattering probability distribution \n\\cite{Weisskopf2011, StokesAnalysis} ($R_{\\rm{src}}$ and $R_{\\rm{bg}}$ \nare the source and background count rates, respectively):\n\n\\begin{equation}\n\\label{eqn:MDP}\n\\rm{MDP} \\simeq \\frac{4.29}{\\mu R_{\\rm{src}}} \\sqrt{\\frac{R_{\\rm{src}} +\nR_{\\rm{bg}}}{T}}.\n\\end{equation}\n\n\n\\subsection{X-Calibur Event Reconstruction and Selection}\n\nEach recorded X-Calibur event contains an event number, a GPS time stamp \n$T_{\\rm{gps}}$, the orientation angle $\\theta_{\\rm{w}}$ of the \nshield/polarimeter with respect to the mounting plate (read by a code \nwheel), and a list of CZT detector pixels that were hit (up to nine) \nincluding their digitized pulse heights. Furthermore, two flags are \nmerged into the data stream: (i) a flag $f_{\\rm{shld}}$ that indicates \nwhether the PMTs reading the active CsI shield got a signal exceeding \nthe defined discriminator threshold, and (ii) a flag $f_{\\rm{sci}}$ that \nindicates if the PMT reading the central scintillator rod of the \npolarimeter (see Fig.~\\ref{fig:Design}, left) exceeded its discriminator \nthreshold. The average analog rise/fall times \n$\\tau_{\\rm{r}}$/$\\tau_{\\rm{f}}$ (time for a signal rise from $10-90\\%$ \nof the amplitude) of the folded scintillator/PMT response were measured \nfor the shield and for the scintillator. For the shield we find \n$\\tau_{\\rm{r}}^{\\rm{shld}} \\approx 70 \\, \\rm{ns}$ and \n$\\tau_{\\rm{f}}^{\\rm{shld}} \\approx 2.6 \\, \\mu\\rm{s}$, respectively. The \nresponse of the scintillator rod of the polarimeter is much faster: \n$\\tau_{\\rm{r}}^{\\rm{sci}} \\approx 3.0 \\pm 1.3 \\, \\rm{ns}$ and \n$\\tau_{\\rm{f}}^{\\rm{sci}} \\approx 13.7 \\pm 5.1 \\, \\rm{ns}$ for cosmic \nrays, and $\\tau_{\\rm{r}}^{\\rm{sci}} \\approx 4.7 \\pm 2.7 \\, \\rm{ns}$ and \n$\\tau_{\\rm{f}}^{\\rm{sci}} \\approx 13.5 \\pm 5.0 \\, \\rm{ns}$ for a \nCs$^{137}$ source placed at configuration $C_{\\rm{wu2c}}^{3}$, \nrespectively. Upon a CZT trigger, the event readout is delayed by \n$2.5-3.3 \\, \\mu\\rm{s}$ (jitter) to allow the shield and scintillator PMT \nsignals to built-up and being converted into the corresponding flags. \nThe flags $f_{\\rm{shld}}$ and $f_{\\rm{sci}}$ are kept up for a duration \nof $6 \\, \\mu\\rm{s}$.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{MultiplicityDistribution}\n\\end{center}\n\n\\caption{Distribution of pixel multiplicities $m$ for events taken with \ndifferent sources (recorded in detector ring {\\it R3}): Cosmic ray (CR) \nbackground in the laboratory, energy calibration using Eu$^{152}$, \nEu$^{152}$ calibration restricting event energies to $E < 80 \\, \n\\rm{keV}$, and data taken at the $40 \\, \\rm{keV}$ CHESS beam (see \nSec.~\\ref{subsec:CHESS}).}\n\n\\label{fig:MultiplicityDistribution}\n\\end{figure}\n\nFor each event, the pulse heights of all contributing channels/pixels \n$i=1...m$ are transformed into energies $E_{i}$ using the channel \ncalibration, see Eq.~(\\ref{eq:EnergyCalibration}) in \nSec.~\\ref{subsec:CZT_Calibration}. The total energy of the CZT event is \n$E = \\sum_{i} E_{i}$. The number of pixels $m$ participating in the \nevent is referred to as the pixel multiplicity. Selection cuts can be \napplied to the data based on the event properties mentioned above: \n$T_{\\rm{gps}}$, $\\theta_{\\rm{w}}$, $E$, $m$, $f_{\\rm{shld}}$, and \n$f_{\\rm{sci}}$. Figure~\\ref{fig:MultiplicityDistribution} shows the \ndistribution of $m$ for different source types (measured in detector \nring {\\it R3}). It can be seen that sources with high energy \ncontributions (such as the cosmic ray background or Eu$^{152}$ with its \n$E> 100 \\, \\rm{keV}$ energy lines) have $m=1$ pixel event contributions \nof $\\simeq 70 \\%$. Sources with energies concentrated in the $20-80 \\, \n\\rm{keV}$ interval, relevant for X-Calibur, have $m=1$ contributions of \n$> 90 \\%$ (e.g. the $40 \\, \\rm{keV}$ CHESS beam, see \nSec.~\\ref{subsec:CHESS}). This is explained by the fact that low energy \nX-rays deposit smaller and more concentrated charge clouds in the CZT \nwith a reduced chance of charge sharing between pixels (that would cause \n$m \\geq 2$ events). Therefore, an event selection cut on $m=1$ is a \nreasonable way for background subtraction without loosing signal events \nat low energies. Cuts on the energy $E$ and the shield flag \n$f_{\\rm{shld}}$ will further reduce the background (see \nSec.~\\ref{subsec:BG_Data}). A selection cut on the scintillator flag \n$f_{\\rm{sci}}$ selects a very clean sample of events that \nCompton-scattered in the scintillator. This has the potential to further \nreduce the background~-- however, with a loss in efficiency at low \nenergies (see Sec.~\\ref{subsec:ScintillatorEfficiency}). However, a cut \non $f_{\\rm{sci}}$ is optional and not a requirement for sensitive \npolarization measurements with X-Calibur.\n\n\n\\subsection{Energy Spectra}\n\nEnergy spectra are used to study the scattering properties of the \npolarimeter and mark an intermediate step to derive the energy-dependent \npolarization properties. Energy spectra can be derived for individual \npixels or for groups of pixels (e.g. a CZT detector or a detector ring \n{\\it R}). The energy spectra shown in this paper are normalized to the \nacquisition time (dead time corrected), the anode detector surface \ncovered by the corresponding pixel group and the width of the energy \nbins. Note, the limited dynamical range of the charge digitization of \nindividual pixel channels will lead to discrete energies. Given the \ndifferent energy calibrations of the channels, these will differ from \npixel to pixel. This difference can lead to binning artifacts in energy \nspectra that are obtained from a small group of pixels.\n\n\n\\subsection{Polarization Properties} \\label{subsec:AnalysisPolarization}\n\nThe signature of a polarized beam will be imprinted in the azimuthal \nscattering distribution of recorded events (see for example \nFigs.~\\ref{fig:AzimuthDistribution} or \\ref{fig:CHESS_2DAzimuthData} in \nSec.~\\ref{subsec:CHESS}). Either the scattering distribution itself, or \na method involving the Stokes parameters, can be used to extract the \npolarization fraction $r$ and polarization direction $\\Omega$. First, \nthe details of the detector geometry have to be carefully considered in \nthe analysis, as outlined below.\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{X-Calibur_CZT_BeamOffset}\n\\hfill\n\\includegraphics[width=0.48\\textwidth]{StokesInterpol}\n\\end{center}\n\n\\caption{{\\bf Left:} Angular coverage of two selected CZT pixels within \na {\\it single-pixel ring}\\xspace for different beam positions $P$. For a non-rotating detector \n(left) the angular coverage for each pixel only needs to be calculated \nonce. In the rotating system (right) the beam offset in the horizon \nsystem (solid arrow) leads to varying angular coverage per pixel \ndepending on the current rotation $\\alpha$ that have to be updated on an \nevent-by-event basis. Note, the pixel contacts are physically deposited \non the backside of the detectors but are shown on the front side (facing \nthe scintillator) for a better illustration of the angular measurement. \nThe cyan dashed arrows indicate the $x/z$ detector coordinate system. \n{\\bf Right:} Stokes parameters $Q$, $U$, and $I$ as measured within a \n{\\it single-pixel ring}\\xspace for a polarized beam with $\\Omega = 90^{\\circ}$. The pixels $j$ \nhave varying angular coverage $\\Delta \\Phi_{j}$ (indicated by the \nhorizontal error bars). The gray bands indicate the gaps between \ndetector boards (see left panel). The filled data points show the \nmeasured values. The open data points show the interpolated Stokes \nparameters to recover the contributions missed by the gaps. Note, the \nentries in the figure are normalized to the solid angle $\\Delta \\Phi$ \ncovered by each pixel. Therefore, the normalization leads to apparent \njumps from the pixels to the values shown for the gaps. The open symbol \nat $\\Phi \\simeq 360^{\\circ}$ corresponds to a dead pixel.}\n\n\\label{fig:SchematicBeamOffset_AndStokes}\n\\end{figure*}\n\n{\\it Azimuthal pixel coverage.} The $360^{\\circ}$ range in azimuth is \ncovered by the $4 \\times 8 = 32$ pixels per {\\it single-pixel ring}\\xspace (see \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). Different pixels \ncover different azimuthal ranges $\\Delta \\Phi$ with respect to the \ncenter of the scattering rod which defines the optical axis. Therefore, \nthe four sides of detector boards lead to a 4-fold symmetry in the \nscattering distribution (Fig.~\\ref{fig:AzimuthDistribution}). This \npurely geometrical effect can be corrected for by dividing the counts \n$C_{j}$ in each pixel $j$ by its azimuthal coverage $\\Delta \\Phi_{j}$.\n\n{\\it Azimuthal pixel coverage for beam offsets.} A special situation \narises if the optical axis of the X-ray beam is not aligned with the \ngeometrical axis of the polarimeter (see $P_{1}$ in \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). Such an offset, if \nnot corrected, will introduce asymmetries in the azimuthal scattering \ndistributions and can mimic a wrong polarization signature (see \nSec.~\\ref{subsec:Systematics} for a detailed study). Assuming a known \noffset vector $\\vec{P}1$, the azimuthal coverage $\\Delta \\Phi_{p1, j}$ \ncan be recalculated for each pixel $j$. This correction re-aligns the \norigin of the detector system with the beam axis and strongly reduces \nthe systematic effect introduced by the offset. In the case of \nsufficient event statistics, first moments can be used to estimate the \nbeam offset from the data itself (see Sec.~\\ref{subsec:Systematics}). If \nthe polarimeter is rotating, the beam offset in the horizon system will \nrotate in the detector system (see \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). In this case \n$\\Delta \\Phi_{p1,j}$ has to be updated on an event-by-event basis, \ntaking into account the current polarimeter orientation $\\alpha$ \nmeasured by the code wheel.\n\n{\\it Pixel acceptance and flat fielding.} Individual pixels have \ndifferent trigger efficiencies, energy thresholds and energy resolutions \naffecting the number of counts derived from a given energy interval. To \ncorrect for these differences, the pixels are flat fielded using \nCompton-scattered events recorded from a non-polarized X-ray beam that \nresults in a flat azimuthal scattering distribution. For a {\\it single-pixel ring}\\xspace the \nevent counts per azimuthal coverage $C_{j} / \\Delta \\Phi_{j}$ are \naveraged for all $j=1...32$ pixels and are used to determine a relative \nazimuthal acceptance $a_{j}$:\n\n\\begin{equation}\nw_{j} = \\frac{1}{a_{j}} = \\frac{\\Delta \\Phi_{j}}{C_{j}} \\frac{1}{N}\\sum_{i=1}^{N}\\frac{C_{i}}{\\Delta \\Phi_{i}}.\n\\label{eqn:Acceptance}\n\\end{equation}\n\nThe pixel acceptance $a_{j}$ can in turn be used to weight individual \nevents with $w_{j}=1/a_{j}$. Implicitly, $a_{j}$ depends on the event \nselection cuts~-- so that it has to be computed for the particular set \nof cuts applied to the data. Dead pixels cannot be recovered by a \ncorresponding weight since they contribute zero events. However, once \nthe polarimeter/shield assembly is rotating with respect to the \npolarization plane, the pixel acceptances can be ignored since they \naverage out throughout the measurement~-- this includes the treatment of \ndead pixels.\n\n{\\it Polarization properties derived from the azimuthal scattering \ndistribution.} Integrating $C_{j} / \\Delta \\Phi_{j}$ for each pixel $j$ \nin a certain energy range and weighting the individual counts with \n$w_{j}$ will result in the azimuthal scattering distribution. Here, the \nangular coverage $\\Delta \\Phi_{j}$ determines the horizontal error bar \nof the data point. The scattering distribution can be derived for \nindividual detector rings. Fitting a sinusoidal function to the \ndistribution (e.g. left panel in Fig.~\\ref{fig:AzimuthDistribution}, \nSec.~\\ref{subsec:CHESS}) allows one to reconstruct (i) the orientation \nof the polarization plane (minimum), as well as (ii) the modulation of \nthe data $\\mu_{\\rm{data}}$ following Eq.~(\\ref{eq:ModulationFactor}). \nThe corresponding modulation factor $\\mu_{\\rm{sim}}$ is derived from \nsimulations of a $100 \\%$ polarized beam (see \nSec.~\\ref{sec:Simulations}) that are analyzed using the same set of \nevent selection cuts. The polarization fraction $r$ of the measured beam \nis in turn calculated to be:\n\n\\begin{equation}\nr = \\frac{\\mu_{\\rm{data}}}{\\mu_{\\rm{sim}}}.\n\\label{eqn:PolFracPhiDistri}\n\\end{equation}\n\nThe effect of dead detector pixels and gaps between the detector boards \n(see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left) is accounted \nfor automatically, since the corresponding points do not show up in the \ndistribution and will not affect the fit.\n\n{\\it Polarization properties derived from Stokes parameters.} An \nalternative approach to reconstruct the polarization properties is based \non the Stokes parameters \\cite{Chandrasekhar1960, StokesAnalysis} that \nare calculated for each event $k$:\n\n\\begin{equation}\nq_{k} = \\cos(2 \\Phi_{k}), \\quad u_{k} = \\sin(2 \\Phi_{k}), \\quad i_{k} = 1.\n\\label{eqn:StokesParams}\n\\end{equation}\n\nThe Stokes parameters can be summed for a subset of the data consisting \nof $N$ events (e.g. over a specific energy interval and detector ring):\n\n\\begin{equation}\nQ = \\sum_{k=1}^{N} w_{j(k)} q_{k}, \\quad U = \\sum_{k=1}^{N} w_{j(k)} u_{k}, \\quad I = \\sum_{k=1}^{N} w_{j(k)}.\n\\label{eqn:StokesSum}\n\\end{equation}\nHere, each event $k$ (originating from pixel $j$) is weighted with \n$w_{j(k)} = 1/a_{j(k)}$. Since each pixel $j$ covers a range in azimuth \n$\\Delta \\Phi_{j} = \\Phi_{j,\\rm{max}} - \\Phi_{j,\\rm{min}}$, the values \n$q_{k}$ and $u_{k}$ derived from the mean angle $\\Phi_{j(k)}$ in a \nnon-rotating system may lead to inaccurate and/or biased results. \nTherefore, the mean Stokes parameters $\\left< q \\right>_{j(k)}$ and \n$\\left< u \\right>_{j(k)}$ are calculated based on the covered range in \nazimuth of the corresponding pixel $j$:\n\n\\begin{eqnarray}\n\\begin{split}\n\\left< q \\right>_{j(k)} & = \\frac{1}{\\Delta \\Phi_{j(k)}} \\int_{\\Phi_{j(k),\\rm{min}}}^{\\Phi_{j(k),\\rm{max}}} \\cos(2 \\Phi) \\, \\rm{d}\\Phi, \\\\\n\\left< u \\right>_{j(k)} & = \\frac{1}{\\Delta \\Phi_{j(k)}} \\int_{\\Phi_{j(k),\\rm{min}}}^{\\Phi_{j(k),\\rm{max}}} \\sin(2 \\Phi) \\, \\rm{d}\\Phi.\n\\end{split}\n\\label{eqn:MeanStokes}\n\\end{eqnarray}\n\nThe values of $\\left< q \\right>_{j(k)}$ and $\\left< u \\right>_{j(k)}$ \nare in turn used in Eq.~(\\ref{eqn:StokesSum}). The proper treatment of \ndead pixels and gaps between the detector boards {\\it Bd0}-{\\it Bd3} \n(see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left) is crucial when \nworking with the Stokes parameters in a non-rotating coordinate system, \nsince a lack of events from a particular azimuthal direction \n$\\tilde{\\Phi}$ will lead to an increase/decrease in $\\sum_{k} \\left< q \n\\right>_{j(k)}$ and/or $\\sum_{k} \\left< u \\right>_{j(k)}$, \nsystematically affecting the reconstructed polarization properties (see \nFig.~\\ref{fig:SchematicBeamOffset_AndStokes}, right). The amplitude and \nsign of the resulting effect depends on the orientation between the \npolarization plane and the detector plane (assuming the gaps are located \nat angles of $45^{\\circ}$, $135^{\\circ}$, $225^{\\circ}$, and \n$315^{\\circ}$, see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). \nIf the planes are exactly parallel or exactly perpendicular, the effect \nleads to a maximal underestimation of $r$. An angle of $45^{\\circ}$ \n(polarization plane being aligned with two of the four gaps) leads to \nthe maximal overestimation of $r$. At angles of $45/2 = 27.5^{\\circ}$ \n(and multiples thereof) the effect of detector gaps cancels. A \nsimulation for the X-Calibur detector layout was performed and resulted \nin a systematic effect of $\\pm6 \\%$ for a $100\\%$ polarized beam. \nTherefore, a correction has to be applied: (i) the $m=1,2,...$ azimuthal \nranges $\\delta \\tilde{\\Phi}_{m}$ covered by the gaps have to be \nidentified in which the detector is not sensitive (ii) The distributions \nof $\\delta Q / \\delta \\Phi$, $\\delta U / \\delta \\Phi$, and $\\delta I / \n\\delta \\Phi$ have to be collected as a function of $\\Phi$ \n(Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, right). (iii) These \ndistributions are in turn used to interpolate the data gaps $\\delta \n\\tilde{\\Phi}_{m}$ from neighboring pixel to recover the `missing' \ncontributions in Eq.~(\\ref{eqn:StokesSum}):\n\n\\begin{eqnarray}\n\\begin{split}\nQ & \\rightarrow Q + \\sum_{m} \\Delta \\tilde{Q}_{m} \\delta \n\\tilde{\\Phi}_{m}, \\\\ U & \\rightarrow U + \\sum_{m} \\Delta \\tilde{U}_{m} \n\\delta \\tilde{\\Phi}_{m}, \\\\ I & \\rightarrow I + \\sum_{m} \\Delta \n\\tilde{I}_{m} \\delta \\tilde{\\Phi}_{m}.\n\\end{split} \n\\label{eqn:StokesGapCorrection} \n\\end{eqnarray}\nThe errors on the added sums are calculated using error propagation \nduring the interpolation. Given the scalar nature of the Stokes \nanalysis, there is no simple way of representing the azimuthal \nmodulation of the scattering distribution. Therefore, it is useful to \ncompare the Stokes results with the results obtained from the azimuthal \nscattering distribution (previous paragraph) in order to identify \npossible systematic effects. Note again, that a rotating polarimeter \nwill not require a correction for dead pixels or detector gaps.\n\nThe Stokes sums in (\\ref{eqn:StokesSum}) or \n(\\ref{eqn:StokesGapCorrection}) can be used to reconstruct the \npolarization fraction $r$ and polarization angle~$\\Omega$:\n\n\\begin{eqnarray}\n\\begin{split}\nr & = \\frac{2}{\\mu_{\\rm{sim}}} \\frac{\\sqrt{Q^{2} + U^{2}}}{I}, \\\\\n\\Omega & = \\frac{1}{2} \\rm{atan} (U / Q).\n\\end{split}\n\\label{eqn:PolarizationFromStokes}\n\\end{eqnarray}\n\nResults from $l$ independent measurements (e.g. from different detector \nrings) can be combined in a weighted average. Since the polarization \nfraction is always positive it can lead to an overestimation if the true \npolarization fraction is in the MDP regime of the data set, see \nEq.~(\\ref{eqn:MDP}), where the error bars are highly asymmetric. This \nsystematic effect can be avoided by averaging a set of modified Stokes \nparameters:\n\n\\begin{eqnarray}\n\\begin{split}\nQ' & = \\sum_{l} \\frac{2 w_{l}}{\\mu_{\\rm{sim,l}}} \\frac{Q_{l}}{I_{l}}, \\quad U' = \\sum_{l} \\frac{2 w_{l}}{\\mu_{\\rm{sim,l}}} \\frac{U_{l}}{I_{l}}, \\\\\nr' & = \\frac{1}{L'} \\sqrt{Q'^{2} + U'^{2}} \\quad {\\rm with} \\quad L' = \\sum_{l} w_{l}.\n\\end{split}\n\\label{eqn:Stokes_Q_U_Average}\n\\end{eqnarray}\nThe weights $w_{l} = 1/\\sigma_{l}^{2}$ account for the statistical \nuncertainties of the individual measurements $l$.\n\n{\\it Unfolding analysis.} An unfolding analysis that takes into account \nthe energy-dependent detector response and photon detection efficiencies \nand that can be used to reconstruct polarization fraction and angle as a \nfunction of true photon energy will be described in a separate paper \n\\cite{XCLB_LogLikelihoodAnalysis}.\n\n{\\it Forward folding.} Recording the azimuthal scattering distributions \nwith planar detectors will lead to projection effects that depend on the \npolar angle of the scattering in the scintillator. For most parts of the \npolarimeter these effects are negligible, since (i) a particular \ndetector ring sees the superposition of different polar scattering \nangles canceling the effect, and (ii) the same effect is present in the \nsimulations that are used to determine the polarization fraction \nfollowing Eq.~(\\ref{eqn:PolFracPhiDistri}). Only for polarization \nfractions $r \\ll 1$ measured in detector ring {\\it R1}, which sees only \nback-scatter events, a second order correction may be \nneeded\\footnote{Here, the $r \\propto \\mu_{\\rm{data}}$ relation in \nEq.~(\\ref{eqn:PolFracPhiDistri}) will no longer be exactly linear.}. To \nfully take into account these effects, the data can be analyzed with a \nforward folding method (which is beyond the scope of this paper). The \nmodeling of the pixel acceptance in {\\it R1}, only affecting \nmeasurements with the non-rotating polarimeter, will also be slightly \naffected by the effect.\n\n\n\\section{Simulations} \\label{sec:Simulations}\n\n\nSimulations of the energy-dependent response of the X-Calibur \npolarimeter are needed in order to reconstruct the polarization \nproperties from measured data (see \nSec.~\\ref{subsec:AnalysisPolarization}). The detector geometry is \nmodeled and the X-ray flux/spectrum was simulated for the different \nexperimental setups in which the data presented in this paper were \ntaken. The simulations were performed in the following steps.\n\n\\begin{enumerate}\n\n\\item Physics interactions, scatterings, and energy depositions in the \nscintillator and the CZT detectors were simulated using {\\it \nGEANT4}\\footnote{{\\tt http://geant4.cern.ch/}} with the Livermore \nlow-energy electromagnetic model list.\n\n\\item The charge collection efficiency as a function of \ndepth-of-interaction in the CZT detectors was determined using an \nin-house developed software to (i) calculate the 2D electric potential \ninside the detector crystals followed by (ii) the integration of the \nweighting potential \\cite{Jung2007} along the charge transport tracks, \nresulting in the collected charge for each energy deposition (using a \ndielectric constant for CZT of $10$). The simulations are valid for \ndetectors with strip anode contacts but will roughly resemble the \nresponse for pixelated detectors, as well. A mobility of electrons/holes \nof $\\mu_{\\rm{e}} = 1000 \\, \\rm{cm}^{2} / \\rm{V} / \\rm{s}$, and \n$\\mu_{\\rm{h}} = -120 \\, \\rm{cm}^{2} / \\rm{V} / \\rm{s}$ is assumed, as \nwell as life times of $\\tau_{\\rm{e}} = 10^{-6} \\, \\rm{s}$ and \n$\\tau_{\\rm{h}} = 4 \\cdot 10^{-6} \\, \\rm{s}$, respectively. This \ncorresponds to a mobility-lifetime product of $\\mu_{\\rm{e}}\\tau_{\\rm{e}} \n= 10^{-3}$ which is in reasonable agreement with the values measured for \na selection of the detectors used in X-Calibur (see \nFig.~\\ref{fig:TempStudies}, bottom).\n\n\\item The energy resolution (asymmetric Gaussian function, implicitly \nincluding the electronic readout noise) and energy threshold were \nmeasured from real data (Sec.~\\ref{subsec:ThreshAndResolution}) and were \nfolded into the simulations on a pixel-by-pixel basis. However, \ndifferences in channel trigger efficiencies were not simulated. Using \nthe reversed energy calibration in Eq.~(\\ref{eq:EnergyCalibration}), the \nsimulated events were in turn converted into the X-Calibur data format \n(ASIC/channel ID and digitized raw pulse height) and can be analyzed in \nthe same way as the measured data.\n\n\\item Since we find that the simulations do not properly account for low \nenergy tails in the detector response (see next paragraph), an empirical \nmodel was used to scatter an exponential tail $T(E) = A \\cdot \n\\exp(E/E_{\\rm{t}})$ into the simulations with a relative fraction of \n$f=0.45$ per energy deposit. The parameter $E_{\\rm{t}} = 15 \\, \\rm{keV}$ \n(at $E=40 \\, \\rm{keV}$) and $E_{\\rm{t}} = 30 \\, \\rm{keV}$ (at $E = 120 \n\\, \\rm{keV}$) was interpolated in $\\log E$ for the energy range covered.\n\n\\item The scintillator trigger flag $f_{\\rm{sci}}$ was generated for \neach event based on the scintillator trigger efficiency determined from \nreal data (see Fig.~\\ref{fig:Scintillator_TriggerEfficiency} in \nSec.~\\ref{subsec:ScintillatorEfficiency}).\n\n\\end{enumerate}\n\nDifferent scenarios/setups were simulated, reflecting the measurements \nand studies presented in this paper.\n\n{\\it CZT detector response.} To test the validity of the simulation \nchain, the direct illumination of a single CZT detector with a \nEu$^{152}$ point source was simulated\\footnote{All lines with \nintensities above $2 \\%$ were generated according to their relative \nemission intensities.}~-- corresponding to the experimental setup used \nfor the detector calibration measurements presented in \nSec.~\\ref{subsec:CZT_Calibration}. The comparison between the \nsimulations and data (Fig.~\\ref{fig:EnergyCalibration}, right) shows \nreasonable agreement in terms of line positions, widths, and threshold \neffects. However, if ignoring step (4) of the simulation chain (`Sim' in \nthe legend of Fig.~\\ref{fig:EnergyCalibration}), a lack in continuum \nemission can be seen in the simulations. This motivated the introduction \nof step (4) which leads to a reasonable agreement between data and \nsimulations over the whole energy band relevant for X-Calibur (`Sim+' in \nthe figure legend). Possible reasons for the continuum in the data may \nbe related to details in the detector response or back-reflection of \nemitted X-rays from the source off the surrounding fixture that is not \nsimulated\\footnote{For similar CZT detectors we find a photo-peak \ndetection efficiency of order unity \\cite{VarPitch}.}. It should be \nnoted that the goal of the CZT simulations is not to find an accurate \nmodel for the detector response~-- but rather a suited parameterization \nthat reproduces the integral spectral response for the different energy \nbins.\n\n{\\it CHESS beam.} X-Calibur performance measurements were performed at \nthe highly polarized synchrotron X-ray beam at the CHESS facility~-- \nproviding a strong, mono-energetic X-ray beam (Sec.~\\ref{subsec:CHESS}). \nA corresponding set of simulations was performed using steps (1)-(5), \nresembling the CHESS setup of pencil-beam X-rays (polarized and \nnon-polarized) at $40 \\, \\rm{keV}$, $80 \\, \\rm{keV}$, and $120 \\, \n\\rm{keV}$ entering the polarimeter along the optical axis of the \nscintillator. Detector pixels that were excluded during the CHESS data \nruns were also excluded in the simulations to resemble a configuration \nclose to the one used for the measurements. The trigger efficiency of \nthe scintillator as a function of energy deposition was derived from the \nCHESS data (see Fig.~\\ref{fig:Scintillator_TriggerEfficiency} in \nSec.~\\ref{subsec:ScintillatorEfficiency}) and was fed into the \nsimulations in step (5) to generate the trigger flag $f_{\\rm{sci}}$ on \nan event-by-event basis. No backgrounds were simulated in the case of \nthe CHESS measurements since the measurements were completely \nsignal-dominated.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CrabSimulations}\n\\end{center}\n\n\\caption{Simulated X-Calibur observations of the Crab nebula ($5.6 \\, \n\\rm{h}$), taken from \\citet{Guo2010}. The data points show the \nreconstructed flux (top), the fraction of polarization (middle), and \npolarization angle (bottom). The lines show the assumed flux, \npolarization fraction and polarization direction.}\n\n\\label{fig:XCLB_Sims}\n\\end{figure}\n\n{\\it Balloon flight.} A balloon flight in the focal plane of the \n{\\it InFOC$\\mu$S}\\xspace mirror assembly was assumed in an earlier simulation \n\\cite{Guo2010} that only involved step (1) in the above chain. The \neffective detection areas of the X-ray mirror are $95/60/40 \\, \n\\rm{cm}^{2}$ at $20/30/40 \\, \\rm{keV}$, respectively. We accounted for \natmospheric absorption at a floating altitude of $130,000$ feet using \nthe NIST XCOM attenuation coefficients\\footnote{{\\tt \nhttp://www.nist.gov/pml/data/xcom/index.cfm}} and an atmospheric depth \nof $2.9 \\, \\rm{g/cm}^{2}$ (observations performed at zenith); the \natmospheric transmissivity rapidly increases from $0$ to $0.6$ in the \n$20 - 80 \\, \\rm{keV}$ range. The trigger efficiency of the scintillator \nscatterer was assumed to be $f_{\\rm{sci}} = 1$ above an energy \ndeposition of $2 \\, \\rm{keV}$ and $f_{\\rm{sci}} = 0$ below.\n\nWe simulated the most important backgrounds such as the cosmic X-ray \nbackground \\cite{Ajello2008}, albedo photons and cosmic ray protons and \nelectrons \\cite{Mizuno2004}. The neutron background was not modeled \nsince a detailed study of \\citet{Parsons2004} showed that the \ncontribution in CZT can be neglected. Different shield configurations \nand shield thicknesses were simulated. The configuration shown in \nFig.~\\ref{fig:ActiveShieldAndTelescope} (left) represents an optimized \ncompromise balancing the background rejection power and the \nmass/complexity of the shield. A Crab-like source was simulated for a \n$5.6 \\, \\rm{hr}$ balloon flight. We assumed a power law energy spectrum, \nand a continuous change of the polarization fraction and angle between \nthe values measured at $5.2 \\, \\rm{keV}$ with OSO-8 \\cite{Weisskopf1978} \nand at $E>100 \\, \\rm{keV}$ with {\\it INTEGRAL}\\xspace \\cite{Dean2008} by modeling a \ntransition following a Fermi distribution.\n\nFor a Crab-like source the simulations predict an event rate of $1.1 \n(3.2) \\, \\rm{Hz}$ with (without) requiring a triggered scintillator \ncoincidence ($f_{\\rm{sci}} = 1$). Figure~\\ref{fig:XCLB_Sims} compares \nthe simulation results with the assumed model curves; the errors were \ncomputed in a similar way as described by \\citet{Weisskopf2010}. \nSimulations performed at different zenith angles $\\theta$ show that the \nsource rate scales with $(\\cos \\theta)^{1.3}$ which is taken into \naccount for simulating astrophysical observations. More details about \nthese simulations are discussed in \\citet{Guo2010} and \\citet{Guo2013}.\n\n\n\\section{CZT Detector Characterization} \\label{sec:DetectorCharacterization}\n\nThe performance of the individual CZT detectors used in X-Calibur is \ncoupled to the performance of the polarimeter as a whole~-- including \nits energy threshold and its energy resolution. This section describes \nthe energy calibration of the individual CZT detectors \n(Sec.~\\ref{subsec:CZT_Calibration}), as well as measurements of the \nenergy threshold and energy resolution \n(Sec.~\\ref{subsec:ThreshAndResolution}). The CZT performance serves as \nimportant input for the simulations described in \nSec.~\\ref{sec:Simulations}. In contrast to the laboratory, the \npolarimeter will be operated in an environment of varying temperature \nconditions during the balloon flight which motivates the study of the \ntemperature dependence of the CZT detector performance which will be \ndescribed in Sec.~\\ref{subsec:TempStudies}.\n\nIn order to quantify the characteristics of individual pixels, the \nemission lines in the calibrated energy spectra are fitted with a \nGaussian function. The peak position is described by the mean \n$E_{\\rm{p}}$. To account for the asymmetric shape of the peaks (see for \nexample Fig.~\\ref{fig:EnergyCalibration}), the fitted function allows \nfor asymmetric spectral continua levels ($c_{1}$, $c_{2}$) and \nasymmetric peak widths ($\\sigma_{1}$, $\\sigma_{2}$) for the $E < \nE_{\\rm{p}}$ (1) and $E > E_{\\rm{p}}$ (2) regimes, respectively. The fit \nparameters are used to characterize the measured peak. The energy \nresolution is calculated as the full width half maximum, $\\Delta E = \n\\rm{FWHM} = 2 \\sqrt{2 \\ln 2} \\cdot \\frac{1}{2} (\\sigma_{1} + \n\\sigma_{2})$. The peak rate is determined by counting the events in the \ninterval $\\pm 2 \\, \\rm{FWHM}$ centered around $E_{\\rm{p}}$, normalized \nby the observation time.\n\nA Eu$^{152}$~source is used as calibration/test source in a variety of \nstudies presented in this paper. Eu$^{152}$~emits X-ray lines at \n$39.5\\,\\rm{keV}$ ($\\rm{K}_{\\alpha2}$), $40.12\\,\\rm{keV}$ \n($\\rm{K}_{\\alpha1}$), $45.7 \\, \\rm{keV}$, $122.78 \\, \\rm{keV}$, $244.7 \n\\, \\rm{keV}$, $344.28 \\, \\rm{keV}$, and at higher energies. The line at \n$40.12\\,\\rm{keV}$ is the strongest in the low energy triplet; relative \nto the $40.12\\,\\rm{keV}$ line, the $39.5 \\, \\rm{keV}$ line is emitted at \n$55 \\%$ intensity and the $45.5 \\, \\rm{keV}$ line at $31 \\%$ intensity. \nThe $40.12\\,\\rm{keV}$, used as low-energy performance marker, is \ntherefore fitted jointly with the two close-by lines that are set at \nfixed distance $\\Delta E$ and fixed intensity relative to the \n$40.12\\,\\rm{keV}$ peak (with the free fit parameter $\\sigma$ being the \nsame for all three lines since the energy resolution of a detector pixel \nis not expected to change within a few keV). Including the two \nneighboring lines avoids systematic shifts and artificial broadening of \nthe fitted peak (see Fig.~\\ref{fig:EnergyCalibration}, left).\n\n\n\\subsection{Energy Calibration} \\label{subsec:CZT_Calibration}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{ZoomChannelEu152_Asic19_Channel_10}\n\\hfill\n\\includegraphics[width=0.50\\textwidth]{EnergyCalibrationSpectraEu152_Mosaic}\n\\end{center}\n\n\\caption{Eu$^{152}$ energy spectra (direct CZT illumination). The \nvertical lines indicate the nominal X-ray line energies; relative \nheights indicate the emission intensity folded with the absorption \nprobability in $5 \\, \\rm{mm}$ CZT and $2 \\, \\rm{mm}$ CZT (small tick), \nrespectively. Data were taken with X-Calibur configuration \n$C_{\\rm{wu1}}$ (see Fig.~\\ref{fig:X-Calibur_Configurations}). {\\bf \nLeft:} Spectrum of an individual detector pixel (linear energy axis, \n$C_{\\rm{wu1a}}^{1}$). The dashed vertical line indicates the trigger \nenergy threshold. The fitted functions ($40.1 \\, \\rm{keV}$ and $121.8 \\, \n\\rm{keV}$) are used to determine the peak position and energy resolution \n(FWHM). {\\bf Right:} Energy spectra summed over all pixels of the \ndetectors in ring {\\it R3} ($5 \\, \\rm{mm}$ thickness) and ring {\\it R7} \n($2 \\, \\rm{mm}$ thickness), respectively. Also shown are the simulated \nspectra with/without (Sim+/Sim) additional continuum (see \nSec.~\\ref{sec:Simulations}). A CR background spectrum (scaled by a \nfactor of $10$) is shown for reference. The horizontal dotted line \nindicates the energy range relevant for X-Calibur.}\n\n\\label{fig:EnergyCalibration}\n\n\\end{figure*}\n\nThe energy calibration of the individual CZT detector pixels is done \nwith a compact Eu$^{152}$ source (cylindrical emitting volume with a \ndiameter of $\\simeq$$3 \\, \\rm{mm}$) in the X-Calibur configuration \n$C_{\\rm{wu1b}}^{2}$ in which the the scintillator rod is not installed \n(see Fig.~\\ref{fig:X-Calibur_Configurations}). The source was \nsuccessively placed at the centers of the detector rings~-- allowing to \ncalibrate rings {\\it R1} to {\\it R8}, one at a time. In a first step, an \nautomatic routine is used to optimize the pixel trigger thresholds: data \nare taken in a special acquisition mode that adjusts ASIC/channel \ndiscriminators based on measured event rates for each pixel, such that \nthe trigger threshold is as low as possible (maximizing the integral \ntrigger rate), but at the same time is safely above the electronic noise \nregime. The noise regime leads to very high (artificial) trigger rates \nand differs from channel to channel. The routine works reliably for most \nchannels. However, a visual inspection of all 2048 recorded energy \nspectra was performed to assure that channels with trigger thresholds \nset too high or too low (failed automatic detection of the noise regime) \nwere adjusted manually.\n\nAbout 5 million events were taken for each CZT detector (20 million \nevents per ring {\\it R}), and the known energy lines at $40.1 \\, \n\\rm{keV}$ and $122 \\, \\rm{keV}$ were used to determine the pedestal \n$p_{0}$ and amplification slope $a$ for each channel\\footnote{The \nlinearity of the channels was confirmed using the internal test pulse \ngenerator of the ASIC.}. The energy of a measured pulse height $p$ is in \nturn calculated using\n\n\\begin{equation}\n\\label{eq:EnergyCalibration} \nE(p) = (p-p_{0})/a.\n\\end{equation}\n\nThe left panel of Figure~\\ref{fig:EnergyCalibration} shows the \ncalibrated energy spectrum of a single pixel. The right panel shows the \naveraged calibration spectra of two chosen detector rings ($4 \\times 64$ \npixels each). Also shown are the energy spectra obtained from the \nsimulations of the corresponding setup (see Sec.~\\ref{sec:Simulations}). \nIt can be seen, that the $2 \\, \\rm{mm}$ detectors (ring {\\it R7}) loose \nperformance at energies $E> 100 \\, \\rm{keV}$, as compared to the \ndetectors with $5 \\, \\rm{mm}$ thickness (ring {\\it R3}). However, in the \n$20-80 \\, \\rm{keV}$ energy band, relevant for X-Calibur, both types of \ndetector thickness perform at a similar level.\n\nIn addition to the temperature-dependence of the detector performance \ndiscussed in Sec.~\\ref{subsec:TempStudies}, another consideration has to \nbe made when applying the calibration to the data. The CZT detectors \nused in X-Calibur are not setup for measuring the depth position of the \nX-ray interaction/absorption between the detector cathode and anode~-- \nreferred to as the depth-of-interaction (DOI). The energy calibration in \nEq.~(\\ref{eq:EnergyCalibration}), however, depends on the average DOI of \na given energy. The mean DOI, however, changes with the cosine of the \ninclination angle measured between the absorbed X-ray and the detector \nplane. The calibration was determined with the X-ray source located $c \n\\simeq 11 \\, \\rm{mm}$ above the center of the detector cathode (see \nFig.~\\ref{fig:X-Calibur_Configurations}, top). Depending on their \ngeometrical locations, the pixels are hit under angles between \n$0-45^{\\circ}$. X-rays Compton-scattering in the X-Calibur scintillator, \non the other hand, can hit the CZT detectors at angles between \n$0-90^{\\circ}$, which adds a systematic error/uncertainty to the \nreconstructed energy for small incident angles. However, in the $20-80 \n\\, \\rm{keV}$ band the DOI distribution is very narrow and localized \nclose to the cathode. Therefore, the angle dependence is negligible~-- \nthe systematic shift of the reconstructed $40.1 \\, \\rm{keV}$ line was \nexperimentally constrained to be less than $1\\%$ for shallow inclination \nangles.\n\n\nA $< 3\\%$ fraction of ASIC channels were found to be dead, too noisy, or \ndid not make contact to the detector pixel. These channels were excluded \nfrom the analysis and are marked with `x' in the corresponding 2D plots \nshown in this paper (e.g. Fig.~\\ref{fig:PixelThreshold}).\n\n\n\\subsection{Energy Resolution and Threshold} \n\\label{subsec:ThreshAndResolution}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Results_CFG1_Threshold}\n\\end{center}\n\n\\caption{2D distribution of energy thresholds of all X-Calibur pixels. \nIn this representation, the four detector sides surrounding the \nscintillator ({\\it Bd0} to {\\it Bd3}, see Fig.~\\ref{fig:Design}, left) \nare unfolded into a plane. The segments enclosed by dashed and dotted \nlines indicate the CZT detectors ($8 \\times 8$ pixels each). The \nelongated box (solid line in {\\it Bd0} to {\\it Bd3}, each) indicates the \nposition of the scintillator, even though it was not installed for this \nparticular data set (configuration $C_{\\rm{wu1b}}^{2}$, see \nFig.~\\ref{fig:X-Calibur_Configurations}, top). Pixels marked with `x' \nwere not included in the analysis. The left panels show the threshold \ndistributions per detector ring {\\it R} (the gray distribution shows the \nscaled average of all pixels).}\n\n\\label{fig:PixelThreshold}\n\n\\end{figure}\n\nTo study the performance of the X-Calibur CZT detectors, the calibration \ndata (Sec.~\\ref{subsec:CZT_Calibration}) were used to characterize the \nenergy spectra of individual pixels (see left panel of \nFig.~\\ref{fig:EnergyCalibration} for reference). The relevant properties \nstudied in this section are the energy threshold, the fitted line/peak \nposition, and the energy resolution (FWHM). As for the energy \ncalibration, the data were taken with the configuration \n$C_{\\rm{wu1b}}^{2}$ (see Fig.~\\ref{fig:X-Calibur_Configurations}).\n\nThe energy threshold of a pixel is defined as the reconstructed energy \n$E_{\\rm{thr}}$ above which the corresponding ASIC channel starts to \ntrigger on events (see Fig.~\\ref{fig:EnergyCalibration}, left). The \ndistribution of thresholds for all X-Calibur pixels is shown in \nFig.~\\ref{fig:PixelThreshold}. The thresholds vary from pixel to pixel, \nbut no significant geometrical trends can be identified if comparing \nedge pixels versus central pixels, or pixels of detectors with different \nthickness ($5 \\, \\rm{mm}$, rings {\\it R1-R5} versus $2 \\, \\rm{mm}$, \nrings {\\it R6-R8}). About $50\\%$ of all pixels have an energy threshold \nof $E<20 \\, \\rm{keV}$. It should be noted that one has to account for \nthe energy resolution of a pixel in order to determine the analysis \nthreshold that is about $3-4 \\, \\rm{keV}$ higher. The thresholds shown \nin Fig.~\\ref{fig:PixelThreshold} reflect the status of the compact \nX-Calibur configuration. The thresholds of the detectors operated as a \nsingle unit are up to $5 \\, \\rm{keV}$ lower (see \nFig.~\\ref{fig:TempStudies}).\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.42\\textheight]{Results_CFG1_40keV_Resolution}\n\\hfill\n\\includegraphics[height=0.42\\textheight]{Results_CFG1_Noise_Resolution_Zoom}\n\\hfill\n\\includegraphics[height=0.42\\textheight]{Results_CFG1_122keV_Resolution}\n\\end{center}\n\n\\caption{2D distribution of the pixel-by-pixel energy resolution \n(compare with Fig.~\\ref{fig:PixelThreshold}). {\\bf Left:} Energy \nresolution at $40.1 \\, \\rm{keV}$. {\\bf Middle:} Electronic readout noise \nat $40 \\, \\rm{keV}$ as determined using the ASICs internal test pulse \ngenerator (only shown for {\\it Bd2}). The same range of the color scale \nas in the left panel is used. {\\bf Right:} Energy resolution at $121.8 \n\\, \\rm{keV}$. This energy line is not well-defined in the energy spectra \nmeasured with the $2 \\, \\rm{mm}$ detectors (see \nFig.~\\ref{fig:EnergyCalibration}) so that results are only shown for the \n$5 \\, \\rm{mm}$ rings {\\it R1} to {\\it R5}.}\n\n\\label{fig:PixelResolution}\n\n\\end{figure*}\n\n\nFigure~\\ref{fig:PixelResolution} shows the energy resolutions at $40.1 \n\\, \\rm{keV}$ and $121.8 \\, \\rm{keV}$, respectively. Detector rings {\\it \nR2} and {\\it R3} are the most sensitive ones when it comes to detecting \nthe Compton-scattered X-rays in the polarization measurements (see \nSec.~\\ref{subsec:CHESS}). Therefore, the best performing $5 \\, \\rm{mm}$ \ndetectors were positioned in these rings accordingly. Some of the lower \ndetector rings ({\\it R4} to {\\it R8}) show regions with clearly \npoorer-than-average energy resolution. However, any asymmetry in \nazimuthal detector performance will cancel out due to the rotation of \nthe polarimeter in the final mode of operation. The energy line at \n$121.8 \\, \\rm{keV}$ is not well-defined in the spectra measured with the \n$2 \\, \\rm{mm}$ detectors (see Fig.~\\ref{fig:EnergyCalibration}, right). \nTherefore, Fig.~\\ref{fig:PixelResolution} only shows the $121.8 \\, \n\\rm{keV}$ energy resolutions for the $5 \\, \\rm{mm}$ detectors. The \naverage energy resolution in rings {\\it R1} to {\\it R3} at $40.1 \\, \n\\rm{keV}$ amounts to $4.0 \\, \\rm{keV}$ ($10.0\\%$); the corresponding \nvalue at $121.8 \\, \\rm{keV}$ is $4.9 \\, \\rm{keV}$ ($\\simeq 4\\%$). The \naverage performance of rings {\\it R4} to {\\it R7} at $40.1 \\, \\rm{keV}$ \namounts to $4.6 \\, \\rm{keV}$ ($\\simeq 11\\%$). The performance of the \ndetectors in ring {\\it R8} is modest.\n\nThe energy resolution of a detector pixel is mainly determined by two \nfactors~-- the quality of the CZT crystal and the noise of the read-out \nelectronics. The electronic readout noise was determined for all \nchannels using the ASICs internal pulse generator (see \nSec.~\\ref{sec:XCLB}). The generator injects charge into the amplifier of \nthe corresponding channel and allows one to test the \ntrigger/digitization chain on a chennel-by-channel basis. 1000 events \nwere taken per channel with the detectors connected and biased at \nnominal operation voltage (to also account for noise introduced by dark \ncurrents in the CZT). The results are shown in the middle panel of \nFig.~\\ref{fig:PixelResolution} for one of the four boards. More details \non the electronic noise studies are presented in \nSec.~\\ref{subsec:TempStudies}. Note, that the internal ASIC capacitor \ndoes not allow to inject charges that correspond to energies lower than \n$\\approx 200 \\, \\rm{keV}$. However, the noise versus energy trend seems \nto level off for energies lower than $\\simeq 500 \\, \\rm{keV}$ (see \nFig.~\\ref{fig:TempStudiesNoise}). Therefore, the absolute noise \nresolution measured at $\\simeq 200 \\, \\rm{keV}$ was used to estimate the \nrelative noise contribution at $40 \\, \\rm{keV}$, as shown in the middle \npanel of Fig.~\\ref{fig:PixelResolution}.\n\nA comparison between the electronic noise and the energy resolution \ndetermined from the spectral lines shows that the low-energy resolution \nis dominated by the electronic noise. This kind of comparison can in \ngeneral assist in localizing the cause for modestly performing \ndetectors, e.g. by disentangling the contributions of electronic noise \nversus CZT crystal quality. The detector located in ring {\\it R5} of \n{\\it Bd2}, for example, shows noisy regions in both, the Eu$^{152}$ \ndata, as well as in the electronic noise measurement~-- indicating a \nhigh leakage current as the reason for the sub-optimal performance. The \nring {\\it R8} detector in the same board, on the other hand, does not \nexhibit a poorer energy resolution than others in terms of noise~-- \ntherefore, the poor performance visible in the Eu$^{152}$ data is \nprobably related to a low CZT crystal quality.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{PolarimeterTemperature}\n\\end{center}\n\n\\caption{Temperature-dependent performance of individual CZT detectors. \nEach data point represents the average of $64$ pixels of a detector. \nVertical arrows indicate the shift of the curve if considering only edge \npixels of a detector (dashed direction), or only central pixels of a \ndetector (solid direction). {\\bf Top:} Energy resolution at $122 \\, \n\\rm{keV}$ (only shown for the $5 \\, \\rm{mm}$ detectors). {\\bf \n2$^{\\rm{\\bf{nd}}}$ from top:} Energy resolution at $59 \\, \\rm{keV}$. \n{\\bf 3$^{\\rm{\\bf{rd}}}$ from top:} Peak detection rate of the $59 \\, \n\\rm{keV}$ line normalized to the rate measured at $T=15^{\\circ} \\, \n\\rm{C}$. {\\bf 4$^{\\rm{\\bf{th}}}$ from top:} Energy threshold as \ndetermined after threshold optimization performed at the corresponding \ntemperature. {\\bf Bottom:} Mobility-lifetime product \n$\\mu_{\\rm{e}}\\tau_{\\rm{e}}$ measured based on the $59 \\, \\rm{keV}$ \nline.}\n\n\\label{fig:TempStudies}\n\\end{figure}\n\n\nOn average, the side pixels of individual detectors exhibit a poorer \nthan average energy resolution. At low energies, this shows up as a \nperiodic structure on the left and right side of each detector. The same \npattern can be seen in the electronic noise measurement \n(Fig.~\\ref{fig:PixelResolution}, left versus middle), and can therefore \nbe attributed to the readout noise. This hypothesis is supported by the \nfact that the leads between the corresponding edge pixels and the ASIC \nchannels are located on the `outside' region of the printed circuit \nboard~-- being more susceptible to noise pick up from the surrounding \nelectronics. Subtracting the readout noise, these edge pixels do no \nlonger show poorer energy resolution at low energies as compared to the \nother pixels. For energies $E > 100 \\, \\rm{keV}$ (not relevant for \nX-Calibur), the detector edge pixels show a reduced resolution in \naddition to the electronic noise component (`frame-like' structure \nsurrounding each detector in Fig.~\\ref{fig:PixelResolution}, right). \nThis is a known issue with CZT detectors and can be explained by a less \nhomogeneous electric field in the edge regions of a detector, affecting \nthe charge collection. The effect is less prominent (if visible at all) \nfor the horizontal edge pixels, for which the field is stabilized by the \nneighboring detectors located in the same plane.\n\n\n\\subsection{CZT Performance at different Temperatures} \n\\label{subsec:TempStudies}\n\nDuring a balloon flight the pressure vessel housing the polarimeter will \nundergo several changes in temperature that can potentially affect its \nperformance. At float altitude, the vessel will be in an outside \ntemperature environment of around $-20^{\\circ} \\rm{C}$. During daytime, \nthe thermal radiation fields of the sun and the earth will provide \nadditional sources of energy. During the night, the thermal radiation of \nthe earth is the only external source of heat flow. The electronics \ninside the vessel generate heat at a rate of less than $100 \\, \\rm{W}$. \nThe outside of the vessel will be insulated using a layer of aluminized \nmylar (reflection of sun light and high emittance of thermal radiation) \nthat will be contact-separated from the surface of the vessel with a \nlayer of Dacron mesh. Furthermore, heater bands with a total power of \n$175 \\, \\rm{W}$ are installed inside the vessel to guarantee a \ncontrollable temperature in the range of $0$ to $25^{\\circ} \\rm{C}$. \nThis thermal design will prevent overheating during day-time and will \navoid cold temperatures during the night. Nonetheless, variations in \ntemperature of the polarimeter during flight are expected to some \nextent. Therefore, it is important to understand the \ntemperature-dependent performance of the CZT detectors.\n\nIndividual CZT detectors were used to quantify the temperature \ndependence of the energy resolution, the energy threshold and the \ndetection rate. Data were taken in a temperature chamber in the range of \n$T \\in [-25; +35]^{\\circ} \\, \\rm{C}$ with the detectors being \nilluminated with an Am$^{241}$ source ($59 \\, \\rm{keV}$) and a Co$^{57}$ \nsource ($122 \\, \\rm{keV}$). The sources were located at a distance of \n$\\simeq 2 \\, \\rm{cm}$ above the detector cathode. The data were used to \nre-calibrate each detector (pixel-by-pixel) for each environment \ntemperature in order to cancel temperature-dependent calibration \neffects. Since these measurements were time-intensive and could only be \ndone for one detector at a time, the study was limited to a \nrepresentative subset of the detectors shown in \nTab.~\\ref{tab:Detectors}.\n\n{\\it Temperature-dependent detector characteristics.} The results of the \nmeasurements are shown in Fig.~\\ref{fig:TempStudies}, averaged over all \n64 pixels per detector. The $122 \\, \\rm{keV}$ line is not well defined \nin the $2 \\, \\rm{mm}$ detectors, so that the high-energy results are \nonly shown for the $5 \\, \\rm{mm}$ detectors. Given the limited sample of \ndetectors, we cannot expect to attribute observed trends to a specific \ndetector class (such as the brand); however, some general findings can \nbe identified and are described in the following.\n\nThe energy resolution generally improves if reducing the temperature \nfrom $T = 35^{\\circ} \\, \\rm{C}$ to $T = 0^{\\circ} \\, \\rm{C}$. This \neffect is most prominent for the two tested Quikpak detectors and \namounts to an improvement of up to $50\\%$/$30\\%$ at $59/122 \\, \n\\rm{keV}$, respectively. The effect is much weaker for the Creative \nElectron detectors. For temperatures $T < 0^{\\circ} \\, \\rm{C}$, the \nresolution levels out. For some of the Endicott detectors ($2 \\, \n\\rm{mm}$) it even gets poorer again; however, the two detectors showing \nthis effect (EN$_{2}$4 and EN$_{2}$5) show very poor performance in \ngeneral. All three $5 \\, \\rm{mm}$ detectors show a much better energy \nresolution at $122 \\, \\rm{keV}$ in the central pixels (indicated by the \ndownward arrow in the top panel of Fig.~\\ref{fig:TempStudies}) as \ncompared to the edge pixels (upward arrow). Here, the aspect ratio \nallows for E-field lines to bulge out of the sides of the detector (see \nalso Fig.~\\ref{fig:PixelResolution}). This difference is clearly less \npronounces at $59 \\, \\rm{keV}$ where the charge collection is much more \nconcentrated in the cathode region of the detector crystal.\n\nThe peak detection rate at $59 \\, \\rm{keV}$ (3$^{\\rm{rd}}$ panel from \nthe top in Fig.~\\ref{fig:TempStudies}) shows a clear trend for all \ndetectors: a reduced detection efficiency with decreasing temperature. \nThis effect is not understood. Although the measurements were carefully \nset up, it cannot be excluded that a temperature-dependent contraction \nof the casing/fixture could have lead to a slight change in distance \nbetween the source and the detector as a function of environment \ntemperature.\n\nFor each temperature, the energy thresholds were re-optimized. The \nresults are shown in the 4$^{\\rm{th}}$ panel of \nFig.~\\ref{fig:TempStudies}. No clear trends can be identified~-- the \naverage energy threshold of the studied detectors lies between $15-20 \\, \n\\rm{keV}$.\n\nMeasurements at different bias voltages $V_{\\rm{bi}}$ were used to \ndetermine the mobility lifetime product $\\mu_{\\rm{e}}\\tau_{\\rm{e}}$. \nData were taken at $V_{\\rm{bi}} = -200 \\, \\rm{V}$, $-500 \\, \\rm{V}$, and \n$-700 \\, \\rm{V}$ ($5 \\, \\rm{mm}$ detectors) and at $V_{\\rm{bi}} = -100 \n\\, \\rm{V}$, $-150 \\, \\rm{V}$, and $-200 \\, \\rm{V}$ ($2 \\, \\rm{mm}$ \ndetectors), respectively. The shift of the line position with increasing \n$V_{\\rm{bi}}$ was used to calculate $\\mu_{\\rm{e}}\\tau_{\\rm{e}}$ \n\\cite{MuTauImproved}. The results are shown in the bottom panel of \nFig.~\\ref{fig:TempStudies}. No significant change in \n$\\mu_{\\rm{e}}\\tau_{\\rm{e}}$ can be identified for the temperature range \nstudied. \\citet{Jung2007} discuss the temperature dependence of Imarad \nHigh-Pressure Bridgman CZT detectors and find a decreasing trend of \n$\\mu_{\\rm{e}} \\tau_{\\rm{e}}$ for temperatures $T > 10^{\\circ} \\, \\rm{C}$ \nand for $T < -25^{\\circ} \\, \\rm{C}$.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PeakPosVsTemp}\n\\end{center}\n\n\\caption{Reconstructed line energy for different temperatures using the \ncalibration obtained at $T_{\\rm{cal}} = 25^{\\circ} \\, \\rm{C}$ (dashed \nvertical line). The data points indicate the mean of the peak positions \nof all 64 pixels and the error bars represent the standard deviation of \nthat distribution (separately derived for each temperature). The dashed \nhorizontal line indicates the nominal line energy.}\n\n\\label{fig:PeakPosVsTemp}\n\\end{figure*}\n\n{\\it Temperature-stability of the calibration.} In the studies shown in \nFig.~\\ref{fig:TempStudies} all detector pixels were re-calibrated at \neach temperature. It is important to understand how the calibration \nitself changes with temperature. To study this effect, a reference \ncalibration at $T_{\\rm{cal}} = 25^{\\circ} \\, \\rm{C}$ was applied to the \nmeasurements taken at the different temperatures. The line positions at \n$59.5 \\, \\rm{keV}$ and $122.1 \\, \\rm{keV}$ were determined for the \nindividual pixels. Each distribution (one per temperature) of \nreconstructed line positions was in turn characterized by its mean and \nits standard deviation. The results are illustrated in \nFig.~\\ref{fig:PeakPosVsTemp}, where the standard deviation (spread of \nthe corresponding distribution) is represented as error bar. For \nreference, the spectra of detector QP$_{5}$1 were corrected with the \ncalibration obtained at $T_{\\rm{cal}} = 5^{\\circ} \\, \\rm{C}$. While the \nmean reconstructed line energy does not change significantly, the \nwidening of the error band indicates that individual channels show a \ntemperature-dependent upward/downward drift of the line position. This \nmakes it difficult to globally correct for temperature-dependent changes \nin the calibration~-- unless calibration data are taken for all 2048 \nX-Calibur channels at a variety of temperatures. The calibration changes \nby up to $\\simeq 2\\%$ for temperatures varying around $\\pm 10^{\\circ} \\, \n\\rm{C}$ relative to the reference calibration $T_{\\rm{cal}}$.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{TempNoise}\n\\end{center}\n\n\\caption{Electronic readout noise measured with the ASIC's internal test \npulser for different temperatures and different energies (detector \nCE$_{5}$1 and EN$_{5}$8). For reference, the gray asterisk marker \nindicates the 1~std.dev. range of all 2048 channels as operated in the \nX-Calibur assembly at room temperature of $T \\approx 25^{\\circ} \\, \n\\rm{C}$ with/without biased detectors (solid/dashed line). {\\bf Top:} \nChannel-by-channel noise (vertical dotted lines) and detector average \nwith the detector being biased at $V_{\\rm{bi}}=-500 \\, \\rm{V}$. {\\bf \nMiddle:} Detector averages for three different temperatures with and \nwithout the cathode bias $V_{\\rm{bi}}$. {\\bf Bottom:} A series of $T \n\\approx 25^{\\circ} \\, \\rm{C}$ measurements with different configurations \nas described in the text.}\n\n\\label{fig:TempStudiesNoise}\n\n\\end{figure}\n\n{\\it Electronic noise.} The read-out electronic contributes a certain \namount of jitter to the measured energy resolution of a detector pixel. \nIn order to quantify this contribution, a series of measurements was \ntaken with detectors CE$_{5}$1 and EN$_{5}$8 (see \nTab.~\\ref{tab:Detectors}) at different temperatures using the internal \npulse generator of the ASIC (see Sec.~\\ref{sec:XCLB}). The detectors \nwere operated in an electrically shielded copper box. With the detector \nbeing plugged into the ASIC, the measurements actually reflect the \nreadout noise of the ASIC/detector assembly, rather than the noise of \nthe ASIC alone. Different amounts of charge were injected (corresponding \nto different energies)\\footnote{Note, that given the differences in \npixel acceptance and pixel calibration, a fixed amount of charge \ninjected into the ASIC translates to slightly varying reconstructed \nenergies (if comparing different channels).}. The measured pulse heights \nwere transformed to energies using the corresponding energy calibration \ndetermined for each temperature. For each configuration and channel, a \ntotal of 1000 events were injected. The calibrated distribution was \nfitted in order to determine the corresponding mean energy and energy \nresolution.\n\nThe results are shown in Fig.~\\ref{fig:TempStudiesNoise}. At energies of \naround $200 \\, \\rm{keV}$ the electronic noise of the ASIC/detector unit \nincreases by $(21 \\pm 6)\\%$ in the studied temperature range of \n$-25^{\\circ} \\, \\rm{C}$ to $+35^{\\circ} \\, \\rm{C}$. The shape of the \nenergy-dependent noise curve depends on the temperature. At room \ntemperature, the noise increases by $(20 \\pm 4)\\%$ if going from $200 \\, \n\\rm{keV}$ to $1.7 \\, \\rm{MeV}$ and levels out around $3 \\, \\rm{keV}$ at \nlow energies. This suggests, that the energy resolution in the X-Calibur \nrange ($\\approx 4 \\, \\rm{keV}$ at $40 \\, \\rm{keV}$, see \nFig.~\\ref{fig:PixelResolution}) is dominated by the electronic readout \nnoise, rather than charge transport properties in the CZT crystal. The \ngray asterisk marker in Fig.~\\ref{fig:TempStudiesNoise} shows the \n1~std.dev.~range of the noise distribution of all 2048 data channels as \nmeasured at room temperature ($\\approx 25^{\\circ} \\, \\rm{C}$) in the \nfinal X-Calibur configuration, compare with \nFig.~\\ref{fig:PixelResolution} (middle).\n\nThe middle panel of Fig.~\\ref{fig:TempStudiesNoise} illustrates the \neffect of the bias voltage of the detector. A biased cathode at $T = \n25^{\\circ} \\, \\rm{C}$ increases the low-energy noise by $(8 \\pm 5)\\%$, \nwhereas no noticeable change can be measured for temperatures lower than \nthat. Therefore, the cathode bias only seems to systematically affect \nthe readout noise for temperatures higher than $\\simeq 15^{\\circ} \\, \n\\rm{C}$.\n\n\nThe bottom panel in Fig.~\\ref{fig:TempStudiesNoise} shows the electronic \nnoise measured at room temperature for different configurations: (i) the \ndetector/ASIC unit with biased cathode, (ii) the detector/ASIC unit with \nunbiased cathode, (iii) the ASIC with a ceramic chip carrier but no \ndetector bonded to it, and (iv) only the plain ASIC. It can be seen that \nsteps (i)--(iii) each add $\\approx 0.2 \\, \\rm{keV}$ readout noise to the \nsingle-detector system. The average readout noise of the whole X-Calibur \nassembly (gray asterisk in Fig.~\\ref{fig:TempStudiesNoise}) is shown for \nreference. Another series of measurements was performed with the plain \nASIC at different temperatures (not shown). No significant noise trend \ncould be identified in the $T = -20^{\\circ} \\, \\rm{C}$ to $+25^{\\circ} \n\\, \\rm{C}$ temperature range which leads to the conclusion that the \ntemperature dependence of the readout noise of the ASIC/detector unit \n(top panel of Fig.~\\ref{fig:TempStudiesNoise}) is mostly a result of the \ntemperature dependence of the dark currents in the CZT crystal.\n\n{\\it Caveats:} The electronic readout noise varies from ASIC to ASIC and \ndepends on the electronic shielding environment in which the ASIC is \noperated. Therefore, the comparison between the absolute noise levels of \nthe single ASIC system shown in Fig.~\\ref{fig:TempStudiesNoise} and the \naverage readout noise in the X-Calibur assembly should be treated with \ncare; the relative noise trends found, however, can likely be applied to \nwhole X-Calibur assembly. It should also be mentioned, that the cooling \naggregates of the temperature chamber (increased activity at low \ntemperatures) can potentially introduce external noise pick-up in the \nASIC.\n\n\n\\subsection{Summary of the Detector Calibration and Tests}\n\nEach detector pixel has been energy calibrated according to \nEq.~(\\ref{eq:EnergyCalibration}). With the current readout electronics \nand the compact X-Calibur configuration, the CZT detectors achieve a \nmean trigger threshold of $\\simeq 21 \\, \\rm{keV}$. The mean energy \nresolution at $40 \\, \\rm{keV}$ in the three front-side detector rings \n{\\it R1}-{\\it R3} (detecting most of the scattered events in the \npolarization measurements) is found to be $\\Delta E_{\\rm{czt}} \\simeq 4 \n\\, \\rm{keV}$ FWHM, when operated at room temperature. The energy \nresolution of the polarimeter as a whole is determined by the energy \nresolution of the individual detectors and by the energy deposited/lost \nin the scintillator ($\\Delta E_{\\rm{sci}} = 0 - 5.4 \\, \\rm{keV}$ at $40 \n\\, \\rm{keV}$). The energy resolution of the detectors is thus not \nentirely negligible and X-Calibur would benefit from an optimized \nreadout ASIC. We are currently working on modifying and adopting the \nHD-3 ASIC \\cite{DeGeronimo2003, Vernon2010}. Using a pre-amplifier chain \noptimized for the $2-100 \\, \\rm{keV}$ energy range, we expect a trigger \nthreshold of $1.7 \\, \\rm{keV}$ and electronic readout noise of $\\simeq \n550 \\, \\rm{eV}$ RMS. The noise contribution of the new ASIC to the \nenergy resolution of the polarimeter would be negligible for all \nenergies above $20 \\, \\rm{keV}$. The low energy threshold can \npotentially be used on a satellite-borne version of the polarimeter.\n\nThe effective energy threshold of the polarimeter is slightly higher \nthan the energy threshold of the individual CZT detectors, as a $25 \\, \n\\rm{keV}$ photon looses up to $2.2 \\, \\rm{keV}$ in the scatterer. \nHowever, the polarimeter will detect a large fraction of the X-rays at \n125,000 feet flight altitude, as the residual atmosphere only transmits \nphotons above $\\simeq 25 \\, \\rm{keV}$.\n\nEven though the temperature-dependent trends in energy resolution would \nfavor operating the detectors at $T \\leq 0^{\\circ} \\, \\rm{C}$ \n(Fig.~\\ref{fig:TempStudies}, top), the thermal design of the X-Calibur \nassembly and local internal heat built-up of the polarimeter during the \nballoon flight makes an operation at $T \\simeq (15 \\pm 10)^{\\circ} \\, \n\\rm{C}$ a more likely scenario~-- still guaranteeing a reasonable energy \nresolution for most of the detectors. A change in temperature, which \nwill be monitored during flight, leads to a shift in reconstructed \nenergy~-- which can go either way (Fig.~\\ref{fig:PeakPosVsTemp}). \nIdeally, one should use a data base of temperature-dependent calibration \nvalues on a pixel-by-pixel basis to correct for the temperature trends. \nIf ignoring the temperature-dependence of the calibration, a systematic \nerror on the reconstructed energy of a few percent has to be accounted \nfor in the temperature interval of $\\pm 10^{\\circ} \\, \\rm{C}$ around the \ncalibration temperature. For the first X-Calibur flight, we will choose \nthe second option.\n\n\n\n\\section{X-Calibur: Instrument Characterization} \\label{sec:XCLB_Characterization}\n\nThis section describes measurements of the fully assembled polarimeter \ninstalled in the CsI shield. The goal of the measurements is to \ncharacterize the efficiency of the shield, and to estimate the \nbackground levels in the different (ground-based) environments the \npolarimeter was operated in (Sec.~\\ref{subsec:BG_Data}). The reduction \nof the background is crucial in order to perform sensitive measurements \nof the polarization properties of astrophysical sources, see \nEq.~(\\ref{eqn:MDP}).\n\nThe X-rays that enter the polarimeter along the optical axis will \nproduce a certain amount of scintillation light when scattering in the \nscintillator rod which is read out by a PMT. The efficiency curve, \ndescribing the trigger probability for different energy depositions in \nthe scatterer, is discussed in Sec.~\\ref{subsec:ScintillatorEfficiency}. \nAs will be shown in Sec.~\\ref{subsec:BG_Data}, a high trigger efficiency \nof the scintillator will allow a further reduction of the background. \nThe efficiency curve is fed into the simulations that were discussed in \nSec.~\\ref{sec:Simulations}. The measurements described in this chapter \nwere done with disk-like radioactive sources with a diameter of $\\simeq \n0.5 \\, \\rm{cm}$.\n\n\n\\subsection{Shield Performance and Cosmic Ray Background} \\label{subsec:BG_Data}\n\nThe shielding and suppression of backgrounds is a crucial task for \nsensitive polarimetry measurements. The background in the ground-based \nmeasurements presented in this paper (non-flight) results mostly from \nsecondary particles produced in air showers in the earth's atmosphere, \ninduced by cosmic rays (CRs). The flux of the secondary particles \ndepends on the geographical location at which the measurement is \nperformed, and on the structure/materials of the building in which the \npolarimeter is operated (partly shielding the secondary particles). As \ncan be seen in Fig.~\\ref{fig:MultiplicityDistribution}, the CR \nbackground has a higher fraction of multiplicity $m \\geq 2$ pixel events \nas compared to Compton-scattered X-rays in the $E < 100 \\, \\rm{keV}$ \nregime, relevant for the X-Calibur polarimetry measurements. In general, \nthe distribution of $m$ depends on the energy and the kind of \ninteraction; high-energy muons (ionization), for example, trigger events \nalong a row of pixels (unless they cross the detector perpendicular to \nthe pixel plane). Low energy X-rays, on the other hand, are \nphoto-absorbed with a charge deposition usually contained well within \none pixel. Therefore, a requirement of $m = 1$ pixel events already \nsuppresses the background for the polarization measurements by a certain \namount.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ShieldScalerVsDiscriminator}\n\\end{center}\n\n\\caption{Optimization of the shield discriminator using three different \ndata runs (a)-(c), as described in the text. The background contribution \nis subtracted for the runs with Cs$^{137}$ outside shield (b), and for \nAm$^{241}$/Eu$^{152}$ on the optical axis (c). The dashed vertical line \nindicates the optimal discriminator setting. {\\bf Top:} Shield trigger \nrate for different threshold settings. The horizontal lines indicate the \nrates corresponding to $10/5/2\\%$ (from top to bottom) dead time \nproduced by the shield vetoes. The noise regime starts to dominate \naround DAC $<20$; this in turn leads to an underestimation of the \nCs$^{137}$/Eu$^{152}$ trigger contribution since the CR background is \nsubtracted. In the case of the Am$^{241}$ source, no significant trigger \nrate above background was measured. {\\bf Bottom:} Fraction of detected \nCZT events that are vetoed with $f_{\\rm{shld}} = 1$.}\n\n\\label{fig:ShieldOptimization}\n\\end{figure}\n\n{\\it Optimization of the shield trigger threshold.} A particle crossing \nthe active shield (see Fig.~\\ref{fig:ActiveShieldAndTelescope, left) \nproduces scintillation light in the CsI crystal. The crystal is read by \nfour PMTs which signals are summed and digitized.} A programmable \ndiscriminator decides whether the shield veto flag $f_{\\rm{shld}}$ is \nactivated ($f_{\\rm{shld}}$ is kept active for $6 \\, \\mu\\rm{s}$ per \ntrigger) and merged into the data stream. Events with the corresponding \nflag can in turn be filtered out in the data analysis. In order to \noptimize the shield trigger efficiency, a series of measurements was \nperformed with different settings of the discriminator. The measurements \ncomprise: (a) cosmic ray background only, (b) a collimated Cs$^{137}$ \nsource aimed from outside the shield at the X-Calibur CZT detector \nassembly (configuration $C_{\\rm{wu2b}}^{3}$ in \nFig.~\\ref{fig:X-Calibur_Configurations}), and (c) collimated \nAm$^{241}$/Eu$^{152}$ sources placed on the optical axis of the \npolarimeter (configuration $C_{\\rm{wu2a}}^{3}$) to simulate X-rays from \nthe X-ray mirror entering the polarimeter without interaction in the \nshield\\footnote{Obviously, the lead used to collimate the sources leads \nto indirect scatterings and shield contamination, in particular in the \ncase of the high energy lines of Eu$^{152}$.}. Two shield \ncharacteristics were measured: (i) the raw trigger rate of the shield, \nand (ii) the fraction of triggered CZT events with $f_{\\rm{shld}} = 1$. \nThe results are shown in Fig.~\\ref{fig:ShieldOptimization}. Since the \ncosmic ray background was present in all three runs, the X-ray data runs \n(b) and (c) were corrected for the rates measured in run (a). The \nsetting of the discriminator was optimized according to the following \ncriteria.\n\n\\begin{enumerate}\n\n\\item The dead time produced by noise triggers should not exceed a few \npercent (horizontal lines in Fig.~\\ref{fig:ShieldOptimization}). The \nAm$^{241}$/Eu$^{152}$ sources located on the optical axis (c) should \nresult in CZT events with a $f_{\\rm{shld}} = 1$ contribution as low as \npossible (no signal suppression).\n\n\\item For the CR (a) and Cs$^{137}$ (b) runs, the fraction of CZT \ntriggers with $f_{\\rm{shld}} = 1$ should be as high as possible, \nreflecting a high rejection power.\n\n\\end{enumerate}\n\nWe determined an optimal shield discriminator setting of $\\rm{DACQ} = \n30$ which is indicated by the vertical line in \nFig.~\\ref{fig:ShieldOptimization}. Note, the fraction of Eu$^{152}$ \nevents with $f_{\\rm{shld}} = 1$ is around $25 \\%$, which is probably due \nto X-rays that Compton scatter and interact with the shield and CZT~-- a \nresult of not having a well collimated X-ray beam entering the \npolarimeter in this measurement.\n\n\n\\begin{table}[t!]\n\n\\begin{tabular}{lrr}\n\nData set & $\\left< T_{5 \\, \\rm{mm}} \\right>$ & $\\left< T_{2 \\, \\rm{mm}} \\right>$ \\\\\n & [mHz] & [mHz] \\\\\n\\hline \\hline\n\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{CZT initial calibration (Washington Univ.): $C_{\\rm{wu1a}}^{1}$} \\\\\n\\hline\n\nEu$^{152}$ illumination & $7100$ & $6200$ \\\\\nBackground & $59$ & $35$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{Scintillator characterization (Wash. Univ.): $C_{\\rm{wu2a}}^{3}$} \\\\\n\\hline\n\nEu$^{152}$ on optical axis$^{\\rm{*}}$ & $31$ & $8.7$ \\\\\nBackground in shield (active) & $9.9(5.4)$ & $4.9(2.0)$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{CHESS synchrotron beam (Cornell Univ.): $C_{\\rm{ch}}^{3}$} \\\\\n\\hline\n\n$40 \\, \\rm{keV}$ X-ray beam & $1100$ & $300$ \\\\\nBackground & $13$ & $8.4$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{X-Calibur/shield/InFocus (Ft.~Sumner): $C_{\\rm{ft}}^{3}$} \\\\\n\\hline\n\nX-ray source$^{\\rm{*}}$ & $853$ & $190$ \\\\\nBackground in shield (active) & $8.5(3.9)$ & $4.7(1.6)$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{X-Calibur flight (simulations)} \\\\\n\\hline\n\nCrab$^{\\rm{*}}$ & $1.25$ & $0.3$ \\\\\n\n\\end{tabular}\n\n\\caption{Average event trigger rates per CZT detector pixel for $5 \\, \n\\rm{mm}$ detectors $\\left< T_{5 \\, \\rm{mm}} \\right>$ and for $2 \\, \n\\rm{mm}$ detectors $\\left< T_{2 \\, \\rm{mm}} \\right>$. Rates are shown \nfor the different measurements/environments presented in this paper (see \nFig.~\\ref{fig:X-Calibur_Configurations}). No event selection cuts are \napplied. Rates marked with a `*' are background subtracted. The \ncorresponding background spectra for the different data sets are shown \nin Fig.~\\ref{fig:BG_Spectra}.}\n\n\\label{tab:PixelRates}\n\n\\end{table}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{BackgroundSpectra_Mosaic}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{BackgroundSpectra2_Mosaic}\n\\end{center}\n\n\\caption{Cosmic ray background spectra (events with pixel multiplicity \n$m=1$, if not mentioned otherwise). The dotted vertical lines indicate \nthe ranges of the $\\rm{KL}_{1-3}$ transition energies of tungsten (W) \nand lead (Pb) which show up in some of the spectra. The \nX-Calibur/{\\it InFOC$\\mu$S}\\xspace sensitive energy range is indicated, as well. {\\bf \nLeft:} Spectra measured in different environments (see \nFig.~\\ref{fig:X-Calibur_Configurations}). Date and duration of the runs \nis given in square brackets. The (passive/active) CsI shield strongly \nreduces the background. The three dashed vertical lines on the \n$C_{\\rm{wu1a}}^{1}$ spectrum represent the high-energy lines of \nEu$^{152}$ ($244.7 \\, \\rm{keV}$ and $344.3 \\, \\rm{keV}$) and Cs$^{137}$ \n($661.7 \\, \\rm{keV}$) for which an onset indication can be seen; the \nsources were stored in a lead bunker $\\sim$$2 \\, \\rm{m}$ away during the \nbackground measurements and were probably contributing to the spectrum \nat a low level. {\\bf Right:} Energy spectra for individual detector \nrings measured at Washington University in the active shield (after \nshield veto, $f_{\\rm{shld}} = 0$). We also show energy spectra (scaled \nby a factor of 0.1) from different pixels of the detectors on the \nreadout board {\\it Bd2} to illustrate the difference between central \npixels and pixels located on the edge of detectors, see also \nFig.~\\ref{fig:BG_Maps2D} for reference.}\n\n\\label{fig:BG_Spectra}\n\n\\end{figure*}\n\n{\\it Cosmic ray background levels.} Figure~\\ref{fig:BG_Spectra} shows \nthe CR energy spectra ($m=1$ pixel multiplicity) measured in the CZT \ndetectors at the different locations: the laboratory at Washington \nUniversity ($C_{\\rm{wu}}$), the CHESS X-ray beam facility \n($C_{\\rm{ch}}$, Sec.~\\ref{subsec:CHESS}), and in Ft.~Sumner \n($C_{\\rm{ft}}$, Sec.~\\ref{subsec:FtSumnerData}). For reference, the \ncorresponding event trigger rates ($m \\geq 1$) per pixel are summarized \nin Tab.~\\ref{tab:PixelRates}. Note, that the actual pixel rates can vary \nquite substantial, since different pixels see different income fluxes \ndepending on the type of measurement (e.g. an X-ray beam Compton \nscattered in the scintillator leads to a strongly depth-dependent \nillumination of CZT detectors, compare with \nFig.~\\ref{fig:CHESS_2DAzimuthData}). However, it can be seen that for \nmost measurements presented in this paper the background can be \nneglected. During the balloon flight, however, the expected \nsource-to-background ratio will be much lower and a proper understanding \nand the background suppression will be crucial \\cite{Guo2010, Guo2013}. \n\nEven though the background at flight altitude will be different as \ncompared to the ground-based CR background, some characteristics in the \ndetector response can be discussed qualitatively based on the spectra \nshown in Fig.~\\ref{fig:BG_Spectra}. While $\\alpha$-particles interact \nclose to the surface of the detector, muons as well as primary and \nsecondary high-energy gamma rays will penetrate deeper and their energy \ndeposition is proportional to the detector volume. This volume-dependent \nbackground rate can be seen by comparing the spectra (and trigger rates) \nmeasured with $5 \\, \\rm{mm}$ versus $2 \\, \\rm{mm}$ detectors. Note, that \nthe spectra shown in Fig.~\\ref{fig:BG_Spectra} represent particle fluxes \nfolded with the energy-dependent response of the CZT detectors (with an \nenergy calibration derived from, and valid for, X-rays). The drop in \nevent rate below $30 \\, \\rm{keV}$, for example, is an effect of the \nsuperposition of the different trigger thresholds of the pixels \ncontributing to the spectrum. The dynamical energy range covered by a \nsingle pixel saturates around $2000 \\, \\rm{keV}$ (differing from pixel \nto pixel), so that the combined $m=1$ pixel spectra shown in \nFig.~\\ref{fig:BG_Spectra} drop off around this energy. Allowing events \nwith $m \\geq 1$ multiplicities will extend the energy range (see \nFig.~\\ref{fig:BG_Spectra}, right) which is, however, not relevant for \nthe operation of X-Calibur.\n\n{\\it The CsI shield efficiency.} Figure~\\ref{fig:BG_Maps2D} shows the 2D \ndistribution of event count rates from background data taken with \nX-Calibur installed in the CsI shield at Washington University. The \nrates are shown for two different energy bands. The left panels show the \nraw rates and the right panels show the rates after rejecting events \nthat triggered the active shield, only allowing non-vetoed events \n($f_{\\rm{shld}} = 0$). It again becomes obvious that the $5 \\, \\rm{mm}$ \ndetectors ({\\it R1}-{\\it R5}) collect more background compared to the $2 \n\\, \\rm{mm}$ detectors ({\\it R6}-{\\it R8}). Furthermore, the edge pixels \nof the detector rows on the individual boards {\\it Bd0-Bd3} see a higher \nbackground rate as compared to central pixels (likely because of the \nhigher exposed surface area detecting charged particles and low energy \nX-rays). The energy-resolved difference between edge and central pixels \ncan be seen in Fig.~\\ref{fig:BG_Spectra}, right ($5 \\, \\rm{mm}$ \ndetectors of boards {\\it Bd2}): the central pixels show an almost two \ntimes lower background compared to edge pixels in the energy regime \nrelevant for X-Calibur.\n\nThe background distribution after shield veto (Fig.~\\ref{fig:BG_Maps2D}, \nright) exhibits a spatial gradient with more background events being \ndetected closer to the front side of the experiment, as that side is \nonly shielded by the passive tungsten cap (see \nFig.~\\ref{fig:ActiveShieldAndTelescope}, left). In particular, the \nfront-side (top) pixel row of ring {\\it R1}, with its exposed detector \nside walls, suffers strongly from primary radiation leaking through the \ntungsten, as well as secondary particles being produced in the tungsten. \nHere, spectral signatures of the tungsten KL transitions can be \nidentified in the measured spectra shown in Fig.~\\ref{fig:BG_Spectra} \n(right). Since this single pixel row is not crucial for the polarimetry \nsensitivity it can be excluded from the analysis.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{BG_Maps2D_35-150keV} \\\\\n\\includegraphics[width=0.49\\textwidth]{BG_Maps2D_200-1000keV}\n\\end{center}\n\n\\caption{2D maps of the CR count rates (pixel-by-pixel, representation \nsame as in Fig.~\\ref{fig:PixelThreshold}) of a $192 \\, \\rm{h}$ \nbackground run with X-Calibur being installed in the active CsI shield \n($C_{\\rm{wu2c}}^{3}$ in Fig.~\\ref{fig:X-Calibur_Configurations}). The \ntungsten cap is located at the top of each panel. The count rate is \nderived by integrating the corresponding energy spectra ($m=1$ events) \nin the given energy range: $35-150\\, \\rm{keV}$ (top) and $200-1000 \\, \n\\rm{keV}$ (bottom). The left panel shows the measured rate (passive \nshield rejection only) and the right panel shows the non-vetoed events \nwith $f_{\\rm{shld}} = 0$ (active and passive rejection). Left and right \npanels are shown with the same axis/color ranges, each.}\n\n\\label{fig:BG_Maps2D}\n\\end{figure}\n\nThe passive/active rejection efficiency of the CsI shield becomes \nobvious if comparing the different background spectra in \nFig.~\\ref{fig:BG_Spectra}, left. Moving the polarimeter from its copper \nhousing (used in the initial test measurements, $C_{\\rm{wu1}}^{1}$) into \nthe CsI shield ($C_{\\rm{wu2c}}^{3}$) reduces the background by roughly \none order of magnitude due to passive shielding. Applying the \n$f_{\\rm{shld}} = 0$ veto from the active shield leads to an additional \nbackground rejection by a factor of $\\sim$$2$. Note, that the \nrecombination of the tungsten and lead $\\rm{KL}_{1,2,3}$ transition \nenergies\\footnote{http://www.nist.gov/pml/data/xraytrans/} (probably \nactivated by CRs) can be identified in the background spectra. This \nfeature is most prominent in the spectrum of ring {\\it R1} \n(Fig.~\\ref{fig:BG_Spectra}, right) which is located closest to the \ntungsten cap.\n\nIt should be noted that an additional cut on the scintillator rod \ncoincidence flag $f_{\\rm{sci}} = 1$ reduces the background by another \n$1.5$ orders of magnitude (Fig.~\\ref{fig:BG_Spectra}, left)~-- strongly \nrejecting events that did not enter the polarimeter along the optical \naxis and interacted in the scintillator, see \nSec.~\\ref{subsec:ScintillatorEfficiency}.\n\n\n\\subsection{Compton-scattering and Scintillator Trigger Efficiency} \\label{subsec:ScintillatorEfficiency}\n\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{SpectraScintillatorVsDAC_R1b_Mosaic} \\\\\n\\includegraphics[width=0.47\\textwidth]{SpectraScintillatorVsDAC_R2t_Mosaic} \\\\\n\\includegraphics[width=0.47\\textwidth]{SpectraScintillatorVsDAC_R3_Mosaic}\n\\end{center}\n\n\\caption{A lead-collimated Eu$^{152}$ source enters the polarimeter \nalong the optical axis and Compton-scatterers in the scintillator rod \n($C_{\\rm{wu2a}}^{3}$ in Fig.~\\ref{fig:X-Calibur_Configurations}). The \nresulting CZT spectra ($m=1$, background subtracted) are shown for \ndifferent rings. The horizontal lines indicate the complete range of \npossible energies $E_{\\rm{czt}}$ after Compton-scattering (dotted: \n$180-90^{\\circ}$, solid: $90-0^{\\circ}$). The attached vertical dotted \nlines indicate the sub range for the particular ring~-- given its \ngeometrical coverage of the scintillator. Spectra are measured with \ndifferent scintillator trigger thresholds settings (Digital to Analog \nConverter, DAC) and are compared to the spectra without coincidence \nrequirement (no cuts). The lead collimator prevents direct source hits \n(rings {\\it R1-R3}), but scattered X-rays and/or activation of the \n$\\rm{KL}_{1-3}$ transitions in tungsten (W) and lead (Pb) partly \ncontaminate the spectra.}\n\n\\label{fig:Scintillator_Spectra}\n\\end{figure}\n\nThe central scintillator rod of the polarimeter (Fig.~\\ref{fig:Design}) \nacts as a Compton-scatterer for the incoming X-ray beam. At the same \ntime, it produces scintillation light that is read out by a PMT. In \norder to characterize the Compton-scattering and the trigger response of \nthe scintillator, a collimated Eu$^{152}$ source was placed at the \npolarimeter entrance, illuminating the scintillator along the optical \naxis (see setup $C_{\\rm{wu2a}}^{3}$ in \nFig.~\\ref{fig:X-Calibur_Configurations}). The Compton-scattered X-rays \nare recorded by the surrounding CZT detector rings. An X-ray enters the \npolarimeter with a certain energy $E_{\\rm{line}}$, deposits the energy \n$\\Delta E_{\\rm{sci}}$ in the scintillator and is absorbed in a CZT \ndetector with its post-scattering energy of $E_{\\rm{czt}} = \nE_{\\rm{line}} - \\Delta E_{\\rm{sci}}$. The amount of scintillation light \ncan be assumed to be proportional to $\\Delta E_{\\rm{sci}}$.\n\nData were taken with different settings of the scintillator/PMT \ndiscriminator threshold (DAC). As can be seen in \nTab.~\\ref{tab:PixelRates}, the background level is not negligible in \nthis measurement, so that background spectra without a X-ray source were \nrecorded and subtracted. It should be noted, however, that X-rays \nscattered in the lead collimator and activation/recombination of the \n$\\rm{KL}_{1,2,3}$ electron transitions in the lead collimator and the \ntungsten cap lead to a slight contamination of the measured spectra. \nThese corresponding spectral features do not cancel out after CR \nbackground subtraction since they are (indirectly) introduced by the \nEu$^{152}$ source itself. Since those features will over-proportionally \naffect the data without the $f_{\\rm{sci}}=1$ coincidence (X-ray paths \nthat do not necessarily cross the scintillator), this may lead to an \nunderestimation of the trigger efficiency (which in the ideal case would \nonly compare Compton-scattered events in the scintillator with and \nwithout the $f_{\\rm{sci}} = 1$ trigger coincidence). The scintillator \ntrigger efficiency for different energies $\\Delta E_{\\rm{sci}}$ is \nderived as follows.\n\n{\\it Compton spectra.} The resulting Compton-scattered energy spectra \n$E_{\\rm{czt}}$ are shown in Fig.~\\ref{fig:Scintillator_Spectra} for \ndifferent CZT detector rings. Data are shown with and without the \nscintillator coincidence requirement. The energy $\\Delta E_{\\rm{sci}}$ \ndeposited in the scintillator depends on the scattering kinematics, \nincluding the primary energy of the X-ray $E_{\\rm{line}}$ and its \nscattering angle. The theoretical range of energies after \n$0-180^{\\circ}$ Compton scattering is indicated by horizontal lines for \nthe four main Eu$^{152}$ energy lines. Due to geometrical reasons, some \nof the detector rings can only detect X-rays originating from a limited \nrange of Compton-scattering angles. With the scintillator starting at \nring {\\it R2} (Fig.~\\ref{fig:Design}, left), the sub rings {\\it \nR1$_{\\rm{t}}$}, and {\\it R1$_{\\rm{b}}$}, for example, can only detect \nback-scattered X-rays which deposit/loose the highest amount of energy \n$\\Delta E_{\\rm{sci}}$ in the scintillator~-- leading to a higher trigger \nprobability. For a given incoming energy $E_{\\rm{line}}$, the range of \npossible scattering angles translates into a range of possible \nCompton-scattered energies $E_{\\rm{czt}}$. These ring-dependent sub \nranges in scattered energy are indicated by the vertical lines in \nFig.~\\ref{fig:Scintillator_Spectra}. Note, however, that the scattering \nangles/energies will not be equally distributed within these boundaries. \nOne can qualitatively discuss the X-Calibur response to \nCompton-scattering for the different scattered energy lines.\n\n\\begin{itemlist}\n\n\\item $E_{\\rm{line}} = 40 \\, \\rm{keV}$: The Compton-scattered energy \nrange of the Eu$^{152}$ line doublet ($39.52\\,\\rm{keV}$ and \n$40.12\\,\\rm{keV}$) only covers $E_{\\rm{czt}} = 34.2 - 40.1\\,\\rm{keV}$ at \nan energy resolution of $\\Delta E_{40} \\simeq 4 \\, \\rm{keV}$. This makes \nit difficult to resolve structures in the scattered energy spectrum. \nHowever, the geometrical coverage of ring {\\it R1$_{\\rm{b}}$} \nconstraints the energy deposition in the scintillator to $\\Delta \nE_{\\rm{sci}} = 2.4-5.4 \\, \\rm{keV}$ since only back-scattering angles of \n$80-180^{\\circ}$ are possible (top panel in \nFig.~\\ref{fig:Scintillator_Spectra}). A clear improvement in trigger \nefficiency is obvious when lowering the scintillator DAC threshold from \n170 to 10.\n\n\\item $E_{\\rm{line}} = 121.8 \\, \\rm{keV}$: This well-defined line at \nhigher energy leads to a broader Compton-continuum which can be tested \nat different energies $\\Delta E_{\\rm{sci}}$. While ring {\\it \nR1$_{\\rm{b}}$} again only sees the back-scattered X-rays, the rings \nfurther down cover a broad continuum range. The trigger efficiency for \nthe back-scattering of the $121.8 \\, \\rm{keV}$ line ($\\Delta \nE_{\\rm{sci}} = 23-39 \\, \\rm{keV}$) does not seem to depend on the \nscintillator DAC setting and is therefore already at its maximum. In the \nclose-to forward scattering regime ($\\Delta E_{\\rm{sci}}$ a few keV) in \nring {\\it R3} (bottom panel in Fig.~\\ref{fig:Scintillator_Spectra}) a \nlower DAC threshold leads to a broader Compton continuum in the \n$f_{\\rm{sci}} = 1$ data, reflecting the increase in trigger efficiency. \nStarting at ring {\\it R3}, however, the effective thickness of the lead \ncollimator at the polarimeter entrance is not high enough to fully \nabsorb all X-rays; this leads to some direct detector hits in the raw \nenergy spectrum~-- and therefore an underestimation of the trigger \nefficiency at the corresponding value of $\\Delta E_{\\rm{sci}}$.\n\n\\item $E_{\\rm{line}} = 244.7 \\, \\rm{keV}$ and $344.3 \\, \\rm{keV}$: These \ntwo high-energy (but lower intensity) lines lead to partially \noverlapping Compton-continua. This makes it difficult to study the \nresponse in more detail. It can also be seen, that $344.3 \\, \\rm{keV}$ \ndirect CZT hits are not fully prevented by the lead collimator (peak in \nraw spectra).\n\n\\end{itemlist}\n\n{\\it Scintillator efficiency.} The scintillator trigger efficiency as a \nfunction of $\\Delta E_{\\rm{sci}} = E_{\\rm{line}} - E_{\\rm{czt}}$ was \nestimated from the flux ratio at energy $E_{\\rm{czt}}$ with and without \nthe $f_{\\rm{sci}} = 1$ coincidence requirement (after continuum \nsubtraction). This can be done independently for the different line \nenergies and for different detector rings. Although the different \nEu$^{152}$ energies are ideal to test different values of $\\Delta \nE_{\\rm{sci}}$, the reflection and re-processing of the high energy lines \ncontaminates the raw spectrum hampering the quantitative determination \nof the trigger efficiency (underestimation). Therefore, additional data \nwere taken with an Am$^{241}$ source which has a line at $59.5 \\, \n\\rm{keV}$, but no significant emission above. The trigger efficiencies \ndetermined from the different source lines and detector rings are \nsummarized in Fig.~\\ref{fig:Scintillator_TriggerEfficiency} for the \noptimized discriminator threshold of DAC=10. The trigger distribution \nwas fitted by a function\n\n\\begin{equation}\nf(\\Delta E_{\\rm{sci}}) = \\frac{a}{1 + e^{-b (\\Delta E_{\\rm{sci}}-E_{0})} + e^{-c \n(\\Delta E_{\\rm{sci}}-E_{0})}}.\n\\label{eqn:ScintTriggEffFit}\n\\end{equation}\n\nFor the optimized setting of DAC=10, the trigger efficiency reaches \n$50\\%$ at $\\Delta E_{\\rm{sci}} \\simeq 4.6 \\, \\rm{keV}$. Note, that the \ntrigger fraction does not level out at $a = 1$ ($100 \\%$). This is \npotentially a result of the contamination of the spectrum (direct \ndetector hits)~-- caused by the non-ideal measurement setup that leads \nto a systematic underestimation of the efficiency. For the simulations \nin Sec.~\\ref{sec:Simulations}, we therefore assumed the same trigger \nfunction, however with a value of $a=1$. Note, that we currently do not \nmodel any depth-dependence of $f(\\Delta E_{\\rm{sci}})$ which becomes \nimportant when looking at the different detector rings. In a detailed \n{\\it Monte Carlo} study, \\citet{PogoScintSims} simulate the trigger \nefficiency of a similar scintillator material (EJ-204) and find a \nthreshold of $2-3 \\, \\rm{keV}$. The authors also find a $13 \\%$ drop in \nlight yield along a $20 \\, \\rm{cm}$ long scintillator rod (from close to \nthe PMT towards the distant end).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ScintillatorTriggerFraction}\n\\end{center}\n\n\\caption{Fraction of CZT events with a scintillator coincidence trigger \n($f_{\\rm{sci}}=1$) as a function of energy deposition $\\Delta \nE_{\\rm{sci}}$. Data were taken with Am$^{241}$, Eu$^{152}$, and at the \nCHESS beam (Sec.~\\ref{subsec:CHESS}, however with a much higher \nscintillator threshold). The Compton-scattered continua (above \nbackground, see for example Fig.~\\ref{fig:Scintillator_Spectra}) were \nused to derive the trigger efficiency for different $\\Delta \nE_{\\rm{sci}}$. The distribution was fitted by a function shown in \nEq.~(\\ref{eqn:ScintTriggEffFit}). The dashed horizontal line indicates \nthe range of scintillator energy depositions for incoming X-rays in the \n$20-80 \\, \\rm{keV}$ band.}\n\n\\label{fig:Scintillator_TriggerEfficiency}\n\\end{figure}\n\nThe raw scintillator trigger rate (independent of CZT event triggers) of \nthe optimized threshold setting of DAC=10 was measured to be only a few \nhundred Hz~-- a further reduction in trigger threshold by updating the \nsignal amplifier seems possible and is work in progress.\n\nA cleaner setup to measure the scintillator response is the \nwell-collimated X-ray beam at CHESS (see Sec.~\\ref{subsec:CHESS}). The \nresults are also shown in Fig.~\\ref{fig:Scintillator_TriggerEfficiency} \nfor a non-polarized beam at $40 \\, \\rm{keV}$ and $120 \\, \\rm{keV}$, \nrespectively. However, the amplifier for the PMT reading the \nscintillator was not optimized at the time of the CHESS measurements, so \nthat the $50\\%$ trigger probability is only reached around $\\Delta \nE_{\\rm{sci}} \\simeq 15 \\, \\rm{keV}$. The corresponding fitted trigger \nefficiency (again with the $a=1$ assumption)\\footnote{Also the CHESS \nmeasurements suffered from indirect contamination due to beam absorber \nfoils, see Sec.~\\ref{subsec:CHESS}.} was fed into the simulations that \nwere done for the CHESS measurements discussed in \nSec.~\\ref{subsec:CHESS}.\n\n\n\\subsection{Summary of the Instrument Characteristics}\n\nThe rejection power of the active shield was studied in the laboratory. \nIt reduces the background by more than one order of magnitude in the \nenergy range relevant for X-Calibur.\n\nWhen we require a scintillator trigger ($f_{\\rm{sci}} = 1$), the \neffective energy threshold of the polarimeter increases to $\\simeq 25 \\, \n\\rm{keV}$ (with the current scintillator threshold of $\\sim$$5 \\, \n\\rm{keV}$). We are working on a further reduction of the scintillator \nthreshold \\cite{Fabiani2013}. Above an energy of $30 \\, \\rm{keV}$, we \ncan detect a high fraction of events with a coincident scintillator \nsignal. For ground-based background measurements, the scintillator \ncoincidence reduces the background by another factor of $\\sim$$30$. On \nupcoming balloon flights, we will use events with (for best high-energy \nsignal to noise ratio) and without (for best low-energy response) \nscintillator coincidences.\n\n\n\\section{X-Calibur: Polarization Measurements} \\label{sec:XCLB_PolarimetryMeasurements}\n\nThis section discusses measurements performed with the fully assembled \nand calibrated X-Calibur polarimeter. Measurements of non-polarized and \npolarized X-ray beams at the CHESS facility at Cornell University are \ndescribed in Sec.~\\ref{subsec:CHESS} and are compared to simulations. \nThe results illustrate the functionality of X-Calibur as an X-ray \npolarimeter. Section~\\ref{subsec:FtSumnerData} describes X-Calibur \nmeasurements that were taken with the fully integrated \nX-Calibur/{\\it InFOC$\\mu$S}\\xspace X-ray telescope in Ft.~Sumner during a test \nintegration of a flight-ready balloon setup. \n\nSystematic effects and asymmetries in the detector response can, under \ncertain circumstances, cause measurements of aparent polatization \nfractions, even if the measured X-ray beam itself is non-polarized. It \nis therefore crucial to understand and control these kinds of systematic \neffects in order to correctly study the polarization properties of \nastrophysical sources. The systematic effects (and corrections thereof) \ncaused by a X-ray beam hitting the polarimeter off-center are discussed \nin Sec.~\\ref{subsec:Systematics}.\n\n\n\\subsection{Polarized X-rays from the CHESS Beam} \\label{subsec:CHESS}\n\n\nIn order to measure the response to a polarized X-ray beam, the \nX-Calibur polarimeter was operated at the Cornell High Energy \nSynchrotron Source (CHESS)\\footnote{http://www.chess.cornell.edu/} for \none week in March 2013. CHESS provides a highly collimated and highly \npolarized beam of synchrotron X-rays. Using Bragg reflection from a \n2-bounce silicon (220) monochromator, a $40 \\, \\rm{keV}$ beam was \ngenerated with the 2$^{\\rm{nd}}$/3$^{\\rm{rd}}$ harmonics at $80 / 120 \\, \n\\rm{keV}$, respectively. The measurements were performed in hutch {\\it \nC1}. The polarimeter was mounted on an adjustable X/Y/Z stage table with \nthe scintillator being aligned with the X-ray beam (using X-ray \nfluorescence paper). The setup, referred to as $C_{\\rm{ch}}$, is shown \nin Fig.~\\ref{fig:X-Calibur_Configurations}. X-Calibur was mounted in a \nfixture that allowed us to rotate the polarimeter around the \noptical/beam axis, in order to test the response at different \norientations between the polarization plane and the detector. The \nrotation angle is referred to as $\\alpha$ with detector board {\\it Bd0} \nlocated at the top position at $\\alpha = 0^{\\circ}$ (see $C^{3}$ in the \ntop panel of Fig.~\\ref{fig:X-Calibur_Configurations}). Looking into the \nbeam, a positive angle $\\alpha$ corresponds to a counter clock-wise \nrotation. The accuracy of setting the angle was estimated to be $\\Delta \n\\alpha \\simeq 2^{\\circ}$. It should be noted that we did not use a high \nprecision rotation mechanism (as will be used during the balloon \nflight). Therefore, an $\\alpha$ dependent mis-alignment/tilt of the \nscintillator during the measurements cannot be excluded.\n\n{\\it The CHESS X-ray flux.} A total of 30 synchrotron-emitting electron \nbunches cycle the CHESS accelerator ring. The relative intensity of the \nX-ray beam entering the {\\it C1} hutch is controlled by a system of \nslits, as well as the tunable orientation of the Bragg reflection \ncrystals. The intensity, however, varies with (i) decaying electron \npopulation in the ring during a run (re-charged every $\\sim$$2 \\, \n\\rm{h}$), (ii) local heat built-up on the Bragg crystals, and (iii) the \nposition stability of the electron beam in the ring. The intensity of \nthe monochromatic beam entering the hutch was monitored using an argon \nion chamber ($6 \\, \\rm{cm}$ in length along the beam, set to a counter \nrange of $10^{-8} \\, \\rm{A}/\\rm{V}$). The absolute flux calibration of \nX-Calibur was not a major objective of the measurements; therefore, we \ndid not setup a system to automatically stabilize the beam intensity \nentering the hutch~-- but rather kept it manually in the ballpark of \n$1-2 \\cdot 10^{8} \\, \\rm{cts}/\\rm{s}$ (ion chamber) throughout the \nmeasurements and logged the average intensity for each data run. \nAbsorber foils placed behind the ion chamber further reduced the beam \nintensity by several orders of magnitude. This brought the X-ray flux \nhitting the polarimeter down to the level of $1-2 \\, \\rm{kHz}$, a regime \nthat can be well handled by the readout electronics of the polarimeter.\n\n{\\it Configurations.} An aluminum collimator plate of thickness $0.5''$ \nwas placed directly in front of the X-Calibur polarimeter with an \nentrance hole of $0.25''$ in diameter. The CZT detector configuration \nused for the CHESS measurements is listed in Tab.~\\ref{tab:Detectors}. \nCHESS data presented in this paper were taken with three configurations \n(variations of $C_{\\rm{ch}}$ as depicted in \nFig.~\\ref{fig:X-Calibur_Configurations}).\n\n\\begin{itemlist}\n\n\\item {\\bf $C_{\\rm{ch1}}$}: the absorber foils consist of platinum (Pt), \na $125 \\, \\mu\\rm{m}$ layer placed directly after the ion chamber and a \nstack of four $90 \\, \\mu\\rm{m}$ layers placed between the ion chamber \nand X-Calibur. This setup is optimized for a band-pass at $40 \\, \n\\rm{keV}$.\n\n\n\\item{\\bf $C_{\\rm{ch3}}$}: One layer of $125 \\, \\mu\\rm{m}$ Pt absorber \nfoil is placed directly behind (downstream) the ion chamber, and a $1.27 \n\\, \\rm{mm}$ lead foil (Pb) is placed between the ion chamber and \nX-Calibur. This setup is optimized for a band-pass at $80 \\, \\rm{keV}$.\n\n\\item{\\bf $C_{\\rm{ch4}}$}: Same as $C_{\\rm{ch3}}$, but the ion chamber \nwas removed from the beam path (in order to extend the range of beam \noffsets without interfering with the housing of the ion chamber).\n\n\\end{itemlist}\n\nX-Calibur detects individual events, but only the ASIC that triggered \nthe event (corresponding to $1/64$ of the polarimeter) is dead during \nthe read-out which takes $\\sim$$130 \\,\\mu\\rm{s}$. The dead-time during \ndata runs is therefore usually not higher than $5\\%$. In order to \nguarantee a homogeneous data set, some additional pixels (as compared to \nthe calibration runs) were excluded from the analysis which had high \nthresholds or were too noisy~-- even though the azimuthal acceptance \n(see Sec.~\\ref{sec:Analysis}) is in general capable of correcting for \nmost of these effects. Only events with multiplicity $m=1$ pixels were \nselected. X-Calibur was not operated in its active CsI shield, and the \nscintillator PMT discriminator was not yet optimized at the time of the \nCHESS measurements.\n\n{\\it Data runs.} A series of measurements was taken with configuration \n$C_{\\rm{ch1}}$ (optimized for $40 \\, \\rm{keV}$) and $C_{\\rm{ch3,4}}$ \n(optimized for $80 \\, \\rm{keV}$). Data were taken with different \npolarimeter orientations $\\alpha$. The $C_{\\rm{ch1}}$ data runs span a \nrange of $\\alpha \\in [-90; +90]^{\\circ}$ in steps of $10^{\\circ}$ with \n$4 \\, \\rm{Mio}$ events per orientation. A non-polarized beam was \n`generated' by the superposition of data from two perpendicular \norientations. A higher number of events were taken for these sets: $20 \n\\, \\rm{Mio}$ events for the $\\alpha = 0/-90^{\\circ}$ pair, and $10 \\, \n\\rm{Mio}$ events for the $\\alpha = \\pm 45^{\\circ}$ reference pair. The \n$C_{\\rm{ch3,4}}$ series scanned a range of $\\alpha \\in \n[-90;+10]^{\\circ}$ with a non-polarized beam composed from $10 \\, \n\\rm{Mio}$ events derived from the $\\alpha = 0/-90^{\\circ}$ orientations.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CHESS_AllDetectorUnPolarized_Mosaic}\n\\end{center}\n\n\\caption{Energy spectra of the Compton-scattered non-polarized CHESS \nX-ray beam (superposition of perpendicular polarization planes). The \nmeasurements were performed with the two configurations $C_{\\rm{ch1}}$ \nand $C_{\\rm{ch3}}$ (see text and \nFig.~\\ref{fig:X-Calibur_Configurations}). Shown are the background \nsubtracted integral spectra of all detectors~-- only ring R1 was \nexcluded due to flux contamination (see text and \nFig.~\\ref{fig:CHESS_ComtponSpectra_UnPol}, right, for the reason). The \nbackground level is shown for reference. The horizontal lines represent \nthe nominal energies of the three harmonics for (left to right): $180 - \n90^{\\circ}$ Compton-scattering (dotted), and $90 - 0^{\\circ}$ \nCompton-scattering (solid). Spectra are shown with and without the \n$f_{\\rm{sci}} = 1$ scintillator coincidence requirement. Simulated \nspectra are shown, as well.}\n\n\\label{fig:CHESS_ComtponSpectra_AllDetector_UnPol}\n\\end{figure}\n\n\nSeveral background runs were taken without the X-ray beam entering the \nhutch. The over-all background spectrum is shown in the left panel of \nFig.~\\ref{fig:BG_Spectra} (red data points). The {\\it C1} hutch is \nshielded by (partly) lead-enforced walls which likely contribute to the \nlower background level as compared to the laboratory at Washington \nUniversity. In fact, a signature at the lead $\\rm{KL}_{1-3}$ transition \nenergies can be identified in the CHESS background spectrum. Although \nthe background is negligible compared to the strong X-ray signal (see \nFigs.~\\ref{fig:CHESS_ComtponSpectra_UnPol} and \n\\ref{fig:CHESS_ComtponSpectra_Pol} for reference), it was subtracted \nfrom the spectra shown in this section. It should be noted that the beam \nin the hutch may introduce an additional implicit/diffuse background by \nscattering off several components (slits, absorber foils, etc.). This \nkind of background was not determined in a dedicated measurement, but \ncan potentially be higher than the CR background.\n\n{\\it Synchronization between simulations and data.} The Bragg \nmonochromator only allows the $40/80/120 \\, \\rm{keV}$ harmonics of the \nCHESS white beam to enter the C1 hutch. The spectral intensities of the \nwhite beam, as well as the Bragg-reflected intensities of the \nmonochromator, were calculated using the {\\it X-ray Oriented Programs} \n(XOP) software \npackage\\footnote{http://www.esrf.eu/computing/scientific/xop2.1/}. The \nabsorber foils further change the relative intensities. Given the \ndensity of platinum of $\\rho_{\\rm{Pt}} = 21.45 \\, \\rm{g}/\\rm{cm}^{3}$, \nits mass attenuation coefficient at $40 \\, \\rm{keV}$ of \n$(\\mu/\\rho)_{\\rm{Pt}} = 12.45 \\rm{cm}^{2}/\\rm{g}$, and the thickness of \nthe Pt foils $d$, one can calculate the beam intensity $I$ entering the \npolarimeter by $I/I_{0} = \\exp(-\\mu/\\rho \\cdot \\rho \\cdot d)$. The mass \nattenuation coefficients $\\mu/\\rho$ were taken from the NIST X-ray \ndatabase\\footnote{http://www.nist.gov/pml/data/xraycoef/index.cfm}. \nHowever, an accurate prediction of the relative flux intensities \nentering the polarimeter requires a detailed modeling of all \nenergy-dependent X-ray absorption/transmissions on the beam path \n(including the entrance windows, the Bragg monochromator, and the \nabsorber foils).\n\nTherefore, we instead used the overall spectrum measured by the whole \npolarimeter to synchronize the relative $40/80/120 \\, \\rm{keV}$ flux \nintensities with our Monte Carlo simulations. \nFigure~\\ref{fig:CHESS_ComtponSpectra_AllDetector_UnPol} shows the \noverall X-Calibur responses (all detector rings except for R1, see \nbelow) to the non-polarized CHESS beam, measured with configurations \n$C_{\\rm{ch1}}$ and $C_{\\rm{ch3}}$. The corresponding relative \nintensities of the three mono-energetic energies in the simulations were \nmatched accordingly and are in turn applied for all data versus \nsimulation comparisons that follow. Note, however, that the CHESS white \nbeam intensity varies. Therefore, the simulations were scaled according \nto the relative difference in the integral X-Calibur trigger rate with \nrespect to the data run used to normalize the simulations. We estimate \nthe systematic error on the absolute flux normalization of the \nsimulations to be around $20 \\%$. \nFigure~\\ref{fig:CHESS_ComtponSpectra_AllDetector_UnPol} also shows the \nenergy spectra with a coincident scintillator trigger ($f_{\\rm{sci}} = \n1$), although obtained with the non-optimal scintillator trigger \nthreshold.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CHESS_ComptonSpectra_Mosaic}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{CHESS_ComptonSpectra_Ring1t_Mosaic}\n\\end{center}\n\n\\caption{Energy spectra of the Compton scattered, non-polarized $40/80/120 \n\\, \\rm{keV}$ CHESS X-ray beam (averaged over the complete azimuthal \nscattering range, $m=1$ events). The horizontal lines represent the \nnominal energies after Compton-scattering. For reference, the CR \nspectrum is shown for one of the $5 \\, \\rm{mm}$ rings. {\\bf Left:} \nSpectra from selected sub rings. In the case of ring {\\it \nR1$_{\\rm{t}}$}, the first pixel row was removed (see text and right \npanel). {\\bf Right:} Special case of ring {\\it R1$_{\\rm{t}}$} which is \nlocated at the polarimeter entrance; individual pixel rows are shown. It \ncan be seen that the first pixel row has significantly higher flux at \n$E>40 \\, \\rm{keV}$ as compared to the other rows~-- likely due to \nexternal X-ray contamination. The same spectra are shown with the \nscintillator trigger condition ($f_{\\rm{sci}} = 1$). Simulated spectra \nare shown for the pixel rows 1 and 4. The spectrum obtained from the \nfirst pixel row in Ring {\\it R2$_{\\rm{t}}$} is shown for reference.}\n\n\\label{fig:CHESS_ComtponSpectra_UnPol}\n\\end{figure*}\n\n\n{\\it Compton spectra of a non-polarized beam.} The energy spectra of the \nnon-polarized Compton scattered X-ray beam are shown for individual \ndetector rings in Fig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} (left). The \nspectra are averaged over all azimuthal scattering angles $\\Phi$ within \neach ring. Ring {\\it R1$_{\\rm{t}}$} mostly detects back-scattered events \n(the scintillator geometrically covering rings {\\it R2}-{\\it R8}, see \nFig.~\\ref{fig:Design}, left) which deposit the maximal amount of energy \nin the scintillator: $\\Delta E_{\\rm{sci}}$ up to $5.4 \\, \\rm{keV}$ for \nthe $40 \\, \\rm{keV}$ beam, depositing $E_{\\rm{czt}} = 34.6 \\, \\rm{keV}$ \nin the CZT detector. The rings further downstream detect a superposition \nof X-rays that underwent Compton-scattering under different angles. The \nrear-side rings ({\\it R7} and {\\it R8}), to a larger extent, detect \nX-rays which Compton scattered nearly in the forward direction~-- \ndepositing only a small amount of energy in the scintillator, with an \nenergy deposition in the CZT ($E_{\\rm{czt}}$) close to the energy of the \nprimary X-ray ($E_{\\rm{line}}$). This explains the shift of the peak \npositions in the spectra towards higher energies with increasing ring \nnumber. The theoretical range expectations of $E_{\\rm{czt}}$ are \nindicated by the horizontal lines in \nFig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} (assuming an idealized energy \nresolution).\n\nThe right panel of Fig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} shows the \nenergy spectra of individual pixel rows of ring {\\it R1$_{\\rm{t}}$}. It \ncan be seen that the first pixel row shows a strong anomaly at energies \n$E_{\\rm{czt}} > 50 \\, \\rm{keV}$ as compared to the pixel rows 2-4. The \ncomparison with simulations further underlines the difference. A likely \nexplanation of the additional continuum would be the following. The \nfirst row of pixels has a rather high effective area (pixel sides) in \nthe plane perpendicular to the X-ray beam, as compared to other pixel \nrows. These `side' pixels will detect X-rays that underwent diffuse \nscattering in the Pt/Pb absorber foils or the polarimeter entrance plate \n(see Fig.~\\ref{fig:X-Calibur_Configurations}), and succeeded in entering \nthe polarimeter along a path not along the optical axis. Since this \ninvolves X-ray transmission through some of the material of the \npolarimeter fixture, it is more likely to happen for higher energies, \nwhich is where the anomaly in the spectrum becomes more prominent. The \nfirst pixel row was therefore excluded from the analysis of the data \npresented in this section. It should be noted, that the contamination is \nalso seen in the pixel rows further downstream~-- however, with strongly \ndecreasing strength.\n\nThe right panel of Fig.~\\ref{fig:CHESS_ComtponSpectra_UnPol} also shows \nthe spectra obtained from the same data with the additional requirement \nof a scintillator trigger ($f_{\\rm{sci}} = 1$). This favors high-energy \nand/or back-scattered events which deposit a high amount of energy \n$\\Delta E_{\\rm{sci}}$ in the scintillator. With the $f_{\\rm{sci}} = 1$ \ncut, the first pixel rows no longer show the anomaly discussed above~-- \nunderlining the hypothesis of an external contamination not interacting \nwith the scintillator. A more detailed study of the trigger efficiency \nof the scintillator is presented in \nSec.~\\ref{subsec:ScintillatorEfficiency}. Despite the contamination \nissue seen in the first front-side pixel rows, the Compton-spectra \nmeasured with X-Calibur are in good agreement with the simulations \n(although not explicitly shown for all rings in \nFig.~\\ref{fig:CHESS_ComtponSpectra_UnPol}, right, for reasons of \nvisibility.).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{CHESS_ComptonSpectraPol_Mosaic}\n\\end{center}\n\n\\caption{Compton spectra of the polarized $40/80/120 \\, \\rm{keV}$ CHESS \nbeam. Spectra are shown for two of the four detector sides ({\\it Bd0} \nand {\\it Bd1}, see Fig.~\\ref{fig:X-Calibur_Configurations}, top) in ring \n{\\it R2$_{\\rm{b}}$}, with {\\it Bd0} aligned with the plane of \npolarization ($\\alpha = 0^{\\circ}$). The {\\it Bd1} spectrum is also \nshown with the requirement of a scintillator trigger (the discriminator \nthreshold not being optimized at this point). The horizontal lines \nrepresent the nominal energy ranges after Compton-scattering. The \nsimulated spectra are shown, as well.}\n\n\\label{fig:CHESS_ComtponSpectra_Pol}\n\\end{figure}\n\n{\\it Compton spectra of a polarized beam.} A polarized X-ray beam \nintroduces an azimuthal modulation with a $180^{\\circ}$ periodicity to \nthe measured spectra. Figure~\\ref{fig:CHESS_ComtponSpectra_Pol} shows \nthe Compton spectra measured on two of the four detector sides in ring \n{\\it R2$_{\\rm{b}}$} (with the polarimeter being oriented at an angle of \n$\\alpha = 0^{\\circ}$). As expected, and essential for the functionality \nof the polarimeter, the detector side located perpendicular to the \npolarization plane ({\\it Bd0}) detects a higher number of \nCompton-scattered X-rays. The energy spectra obtained from simulations \nare shown in Fig.~\\ref{fig:CHESS_ComtponSpectra_Pol}, as well, and are \nfound to be in reasonable agreement with the data. For the data versus \nsimulation comparison one has to keep in mind that the simulations were \nperformed for a $100 \\%$ polarized beam, corresponding to the maximal \nmodulation and therefore a maximal scattering concentration in {\\it \nBd0}. The polarization fraction of the CHESS beam, on the other hand, is \n$r < 100 \\%$. Therefore, the azimuthal asymmetry ({\\it Bd0} versus {\\it \nBd1}) in the data is expected to be smaller compared to the simulations. \nDividing the spectra into distinct bins in azimuth, and integrating \ncounts in a given energy interval, will lead to the azimuthal scattering \ndistributions that are discussed in the next paragraph.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{AzimuthDistribution_40keV}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{AzimuthDistribution_SystematicExamples}\n\\end{center}\n\n\\caption{{\\bf Left:} Azimuthal scattering distribution (CHESS at $40 \\, \n\\rm{keV}$, ring R2$_{\\rm{b}}$). The horizontal error bars reflect the \nazimuthal ranges $\\Delta \\Phi_{j}$ covered by the corresponding pixels \n$j$ (see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left). The raw \nnon-polarized beam shows the 4-fold symmetry of the four detectors boards \n{\\it Bd0-Bd3} (see Fig.~\\ref{fig:X-Calibur_Configurations}). The \npolarized beam has the additional $180^{\\circ}$ amplification imprinted \nwhich is clearly extracted once corrected for the azimuthal coverage \n$\\Delta \\Phi_{j}$ and the pixel acceptance $a_{j}$. A sine function is \nfitted to extract $C_{\\rm{min}}$, $C_{\\rm{max}}$, the modulation factor \n$\\mu$, and the angle of the polarization plane $\\Omega_{\\rm{p}}$ (dotted \nvertical lines, indicating the error range). The dashed vertical line \nindicates the nominal polarization plane. For reference, the \ndistribution of a simulated $40 \\, \\rm{keV}$ beam is shown ($100 \\%$ \npolarized, reconstructed in the same way). {\\bf Right:} Normalized \nazimuthal scattering distributions derived from different data sets: a\nnon-polarized beam, a polarized beam measured at $\\alpha = 70^{\\circ}$, as \nwell as the response to a cosmic-ray background run.}\n\n\\label{fig:AzimuthDistribution}\n\\end{figure*}\n\n{\\it Azimuthal scattering distribution.} The azimuthal scattering \ndistributions were generated as outlined in \nSec.~\\ref{subsec:AnalysisPolarization} by integrating the energy range \nof Compton-scattered photons for the three harmonics of the X-ray beam \n($34.6 - 40 \\, \\rm{keV}$, $60.9 - 80 \\, \\rm{keV}$, and $81.7 - 120 \\, \n\\rm{keV}$, respectively). The energy intervals include an additional $2 \n\\, \\rm{keV}$ cushion to account for the energy resolution of the CZT \ndetector pixels. All measured event rates were in turn normalized on a \npixel-by-pixel basis using the azimuthal coverage $\\Delta \\Phi_{j}$ and \nacceptances $a_{j}$. Following Eq.~(\\ref{eqn:Acceptance}) described in \nSec.~\\ref{subsec:AnalysisPolarization}, the pixel acceptances $a_{j}$ \nwere determined from the non-polarized beam for each of the above energy \nintervals.\n\nAn example of an azimuthal scattering distribution measured in the \n$34.6-40 \\, \\rm{keV}$ band is shown in \nFig.~\\ref{fig:AzimuthDistribution} (left). The normalized distribution \nis used to derive the plane of polarization $\\Omega_{\\rm{p}}$ (nominal \nvalue of $\\Omega_{\\rm{nom}} = 90^{\\circ}$), as well as the relative \nscattering amplitude $\\mu_{\\rm{ch}}$. The distribution is compared to \nthe one obtained from the simulation of a $100 \\%$ polarized beam at $40 \n\\, \\rm{keV}$. Both are found to be in good agreement~-- except for the \nslightly lower amplitude of the data which is a result of the $r < 100 \n\\%$ polarization fraction of the CHESS beam (see below).\n\nFigure~\\ref{fig:CHESS_2DAzimuthData} illustrates the reconstruction of \nthe azimuthal scattering distributions for the complete polarimeter \n(subrings {\\it R1$_{\\rm{t}}$}-{\\it R8$_{\\rm{b}}$}), based on simulations \n(top panel) and based on measured CHESS data (bottom panel). The \nresponse to a non-polarized $40 \\, \\rm{keV}$ beam is shown on the left \nside of the figure. The second panel shows the polarimeter response to a \npolarized beam at $40 \\, \\rm{keV}$, not yet corrected for pixel \nacceptance $a_{j}$ and azimuthal coverage $\\Delta \\Phi_{j}$. Applying \nthe corrections leads to the smooth distribution shown in the third \npanel, which is in good agreement if comparing the simulations to the \ndata. The pixel acceptances $a_{j}$ depend on (i) the energy threshold, \n(ii) the energy resolution, and (iii) the trigger efficiency of the \npixel $j$. The simulations take into account (i) and (ii), derived from \nthe calibration measurements presented in \nSec.~\\ref{subsec:CZT_Calibration}, but not (iii). The projected count \ndistributions lead to the azimuthal scattering distributions shown in \nthe right panel of Fig.~\\ref{fig:CHESS_2DAzimuthData}. In contrast to \nthe 2D distributions shown in planar pixel coordinates, the $\\Phi$ \npositions and error bars now represent the proper angular coverage of \neach pixel (compare with Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, \nleft). The expected $180^{\\circ}$ modulation is clearly revealed, and \nthe reconstructed orientation of the polarization plane agrees with the \ndirection of the CHESS beam setup~-- confirming the functionality of the \nX-Calibur polarimeter.\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{CHESS_Example_AzimuthDistribution_CHESS1_E_32_6_42_0keV_Angle-1111deg} \\\\\n\\includegraphics[width=0.99\\textwidth]{CHESS_Example_AzimuthDistribution_CHESS1_E_32_6_42_0keV_Angle0deg} \\\\\n\\end{center}\n\n\\caption{X-Calibur 2D scattering distributions of the $40 \\, \\rm{keV}$ \nbeam at CHESS ($C_{\\rm{ch}}^{3}$, see \nFig.~\\ref{fig:X-Calibur_Configurations}, the beam enters from the top). \nEvents are shown for reconstructed energies of $36-40 \\, \\rm{keV}$. The \ntop row shows the results of simulations of a $100\\%$ polarized beam. \nThe bottom row shows the results of the CHESS measurement. The (blue) \nboxes in the 2D maps indicate the projected outline of the scintillator \n(Compton-scatterer). {\\bf Left:} Raw count map (pixel-by-pixel) of a\nnon-polarized beam (neither corrected for $\\Delta \\Phi_{j}$ nor for pixel \nacceptance $a_{j}$). All four detector sides, {\\it Bd0}-{\\it Bd3}, are \nunfolded into a plane. The detector rings {\\it R1} to {\\it R8} are \nindicated. {\\bf Second:} Count map of a raw measurement of a polarized \nbeam. {\\bf Third:} Count map of the polarized beam, corrected for \n$\\Delta \\Phi_{j}$ and $a_{j}$. {\\bf Right:} Normalized azimuthal \nscattering distribution (corrected for $\\Delta \\Phi_{j}$ and $a_{j}$) \nfor different detector rings. The vertical lines indicate the nominal \nplane $\\Omega_{\\rm{n}}$ of the electric field vector of the polarized \nbeam.}\n\n\\label{fig:CHESS_2DAzimuthData}\n\n\\end{figure*}\n\n{\\it Simulated modulation factors.} Each detector ring (see \nFig.~\\ref{fig:CHESS_2DAzimuthData}), or a sub ring thereof, can be seen \nas an independent detector that allows to reconstruct energy-dependent \npolarization properties of the incoming X-ray beam. \nFigure~\\ref{fig:MC_ModulationFactor} shows the simulated modulation \nfactors $\\mu_{\\rm{sim}}$ for the different energies and X-Calibur \nconfigurations as a function of detector ring/depth, measured along the \noptical axis. The values for $\\mu_{\\rm{sim}}$ were obtained using the \nStokes analysis, and are in agreement with the corresponding results \nobtained from the analysis of the azimuthal scattering distribution (not \nshown). The simulations shown reflect the detector status of the CHESS \nmeasurements (dead pixels, energy resolutions, etc.), not an idealized \ndetector. Ring {\\it R1} detects only back-scattered X-rays with a \ncorrespondingly lower $\\mu_{\\rm{sim}}$~-- but with a better energy \nresolution since the scattering kinematics are better defined. The \nstep-like structure in $\\mu_{\\rm{sim}}^{120\\,\\rm{keV}}$ between rings \n{\\it R5} and {\\it R6} can be explained by the strong degradation in \nenergy resolution at high energies in the $2 \\, \\rm{mm}$ detectors \nlocated at rings {\\it R6}-{\\it R8}. The highest modulation is achieved \nin ring {\\it R2}.\n\nThe simulations of the modulation factor were also done involving the \nscintillator flag $f_{\\rm{sci}}$, based on the two scintillator trigger \nefficiencies as measured in \nFig.~\\ref{fig:Scintillator_TriggerEfficiency}. This reflects the \nefficiency during the CHESS measurements (not optimized), as well as the \nefficiency of the optimized scintillator trigger. The scintillator \nimproves the modulation factor for most rings~-- by requiring a minimum \nenergy deposition $\\Delta E_{\\rm{sci}}$ which effectively limits the \nallowed range of polar scattering angles. This, however, also reduces \nthe event statistics (not imprinted in $\\mu_{\\rm{sim}}$). A scintillator \ntrigger efficiency of $100 \\%$ would again lead to the same distribution \nof modulation factors as no cut on $f_{\\rm{sci}}$ does. The main benefit \nof the scintillator trigger lies in its ability to suppress background \nduring a measurement (see for example Fig.~\\ref{fig:BG_Spectra}, left). \nThe modulation factor $\\mu$, together with the detection rate after \nbackground subtraction, are the crucial characteristics that determine \nthe MDP detection sensitivity of the polarimeter following \nEq.~(\\ref{eqn:MDP}).\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{MC_ModulationFactors}\n\\end{center}\n\n\\caption{Simulated modulation factors $\\mu_{\\rm{sim}}$ as a function of \ndetector ring {\\it R}, shown for different energies. The geometrical \npositions (along the optical axis $y$) are indicated for the \nscintillator and the CZT ring assembly. The simulated data were analyzed \nwith the Stokes method. Results are also shown for the scintillator \ntrigger requirement $f_{\\rm{sci}}=1$, assuming the PMT trigger \nefficiency during the CHESS measurements (PMT$_{\\rm{ch}}$) and the \nefficiency of the optimized PMT threshold (PMT+), see \nFig.~\\ref{fig:Scintillator_TriggerEfficiency}.}\n\n\\label{fig:MC_ModulationFactor}\n\\end{figure}\n\n{\\it Polarization fraction of the CHESS beam.} The azimuthal scattering \ndistributions for each sub ring and each data set were fitted to \ndetermine $\\mu_{\\rm{ch}}$. The simulated modulation factors \n$\\mu_{\\rm{sim}}$ (Fig.~\\ref{fig:MC_ModulationFactor}) were in turn used \nto determine the polarization fraction $r$ of the CHESS beam using \nEq.~(\\ref{eqn:PolFracPhiDistri}). This is shown in \nFig.~\\ref{fig:AzDistri_RecoMu} for the $40 \\, \\rm{keV}$ beam for \ndifferent orientations $\\alpha$ of the polarimeter relative to the \npolarization plane. Also shown are the residuals between the \nreconstructed polarization plane and the nominal polarization plane. \nNote, that for large values of $\\alpha$ the azimuthal scattering \ndistribution shows an additional global slope, possibly caused by an \n$\\alpha$-dependent tilt of the rotation fixture (see \nFig.~\\ref{fig:AzimuthDistribution}, right, for an example orientation of \n$\\alpha = 70^{\\circ}$). This will affect the reconstructed polarization \nproperties and is further discussed in Sec.~\\ref{subsec:Systematics}. To \nreduce this systematic effect, the CHESS azimuthal distributions were \nfolded back into the $[0;180]^{\\circ}$ interval before the sinusoidal \nfunction was fitted to the data points.\n\n\n\\begin{table}[t!]\n\n\\begin{tabular}{lrr}\n\nSetup & $\\Delta \\Omega \\, [\\rm{deg}]$ & $r_{\\rm{rec}} \\, [\\%]$ \\\\\n\\hline \\hline\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{{beam energy: $40 \\, \\rm{keV}$}} \\\\\n\\hline\n\n$C_{\\rm{ch1}}$ & $-4.7 \\pm 1.9$ & $91.0 \\pm 0.6$ \\\\\n & $-4.6 \\pm 1.7$ & $88.6 \\pm 0.4$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{{beam energy: $80 \\, \\rm{keV}$}} \\\\\n\\hline\n\n$C_{\\rm{ch1}}$ & $-2.6 \\pm 1.7$ & $83.5 \\pm 0.8$ \\\\\n & $-2.0 \\pm 2.1$ & $78.3 \\pm 0.8$ \\\\\n$C_{\\rm{ch1}}$, $f_{\\rm{sci}} = 1$ & $-2.1 \\pm 2.5$ & $86.5 \\pm 1.1$ \\\\\n & $-2.1 \\pm 2.7$ & $82.8 \\pm 1.0$ \\\\\n$C_{\\rm{ch4}}$ & $-2.5 \\pm 1.8$ & $90.4 \\pm 0.5$ \\\\\n & $-2.0 \\pm 2.2$ & $85.5 \\pm 0.6$ \\\\\n\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{{beam energy: $120 \\, \\rm{keV}$}} \\\\\n\\hline\n\n$C_{\\rm{ch1}}$ & $-0.5 \\pm 2.9$ & $89.2 \\pm 1.5$ \\\\\n & $-0.2 \\pm 3.3$ & $71.5 \\pm 1.3$ \\\\\n$C_{\\rm{ch1}}$, $f_{\\rm{sci}} = 1$ & $-0.7 \\pm 4.6$ & $86.3 \\pm 1.2$ \\\\\n & $-0.8 \\pm 3.7$ & $80.5 \\pm 1.5$ \\\\\n\n\\end{tabular}\n\n\\caption{Residuals between the reconstructed polarization plane \n$\\Omega_{\\rm{p}}$ and the nominal plane at $\\Omega_{\\rm{n}} = \n90^{\\circ}$: $\\Delta \\Omega = (\\Omega_{\\rm{n}}-\\alpha) - \n\\Omega_{\\rm{p}}$. The second column shows the reconstructed polarization \nfractions $r_{\\rm{rec}}$ of the CHESS beam at different energies and \ndifferent configurations/setups (partly with the $f_{\\rm{sci}} = 1$ \nrequirement). See Figs.~\\ref{fig:AzDistri_RecoMu} and \n\\ref{fig:CHESS_MorePolFractions} for an illustration. The first row per \ndata set shows the results obtained from the analysis of the azimuthal \n$\\Phi$-scattering distribution following \nEq.~(\\ref{eqn:PolFracPhiDistri}). The second row represents the results \nobtained from the Stokes analysis following \nEq.~(\\ref{eqn:PolarizationFromStokes}). The results are averaged over \nall orientations $\\alpha$, each.}\n\n\\label{tab:CHESS_PolFrac}\n\n\\end{table}\n\nFigure~\\ref{fig:CHESS_MorePolFractions} shows the corresponding \npolarization fractions reconstructed from the $80 \\, \\rm{keV}$ harmonic. \nThe reconstructed polarization fractions obtained from all measurements \nare summarized in Tab.~\\ref{tab:CHESS_PolFrac}~-- in all cases averaged \nover all measured orientations $\\alpha$. Table~\\ref{tab:CHESS_PolFrac} \nalso shows the results derived from the Stokes analysis following \nEq.~(\\ref{eqn:PolarizationFromStokes}). Both methods are in reasonable \nagreement, whereas the Stokes analysis seems to slightly underestimate \n$r$ compared to the results obtained with the azimuthal scattering \ndistribution. The systematic error on the reconstructed polarization \nfraction was estimated as described in Sec.~\\ref{subsec:Systematics}. \nThe polarization of the CHESS beam as reconstructed using the \n$\\Phi$-distribution method is measured to be $r_{\\rm{ch}}^{\\Phi} = (87.8 \n\\pm 0.4_{\\rm{stat}} \\pm 4.6_{\\rm{sys}}) \\, \\%$ with no indication of an \nenergy dependence in the $40-120 \\, \\rm{keV}$ range. The corresponding \nvalue reconstructed from the Stokes analysis is $r_{\\rm{ch}}^{\\rm{st}} = \n(81.2 \\pm 0.4_{\\rm{stat}} \\pm 10_{\\rm{sys}}) \\, \\%$. The difference in \nthe systematic error is explained by the fact that the Stokes analysis \nis more sensitive to the detector configuration (which was not perfectly \nconstrained during the CHESS measurements). A better defined geometry \nand the rotation of the polarimeter will reduce the systematic error for \nmeasurements performed during the balloon flight.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{PhiDistriFitResult_CHESS1_E_32_6_42_0keV}\n\\end{center}\n\n\\caption{Reconstructed polarization properties of the CHESS beam at $40 \n\\, \\rm{keV}$ (configuration $C_{\\rm{ch1}}$). {\\bf Top:} Modulation \nfactor $\\mu$ versus polarimeter sub rings {\\it Ri$_{\\rm{t,b}}$}. The \nresults were derived from simulations (MC), as well as from data taken \nunder different X-Calibur orientations $\\alpha$. For each $\\alpha$ and \neach ring, $\\mu_{\\rm{ch}}$ is derived from fits to acceptance corrected \n$\\Phi$-distributions (see Fig.~\\ref{fig:AzimuthDistribution}, left). The \nscintillator starts covering detector rings at $y \\geq 1.8 \\, \\rm{cm}$. \nThe green error band reflects the $1 \\, \\rm{std.dev}$ range of measured \npoints in the particular $y$ slice. {\\bf Middle:} Reconstructed \npolarization fraction following Eq.~(\\ref{eqn:PolFracPhiDistri}). The \ngreen band reflects the $1 \\, \\rm{std.dev}$ range of all reconstructed \nfractions in the corresponding $y$ slice. {\\bf Bottom:} Residual between \nreconstructed polarization plane $\\Omega_{\\rm{p}}$ and true polarization \nplane $\\Omega_{\\rm{n}} = 90^{\\circ}$, corrected for the X-Calibur \norientation $\\alpha$: $\\Delta \\Omega = (\\Omega_{\\rm{n}} - \\alpha) \n-\\Omega_{\\rm{p}}$.}\n\n\\label{fig:AzDistri_RecoMu}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{PhiDistriFitResult_PolFrac_CHESS4_E_58_9_82_0keV} \\\\\n\\end{center}\n\n\\caption{Reconstructed polarization fractions of the CHESS beam (compare \nwith Fig.~\\ref{fig:AzDistri_RecoMu}) derived from the $80 \\, \\rm{keV}$ \nharmonic (configuration $C_{\\rm{ch3}}$).}\n\n\\label{fig:CHESS_MorePolFractions}\n\\end{figure}\n\n{\\it Polarization plane of the CHESS beam.} \nTable~\\ref{tab:CHESS_PolFrac} also summarizes the average reconstructed \npolarization planes $\\Omega_{\\rm{p}}$ (corrected for the X-Calibur \norientation $\\alpha$). Note, that the uncertainty in setting the \nX-Calibur orientation was estimated to be $\\Delta \\alpha = \\pm \n2^{\\circ}$. Within this systematic error, as well as the statistical \nerrors, the average of reconstructed polarization planes (all rings and \nall X-Calibur orientations $\\alpha$) is compatible with the nominal \nplane $\\Omega_{\\rm{n}}$ of the CHESS beam. However, the measurements \nconsistently reconstruct $\\Delta \\Omega < 0^{\\circ}$, which may indicate \na slight offset in the way the rotation mechanism was \ninstalled/calibrated. The reconstructed polarization planes shown in \nFig.~\\ref{fig:AzDistri_RecoMu} indicate a slight dependence on $\\alpha$ \nand $y$. The left panel in Fig.~\\ref{fig:AzimuthDistribution} shows a \ncorresponding azimuthal distribution measured with the X-Calibur \norientation of $\\alpha = 70^{\\circ}$, revealing a slight asymmetry. This \nresult can be seen as another indication of a slight mis-alignment \n(offset and/or tilt) between the rotation axis of the scintillator and \nthe optical axis of the X-ray beam. A more detailed discussion of this \nkind of systematic effect can be found in Sec.~\\ref{subsec:Systematics}.\n\n\n\\subsection{Measurements with the X-Calibur/{\\it InFOC$\\mu$S}\\xspace Assembly} \n\\label{subsec:FtSumnerData}\n\nThe full X-Calibur/{\\it InFOC$\\mu$S}\\xspace experiment was assembled and tested during a \nflight preparation campaign at the NASA {\\it Columbia Scientific Balloon \nFacility} (CSBF) site in Ft.~Sumner, NM, in the fall of 2014. The \npolarimeter was installed in the rotating CsI shield assembly, which in \nturn was installed in the pressure vessel as part of the {\\it InFOC$\\mu$S}\\xspace X-ray \ntelescope (see Fig.~\\ref{fig:ActiveShieldAndTelescope}). This setup, \n$C_{\\rm{ft}}$, is illustrated in \nFig.~\\ref{fig:X-Calibur_Configurations}. The CZT detector configuration \nof these measurements is listed in Tab.~\\ref{tab:Detectors} and differs \nslightly from the one used in the CHESS measurements (described in \nSec.~\\ref{subsec:CHESS}). The X-ray mirror was installed on the optical \nbench located at the front part of the {\\it InFOC$\\mu$S}\\xspace telescope. The pressure \nvessel is located at a focal distance of $8 \\, \\rm{m}$.\n\nBefore the balloon flight, a beryllium (Be) window is installed in the \ntop dome of the pressure vessel, to assure a pressure-sealed entrance to \nthe polarimeter with high transmissivity to hard X-rays. For the \nground-based measurements, presented in this section, the Be window was \nnot installed to allow to visually align the polarimeter with the \noptical axis of the X-ray mirror. Throughout the measurements presented \nin this section, the polarimeter was rotated at $4 \\, \\rm{rpm}$. Due to \nthe rotation, the application of the pixel acceptances $a_{j}$, defined \nin Eq.~(\\ref{eqn:Acceptance}), is no longer required, since each pixel \ntests the complete, time-averaged azimuthal scattering range with \nrespect to the polarization plane. As another consequence, the azimuthal \nbinning can be chosen finer than in the non-rotating system in which it \nwas limited to the number of $32$ CZT pixels per row. For each event, \nthe hit pixel is de-rotated into the laboratory/horizon coordinate frame \n(see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, left) to determine \nthe azimuthal scattering angle $\\Phi$.\n\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.32\\textheight]{FtSumner_MirrorScanSetup}\n\\hfill\n\\includegraphics[height=0.32\\textheight]{FtSumnerMirrorScan2D_Rotating}\n\\hfill\n\\includegraphics[height=0.32\\textheight]{FtSumnerPolFraction}\n\\end{center}\n\n\\caption{{\\it InFOC$\\mu$S}\\xspace/X-Calibur X-ray mirror scan, configuration \n$C_{\\rm{ft}}$ in Fig.~\\ref{fig:X-Calibur_Configurations}. {\\bf Left:} \nThe collimated X-ray source is aligned with the optical axis of the \nmirror and the polarimeter which is situated in the focal plane at a \ndistance of $8 \\, \\rm{m}$ (not visible in this photograph). The X-ray \nsource can be automatically moved in the $X$/$Z$ plane to scan the \nmirror with the X-rays traveling along $Y$. {\\bf Middle:} The \nCompton-scattered event distribution measured with X-Calibur (horizon \nsystem, de-rotated) during the scan in the energy range of $30-50 \\, \n\\rm{keV}$. {\\bf Right:} Background subtracted measurements of the \ncollimated X-ray beam. Three data runs where taken with (i) the beam \nhitting the center of the scintillator ($\\Delta d = 0 \\, \\rm{mm}$), (ii) \nthe beam hitting at an offset of $\\Delta d = -4 \\, \\rm{mm}$, and (iii) \nthe beam scanning the X-ray mirror. The top panel shows the \nCompton-scattered energy spectra (all-detector average, each). The \nbottom panel shows the energy-dependent polarization fraction \nreconstructed from the azimuthal $\\Phi$-scattering distribution, as well \nas using the Stokes parameters. The given errors are statistical only.}\n\n\\label{fig:FtSumner_MirrorScan}\n\\end{figure*}\n\n{\\it X-ray mirror/X-Calibur alignment.} The alignment of the polarimeter \nand the mirror are crucial in order to gain the maximal sensitivity for \npolarization measurements and to reduce systematic effects. The optical \naxis and the on-axis image location of the X-ray mirror were measured by \na CCD camera with an optical parallel beam. The X-ray mirror was placed \nin the optical beam, and its tip and tilt were adjusted such that the \noptical axis is parallel to the optical beam. Then, the CCD camera was \nplaced at the center of the mirror, looking along the optical axis \nfacing the polarimeter, and taking a picture of the on-axis image plane. \nThe image location was recorded in pixel coordinates, which determines \nthe optical axis as well as the on-axis image position. After the X-ray \nmirror was installed onto the optical bench (see \nFig.~\\ref{fig:FtSumner_MirrorScan}, left), a picture of the front \nsurface of the polarimeter's scintillator was taken by the camera. The \nmirror tip and tilt were adjusted by shimming at the interface to the \noptical bench, such that the center of the scintillator is at the \nrecorded location of the on-axis image of the X-ray mirror. An alignment \nbetween the scintillator and mirror of better than $1\\, \\rm{mm}$ was \nachieved which guarantees a systematic error on the reconstructed \npolarization fraction of less than $2 \\%$ (see \nSec.~\\ref{subsec:Systematics}).\n\n{\\it Mirror scan.} The proper alignment of the telescope is \ntested/confirmed in a mirror scan. A movable X-ray source scans the \nsurface of the mirror while the polarimeter response is measured. The \nX-ray scanning system consists of an X-ray source with a $60\\, \\rm{cm}$ \nlong collimator, tip and tilt stages, and $X$/$Z$ travel stages to which \nthe X-ray source is mounted. The X-ray source was positioned in front of \nthe X-ray mirror and could be used to either directly illuminate the \npolarimeter (through the central hole in the X-ray mirror), or to scan \nthe whole mirror aperture. The setup is shown in \nFig.~\\ref{fig:FtSumner_MirrorScan}, left. An Oxford 5011 electron impact \nX-ray tube is used to generate the X-rays, with an active source spot \nsize of $0.05 \\, \\rm{mm}$ in diameter (molybdenum target, Mo). The \nproduced X-ray spectrum is made of emission line (Mo-K) as well as \nbremsstrahlung. The X-ray source provides a continuum spectrum up to $50 \n\\, \\rm{keV}$. The current was adjusted to give a reasonable event rate \nat the polarimeter located at $8 \\, \\rm{m}$ distance (see \nTab.~\\ref{tab:PixelRates} for reference). The collimator can be equipped \nwith two interchangeable pin holes with diameters of $0.1 \\, \\rm{mm}$ \nand $1 \\, \\rm{mm}$, respectively. The beam size at $8 \\, \\rm{m}$ \ndistance is around $2 \\, \\rm{mm}$ in diameter for the $0.1 \\, \\rm{mm}$ \npin hole. The $1 \\, \\rm{mm}$ pin hole produces count rates in the \npolarimeter above one kHz at $8 \\, \\rm{m}$ distance ($20-50 \\, \n\\rm{keV}$, after air absorption). The tip and tilt stages change the \ndirection of the collimated X-ray beam. The $X$/$Z$ translation stages \nallow the X-ray beam to scan over the entire aperture of the mirror \n(with the central hole in the mirror blocked by a lead absorber). The \nX-ray beam was aligned with the optical axis of the mirror. The $1 \\, \n\\rm{mm}$ pin hole was used to scan the mirror within $51$ horizontal \nrows along $X$. The response measured with X-Calibur is shown in \nFig.~\\ref{fig:FtSumner_MirrorScan} (middle) in the de-rotated coordinate \nsystem (the horizon intersects at $90^{\\circ}$ and $270^{\\circ}$ in this \nrepresentation). The corresponding energy spectrum is shown in \nFig.~\\ref{fig:FtSumner_MirrorScan}, right. The results illustrate that \nthe mirror was successfully aligned with the polarimeter.\n\n{\\it Direct beam illumination.} As a reference measurement to the mirror \nscan, data were taken with the polarimeter being directly illuminated by \nthe collimated X-ray source ($0.1 \\, \\rm{mm}$ pin hole). The X-rays \nenter the scintillator along its optical axis, but do not pass the X-ray \nmirror in this measurement. The spectrum of the mirror scan shown in \nFig.~\\ref{fig:FtSumner_MirrorScan} (right, top) drops off faster \ncompared to the spectrum measured from the direct beam illumination. \nThis is a result of the energy-dependent effective area of the mirror. \nAn additional run was taken with the X-ray beam hitting the scintillator \noff center by $\\Delta d = -4 \\, \\rm{mm}$, which is discussed in \nSec.~\\ref{subsec:Systematics}.\n\nThe polarization parameters were reconstructed in the same way as \ndescribed in Sec.~\\ref{subsec:CHESS} (except for the acceptance \ncorrection $a_{j}$). The depth-dependent modulation factors (see \nFig.~\\ref{fig:MC_ModulationFactor}) obtained from the $40 \\, \\rm{keV}$ \nCHESS simulation were used to reconstruct the polarization fraction of \nthe X-ray beam in different energy bands (assuming $\\mu_{\\rm{sim}}$ \nbeing independent of energy in the $20-50 \\, \\rm{keV}$ band). The \nreconstruction was done with both methods described in \nSec.~\\ref{subsec:AnalysisPolarization}, namely using the analysis of the \nazimuthal scattering distribution following \nEq.~(\\ref{eqn:PolFracPhiDistri}), as well as using the Stokes parameters \nfollowing Eq.~(\\ref{eqn:PolarizationFromStokes}). Both methods yield \ncomparable results which are shown in Fig.~\\ref{fig:FtSumner_MirrorScan} \n(right, bottom). An energy dependent polarization fraction is found in \nthe data. However, in contrast to the mono-energetic CHESS beam \n(Sec.~\\ref{subsec:CHESS}), the energy distribution of the incoming \nX-rays (before Compton-scattering in the scintillator) follows a \ncontinuum. To properly reconstruct the polarization fraction as a \nfunction of incoming X-ray energy, one would have to utilize \nreconstruction methods of forward folding or unfolding \n\\cite{XCLB_LogLikelihoodAnalysis}, which is beyond the scope of this \npaper. However, a significant increase in polarization fraction can be \nobserved with increasing energy. The data of the mirror scan was also \nused to reconstruct the energy-dependent polarization fraction, and is \nfound to be in reasonable agreement with the direct beam measurement. \nThis confirms the predictions by \\citet{Katsuta2009} that an X-ray \nmirror does not substantially affect the polarization properties.\n\n\n\\subsection{Systematic Effects in Polarization Measurements} \n\\label{subsec:Systematics}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.285\\textheight]{AzimuthDistribution_OffsetsUnpolarized}\n\\hfill\n\\includegraphics[height=0.285\\textheight]{PolFracSystError}\n\\end{center}\n\n\\caption{{\\bf Left:} Azimuthal scattering distributions of a\nnon-polarized X-ray beam (CHESS data and simulations, both in a \nnon-rotating system) hitting the scintillator at different offsets $d$ \nrelative to the optical axis. X-Calibur was oriented at $\\alpha = \n0^{\\circ}$, with the beam `walking' from {\\it Bd1} to {\\it Bd3} with \nincreasing $\\Delta X$ (see Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}, \nleft, for a coordinate system). Scattering distributions are shown with \nand without the offset correction (see \nSec.~\\ref{subsec:AnalysisPolarization}). {\\bf Right:} Apparent \npolarization fraction $r$ of the non-polarized beam. The light blue region \nindicates the diameter of the scintillator rod. The data were analyzed \nusing the Stokes parameters as described by \nEq.~(\\ref{eqn:Stokes_Q_U_Average}). Results are shown without and with \nthe corrected beam offset.}\n\n\\label{fig:PolFracSysError}\n\\end{figure*}\n\nThe understanding and control of systematic effects of different nature \nis crucial for the correct reconstruction of the polarization properties \nfrom measured data. A series of measurements was performed at CHESS (see \nSec.~\\ref{subsec:CHESS}) to study the effects of a mis-alignment \n(offset) between the X-ray beam and the optical axis of the polarimeter. \nThe $X$/$Z$ stage of the table in hutch {\\it C1} was used to \nsystematically scan the polarimeter response for offsets ranging from \n$-5 \\, \\rm{mm}$ to $+5 \\, \\rm{mm}$ in steps of $1 \\, \\rm{mm}$ (the beam \ncomes in along $Y$, see Figs.~\\ref{fig:X-Calibur_Configurations} and \n\\ref{fig:SchematicBeamOffset_AndStokes} (left) for the definition of the \ncoordinate system). The scan along $X$ was performed with a polarimeter \norientation of $\\alpha = 0^{\\circ}$, whereas the scan along $Z$ was \nperformed with $\\alpha = -90^{\\circ}$. This allowed us to pairwise \nsuperimpose the data runs of perpendicular polarization planes that hit \nthe scintillator at the same position~-- being equivalent to a \nnon-polarized beam hitting at that particular offset position. The data \nruns were taken with configuration $C_{\\rm{ch4}}$, testing the response \nof the $80 \\, \\rm{keV}$ harmonic of the CHESS X-ray beam. A set of \nsimulations with an $80 \\, \\rm{keV}$ beam (polarized and non-polarized) \nwere performed resembling the same offsets as measured at CHESS, as well \nas offset simulations at $40 \\, \\rm{keV}$.\n\n{\\it Non-polarized beam.} The left panel of \nFigure~\\ref{fig:PolFracSysError} shows examples of azimuthal scattering \ndistributions measured with different beam offsets for a non-polarized \nbeam. The distributions were corrected for $\\Delta \\Phi_{j}$ (assuming a \ncentral beam) and $a_{j}$, as before. It can be seen that beam offsets \nof $\\simeq 2 \\, \\rm{mm}$ already introduce systematic asymmetries. \nAlthough the offset distributions are not flat, as expected for a \nnon-polarized beam, they also do not resemble the shape expected from a \npolarized beam. Given the high event statistics of the data used in the \nstudy, the sinusoidal fits used to determine the polarization properties \nwill therefore certainly fail~-- allowing one to detect/filter the \nsystematic effect. However, in the case of data with higher statistical \nuncertainty, the observed asymmetries could potentially lead to the \nreconstruction of an artificial polarization fraction of $r_{\\rm{rec}} > \n0$. Note, that folding the distributions into the $[0; 180]^{\\circ}$ \ninterval (not shown), will substantially reduce the asymmetry. The \nStokes analysis is per definition only considering the $[0; \n180]^{\\circ}$ interval. However, it is not sensitive to deviations from \nthe sinusoidal scattering distribution, and therefore does not provide a \ngoodness-of-fit measure that can be used to detect imprints of a beam \noffset. Figure~\\ref{fig:PolFracSysError} also shows examples of \ndistributions that were offset corrected following the description in \nSec.~\\ref{subsec:AnalysisPolarization}. This, however, assumes that the \nbeam offset is known.\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[height=0.285\\textheight]{AzimuthDistribution_OffsetsPolarized}\n\\hfill\n\\includegraphics[height=0.285\\textheight]{PolFracSystErrorPolarized}\n\\end{center}\n\n\\caption{{\\bf Left:} Azimuthal scattering distributions of polarized \nX-ray beams hitting the scintillator at different offsets $d$. Results \nare shown for the CHESS data ($r_{\\rm{ch}} \\simeq 85\\%$ polarized) and \nsimulations ($r = 100\\%$ polarized), both in a non-rotating system. \nDistributions corrected for the beam offset (see \nSec.~\\ref{subsec:AnalysisPolarization}) are shown, as well. Results are \nshown for offsets along $X$ ($\\alpha = 0^{\\circ}$) and $Z$ ($\\alpha = \n-90^{\\circ}$, corresponding to a scan along $X$ with perpendicular \npolarization). {\\bf Right:} Reconstructed polarization fraction $r$ for \nthe off-center beams, analyzed using the Stokes parameters as in \nEq.~(\\ref{eqn:Stokes_Q_U_Average}). The outline of the scintillator is \nindicated. Also shown are data derived from the Ft.~Sumner measurement \nshown in Fig.~\\ref{fig:FtSumner_MirrorScan}, right. Results are shown \nwithout and with the offset correction.}\n\n\\label{fig:PolFracSysErrorPolarized}\n\\end{figure*}\n\nTo quantify the artificial polarization $r_{\\rm rec}$ inferred when \nneglecting the beam offset, the data were analyzed using the Stokes \nparameters following Eq.~(\\ref{eqn:Stokes_Q_U_Average}). \nFigure~\\ref{fig:PolFracSysError} (right) shows $r_{\\rm{rec}}$ as a \nfunction of beam offset. The results are shown for the simulations as \nwell as for the CHESS data~-- both being in reasonable agreement. Also \nshown is an analytic model that simply integrates the events per \nazimuth angle interval in a plane perpendicular to the optical axis. The \nright panel of Fig.~\\ref{fig:PolFracSysError} shows the same \ndistribution after the beam offset correction described in \nSec.~\\ref{subsec:AnalysisPolarization} was applied. The discrepancy \nbetween the data and simulations after correction can be explained by an \nuncertainty in the absolute beam alignment in the data and the fact \nthat, in contrast to the simulations, two separate measurements had to \nbe superimposed to generate the non-polarized beam, amplifying the effect \nof unaccounted offsets or mis-alignments (any uncertainty in the offset \nnot only moves the data points along the beam offset axis, but also \nalong the $r$ axis).\n\nIn general, however, the first-order correction procedure greatly \nreduces the systematic effect to less than a few percent for offsets $d \n\\leq 3 \\, \\rm{mm}$. The simulated data shown in the right panel of \nFig.~\\ref{fig:PolFracSysError} suggest that the correction works better \nfor beam energies of $80 \\, \\rm{keV}$ as compared to $40 \\, \\rm{keV}$. \nThis can be explained by the fact that only geometrical offsets are \ncorrected for. Differences in absorption lengths in the scintillator \nmaterial, originating from positions other than $P_{0}$ (see left panel \nin Fig.~\\ref{fig:SchematicBeamOffset_AndStokes}), will cause \nsecond-order effects that depend on the energy of the scattered X-ray. \nHowever, addressing these additional correction terms is beyond the \nscope of this paper.\n\n{\\it Polarized beam.} For a polarized beam (left panel in \nFig.~\\ref{fig:PolFracSysErrorPolarized}), the offsets lead to either a \nreduction or amplification of the reconstructed polarization fraction, \ndepending on the angle between the offset vector $\\bf{d}$ and the plane \nof polarization $\\Omega_{\\rm{p}}$. The contribution of the beam offset \nto the reconstructed polarization is illustrated in the right panel of \nFig.~\\ref{fig:PolFracSysErrorPolarized} for different energies, \nincluding the results for the offset corrected analysis. The correction \nsubstantially reduces the systematic effect on $r$. A diverging trend \ncan be identified in the corrected CHESS data for $d > 0 \\, \\rm{mm}$ \n(which is also visible in the case of the `non-polarized' beam shown in \nthe right panel of Fig.~\\ref{fig:PolFracSysError}). This indicates an \ninaccuracy in the experimental setup, e.g. the asymmetric/fractional \nscattering off a fixture along the beam path before entering the \npolarimeter.\n\nAn additional offset measurement was performed during the flight \npreparation in Ft.~Sumner (see Sec.~\\ref{subsec:FtSumnerData}) using the \npartly polarized beam of the collimated X-ray source, hitting the \nscintillator at an offset of $d = -4 \\, \\rm{mm}$ (horizon system). The \npolarimeter/shield assembly was continuously rotating during the \nmeasurements. The corresponding fraction of apparent polarization $r$ is \nindicated in Fig.~\\ref{fig:PolFracSysErrorPolarized} (right). Due to the \nrotation of the polarimeter, the offset correction (see \nSec.~\\ref{subsec:AnalysisPolarization}) was calculated on an \nevent-by-event basis. The results illustrate that the correction also \nworks in a rotated system, canceling the artificial fraction of \npolarization introduced by the offset. The offset-corrected polarization \nspectrum is shown in Fig.~\\ref{fig:FtSumner_MirrorScan} (right), \nreproducing the measured results obtained with no beam offset.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Reco_MeanX}\n\\end{center}\n\n\\caption{Correlation between true beam offset and beam offset measured \nfollowing Eq.~(\\ref{eqn:FirstMoments}) for data and simulations.}\n\n\\label{fig:Offset_RecoMeanX}\n\\end{figure}\n\n{\\it Determination of the beam offset.} The geometrical offset \ncorrection described above assumes that the offset vector $\\bf{d}$ is \nknown. For sufficient event statistics and the assumption of a \ntime-independent beam offset, the offset can be estimated from the data \nitself using first moments, essentially substituting $2 \\Phi_{k} \n\\rightarrow \\Phi_{k}$ in Eq.~(\\ref{eqn:StokesParams}):\n\n\\begin{eqnarray} \\begin{split} \\left< x \\right> & = - \\frac{c}{W} \n\\sum_{k=1}^{N} w_{j(k)} \\sin(\\Phi_{k}), \\\\ \\left< z \\right> & = \n\\frac{c}{W} \\sum_{k=1}^{N} w_{j(k)} \\cos(\\Phi_{k}), \\\\ W & = \n\\sum_{k=1}^{N} w_{j(k)}. \\end{split} \\label{eqn:FirstMoments} \n\\end{eqnarray} Here, the constant $c=11 \\, \\rm{mm}$ reflects the \ndistance between the scintillator center and the detector plane (see \nFig.~\\ref{fig:X-Calibur_Configurations}, top). The weights $w_{j(k)}$ \nare the same as in Eq.~(\\ref{eqn:StokesSum}). As shown in \nFig.~\\ref{fig:Offset_RecoMeanX}, the measured offsets are linearly \ncorrelated with the true beam offsets. However, the slope of the \ncorrelation depends to some extent on the true polarization fraction $r$ \nof the beam and the angle between $\\Omega_{\\rm{p}}$ and {\\bf d}. Since \nthe offset of a non-polarized beam itself mimics a polarization fraction, \nthere will be a residual ambiguity that prevents to completely \ndisentangle $\\bf{d}$ and $r$ from the data alone. Therefore, a \ntime-resolved external monitoring of the beam position during the \nballoon flight is preferable.\n\nFigure~\\ref{fig:Offset_RecoMeanX} reveals a slight shift/translation \nbetween the slope of the reconstructed offset measured in the CHESS data \nversus the slope obtained from the simulations. This can be used to \nestimate an accuracy of the alignment achieved during the CHESS \nmeasurements to be $\\Delta X_{\\rm{sys}} \\simeq 0.3 \\, \\rm{mm}$ for the \nX-Calibur orientation of $\\alpha = 0^{\\circ}$. Note, with a rotation \naxis possibly not exactly aligned with the axis of the scintillator, \nthis may translate into larger offsets $\\Delta X$ for different \nX-Calibur orientations $\\alpha$.\n\n{\\it Systematic error on the polarization fraction.} The systematic \nerror on the reconstructed polarization fraction $\\Delta r_{\\rm{sys}}$ \ncan be estimated as follows. For the polarization measurements presented \nin this paper (Sec.~\\ref{subsec:CHESS} and \\ref{subsec:FtSumnerData}), \nwe assume an unaccounted beam/scintillator mis-alignment of $\\Delta d = 1 \n\\, \\rm{mm}$. For a highly-polarized beam of $r = O(100\\%)$ this leads to \nan error of $\\Delta r_{\\rm{sys,\\Delta d}} \\simeq 2\\%$ relative to the \nreconstructed on-axis beam (see simulated curve in \nFig.~\\ref{fig:PolFracSysErrorPolarized}). For a non-polarized beam \n(simulations in Fig.~\\ref{fig:PolFracSysError}) the offset leads to an \noverestimation of $\\simeq 2\\%$. For simplicity, we describe the \nsystematic error introduced by the beam offset as independent of the \ntrue polarization fraction: $\\Delta r_{\\rm{sys,\\Delta d}} = 2\\%$.\n\nThe X-Calibur version used during the CHESS and Ft.~Sumner measurements \n(see Secs.~\\ref{subsec:CHESS} and \\ref{subsec:FtSumnerData}) allowed for \nsome flexibility in adjusting the distance between the plane of the CZT \ndetectors and the optical axis when assembling the polarimeter (see \nFig.~\\ref{fig:X-Calibur_Configurations}, top: $c = 11 \\, \\rm{mm}$). We \nmeasured $c_{\\rm{ch}} = (11.3 \\pm 0.5) \\, \\rm{mm}$ for the CHESS setup \nand $c_{\\rm{ft}} = (10.5 \\pm 0.5) \\, \\rm{mm}$ for the Ft.~Sumner setup. \nConservatively, we assume $c = (11 \\pm 1) \\, \\rm{mm}$. To study how an \nincreased/decreased value of $c$ affects $r_{\\rm{rec}}$, we \ncorrespondingly shifted pixel coordinates when analyzing a data set, but \ndetermined $r_{\\rm{rec}}$ with the modulation factor $\\mu_{\\rm{sim}}$ \nobtained from simulations with the nominal value of $c = 11 \\, \\rm{mm}$. \nFor $\\Delta c = \\pm 1 \\, \\rm{mm}$, a $100\\%$ polarized beam is \nreconstructed to be $r_{\\rm{rec}} = r_{-0.5\\%}^{+1\\%}$ (overestimation \nfor a reduced distance $c$). For a non-polarized beam the effect is \nestimated to be $<0.1\\%$. We therefore assume that the distance related \nsystematic error\\footnote{Note, that asymmetric distances of the \ndifferent detector sides will mimic beam offsets with potentially \nstronger effects.} is proportional to $r$ with $\\Delta r_{\\rm{sys,c}} = \n0.01 \\, r_{\\rm{rec}}$. In the case of the Stokes analysis, the \nuncertainty in $c$ will introduce another systematic error: the angular \ncoverage of the detector gaps, corrected for by \nEq.~(\\ref{eqn:StokesGapCorrection}), will be over/underestimated. For a \n$100\\%$ polarized beam and $\\Delta c = -1 \\, \\rm{mm}$, we find maximal \ndeviation of $\\Delta r_{\\rm{sys,c}}^{\\rm{Stk}} = \\pm 6\\%$, and for \n$\\Delta c = +1 \\, \\rm{mm}$ we find $\\Delta r_{\\rm{sys,c}}^{\\rm{Stk}} = \n\\pm 4.5\\%$, with the sign and strength depending on the orientation of \nthe detector planes relative to the polarization vector (see \nSec.~\\ref{subsec:AnalysisPolarization}).\n\nAn error introduced by the analysis procedure can be estimated by \ncomparing the results obtained with the two analysis methods presented \nin Sec.~\\ref{subsec:AnalysisPolarization}. No significant differences \nwere found when analyzing simulated data of a non-polarized beam. Based \non the modulation factors of a $100\\%$ polarized beam, see \nEq.~(\\ref{eq:ModulationFactor}), we estimate the systematic error to be \non the order of $2\\%$. Therefore, a relative error of $\\Delta \nr_{\\rm{sys,a}} = 0.02 \\, r_{\\rm{rec}}$ is assumed. Reasons for the \ndifference can be the different treatment of dead pixels, detector gaps, \netc. (see Sec.~\\ref{subsec:AnalysisPolarization}). The total systematic \nerror on the polarization fraction is:\n\n\\begin{eqnarray}\n\\begin{split}\n\\Delta r_{\\rm{sys}} & = \\Delta r_{\\rm{sys,\\Delta d}} + \\Delta \nr_{\\rm{sys,c}} + \\left( \\Delta r_{\\rm{sys,c}}^{\\rm{Stk}} \\right) + \n\\Delta r_{\\rm{sys,a}} \\\\ & = 2\\% + 0.01 \\, r_{\\rm{rec}} + \\left(\\Delta \nr_{\\rm{sys,c}}^{\\rm{Stk}} \\right)+ 0.02 \\, r_{\\rm{rec}}.\n\\end{split}\n\\label{eqn:SystErrorPolFrac} \n\\end{eqnarray}\nThis error is estimated and valid for the particular measurements \npresented in this paper. A better alignment in future measurements, a \nrotating polarimeter, and a more detailed study of the analysis methods \nwill allow one to further reduce the error.\n\n{\\it Second-order systematic effects.} The tilt of the scintillator axis \nwith respect to the X-ray beam will introduce another (depth-dependent) \nsystematic effect. It should be mentioned that the simulations presented \nin this section assume a point-like X-ray beam, whereas the CHESS X-ray \nbeam had a square-shaped footprint with side lengths of the order of $1 \n\\, \\rm{mm}$. The systematic effects introduced by the offset/tilt might \nbe weakened by properly taking into account the point spread function of \nthe X-ray mirror that will be used for the astrophysical observations. \nThe temperature dependence of the energy calibration of the CZT \ndetectors (Sec.~\\ref{subsec:TempStudies}), if not corrected for, can \nalso modestly affect the reconstructed modulation factors of the data if \nthe measurements are performed at temperatures other than the \ncalibration temperature. However, the study of the effects discussed in \nthis paragraph are beyond the scope of this paper.\n\n\n\\subsection{Summary of the Polarimeter Performance}\n\nWe used measurements at the CHESS X-ray beam facility to calibrate the \npolarimeter. CHESS gives a highly polarized and well collimated beam \nwith well defined photon energies. We were able to demonstrate the full \nfunctionality of X-Calibur and measured the beam polarization to be \n$r_{\\rm{ch}} = (88 \\pm 5)\\%$. In addition, we used a collimated X-ray \nbeam from an X-ray source to make end-to-end tests of the full \n{\\it InFOC$\\mu$S}\\xspace/X-Calibur assembly in the field.\n\nWhen uncorrected for, the $\\sim$$3\\%$ of defect pixels and gaps between \ndetector boards produce an apparent polarization of a non-polarized beam \nof up to $5\\%$ (Stokes analysis only). After correction, the apparent \npolarization goes down to $< 2 \\%$. Once we rotate the polarimeter, we \nexpect that the systematic effect is reduced by a factor $\\sim \n\\sqrt{n}$, with $n$ being the number of detected events. The systematic \nerror will then be much smaller than the statistical error.\n\nThe detection principle of X-Calibur requires to focus the X-rays onto \nthe center of the scatterer. If uncorrected for, an offset of $1 \\, \n\\rm{mm}$ leads to an apparent polarization of $\\simeq 2\\%$ for a\nnon-polarized beam. For the upcoming balloon flight, our goal is to limit \nthe offset of the focal point from the center of the scatterer to $< 1 \n\\, \\rm{mm}$, and to monitor the offset with a backward looking camera \nlocated close to the X-ray mirror. The camera will monitor a LED cross \nhair. We can in addition use X-ray data to constrain the location of the \nfocal spot. The systematic error on the polarization fraction after \ncorrecting for the offset of the focal point is $< 1 \\%$. Thus, for the \nupcoming balloon flight, we estimate that X-Calibur can detect $\\geq \n3~\\%$ polarization fractions in the $20 - 80 \\, \\rm{keV}$ band.\n\n\n\\section{Summary and Conclusions} \\label{sec:Conclusion}\n\nWe designed, optimized, and built an X-ray polarimeter, X-Calibur, and \nstudied its performance and sensitivity. The fully assembled X-Calibur \npolarimeter was tested (i) in the laboratory at Washington University, \n(ii) with a polarized X-ray beam at the Cornell High Energy Synchrotron \nSource (CHESS), and (iii) during a X-Calibur/{\\it InFOC$\\mu$S}\\xspace flight-integration \ntest in Ft.~Sumner, NM. X-Calibur makes use of the fact that polarized \nphotons Compton scatter preferentially perpendicular to their electric \nfield orientation. It combines a detection efficiency on the order of \n$80 \\%$, with a high modulation factor of $\\mu \\approx 0.5$ averaged \nover the whole detector assembly, and with values up to $\\mu \\approx \n0.7$ for select subsections of the polarimeter. Operated in a mode of \ncontinuous rotation, X-Calibur allows for a good control over systematic \neffects. Scattering polarimetry has the strength that it can operate \nover a wide energy range. The low energy threshold is given by the \ncompetition between photoelectric absorption and scattering processes, \nthe mirror reflectivity limits the sensitivity at high energies.\n\nWe calibrated all 2048 CZT detector pixels of the polarimeter and \nstudied their performance with respect to their energy threshold, energy \nresolution (including the contribution of electronic readout noise), and \ntrigger efficiency. The CZT detectors achieve a mean energy threshold of \n$21 \\, \\rm{keV}$ and a mean $40 \\, \\rm{keV}$ energy resolution of \n$\\Delta E_{\\rm{czt}} \\approx 4 \\, \\rm{keV}$ FWHM. The \ntemperature-dependencies of the pixel responses were studied, as well, \nand we found that the effects can be sufficiently controlled for the \ntemperature range expected during a balloon flight.\n\nWe characterized the performance of the active CsI shield and find a \nbackground suppression by more than one order of magnitude in the energy \nrange relevant for X-Calibur. We also measured the trigger efficiency of \nthe scintillator that is used as Compton scatterer and find a trigger \nthreshold around $5 \\, \\rm{keV}$ with the potential for further \nreduction (using an improved amplification circuit). We used the CHESS \nX-ray beam to test the polarimeter. Detailed comparisons of experimental \nand simulated data allowed us to demonstrate the full functionality of \nthe polarimeter and to measure the polarization of the CHESS beam. We \nstudied different systematic effects that potentially affect the \nreconstructed polarization properties and estimated the systematic error \nto be smaller than $2\\%$ for an upcoming balloon flight.\n\nOur tentative observation program for an upcoming X-Calibur/{\\it InFOC$\\mu$S}\\xspace \nballoon flight includes galactic sources (Crab nebula, Her\\,X-1, \nCyg\\,X-1, GRS\\,1915, EXO\\,0331) and one extragalactic source (Mrk\\,421) \nfor which sensitive polarization measurements will be carried through.\n\nIn principle, a similar space-borne scattering polarimeter could operate \nover the broader $3-80 \\, \\rm{keV}$ energy band. Here, a LiH rod would \nbe used as passive scatterer. In contrast to the plastic scintillator \nused in the balloon-borne polarimeter, the LiH scatterer does not yield \na coincidence signal. However, the lower atomic number of LiH results in \nthe possibility of using the polarimeter down to energies of a few keV.\n\n\n\n\n\\section*{Acknowledgments}\n\nWe are grateful for NASA funding from grants NNX10AJ56G, NNX12AD51G and \nNNX14AD19G, as well as discretionary funding from the McDonnell Center \nfor the Space Sciences to build the X-Calibur polarimeter. Polarization \nmeasurements: This work is based upon research conducted at the Cornell \nHigh Energy Synchrotron Source (CHESS) which is supported by the \nNational Science Foundation and the National Institutes of \nHealth/National Institute of General Medical Sciences under NSF award \nDMR-0936384. We would like to thank Ken Finkelstein for the excellent \nsupport in setting up the experiment at CHESS and for the continuous \ndiscussions thereafter.\n\n\n"}}},{"rowIdx":83599,"cells":{"uuid":{"kind":"number","value":1116691499414,"string":"1,116,691,499,414"},"dataset":{"kind":"string","value":"arxiv"},"text":{"kind":"string","value":"\\section{Introduction}\n\tA region $\\Omega \\subset \\mathbb C$, the boundary of which is a polygonal curve with angles multiple of $\\pi/2$, is considered in this article. This region models the shape of a channel under a dam. The calculation of fluid flow in such channel boils down to a conformal mapping problem of $\\Omega$ onto the upper half-plane. The solution of such problems is given by Christoffel-Schwartz integral (see, e.g., \\cite{ShabatEng} or \\cite{SchwartzChristoffel}), which in this case is naturally defined on an elliptic Riemann surface. In this paper the simple formula, which writes integral in terms of Weierstrass sigma function (see e.g., \\cite{AkhiezerEng} or \\cite{Chandra}), was found. This approach allows to avoid numerical integration and the mapping parameters can be found from a simple nonlinear system of equations; therefore, the computation is significantly simplified.\n\t\n\tSimilar problems have been considered in \\cite{SigBog}, \\cite{DamBogEng}, \\cite{BogHept}, \\cite{BogGrig}, and \\cite{BogCond}, where Christoffel-Schwartz integral was efficiently represented by theta functions (see, e.g., \\cite{FarkasKra}). In the paper \\cite{BogCond} the application of Lauricella functions to these problems was studied, and also different approaches were compared.\n\t\n\tThe main advantage of Weierstrass functions over theta functions is that they have limiting values when the surface degenerates. The paper analyzes behavior of the constructed conformal mapping under the condition that the dam width tends to zero. It turns out that conformal mappings have a limit, which is a solution to the limiting problem. Thus, it is shown that the solution is stable under the considered degeneration.\n\t\n\tThe described property of the Weierstrass sigma function loses its value if the standard method is used for calculations, which expresses the sigma function in terms of the theta function (since it does not withstand degeneration). Thus, it is necessary to use an independent method for calculating sigma functions. In this paper, we use the expression for the coefficients of its expansion into Taylor series obtained by Weierstrass (see \\cite{Weier}). Since the proof presented there is apparently not complete (at one point, the analyticity of the sigma function in three variables in a neighborhood of zero is used, which is not obvious), we give a more detailed proof in the appendix. The above formula, however, is not sufficient for the final numerical solution, since Taylor series are not suitable for calculations with large arguments (namely, such a need arises with degeneration). Thus, the problem of constructing an efficient computational method for the sigma function independent of theta functions still remains unsolved. If such a method is available, it will be possible to construct formulas that are stable under various degenerations and use them in calculations. The results of this work illustrate the need in such methods.\n\t\n\tProblems in which hyperelliptic Riemann surfaces of higher genus arise can also be solved using the theory of sigma functions developed by Klein and Baker in \\cite{Klein} and \\cite{Baker} respectively (more detailed exposition can be found in \\cite{Buch}). There is hope that it will be possible to prove the stability of formulas expressing the solution of the above problems in terms of high-order sigma functions. Thus, the construction of Weierstrass-type recurrent formulas (which are known for genus 1 and 2; see \\cite{Buch}) and calculation methods for sigma functions can be extremely useful in applied problems.\n\t\n\t\\section{The statement and the origin of the problem}\n\tConsider region $\\Omega$ in the complex plane pictured on Figure 1. \n\n\n\n\n\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\draw[black,thick] (-5,-1) -- (5,-1);\n\t\t\t\\draw[black,thick] (-5,2) -- (0,2) node [pos=1,below left] {$w_4$};\n\t\t\t\\draw[black,thick] (0,2) -- (0,0) node [pos=1,left] {$w_3$};\n\t\t\t\\draw[black,thick] (0,0) -- (1,0) node [pos=1,below right] {$w_2$};\n\t\t\t\\draw[black,thick] (1,0) -- (1,1) node [pos=1,above] {$w_1$};\n\t\t\t\\draw[black,thick] (1,1) -- (5,1);\n\t\t\t\\draw[<->,black,dashed,thick] (0,-0.2) -- (1,-0.2) node [pos=0.5,below] {$\\delta$};\n\t\t\t\\draw[<->,black,dashed,thick] (1.2,0) -- (1.2,1) node [pos=0.5,right] {$h$};\n\t\t\t\\draw[<->,black,dashed,thick] (-4,-1) -- (-4,2) node [pos=0.5,right] {$h^-$};\n\t\t\t\\draw[<->,black,dashed,thick] (4,-1) -- (4,1) node [pos=0.5,right] {$h^+$};\n\t\t\\end{tikzpicture}\n\t\t\n\t\t\\caption{Region $\\Omega$.}\n\t\\end{figure}\n\tIt is bounded from below by a line and from above by a polygonal curve with four vertices $w_1,w_2,w_3,w_4$ (it is convenient to think that this region also has two vertices at $\\pm \\infty$). Let the line, which bounds the region from below be parallel to the real axis, and let the vertex $w_4$ be at the origin. Then, this region is determined by four real parameters $h^-,h^+,h,\\delta$, where $h$ is equal to the length of segment $[w_1,w_2]$, $\\delta$ is the length of $[w_2,w_3]$, and $h^-$ and $h^+$ are equal to the distance from the line that bounds the region from below to $w_4$ and $w_1$ respectively. These parameters are positive and satisfy inequalities $h^- - h^+ + h > 0$, which corresponds to positivity of the length of $[w_3,w_4]$, and $h < h^+$. The region is determined uniquely by these parameters.\n\n\tRegions similar to $\\Omega$ arise in problems connected with the computation of fluid flow through the porous material under a dam. Since the flow is continuous and satisfies the Darcy's law, the pressure $p$ is a harmonic function in $\\Omega$. Assuming that segments $[w_1, w_2]$, $[w_2,w_3]$, $[w_3, w_4]$, and channel's bottom are impenetrable, we obtain natural boundary conditions: the normal derivative $\\partial p/\\partial n$ vanishes on impenetrable segments of the boundary, while on the remaining segments (i.e. on the half-lines starting from $w_1$ and $w_4$) $p$ is locally constant.\n\n\tConsider a real-valued function $q$ in the region $\\Omega$ such that $f = p + iq$ is holomorphic (such function exists because $\\Omega$ is simply connected). The normal derivative of $p$ vanishing condition is easily equivalent to constancy of $q$ on the corresponding boundary segment. It follows, that, if $f$ is a function that conformally maps $\\Omega$ onto a rectangle in a way, such that $w_1$, $w_4$, and vertices at infinity are mapped to the vertices of a rectangle, then $p = \\RE f$ is a solution to the original problem. $q = \\IM f$ is called the current function. Its level lines are the streamlines of the fluid under the dam. It is clear, that it is enough to solve the problem of conformal mapping of $\\Omega$ onto upper half-plane $\\mathbb C_+$ in order to solve the specified problem.\n\n\tIn what follows, the conformal mapping problem will be solved explicitly using the tools of Weierstrass elliptic functions. Below we show the calculation of streamlines in $\\Omega$ obtained using the method constructed in this work.\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{subfigure}{.5\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.9\\textwidth]{LinesColor2.pdf}\n\t\t\\end{subfigure}%\n\t\t\\begin{subfigure}{.5\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.9\\textwidth]{LinesColorCloser2.pdf}\n\t\t\\end{subfigure}\n\t\t\n\t\t\\caption{Streamlines in the region $\\Omega$.}\n\t\\end{figure}\n\t\\FloatBarrier\n\t\\section{The solution of the conformal mapping problem}\n\t\\subsection{The general form of the solution and parameter determination}\n\tSince $\\Omega$ is simply connected, there is a conformal mapping $W:\\mathbb C_+ \\rightarrow \\Omega$, where $\\mathbb C_+ = \\{z \\in \\mathbb C: \\IM z > 0\\}$ is the upper half-plane (see, e.g., \\cite{KartanEng} or \\cite{ShabatEng}). Using, if necessary, a suitable automorphism of $\\mathbb C_+$, one can make it such that the point $w_4$ is the preimage under $W$ (more precisely, under its continuation to the boundary) of $\\infty$. Then, by Christoffel-Schwartz theorem (see \\cite{SchwartzChristoffel}), there exist $x^-<x^+<x_1<x_2<x_3 \\in \\mathbb R$ and $C \\in \\mathbb C$ such that\n\t\\begin{equation}\n\t\tdW = \\phi = C \\frac{\\sqrt{(x - x_2)(x-x_3)}}{(x-x^-)(x-x^+)\\sqrt{x-x_1}}dx.\n\t\\end{equation}\n\t\\begin{remark}\n\t\tHere $x_i$ is the preimage of $w_i$ under $W$, and $x^-$ and $x^+$ are the points on the boundary of the upper half-plane in which $W$ goes to infinity (preimages of the vertices at infinity).\n\t\\end{remark}\n\n\tThe differential form $\\phi$ can be considered on the hyperelliptic Riemann surface $V$ of genus $1$, defined by equation $y^2 = F(x) = 4(x - x_1)(x-x_2)(x-x_3)$. Using the shift of the upper half-plane we can set $x_1 + x_2 + x_3 = 0$ without loss of generality. Thus, $F(x) = 4x^3 - g_2x - g_3$ for some real $g_2,g_3$ (that are determined by $x_1,x_2,x_3$). We can rewrite $\\phi$ on this surface in the form\n\t\\begin{equation}\n\t\t\\phi = 2C \\frac{(x-x_2)(x-x_3)}{y(x-x^-)(x-x^+)}dx.\n\t\\end{equation}\n\t\n\tLet us fix the branch of $\\sqrt{F(x)}$ in the region that is made from $\\mathbb C$ by throwing off segment $[x_1,x_2]$ and half-line $[x_3,\\infty]$. Let this branch have positive values as the argument tends to half-line $(x_3, \\infty)$ from the upper half-plane. Recalling that $dx/y$ is a holomorphic (everywhere non-zero) form on $V$, we obtain that $\\phi$ has two zeros of multiplicity $2$ at $(x_2,0)$ and $(x_3,0)$ and also four simple poles at $(x^-,\\pm \\sqrt{F(x^-)})$ and $(x^+,\\pm\\sqrt{F(x^+)})$. Note that the residues of this form at these poles are equal to $\\pm h^-/\\pi$ and $\\mp h^+/\\pi$ respectively.\n\n\tNow we shall use Abel map (see, e.g., \\cite{ForsterEng}) which identifies $V$ with $\\Jac(V)$ (as usual, we set the infinity as the initial point and $dx/y$ as the basis of holomorphic forms). Let us introduce the half-periods $$\n\t\\omega = \\int_{x_1}^{x_2} \\frac{dx}{y},\\;\\; \\omega' = -\\int_{x_2}^{x_3} \\frac{dx}{y},\n\t$$ and quantities $\\eta = \\zeta(\\omega)$ and $\\eta' = \\zeta(\\omega')$, where $\\zeta$ is the Weierstrass zeta function (see \\cite{AkhiezerEng}).\n\tIt is easy to see that $\\omega,\\eta \\in \\mathbb R$ and $\\omega', \\eta' \\in i\\mathbb R$. The set of points $(x,\\sqrt{F(x)})$, where $x \\in \\mathbb C_+$, is mapped by this map onto the rectangle with vertices $0,\\omega', \\omega' - \\omega, -\\omega$. Let us denote the images of the points $(x^-, \\sqrt{F(x^-)})$ and $(x^+,\\sqrt{F(x^+)})$ by $z^-$ and $z^+$ respectively (see Figure 3, where the preimages of points are indicated in the brackets).\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale = 2]\n\t\t\t\\draw[black,thick] (0,0) -- (0,2)node [pos=0,right] {$0\\;(\\infty)$};\n\t\t\t\\draw[black,thick] (0,2) -- (-3,2)node [pos=0,right] {$\\omega'\\;(x_1)$};\n\t\t\t\\draw[black,thick] (-3,2) -- (-3,0)node [pos=0,left] {$\\omega' - \\omega\\;(x_2)$};\n\t\t\t\\draw[black,thick] (-3,0) -- (0,0) node [pos=0,left] {$-\\omega\\;(x_3)$};\n\t\t\t\\filldraw [black] (0,0.7) circle (0.5pt) node[right] {$z^-$};\n\t\t\t\\filldraw [black] (0,1.2) circle (0.5pt) node[right] {$z^+$};\n\t\t\\end{tikzpicture}\n\t\t\n\t\t\\caption{Image of the upper half-plane under the Abel map.}\n\t\\end{figure}\n\tThe images of $(x^-, -\\sqrt{F(x^-)})$ and $(x^+,-\\sqrt{F(x^+)})$ in this case are equal to $-z^-$ and $-z^+$. Consider the differential form $\\psi$ on the torus that corresponds to $\\phi$ under this identification of $V$ with $\\Jac(V)$. This form has $4$ simple poles in the points $\\pm z^-$ and $\\pm z^+$ and its residues are equal to $\\pm h^-/\\pi$ and $\\mp h^+/\\pi$ respectively.\n\n\tNow we use the method of representing elliptic functions by Weierstrass functions that is described in \\cite{AkhiezerEng}. Consider meromorphic function\n\t\\begin{equation}\\label{eq413}\n\t\tg(z) = \\frac{h^-}{\\pi}(\\zeta(z - z^-) - \\zeta(z + z^-)) - \\frac{h^+}{\\pi}(\\zeta(z-z^+) - \\zeta(z+z^+)).\n\t\\end{equation}\n\tUsing the quasiperiodic properties of $\\zeta$ (see \\cite{AkhiezerEng}) it is easy to conclude that $g$ is elliptic. Form $g(z)dz$ has the same simple poles as $\\psi$ with the same residues. Therefore, $\\psi - g(z)dz$ is a holomorphic form on the torus. Since the space of holomorphic $1$-forms on the torus is one-dimensional it follows that $\\psi - g(z)dz = Ddz$, where $D$ is a constant (note that $D \\in i\\mathbb R$).\n\t\n\tNow we return to the map $W$. It is clear that $$\n\tW(x) = -\\int_{x}^\\infty \\phi.\n\t$$ In view of that, let \\begin{equation}\n\t\tQ(z) = \\int_0^z \\psi.\n\t\\end{equation}\n\tObviously, $W(x)$ is equal to $Q(z)$, where $z$ is the image of $(x,\\sqrt{F(x)})$ under the Abel map. Thus, $Q$ conformally maps the rectangle with vertices $0,\\omega', \\omega' - \\omega, -\\omega$ onto $\\Omega$, and $\\omega'$ is mapped to $w_1$, $\\omega'-\\omega$ is mapped to $w_2$, and $-\\omega$ to $w_3$ (also $0$ is mapped to $w_4$). Now we can derive the system of equations from the previously obtained relations:\n\t\\begin{equation}\\label{eq415}\n\t\tg(-\\omega) + D = 0,\\;\\;\\; g(\\omega' - \\omega) + D = 0,\n\t\\end{equation}\n\t\\begin{equation}\\label{eq416}\n\t\tQ(\\omega'-\\omega) - Q(\\omega') = -ih,\\;\\;\\; Q(-\\omega) - Q(\\omega'-\\omega) = -\\delta.\n\t\\end{equation}\n\t\\begin{remark}\n\t\tThe first pair of equations follows from the fact that $\\phi$ has zeros in the points $(x_2,0)$ and $(x_3,0)$, and the second pair is a consequence of relations $w_3-w_2 = -\\delta$, $w_2 - w_1 = -ih$.\n\t\\end{remark}\n\tIt remains to derive a reasonable formula for $Q$. Recall that $\\zeta$ is a logarithmic derivative of $\\sigma$. It easily follows that\n\t\\begin{equation}\\label{eq417}\n\t\tQ(z) = Dz + \\frac{h^-}{\\pi}\\ln\\left(\\frac{\\sigma(z-z^-)}{\\sigma(z+z^-)}\\right) - \\frac{h^+}{\\pi}\\ln\\left(\\frac{\\sigma(z-z^+)}{\\sigma(z+z^+)}\\right) - i(h^- - h^+),\n\t\\end{equation}\n\twhere $\\ln$ denotes the branch of logarithm in the plane cut by negative imaginary half-line such that $\\ln(1) = 0$. Substituting in \\eqref{eq415} the formula for $g$ from \\eqref{eq413} and using quasiperiodicity of $\\sigma$ (see, e.g., \\cite{AkhiezerEng}), we obtain the system of equations:\n\\begin{equation}\\label{eq418}\n\t\\begin{dcases}\n\t\t-D\\omega - \\frac{2h^+}{\\pi}\\eta z^+ + \\frac{2h^-}{\\pi}\\eta z^- = -ih, \\\\\n\t\t-D\\omega' - \\frac{2h^+}{\\pi}\\eta'z^+ + \\frac{2h^-}{\\pi}\\eta'z^- = -\\delta, \\\\\n\t\tD + \\frac{h^-}{\\pi}(\\zeta(\\omega - z^-) - \\zeta(\\omega + z^-)) - \\frac{h^+}{\\pi}(\\zeta(\\omega-z^+) - \\zeta(\\omega+z^+)) = 0, \\\\\n\t\tD + \\frac{h^-}{\\pi}(\\zeta(\\omega' + \\omega - z^-) - \\zeta(\\omega' + \\omega + z^-)) \\\\ \\qquad-\\frac{h^+}{\\pi}(\\zeta(\\omega' + \\omega-z^+) - \\zeta(\\omega' + \\omega+z^+)) = 0.\n\t\\end{dcases}\n\\end{equation}\n\nIn this system of equations there are five variables (since the quantities $\\omega,\\omega',\\eta,\\eta'$ are determined by $g_2$ and $g_3$) $g_2,g_3,D,z^+,z^-$ (the first two are real and the other are imaginary) and four equations \\eqref{eq418} (the first, third and fourth equations are imaginary and the second one is real). Thus, it is natural to consider a single parameter family of curves that necessarily contains a suitable one, i.e. to consider functions $g_2 = g_2(\\gamma)$ and $g_3 = g_3(\\gamma)$ and use the system \\eqref{eq418} to determine the parameters $\\gamma, D, z^+, z^-$.\n\nIn what follows we shall use the family of curves that is defined by the roots of the polynomial $F$: $x_1 = \\gamma - 1/2$, $x_2 = -2\\gamma$, $x_3 = \\gamma+1/2$, $\\gamma \\in (-1/6,1/6)$ (more detailed analysis of this family is given during the study of the degeneration $\\delta \\rightarrow 0$). This family corresponds to the normalization condition $x_3 - x_1 = 1$ in addition to the already given relation $x_1 + x_2 + x_3 = 0$.\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{Figure_2.pdf}\n\t\t\n\t\t\\caption{ The rectangle $P$ and contours in it.}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{Figure_1.pdf}\n\t\t\n\t\t\\caption{The image of the rectangle and the contours.}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{Figure_1alt.pdf}\n\t\t\n\t\t\\caption{The behaviour near $[w_2,w_3]$.}\n\t\\end{subfigure}%\n\t\n\t\\caption{The conformal mapping $Q$.}\n\\end{figure}\n\n\\subsection{On the numerical implementation}\n\n\n\nIt was decided to use for numerical implementation the explicit computation of the sigma function depending on parameters $g_2,g_3$ through its Taylor series (see \\cite{Weier} or Theorem~\\ref{tWeierSeries}). It is clear that for the effective solution of the system \\eqref{eq418} it is necessary to compute all the quantities in it and their derivatives with respect to parameters. In the end it reduces to the computation of $\\omega$ and $\\omega'$ and their derivatives with respect to $g_2$ and $g_3$ and, also, $\\zeta$ and its derivatives with respect to $z, g_2,g_3$. Since $$\n\\zeta = \\frac{1}{\\sigma} \\frac{\\partial \\sigma}{\\partial z} ,\n$$\nthe problem of computation of $\\zeta$ and its derivatives can be solved easily. In order to compute $\\omega$ we note that $\\sigma$ has zeros exactly in the points of the lattice $\\{2m\\omega + 2n\\omega':n,m \\in \\mathbb Z\\}$ and these zeros are simple. An effective way to localize a simple zero $z_0$ of a holomorphic function $f$ is to compute integral of $zf'(z)/2\\pi if(z)$ on a contour enclosing $z_0$. To find a suitable contour it is possible to apply a variant of binary search using that $\\omega \\ge \\pi/2$. Using the specified method either directly, or for an approximate calculation of zero and subsequent application of equation solving methods, it is easy to construct an effective and precise algorithm of computation of $\\omega$ (and $\\omega'$). In order to compute their derivatives it is possible to differentiate the integral of $z\\sigma'(z)/\\sigma(z)$ by $g_2$ or $g_3$, and to compute it explicitly by determining the residue in the zero of $\\sigma$. Thus, the solution of the system \\eqref{eq418} can be completely reduced to the computation of the sigma function and its derivatives with respect to $z$, $g_2$, and $g_3$.\n\n\n\nWe demonstrate the solution of a specific problem by this method. Let $h^+ = \\pi$, $h^- = \\pi + 0.5$, $h = 0.5$, $\\delta = 0.2$. We shall search for the solution in the one parameter family of curves defined by $x_1 = \\gamma - 1/2$, $x_2 = -2\\gamma$, $x_3 = \\gamma + 1/2$, $\\gamma \\in (-1/6, 1/6)$. The solution of the system \\eqref{eq418} is $$(\\gamma,D,z^+,z^-) = (0.1051616134,0.0203152915i,1.3043479103i,0.7195735824i).$$ Given this $\\gamma$ we obtain $\\omega = 1.6518996331$, $\\omega' = 2.2939120295i$. On Figure 4 the image of the rectangle $P$ with vertices $0,\\omega',\\omega'-\\omega,-\\omega$ under the map $Q$ is shown.\n\n\n\\section{Stability of the solution under the degeneration of the region}\nHere we shall consider a problem of conformal mapping of the upper half-plane onto the region $\\widetilde{\\Omega}$ \nthat comes from $\\Omega$ with degeneration $\\delta\\rightarrow 0$ (see Figure 5) and analyse behaviour of the solution under the condition that no other degeneration is happening (i.e. quantities $h^-,h^+,h,h^-+h-h^+,h^+-h$ have positive limits).\n\n\\begin{figure}[h!]\n\t\\centering\n\n\t\\begin{tikzpicture}\n\t\t\\draw[black,thick] (-5,-1) -- (5,-1);\n\t\t\\draw[black,thick] (-5,2) -- (0,2) node [pos=1,below left] {$w_4$};\n\t\t\\draw[black,thick] (0,2) -- (0,0) node [pos=1,below] {$w_2= w_3$};\n\t\t\\draw[black,thick] (0,1) -- (5,1) node [pos=0,left] {$w_1$};\n\t\t\\draw[<->,black,dashed,thick] (0.2,0) -- (0.2,1) node [pos=0.5,right] {$h$};\n\t\t\\draw[<->,black,dashed,thick] (-4,-1) -- (-4,2) node [pos=0.5,right] {$h^-$};\n\t\t\\draw[<->,black,dashed,thick] (4,-1) -- (4,1) node [pos=0.5,right] {$h^+$};\n\t\\end{tikzpicture}\n\t\\caption{Region $\\widetilde \\Omega$.}\n\\end{figure}\n$\\widetilde{\\Omega}$ is determined by three parameters\n$h,h^+$ and $h^-$. A conformal mapping of the upper half-plane onto $\\widetilde\\Omega$ can be found by the analogous method (using Christoffel-Schwartz theorem). In this case, since the corresponding Riemann surface has genus $0$, the solution can be expressed in elementary functions. Another method (that is considered here) is to apply formula \\eqref{eq417}, using the fact that $\\sigma$ is defined for $g_2$ and $g_3$ such that $F(x) = 4x^3 - g_2x - g_3$ has multiple roots. It is natural to suppose that the solution can be found by taking the limit under gluing the roots, that are mapped to $w_2$ and $w_3$. Together with that the stability of the solution under $\\delta\\rightarrow 0$ will be proved.\n\n\\subsection{Gluing of the roots}\nAgain consider a family of curves depending on $\\gamma \\in (-1/6, 1/6)$ that is given by $F_\\gamma(x) = 4(x - x_1(\\gamma))(x - x_2(\\gamma))(x - x_3(\\gamma))$, where $x_1(\\gamma) = \\gamma - 1/2$, $x_2(\\gamma) = -2\\gamma$, $x_3(\\gamma) = \\gamma + 1/2$. Under $\\gamma \\rightarrow -1/6$ the roots $x_2$ and $x_3$ glue. The limiting values of $g_2$ and $g_3$ are $4/3$ and $-8/27$ respectively. For each $\\gamma$ we define quantities $\\omega(\\gamma)$, $\\omega'(\\gamma)$, $\\eta(\\gamma)$, $\\eta'(\\gamma)$. In what follows we shall omit dependence on $\\gamma$.\n\n\\begin{lemma}\\label{l41}\n\tUnder $\\gamma \\rightarrow -1/6$ we have\n\t\\begin{equation}\\label{eq441}\n\t\t\\omega,\\eta \\rightarrow \\infty,\\;\\; \\omega' \\rightarrow \\frac{i\\pi}{2},\\;\\;\\eta' \\rightarrow -\\frac{i\\pi}{6},\\;\\; \\frac{\\eta}{\\omega} \\rightarrow -\\frac{1}{3}.\n\t\\end{equation}\n\tMoreover,\n\t\\begin{equation}\\label{eq442}\n\t\t\\begin{gathered}\n\t\t\\sigma(z,\\frac{4}{3}, -\\frac{8}{27}) = e^{-\\frac{z^2}{6}} \\sinh(z),\\;\\; \\zeta(z,\\frac{4}{3}, -\\frac{8}{27}) = \\coth(z) - \\frac{z}{3},\\\\ \\wp(z,\\frac{4}{3}, -\\frac{8}{27}) = \\frac{1}{\\sinh^2(z)} + \\frac{1}{3}.\n\t\t\\end{gathered}\n\t\\end{equation}\n\tFinally, there exists $\\varepsilon > 0$ such that for $\\gamma + 1/6 < \\varepsilon$ the estimation\n\t\\begin{equation}\\label{eq44asympt}\n\t\t-c_1\\ln(\\gamma + 1/6) \\le \\omega(\\gamma) \\le -c_2\\ln(\\gamma + 1/6)\n\t\\end{equation}\n\tholds, where $0 < c_1 < c_2$.\n\\end{lemma}\n\\begin{proof}\n\t\\eqref{eq441} easily follows from integral representations\n\t$$\n\t\\omega(\\gamma) = \\frac{1}{2}\\int_{\\gamma - \\frac{1}{2}}^{-2\\gamma} \\frac{dx}{\\sqrt{(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}}, $$ $$\n\t\\omega'(\\gamma) = \\frac{1}{2}\\int_{-2\\gamma}^{\\gamma+\\frac{1}{2}} \\frac{dx}{\\sqrt{-(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}}, $$ $$\n\t\\eta(\\gamma) = -\\frac{1}{2}\\int_{\\gamma - \\frac{1}{2}}^{-2\\gamma} \\frac{xdx}{\\sqrt{(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}},\n\t$$ $$\n\t\\eta'(\\gamma) = -\\frac{1}{2}\\int_{-2\\gamma}^{\\gamma + \\frac{1}{2}} \\frac{xdx}{\\sqrt{-(x - \\gamma - 1/2)(x - \\gamma + 1/2)(x + 2\\gamma)}}. $$\n\tTo derive \\eqref{eq442} one can pass to the limit $\\gamma \\rightarrow -1/6$ using the formula that represents $\\sigma$ as the infinite product (see \\cite{AkhiezerEng}). We obtain\n\t$$\n\t\\sigma(z,\\frac{4}{3}, -\\frac{8}{27}) = z\\prod_{n \\ne 0} \\left(1 - \\frac{z}{i n \\pi}\\right) e^{\\frac{z}{i n \\pi} - \\frac{z^2}{2 n^2 \\pi^2}}.\n\t$$\n\tUsing classical identities\n\t$$\n\t\\sum_{n = 1}^\\infty \\frac{1}{n^2} = \\frac{\\pi^2}{6},\\;\\; \\prod_{n = 1}^\\infty \\left(1 - \\frac{x^2}{n^2 \\pi^2}\\right) = \\frac{\\sin(x)}{x},\n\t$$\n\twe derive the first equation of \\eqref{eq442}. The rest follow from equalities $\\zeta(z) = \\sigma'(z)/\\sigma(z)$, $\\wp(z) = -\\zeta'(z)$.\n\t\n\tNow we estimate the growth of $\\omega(\\gamma)$. Consider equality\n\t$$\n\t\\omega(\\gamma) = \\frac{1}{2}\\int_0^{1/2 - 3\\gamma} \\frac{dt}{\\sqrt{t(1-t)(1/2 - 3\\gamma - t)}}.\n\t$$\n\tNote that for small $\\gamma$ integral on the segment $[0,1/2]$ is bounded and, therefore,\n\t$$\n\t\\omega(\\gamma) \\sim \\frac{1}{2}\\int_{1/2}^{1/2 - 3\\gamma} \\frac{dt}{\\sqrt{t(1-t)(1/2 - 3\\gamma - t)}}.\n\t$$\n\tThe estimation of the integral in rhs is quite simple since $1/\\sqrt{t}$ is bounded from below and above by positive constants and the remaining integral can be calculated explicitly.\n\\end{proof}\n\nSince $\\omega' \\rightarrow i\\pi/2$ and $\\omega \\rightarrow \\infty$, it is natural to suppose that \\eqref{eq417} can give a formula for a conformal mapping of the half-strip $$S = \\{z \\in \\mathbb C: \\RE z < 0, \\IM z \\in (0,\\pi/2)\\}$$ onto $\\widetilde\\Omega$. Let\n\\begin{equation}\\label{eq443}\n\t\\widetilde Q(z) = Dz + \\frac{h^-}{\\pi}\\ln\\left(\\frac{\\sigma(z-z^-)}{\\sigma(z+z^-)}\\right) - \\frac{h^+}{\\pi}\\ln\\left(\\frac{\\sigma(z-z^+)}{\\sigma(z+z^+)}\\right) - i(h^- - h^+),\n\\end{equation}\nwhere $\\sigma$ is taken at values $g_2 = 4/3$, $g_3 = -8/27$ and $D \\in \\mathbb C$, $z^-, z^+\\in(0,i\\pi/2)$ are the parameters. Substituting \\eqref{eq442} into \\eqref{eq443}, we obtain\n\\begin{equation}\\label{eq444}\n\t\\begin{multlined}\n\t\\widetilde Q(z) = z\\left(D + \\frac{2h^-z^-}{3\\pi} - \\frac{2h^+z^+}{3\\pi}\\right) \\\\+ \\frac{h^-}{\\pi}\\ln\\frac{\\sinh(z - z^-)}{\\sinh(z+z^-)} - \\frac{h^+}{\\pi}\\ln\\frac{\\sinh(z - z^+)}{\\sinh(z + z^+)}- i(h^- - h^+).\n\t\\end{multlined}\n\\end{equation}\nIt is easy to check that if $\\widetilde Q$ has a non-zero linear term it has no limit under $\\RE z \\rightarrow -\\infty$. If$$\nD + \\frac{2h^-z^-}{3\\pi} - \\frac{2h^+z^+}{3\\pi} = 0,\n$$ its limit is equal to $2(h^-z^- - h^+z^+)/\\pi - i(h^- - h^+)$. Thus, if $\\widetilde Q$ conformally maps $S$ onto $\\widetilde \\Omega$, then the conditions\n\\begin{equation}\\label{eq445}\n\t\\begin{dcases}\n\t\tD + \\frac{2h^-z^-}{3\\pi} - \\frac{2h^+z^+}{3\\pi} = 0, \\\\\n\t\th^- z^- - h^+ z^+ = -\\frac{ih\\pi}{2}\n\t\\end{dcases}\n\\end{equation}\nhold. Obviously, these conditions are sufficient under the additional assumption that the derivative of $\\widetilde{Q}$ is not vanishing on $S$ and its boundary. Using tedious but elementary calculations it can be shown that this condition is equivalent to \n\\begin{equation}\\label{eq446}\n\th^-\\sinh(2z^-) = h^+\\sinh(2z^+).\n\\end{equation}\nThus, \\eqref{eq445} and \\eqref{eq446} determine the parameters $D,z^-,z^+$, at which $\\widetilde Q$ is the desired conformal mapping.\n \nWe show that the obtained formula reduces to Christoffel-Schwartz integral in the upper half-plane under the variable change $x = \\wp(z) = 1/\\sinh^2(z) + 1/3$. Changing the variable in \\eqref{eq443} and using that the linear term vanishes we obtain the following formula for the conformal mapping of $\\mathbb C_+$ onto $\\widetilde{\\Omega}$:\n \\begin{equation}\\label{eq448}\n\t\\widetilde{W}(x) = \\frac{h^-}{\\pi}\\ln \\left(\\frac{\\sqrt{x^- + 2/3} - \\sqrt{x + 2/3}} {\\sqrt{x^- + 2/3} + \\sqrt{x + 2/3}}\\right) + \\frac{h^+}{\\pi}\\ln \\left(\\frac{\\sqrt{x^+ + 2/3} - \\sqrt{x + 2/3}} {\\sqrt{x^+ + 2/3} + \\sqrt{x + 2/3}}\\right).\n\\end{equation}\n \n The equations on the parameters $z^-$ and $z^+$ can be rewritten in the form\n \\begin{equation}\\label{eq449}\n \t\\begin{dcases}\n \t\t\\frac{h^-\\sqrt{x^- + 1}}{x^-} = \\frac{h^+\\sqrt{x^++1}}{x^+}, \\\\\n \t\th^- \\ln\\left(\\frac{1+\\sqrt{x^- + 4/3}}{\\sqrt{x^- + 1/3}}\\right) + h^+\\ln\\left(\\frac{1+\\sqrt{x^+ + 4/3}}{\\sqrt{x^+ + 1/3}}\\right) = -\\frac{ih\\pi}{2}.\n \t\\end{dcases}\n \\end{equation}\nDifferentiating $\\widetilde{W}$ and using the first equation from \\eqref{eq449} we obtain $$\nd\\widetilde{W} = \\frac{h^-\\sqrt{x^-+1}- h^+\\sqrt{x^++1}}{\\pi} \\frac{(x - 1/3)dx}{\\sqrt{x+2/3}(x-x^-)(x-x^+)}.\n$$\nTherefore we found the exact form of the constant in Christoffel-Schwartz integral. The remaining parameters $x^-$ and $x^+$ can be found from the system of equations \\eqref{eq449}.\n\n\\subsection{Passing to the limit}\n\nConsider a sequence of regions determined by the parameters $(h^-_n, h^+_n, h_n, \\delta_n)$. Assume that they have limits $(h^-_{\\lim}, h^+_{\\lim}, h_{\\lim}, 0)$. Also we assume that $h^+_{\\lim} - h_{\\lim} > 0$ and $h^-_{\\lim} - h^+_{\\lim} + h_{\\lim} > 0$. We shall prove that the parameters $(D_n, \\gamma_n, z^-_n, z^+_n)$, given by the solution of the system \\eqref{eq418} for the corresponding regions, have limits $(D_{\\lim}, -1/6, z^-_{\\lim}, z^+_{\\lim})$, and, moreover, parameters $(D_{\\lim}, z^-_{\\lim}, z^+_{\\lim})$ satisfy \\eqref{eq445} and \\eqref{eq446}. Thus, in view of the fact that the Weierstrass sigma function is entire, it follows that the constructed solution is stable.\n\nThe following proof is rather long and technical and, therefore, we shall omit most of the calculations.\n\nIn the following estimations we shall also use parameters $x^-_n, x^+_n, x_1^{(n)}, x_2^{(n)}, x_3^{(n)}, C_n$ of the map $W_n$.\n\n\\begin{lemma}\n\tAssume that $\\gamma_n \\rightarrow -1/6$ and $\\delta_n \\omega(\\gamma_n) \\rightarrow 0$.\n\tThen the indicated convergence holds.\n\\end{lemma}\n\\begin{proof}\n\tNote that $\\gamma_n \\rightarrow -1/6$ implies that sequences $z^-_n$ and $z^+_n$ are bounded. The first equation in \\eqref{eq418} then implies that $D_n$ is also bounded. Passing to the subsequences we can assume that all these sequences are convergent (if we succeed to prove that the limits satisfy \\eqref{eq445} and \\eqref{eq446}, then uniqueness of the solution implies that all the subsequences converge to the same limit, and, therefore, the initial sequences converge). The first equation in \\eqref{eq445} is obtained by passing to the limit in the second equation in \\eqref{eq418} in view of Lemma \\ref{l41}. Multiplying the first equation in \\eqref{eq418} by $\\omega'$ and the second one by $\\omega$, subtracting and passing to limit leads to the second equation in \\eqref{eq445} (the term $\\delta \\omega$ by assumption tends to zero). \n\t\n\tNow we derive \\eqref{eq446} for the limits of the sequences. Recall the following notation of the theory of elliptic functions (see \\cite{AkhiezerEng}):\n\t\t$$\n\t\\zeta_2(z) = \\zeta(z + \\omega) - \\eta,\n\t$$\n\t$$\n\t\\zeta_3(z) = \\zeta(z + \\omega + \\omega') + \\eta + \\eta'.\n\t$$\n\tThese functions are connected to $\\sigma_2, \\sigma_3$:\n\t$$\n\t\\zeta_k = \\frac{1}{\\sigma_k} \\frac{d \\sigma_k}{d z} = \\frac{d \\ln \\sigma_k}{d z}.\n\t$$\n\tFinally, $\\sigma_k = \\sigma \\sqrt{\\wp - x_k}$. The last two equations in \\eqref{eq418} can be rewritten as\t\n\t\\begin{equation}\\label{eq447}\n\t\tD + \\frac{h^-}{\\pi}(\\zeta_k(-z^-) - \\zeta(z^-)) - \\frac{h^+}{\\pi}(\\zeta_k(-z^+) - \\zeta(z^+)) = 0, \\;\\; k = 2,3.\n\t\\end{equation}\n\n\tSince\n\t$$\n\t\\begin{multlined}\n\t\t\\zeta_2(z) - \\zeta_3(z) = \\frac{\\sigma'(z) \\sqrt{\\wp - x_2} + \\wp'(z) \\sigma(z) (\\wp - x_2)^{-1/2}}{\\sigma(z) \\sqrt{\\wp - x_2}} \\\\ - \\frac{\\sigma'(z) \\sqrt{\\wp - x_3} + \\wp'(z) \\sigma(z) (\\wp - x_3)^{-1/2}}{\\sigma(z) \\sqrt{\\wp - x_3}},\n\t\\end{multlined}\n\t$$\n\tit follows that\n\t\\begin{equation}\\label{zetadiff}\n\t\t\\zeta_2(z) - \\zeta_3(z) = \\frac{\\wp'(z)(x_2 - x_3)}{(\\wp - x_2)(\\wp - x_3)}.\n\t\\end{equation}\n\tEquation \\eqref{eq446} is derived by passing to the limit (using Lemma \\ref{l41}) from equations \\eqref{eq447} (from which the constant $D$ can be eliminated) and substitution the formula for $\\zeta_2 - \\zeta_3$ from \\eqref{zetadiff}.\n\\end{proof}\n\\begin{lemma}\n\tInequalities $|C_n| \\ge a_1$, $|C_n| \\le a_2 \\sqrt{x_3^{(n)} - x^-_n}$ hold for some positive constants $a_1,a_2$. Moreover, the sequence $x^+_n$ is bounded from below.\n\\end{lemma}\n\\begin{proof}\n\tThe estimations for $C^{(n)}$ easily follow from\n\t$$|C_n|\\bigintss_{x_3^{(n)}}^{+\\infty} \\frac{\\sqrt{\\left(x - x_2^{(n)}\\right)\\left(x - x_3^{(n)}\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h^-_n - h^+_n + h_n.\n\t$$\n\tTo prove the boundedness from below for $x^+_n$ it suffices to consider equality\n\t$$\n\t|C_n|\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)}-x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h_n.\n\t$$\n\\end{proof}\n\\begin{lemma}\n\tAssume that sequence $x^-_n$ is bounded from below. Then there exist constants $0 < b_1 < b_2$ such that $b_1 \\sqrt{\\delta_n} \\le |x_2^{(n)} - x_3^{(n)}| \\le b_2 \\sqrt{\\delta_n}$. \n \n\\end{lemma}\n\\begin{proof}\n\tIt follows from easy estimation for the integral in equality\n\t$$\n\t|C_n|\\bigintss_{x_2^{(n)}}^{x_3^{(n)}} \\frac{\\sqrt{\\left(x - x_2^{(n)}\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = \\delta_n.\n\t$$\n\\end{proof}\n\nThe foregoing Lemmas imply that it remains to prove that sequence $x^-_n$ is bounded from below (in view of the asymptotics \\eqref{eq44asympt}). Assume that this is not true. Passing to the subsequences we can assume that $x^-_n \\rightarrow -\\infty$ and $x_1^{(n)},x_2^{(n)},x_3^{(n)}$, $x^+_n$ are convergent.\n\n\\begin{lemma}\n\tIn the foregoing assumptions statements $x_3^{(n)} - x_2^{(n)} \\rightarrow 0$, $x_1^{(n)} - x^+_n \\rightarrow 0$ hold.\n\\end{lemma}\n\\begin{proof}\n\tEquality\n\t$$\n\t|C_n| \\frac{\\sqrt{\\left(x_2^{(n)} - x^+_n\\right)\\left(x_3^{(n)} - x^+_n\\right)}}{\\sqrt{x_1^{(n)} - x^+_n}(x^+_n - x^-_n)} = \\frac{h^+_n}{\\pi}\n\t$$\n\timplies $x_1^{(n)} - x^+_n \\rightarrow 0$. \n\t\n\tTo prove that $x_3^{(n)} - x_2^{(n)} \\rightarrow 0$ we return to parameters $(D_n, \\gamma_n, z^-_n, z^+_n)$. Assume that $x_2^{(n)} - x_1^{(n)} \\rightarrow 0$. Then $\\omega'(\\gamma_n), \\eta'(\\gamma_n) \\rightarrow \\infty$, and $\\omega$ and $\\eta$ have finite limits. Moreover, Legendre identity (see, e.g., \\cite{AkhiezerEng} or \\cite{Chandra}) implies\n\t$$\n\t\\lim_{n \\rightarrow \\infty} \\frac{\\omega'(\\gamma_n)}{\\eta'(\\gamma_n)} = \\lim_{n \\rightarrow \\infty} \\frac{\\omega(\\gamma_n)}{\\eta(\\gamma_n)}.\n\t$$\n\tThe first two equations from \\eqref{eq418} imply\n\t$$\n\t\\lim_{n \\rightarrow \\infty}\\left(\\frac{2 h^+_n}{\\pi} z^+_n \\left(\\frac{\\omega'}{\\eta'} - \\frac{\\omega}{\\eta}\\right) - i \\frac{h_n}{\\omega}\\right) = 0.\n\t$$\n\tTherefore, \n\t$$\n\t\\lim_{n \\rightarrow \\infty}\\left(\\frac{h^+_n z^+_n}{\\omega'} - h_n\\right) = 0,\n\t$$\n\tand, passing to the limit, we obtain $h_{\\lim} \\ge h^+_{\\lim}$. This contradicts the assumptions made.\n\t\n\tNow assume that $x_2^{(n)} - x_1^{(n)} \\nrightarrow 0$ and$x_3^{(n)} - x_2^{(n)} \\nrightarrow 0$. Then both periods $\\omega$ and $\\omega'$ have finite limits. In this case $z^-_n \\rightarrow 0$, $z^+_n \\rightarrow \\omega'(\\gamma_{\\lim})$. It is obvious that $D_n$ also is convergent and, passing to the limit in the second equation in \\eqref{eq418}, we obtain\n\t$$\n\t-D_{\\lim} \\omega' - \\frac{2h^+_{\\lim} \\omega' \\eta'}{\\pi} = 0.\n\t$$\n\tSubstituting into the first equation we get\n\t$$\n\t\\frac{2h^+_{\\lim}}{\\pi} \\omega \\eta' - \\frac{2h^+_{\\lim}}{\\pi} \\omega' \\eta = -ih_{\\lim},\n\t$$\n\timplying $h^+_{\\lim} = h_{\\lim}$.\n\\end{proof}\n\nNow we have enough preparation to deduce a contradiction from $x^-_n \\rightarrow -\\infty$. In order to do this we shall analyse asymptotics of some sequences (in what follows the equivalence of sequences means that the quotient of them tends to $1$).\n\nEquality\n$$\n|C_n| \\frac{\\sqrt{\\left(x_2^{(n)} - x^-_n\\right)\\left(x_3^{(n)} - x^-_n\\right)}}{\\sqrt{x_1^{(n)} - x^-_n}(x^+_n - x^-_n)} = \\frac{h^-_n}{\\pi}\n$$\nimplies that\n\\begin{equation}\\label{asympt1}\n\t|C_n| \\sim \\frac{h^-_n}{\\pi} \\sqrt{|x^-_n|}.\n\\end{equation}\nOn the other hand\n$$\n|C_n| \\frac{\\sqrt{\\left(x_2^{(n)} - x^+_n\\right)\\left(x_3^{(n)} - x^+_n\\right)}}{\\sqrt{x_1^{(n)} - x^+_n}(x^+_n - x^-_n)} = \\frac{h^+_n}{\\pi},\n$$\nand, therefore,\n\\begin{equation}\\label{asympt2}\n\t\\sqrt{x_1^{(n)} - x^+_n} \\sim \\frac{h^-_n}{h^+_n} \\frac{1}{\\sqrt{|x^-_n|}}.\n\\end{equation}\nNow consider equality\n$$\n|C_n|\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} = h_n.\n$$\nUsing \\eqref{asympt1}, it is easy to show that the sequence in lhs is equivalent to sequence$$\n\\frac{h^-_n}{\\pi \\sqrt{|x^-_n|}} \\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^+_n)}.\n$$\nNow, changing the variable, we obtain\n$$\n\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^+_n)} = \\bigintss_{0}^{x_2^{(n)} - x_1^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x_1^{(n)} - x\\right)(1 - x)}dx}{\\sqrt{x}(x + x_1^{(n)} - x^+_n)}.\n$$\nIt appears that the asymptotics of the last integral is independent of the convergence rate of $|x_2^{(n)} - x_1^{(n)}| \\rightarrow 1$. Namely, for all sequences $\\alpha_n \\rightarrow 1$ and $a_n \\rightarrow 0$ the equivalence\n$$\n\\int_0^{\\alpha_n} \\frac{\\sqrt{(\\alpha_n - x)(1 - x)}dx}{\\sqrt{x}(x + a_n)} \\sim \\int_0^1 \\frac{(1-x)dx}{\\sqrt{x}(x+a_n)} \\sim \\frac{\\pi}{\\sqrt{a_n}}\n$$\nholds. Finally, in view of \\eqref{asympt2}, we obtain\n$$\nh_n = |C_n|\\bigintss_{x_1^{(n)}}^{x_2^{(n)}} \\frac{\\sqrt{\\left(x_2^{(n)} - x\\right)\\left(x_3^{(n)} - x\\right)}dx}{\\sqrt{x - x_1^{(n)}}(x - x^-_n)(x - x^+_n)} \\sim \\frac{h^-_n}{\\pi \\sqrt{|x^-_n|}} \\frac{\\pi}{\\sqrt{x_1^{(n)} - x^+_n}} \\sim h^+_n.\n$$\nIt contradicts the assumption $h_{\\lim} < h^+_{\\lim}$.\n\n\\section{Conclusion}\nThe simple expression through the Weierstrass sigma function for a conformal mapping of a polygonal region $\\Omega$ was obtained. For the specific example the numerical experiment was carried out. The behaviour under degeneration was analyzed and it was shown that the formula is stable and converges to the solution of the limiting problem.\n\nThe future research direction can be connected either with the construction and analysis of the solutions to similar problems corresponding, for example, to Riemann surfaces of genus $2$, or with the development of the sigma function theory: construction of the recurrent formulas for higher genus and elaboration of computational methods independent of the theta function theory.\n\n\\section{Acknowledgements}\n\n\nThe author expresses his gratitude to A. Bogatyrev and O. Grigoriev for posing the problem and useful discussions, and also to K. Malkov for help in the computer implementation of the calculations. The author also thanks the Center for Continuing Professional Education ``Sirius University'' for the invitation to the educational module ``Computational Technologies, Multidimensional Data Analysis, and Modelling'', during which some of the results of this work were obtained.\n\n\\begin{appendices}\n\\section{On the coefficients of the Weierstrass sigma function Taylor series}\n\tHere we prove that the sigma function is an entire function of three variables and derive a recurrent formula for its Taylor series coefficients, that was originally established by Weierstrass in \\cite{Weier}. The proof given there has a gap connected to the analyticity of the sigma function in a neighbourhood of zero. Perhaps, this fact can be proved by an independent argument but, since Weierstrass does not give any references (and omits this issue completely), we decided to provide a complete proof here.\n\n\tThe homogeneity condition\n\t$$\n\t\\sigma(\\frac{z}{\\lambda}, \\lambda^4 g_2, \\lambda^6 g_3) = \\frac{1}{\\lambda} \\sigma(z,g_2,g_3)\n\t$$\n\teasily implies the following differential equation for the $\\sigma$ function:\n\t\\begin{equation}\\label{eq321}\n\t\tz \\frac{\\partial \\sigma}{\\partial z} - 4g_2\\frac{\\partial \\sigma}{\\partial g_2} - 6g_3 \\frac{\\partial \\sigma}{\\partial g_3} - \\sigma = 0.\n\t\\end{equation}\n\tFurther, using the definition of the $\\sigma$ function and the standard differential equation for the $\\wp$ function, one can derive an equation (for a proof see \\cite{Halphen})\n\t\\begin{equation}\\label{eq322}\n\t\t\\frac{\\partial^2 \\sigma}{\\partial z^2} - 12g_3 \\frac{\\partial \\sigma}{\\partial g_2} - \\frac{2}{3} g_2^2 \\frac{\\partial \\sigma}{\\partial g_3} + \\frac{1}{12} g_2 z^2 \\sigma = 0.\n\t\\end{equation}\n\t\n\tLet $f$ be an entire function of three variables $(z,g_2,g_3)$ satisfying \\eqref{eq321} and \\eqref{eq322}. We derive a relation between the Taylor series coefficients $f_{mnk}$ of $f$:\n\t$$\n\tf = \\sum_{m,n,k = 0}^\\infty f_{mnk} g_2^m g_3^n z^k.\n\t$$\n\t\\eqref{eq321} implies that $f_{mnk} = 0$, if $k \\ne 4m + 6n + 1$. Therefore, $f$ can be written in the form\n\t$$\n\tf = \\sum_{m,n = 0}^\\infty a_{mn} g_2^m g_3^n z^{4m + 6n + 1}.\n\t$$\n\tNow, substituting this expression of $f$ into \\eqref{eq322}, we obtain the equality\n\t\\begin{equation}\\label{eq323}\n\t\ta_{mn} = \\frac{12(m+1)a_{m+1,n-1} + \\frac{2}{3}(n+1)a_{m-2,n+1} - \\frac{1}{12}a_{m-1,n}}{(4m + 6n + 1)(4m + 6n)},\n\t\\end{equation}\n\tin which for convenience $a_{mn}$ is defined by zero when $m$ or $n$ is negative. It is easy to see that \\eqref{eq323} uniquely determines sequence $a_{mn}$ for given $a_{00}$. To prove this let us introduce an order relation on pairs of nonnegative integers $(m,n)$: $(m,n) \\le (m',n')$, if $m+n < m'+n'$ or if $m+n = m'+n'$ and $n \\le n'$.\n\tIt is easy to see that we defined a well-order on $\\mathbb Z_+ \\times \\mathbb Z_+$, and in \\eqref{eq323} indices of the terms $a_{mn}$ in rhs are strictly less then $(m,n)$. Thus, it is proved that \\eqref{eq323} determines $a_{mn}$ recursively for given $a_{00}$.\n\t\n\tIf the sigma function was an entire function of three variables, or, at least, holomorphic in some neighbourhood of zero, then the recurrence relation \\eqref{eq323} for its Taylor series coefficients would be proved. The difficulty is that the domain of $\\sigma$ is the set $\\{(z,g_2,g_3) \\in \\mathbb C^3: g_2^3 - 27 g_3^2 \\ne 0\\}$. The following considerations prove the entirety of $\\sigma$ and the recurrence relation \\eqref{eq323}.\n\t\n\t\\begin{remark}\n\t\tIt is known (see, e.g., \\cite{AkhiezerEng} or \\cite{Chandra}) that condition $g_2^3 - 27 g_3^2 \\ne 0$ is equivalent to simplicity of the roots of polynomial $4x^3 - g_2 x - g_3$.\n\t\\end{remark}\n\t\\begin{lemma}\\label{l32}\n\t\tLet $a_{mn}$ satisfy the recurrence relation \\eqref{eq323}. Then for all $q > (28 + \\sqrt{811})/36 \\approx 1.569$ there exists $C > 0$ such that \n\t\t\\begin{equation}\\label{eq324}\n\t\t\t|a_{mn}| \\le C\\frac{q^{2m + 3n}}{(2m+3n)!}.\n\t\t\\end{equation}\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tSubstituting in \\eqref{eq323} this estimation and it is easy to show that for the existence of a constant, it suffices that the inequality\n\t\t$$\n\t\t \\frac{6(m+1)}{4m+6n+1}\\frac{q^{2m+3n-1}}{(2m+3n)!} + \\frac{(n+1)q^{2m+3n-1}}{6(4m+6n+1)(2m+3n)!} + \\frac{q^{2m+3n-2}}{48(2m+3n)!} \\le \\frac{q^{2m+3n}}{(2m+3n)!}\n\t\t$$\n\t\tholds starting from some index $(m,n)$ (in terms of the foregoing ordering). For this, in turn, it suffices to satisfy the inequality\n\t\t$$\n\t\t\\frac{1}{48} + q\\left(\\frac{3}{2} + \\frac{1}{18}\\right) < q^2.\n\t\t$$\n\t\tSolving the quadratic equation, we obtain the required statement.\n\t\\end{proof}\n\t\n\tLemma \\ref{l32} allows to define an entire function\n\t$$h(z,g_2,g_3) = \\sum_{m,n = 0}^\\infty a_{mn}g_2^m g_3^n z^{4m+6n+1},$$\n\twhere $a_{mn}$ are determined by recurrence relation \\eqref{eq323} and initial condition $a_{00} = 1$. We shall prove that $h \\equiv \\sigma$ for $(g_2,g_3)$ such that $g_2^3 - 27 g_3^2 \\ne 0$.\n\t\n\t\\begin{lemma}\n\t\tLet $f$ be a holomorphic function of variables $(z,g_2,g_3)$, defined on a set $\\mathbb C \\times U$, where $U \\subset \\mathbb C^2$ is open, satisfying equation \\eqref{eq322}. Assume that $f$ is odd in variable $z$. Then $f$ can be represented by series \\begin{equation}\\label{eq325}\n\t\t\tf(z,g_2,g_3) = \\sum_{n = 0}^\\infty c_n(g_2,g_3) z^{2n+1},\n\t\t\\end{equation}\n\t\tand in $U$ the recurrence relation\n\t\t\\begin{equation}\\label{eq326}\n\t\t\t(2n+3)(2n+2)c_{n+1} - 12g_3\\frac{\\partial c_n}{\\partial g_2} - \\frac{2}{3} g_2^2 \\frac{\\partial c_n}{\\partial g_3} + \\frac{1}{12}g_2 c_{n-1},\n\t\t\\end{equation}\n\t\tholds, where $n \\ge 0$ (for $n = 0$ we set $c_{n-1} = 0$).\n\t\\end{lemma} \n\t\\begin{proof}\n\t\tIndeed the representability of $f$ by the series follows from entirety of $f$ by $z$. Its coefficients $c_n(g_2,g_3)$ are given by \n\t\t$$\n\t\tc_n(g_2,g_3) = \\frac{1}{(2n+1)!} \\frac{\\partial^{2n+1} f}{\\partial z^{2n+1}}|_{z = 0}.\n\t\t$$\n\t\tIt is easy to see that the series \\eqref{eq325} can be differentiated term-by-term, and therefore we can substitute it in \\eqref{eq322}. Collecting the coefficient at $z^{2n+1}$, we obtain \\eqref{eq326}.\n\t\\end{proof}\n\t\n\tRecurrence relation \\eqref{eq326} can be used to prove, that $\\sigma$ and $h$ coincide on the domain of the $\\sigma$ function. Indeed, if the first terms of their expansions coincide, then these function coincide (note that they are both odd in $z$). Indeed, $\\partial \\sigma/\\partial z |_{z = 0} \\equiv \\partial h/\\partial z |_{z = 0} \\equiv 1$. Thus, $h$ is the analytic continuation of the $\\sigma$ function to an entire function of variables $(z,g_2,g_3)$. This completes the proof of the following theorem.\n\t\\begin{theorem}[Weierstrass]\\label{tWeierSeries}\n\t\tThe $\\sigma$ function is entire and for all $(z,g_2,g_3) \\in \\mathbb C^3$ equality\n\t\t\\begin{equation}\\label{eq327}\n\t\t\t\\sigma(z,g_2,g_3) = \\sum_{m,n = 0}^\\infty a_{mn}g_2^m g_3^n z^{4m+6n+1}\n\t\t\\end{equation}\n\t\tholds, where coefficients $a_{mn}$ are determined by recurrence relation \\eqref{eq323} and initial condition $a_{00} = 1$.\n\t\\end{theorem}\n\\end{appendices}\n\\printbibliography\n\\end{document}\n"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":835,"numItemsPerPage":100,"numTotalItems":87268,"offset":83500,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjI1NTA1NSwianRpIjoiNzIxYmY3MTctN2VjZC00ZDRhLTkxOGYtODZiOWUyOGY3NjUyIiwic3ViIjoiL2RhdGFzZXRzL0NvZmVBSS9OYW5vRGF0YSIsImV4cCI6MTc4NjI1ODY1NSwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.1Y84d_hjzKjeEEZ19AxVNzcc1P59uEdP2bnpkecfBB_OMJb_3MYU7HKsORUS-gE4zBnMQPdbtmPcfr82x5-IDg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":87268,"numBytesParquet":2292871178}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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1,116,691,499,315 | arxiv | \section{Introduction}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{aaai2022figs/annotation.pdf}
\caption{Top left: original image. Bottom left: fully supervised annotation. Top right: scribble annoation. Bottom right: point annotation.}
\label{annotation}
\end{figure}
Salient obj... |
1,116,691,499,316 | arxiv | \section{Introduction}
Question classification (QC) deals with question analysis and question labeling based on the expected answer type. The goal of QC is to assign classes accurately to the questions based on expected answer. In modern system, there are two types of questions \cite{islam2016word}. One is Factoid que... |
1,116,691,499,317 | arxiv | \section{Introduction}
In a standard scenario, big bang nucleosynthesis (BBN) can produce only light elements, up to $^{7}$Li, and
all heavy elements have been synthesized in stars.
However, many phase transitions in the early universe could have printed their trace in a non-standard way.
For example, some baryoge... |
1,116,691,499,318 | arxiv | \section{Introduction}\label{sec:overview}
\IEEEPARstart{I}{ntelligent} reflecting surface (IRS), aka reconfigurable intelligent surface (RIS), is an emerging wireless network device that aims to improve wireless environment by manipulating signal reflections \cite{Wu2018gc,Bjoernson2022,Xu2021}. Owing to its much low... |
1,116,691,499,319 | arxiv | \section{Introduction}
A recent remarkable trend in the physics community is its engagement
in interdisciplinary fields using physics-inspired techniques. Econophysics
is one such area in which financial markets are simulated by agent-based
models in much the same way as other many-body systems in statistical
physics.... |
1,116,691,499,320 | arxiv | \section{Introduction}
The most mature systems for deductive verification of randomized algorithms are
\emph{expectation-based} techniques; seminal examples include \textsc{PPDL}\xspace
\cite{Kozen85} and \textsc{pGCL}\xspace \cite{Morgan96}. These approaches reason about
\emph{expectations}, functions $E$ from states... |
1,116,691,499,321 | arxiv | \section{Introduction}
Accretion-powered Classical T Tauri stars (CTTSs) are young low-mass stars which
often show signs of a strong magnetic field (e.g., \citealt{basr92,john99})
which is expected to have a complex structure (e.g., \citealt{john07}).
The Zeeman-Doppler imaging technique has proven very successful in... |
1,116,691,499,322 | arxiv | \section{Introduction}
The incorporation of individual quantum systems into solid-state platforms,\cite{MichlerScience00, WallraffNature04, KoppensNature06, PlaNature12, BrannyNatCom17, EvansScience18} their coherent control, and interfacing them with external degrees of freedom\cite{PettaScience05, NowackScience07, Lo... |
1,116,691,499,323 | arxiv | \section{Introduction}
High-velocity wind-like outflows are an important part of the quasar system and a potential contributor to feedback to the host galaxy. Many properties of quasar outflows are still poorly understood, including their location and three-dimensional structure. One way to obtain constraints on quasa... |
1,116,691,499,324 | arxiv | \section{Introduction}
\textit{Tropical geometry} is a combinatorial shadow of algebraic geometry.
We take a tropical approach to problems on algebraic classes of cohomology groups (such as the Hodge conjecture, the Tate conjecture, and the standard conjectures).
In this paper,
inspired by Liu's construction of \texti... |
1,116,691,499,325 | arxiv | \section{Introduction}
In recent decades, there has been much interest in modelling and analysing the many networks which appear in the real world, in contexts such as the world wide web or online social networks. This work has drawn heavily on the mathematical study of random graphs, a subject with its origins in th... |
1,116,691,499,326 | arxiv |
\section{Introduction}
Mass is the most crucial input in stellar internal structure modelling. It predominantly influences the luminosity of a star at a given stage of its evolution, and also its lifetime. The knowledge of the mass of stars in a non-interacting binary system, together with the assumption tha... |
1,116,691,499,327 | arxiv | \section{Introduction}
A variety of systems such as supercooled liquids, colloids,
granular matter and foams, exhibit a transition from a flowing
fluid phase to a frozen solid phase. Jamming due to spatial
constraints imposed on the elementary constituents of these
materials has been proposed as a possible common caus... |
1,116,691,499,328 | arxiv | \section{Introduction}
Radial abundance gradients in the galactic disk and their time
variations are among the main constraints of chemical evolution
models for the Milky Way. These gradients can be determined from
a variety of objects, such as HII regions, cepheid variables,
open clusters and planetary nebulae (PN). ... |
1,116,691,499,329 | arxiv | \section{The Problem}
It has been pointed out some time ago \cite{CARTER}
that within the KM ansatz large {\bf CP}~asymmetries have to arise
in B decays involving $B^0 - \bar B^0$ oscillations.
In particular the channel $B_d \rightarrow \psi K_S$ combines
a striking experimental signature with a clean theoretical ... |
1,116,691,499,330 | arxiv | \section{Introduction}\label{sec:Intro}
The discovery of the Standard Model (SM) Higgs boson in the LHC experiment was one of the most important contributions to our understanding of particle physics in recent years~\cite{ATLAS:2012yve,CMS:2012qbp}.
Although the SM is almost complete with this discovery, various astro... |
1,116,691,499,331 | arxiv | \section{Introduction}
\label{s1}
The density of states (DOS) of disordered {\em interacting} electronic systems
exhibits singularity near the Fermi energy.
This phenomenon manifests itself in the suppression of the tunneling conductance
at small external bias and is referred to as zero bias anomaly (ZBA).
It ha... |
1,116,691,499,332 | arxiv | \section{Introduction}
In optical communication, a sender encodes a message in an optical signal and sends it to a receiver who detectes the signal to decode the message \cite{g.cariolaro}. Thus, the success probability of the optical communication is determined by the physical and statistical properties of the optica... |
1,116,691,499,333 | arxiv | \section{Introduction}
\label{sec:introduction}
Cognitive scientists use computational models to understand how humans achieve cognition and how the brain gives rise to the mind.
Cognitive models simulate mental processes by describing how inputs (e.g., visual or auditory stimuli to the brain) are processed by a set o... |
1,116,691,499,334 | arxiv | \section{Introduction}
The dark energy and dark matter problems are the most important unsolved issues in cosmology.
Although many models of dark energy and dark matter have been proposed, none of them are conclusive at the present time.
Phenomenologically, the cold dark matter with a cosmological constant ($\Lambda$CD... |
1,116,691,499,335 | arxiv | \section{Introduction}
Algebras of dihedral, semi-dihedral and quaternion types arises naturally from Erdmann's classical book \cite{13} as a product of classification of tame blocks. Hochschild cohomology on them are well-studied by large amount of mathematicians, such as A. I. Generalov (see \cite{8}, \cite{15}, \cit... |
1,116,691,499,336 | arxiv | \section{Introduction}
Soft gamma-ray repeaters (SGRs, for a recent review see
\cite{woodsrew}) are a small group of peculiar high-energy sources
generally interpreted as ``magnetars'', i.e. strongly magnetised
($B\sim$10$^{15}$ G), slowly rotating ($P\sim$ 5-8 s) neutron
stars powered by the decay of the magnetic ene... |
1,116,691,499,337 | arxiv | \section{Introduction}\label{sec: intro}
Higher category theory is a subject that is currently receiving a lot of interest, with strong links not only with algebraic topology (where we can trace its origins), but with logic, computer science, foundations of mathematics, mathematical physics, general systems' theory an... |
1,116,691,499,338 | arxiv | \section{Introduction}
Many specialists connect the further development of information protection facilities with the application of multiple-valued function of logical algebra (MVFLA), in particular, with the use of pseudorandom sequences (PRS) over GF($q$) ($q>2$), which possess a wider scope of unique features, if... |
1,116,691,499,339 | arxiv | \section{Introduction}
It is well-known that the concept of a measurement in quantum physics challenges our everyday intuition.
In a classical theory objects have properties, whether we look at them or not, and a measurement simply reveals to us their pre-existing values.
In quantum mechanics, on the other hand, perfo... |
1,116,691,499,340 | arxiv | \section{Introduction}
In a recent paper
\cite{Panin:2010fk}
we
defined motivic versions of symplectically oriented cohomology theories $A$ and of
quaternionic Grassmannians $HGr(r,n)$. This $HGr(r,n)$ is the open subscheme of the ordinary
Grassmannian $Gr(2r,2n)$ parametrizing subspaces on which the standard symplec... |
1,116,691,499,341 | arxiv | \section{Introduction \label{s1}}\rm
Throughout this paper $\K$ is an arbitrary field, $\Z_+$ is the
set of non-negative integers and $\N$ is the set of positive
integers. For a set $X$, $\K\langle X\rangle$ stands for the free
associative algebra over $\K$ generated by $X$. We deal with
{\it quadratic algebras}... |
1,116,691,499,342 | arxiv | \section{#1}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\topmargin-20pt
\textwidth520pt
\textheight650pt
\oddsidemargin-25pt
\evensidemargin-25pt
\defW\times H{W\times H}
\def\emph{et al.\ }{\emph{et al.\ }}
\def1/\kern-2pt\sqrt3{1/\kern-2pt\sqrt3}
\def\phantom.\hskip30pt{\phantom.\hskip30pt}
\deft{t... |
1,116,691,499,343 | arxiv | \section{#1}
\setcounter{equation}{0}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{... |
1,116,691,499,344 | arxiv | \section{Introduction}
The shell model of the nucleus has remained its most credible microscopic description through
more than seven decades now. Testing the model across the nuclear chart and refining the inputs,
towards accomplishing better overlap with data, has been an agenda of nuclear structure
studies through... |
1,116,691,499,345 | arxiv | \section{Introduction}
\label{sec:intro}
Gases in astrophysics are commonly multiphase, that is, phases with vastly different temperatures exist co-spatially.
We know, for instance, that the interstellar medium is kept in a stable at three phase state due to thermal feedback processes \citep{McKee1977}. More quiescent... |
1,116,691,499,346 | arxiv | \section{Introduction}
\label{introduction}
Synchronized behavior arising among the constituents of an ensemble is common in
nature. Examples include the synchronized flashing of fireflies, the
blossoming of flowers, cardiac cells giving rise to the pacemaker role of the
sinoatrial node of the heart and the electrical ... |
1,116,691,499,347 | arxiv | \section{Introduction}
Cuprate thin films are a promising avenue for exploring issues of dimensionality, disorder, and free carrier density in these strongly correlated materials\cite{ahn,Konstantinovic2000859,ahn1999, PhysRevB.51.3257}. Mechanical exfoliation has proven to be an effective way of producing a variety of... |
1,116,691,499,348 | arxiv | \section{Introduction}
The study of variational problems with Euclidean symmetry is an old problem, indeed, Euler's 1744 study of elastic beams
is such a case. However, methods to analyse such problems efficiently and effectively, are still of interest.
In this paper, we consider variational problems for curves in... |
1,116,691,499,349 | arxiv | \section{Introduction}
Control of magnetic states by optical excitations in
magnetically ordered materials
has attracted considerable attention
since the demonstration of ultrafast demagnetization
in Ni within 1 ps, explored by time-resolved
magneto-optical Kerr effect studies \cite{Beaurepaire1996}.
Ultra... |
1,116,691,499,350 | arxiv |
\chapter{Background}
\section{Scalability in general-purpose processors}
Assuming continued demand for computers where \emph{new functions can
be defined by the end-user by writing and using their own software}
(cf. \cref{sn:gpdemand}), the question remains of how to make
general-purpose processors that are both f... |
1,116,691,499,351 | arxiv | \section{Introduction}
Imagine you are driving a car on a highway and suddenly an object appears in your visual field, crossing the road. Your initial reaction is to slam on the brakes even before recognizing the object. This highlights a core difference between human vision and current machine learning strategies for ... |
1,116,691,499,352 | arxiv |
\section{Introduction}
\label{sec:intro}
Mobility-on-Demand (MoD) systems have become a fixture in urban transportation networks, with the popularity of ride-hailing services operated by transport network companies (TNCs) such as Uber and Lyft. The popularity of such services has also stimulated the deployment of rid... |
1,116,691,499,353 | arxiv | \section{Introduction}
We consider finite, undirected graphs without loops and multiple edges. For a graph $G$, $V(G)$ and $E(G)$ denotes the vertex-set and the edge-set respectively. A $(p,q)$ graph $G$ is a graph such that $|V(G)|=p$ and $|E(G)|=q$. We refer the reader to \cite{WA} and \cite{WE} for all other terms ... |
1,116,691,499,354 | arxiv | \section{Introduction}
Power electronic loads have found a wider application in power system networks especially after their advancement in the late 1900s. Many loads require essential power electronic converters for stage conversion. Some common examples of power electronic loads include uninterrupted power supply (UP... |
1,116,691,499,355 | arxiv | \section{Introduction}
The study of topological defects has wide applicability in many areas of
physics. In the cosmological arena, defects have been put forward as a possible
mechanism for structure formation~\cite{VS}, and while recent work on global
defects~\cite{ABR,PST} indicates that these were probably not resp... |
1,116,691,499,356 | arxiv | \section*{Abstract}
This manuscript aims to develop and describe gain scheduling control concept for a gas turbine engine which drives a variable pitch propeller. An architecture for gain-scheduling control is developed that controls the turboshaft engine for large thrust commands in stable fashion with good performa... |
1,116,691,499,357 | arxiv |
\section{Introduction}
Conversational Recommender Systems (CRS) aim to make recommendations by learning users' preferences through interactive conversations~\cite{cikm18-saur,wsdm20-ear,nips18-redial}. CRS has become one of the trending research topics for recommender systems and is gaining increasing attention, due t... |
1,116,691,499,358 | arxiv | \section{Introduction} %
\label{introduction}
Type III solar radio bursts are caused by semi-relativistic electrons
streaming through and perturbing the ambient coronal or interplanetary plasma.
A recent review is given by \citet{Reid14}.
The dominant theory, proposed by \citet{Ginzburg58}, invokes a two-step
proce... |
1,116,691,499,359 | arxiv | \section{Introduction}
The central engine of $\gamma$-ray bursts (GRBs, see, \citealp[e.g.][]{kumarandzhang2015} for a review) is
hidden to direct observation. However, its workings may be imprinted in observational signals
of this phenomenon. \cite{bromberg2012,bromberg2013} proposed that the prompt $\gamma$-ray dur... |
1,116,691,499,360 | arxiv | \section{Introduction}
There has been much interest in high-temperature (unconventional) superconductivity (SC) over more than last two decades.
Moreover, the phase separations phenomenon involving superconducting states (SS) is a very current topic after it has been evidenced in a broad range of intensely investigate... |
1,116,691,499,361 | arxiv | \section*{Proof}
Define $f\colon \mathbb{R}^2 \mapsto \mathbb{R}^k$ and $K: \mathbb{R}^2 \mapsto \mathbb{R}^{k\times l}$, where $k$ is the number of the input channels and $l$ is the number of the output channel. The convolution can be expressed by
$(K\ast f)(\vec{v})= \sum_{\vec{w}}f(\vec{v}+\vec{w})K(\vec{w})$, w... |
1,116,691,499,362 | arxiv | \section{\large{S\MakeLowercase{upplemental} M\MakeLowercase{aterials for:} \\
K\MakeLowercase{ondo} C\MakeLowercase{orrelation} I\MakeLowercase{nduced} L\MakeLowercase{ow-}F\MakeLowercase{ield} M\MakeLowercase{agnetoresistance} A\MakeLowercase{nomalies in} I\MakeLowercase{n}S\MakeLowercase{b} N\MakeLowercase{anowire} ... |
1,116,691,499,363 | arxiv | \section{Introduction}
\label{intro}
Consider an undirected graph G=(V, E) with $v=\{1, 2,... , n\}$, where specific vertex 1 is root vertex and edge set is $E=\{(i,j): i,j\in V,i<j\}$, that each edge has an associated cost $c_{ij}\geq 0$. Also, we define a positive integer $H$ to show the maximum allowance distance of... |
1,116,691,499,364 | arxiv | \section{Introduction}\label{sec:1}
We are concerned with numerical methods for Zakai equations,
linear stochastic partial differential equations of the form
\begin{equation}
\label{eq:1.1}
\begin{split}
du(t,x) = L_0u(t,x)dt + \sum_{k=1}^mL_ku(s,x)dW_k(t),
\quad 0\le t\le T,
\end{split}
\end{equation}
wi... |
1,116,691,499,365 | arxiv | \section{Introduction}\label{Intro}
We recall that, in the \textit{Riemannian case}, \textit{harmonic maps} are the critical points of the {\em energy functional}
\begin{equation}\label{energia}
E(\varphi)=\frac{1}{2}\int_{M}\,\|d\varphi\|^2\,dV \, ,
\end{equation}
where $\varphi:M\to N$ is a smooth map between two Rie... |
1,116,691,499,366 | arxiv | \section{Introduction and Related Work}
\label{Sec:Intro}
Dictionary learning is a useful procedure by which dependencies among input features can be represented in terms of suitable bases\cite{aharon2006ksvd, chi2013petrels,DictionaryLearningSurvey,elad2006image,mairal2010online,zou2006sparse,shen2008sparse,lee201... |
1,116,691,499,367 | arxiv | \section{Introduction}
The continual gravitational collapse of a massive
matter cloud within the framework of general relativity was
investigated for the first time by the
classic works of Oppenheimer and Snyder, and Datt (OSD)
\cite{OSD}.
Such a treatment of dynamical collapse would be
essential to determine ... |
1,116,691,499,368 | arxiv | \@startsection{subsubsection}{3}{10pt}{-1.25ex plus -1ex minus -.1ex}{0ex plus 0ex}{\normalsize\bf}{\@startsection{subsubsection}{3}{10pt}{-1.25ex plus -1ex minus -.1ex}{0ex plus 0ex}{\normalsize\bf}}
\def\@startsection{paragraph}{4}{10pt}{-1.25ex plus -1ex minus -.1ex}{0ex plus 0ex}{\normalsize\textit}{\@startsection{... |
1,116,691,499,369 | arxiv | \section{Introduction}
This paper is a continuation of a collection of works about applications of
the theory of quadratic differentials in quantum theory. In fact, quadratic
differentials have provided an important tool in the asymptotic study of some
solutions of algebraic equations. In quantum theory, trajectories ... |
1,116,691,499,370 | arxiv | \section{Background and Overview}\label{sec:bo}
The arc of this paper begins and ends with a discussion of the Prisoner's Dilemma, but it passes through a new result in provability logic. Thus, it will hopefully be of interest to game theorists and logicians alike.
\subsection{Open-source Prisoner's Dilemma}
Consid... |
1,116,691,499,371 | arxiv | \section*{Introduction}
In their paper \cite{bps} M.A. Barja, R. Pardini and L. Stoppino introduce and study the \emph{continuous rank function} associated to a line bundle $M$ on a variety $X$ equipped with a morphism $X\buildrel f\over\rightarrow A$ to a polarized abelian variety.
Motivated by their work, we con... |
1,116,691,499,372 | arxiv | \section{Introduction and summary}
\label{sec:intro}
\input{sections/intro}
\section{Double holography for Interface CFTs}
\label{sec:double_holo}
\input{sections/double}
\section{Correlation functions of heavy operators in ICFT}
\label{sec:corr_heavy}
\input{sections/corr_heavy}
\section{Entanglement entrop... |
1,116,691,499,373 | arxiv | \section{Introduction}
Submillimetre (submm) observations probe the Rayleigh-Jeans side of the blackbody emission of dust in galaxies.
In that regime, the dimming of the submm flux density of a galaxy due to its cosmological distance is counterbalanced by the redshifting of its spectral energy distribution (SED).
Conse... |
1,116,691,499,374 | arxiv | \section{Introduction}%
The analysis of transient and time-dependent queueing models is of great relevance for numerous applications, such as call centres \cite{brown2005statistical} and outpatient wards of hospitals where the server operates only over a finite amount of time \cite{ kim2014callcenter, kim2014choosing}... |
1,116,691,499,375 | arxiv | \section{Introduction}
Statisticians are interested in inferring properties about a population based on independently sampled data. In the parametric regime, the inference problem boils down to constructing point estimates and confidence intervals for a finite number of unknown parameters. When the data-generation pr... |
1,116,691,499,376 | arxiv | \section{Introduction}
One of the most important parameters of QCD is the total number of dynamical quarks. Asymptotic freedom is lost at the one-loop level for $N_f > 33/2$, while chiral symmetry is broken spontaneously for two or more flavors until QCD becomes conformal for a number of flavors that can be determined... |
1,116,691,499,377 | arxiv | \section{Introduction}
The abundance profiles of chemical elements constitute one of the key properties of galactic disks. They depend on the past history of the disk and the various physical effects that affected it: star formation, infall and outflows, radial flows of gas, radial motions of stars and tidal interacti... |
1,116,691,499,378 | arxiv | \section{Introduction}
\IEEEPARstart{B}{LIND} face restoration (BFR) is a typically ill-posed inverse problem and aims at reproducing realistic and reasonable high-quality (HQ) face images from unknown degraded inputs. Low-quality (LQ) face images are often affected by the combination of multiple unknown degradation fa... |
1,116,691,499,379 | arxiv | \section{Introduction}
\setcounter{equation}{0}
The study of localized tachyon condensation \cite{aps,vafa,hkmm,dv,many} has been
considered with many interesting developments.
The basic picture is that
tachyon condensation induces cascade of decays of the orbifolds to less singular ones until the spacetime
s... |
1,116,691,499,380 | arxiv | \section{introduction}
During the recent years, tremendous progress in B physics has been
made through the fruitful interplay between theory and experiment.
The precise measurements of the $B$-meson decays can provide an
insight into very high energy scales via the indirect loop effects
of new physics beyond the standa... |
1,116,691,499,381 | arxiv | \section{Introduction}
Black holes can be described by two observable parameters, mass and
spin. To date there are over twenty stellar mass black holes with
dynamically constrained mass (for a review see McClintock \& Remillard
2006); however for just a handfull of these systems do we have any
measurements of their sp... |
1,116,691,499,382 | arxiv | \section{Introduction}
\label{sec:Introduction}
Spectrum analyses of pre-main sequence stars (PMS) require special techniques, notably for dealing with peculiarities of cool, low-mass members of young clusters.
Optical spectra of such stars may include the presence of veiling, large broadening due to fast rotation, em... |
1,116,691,499,383 | arxiv | \section{Introduction}
In this paper, we characterize the locus of all points $P$ with an isogonal conjugate in a given quadrilateral $ABCD$. This turns out to be a cubic plane curve, which we will call the \textit{isogonal cubic} of $ABCD$. The isogonal cubic is a well-established figure in geometry. However, its pro... |
1,116,691,499,384 | arxiv | \section{Introduction}
\label{sec:Introduction}
Cherenkov radiation is induced in the atmosphere when a cascade of secondary particles propagate in the air from an initial point where an ultra-high energy (UHE) cosmic ray (CR, in other words, astroparticle) enters the atmosphere. The necessary condition for a charged p... |
1,116,691,499,385 | arxiv | \section*{References}
\begin{description}
\bibitem[1] {PR}
G.Ponzano and T.Regge, {\it Semiclassical limit of Racah coefficients},
in: Spectroscopic and Group Theoretical Methods in Physics, ed. F. Block
{\it et al.} ( North Holland, Amsterdam, 1968) pp. 1-58.
\bibitem[2] {R1}
T.Regge, Nuovo Cimento {\bf 19} (1961) 55... |
1,116,691,499,386 | arxiv | \section{Introduction}
In this paper we find a new, and perhaps unexpected, connection between
polyanalytic functions and time-frequency analysis, and use it to obtain a
complete characterization of all lattice sampling and interpolating
sequences in the Bargmann-Fock space of polyanalytic functions or,
equivalently, ... |
1,116,691,499,387 | arxiv | \section{Introduction}
Recently, cosmological theories and observations have provided interesting information about the universe. Data and measurements from the cosmic microwave background (CMB) show that the matter and energy fluctuations are always unstable on a large scale \cite{2,3}. The leading cause of these cos... |
1,116,691,499,388 | arxiv |
\section{Introduction}
\label{sec:intro}
\input{introduction}
\section{Pure exploration, confidence interval, and p-value}
\label{sec:background}
\input{background}
\section{Advances in best-arm identification and hypothesis testing}
\label{sec:advances}
\input{advances}
\section{Challenges of adaptive online... |
1,116,691,499,389 | arxiv | \section{\label{sec:introduction}{INTRODUCTION}}
Free energy is a significant physical property for estimating thermodynamic stability.
It is desirable to estimate free energy as accurately as possible.
Such free energy estimation is becoming important in a variety of research fields;
in particular, biological molec... |
1,116,691,499,390 | arxiv | \subsubsection{The Second-Level Disengagement}
\begin{figure}[t]
\centering
\subfigure[Clusters $Y_{(a,a')}$ and its child-cluster $Y_{(b,b')}$ (clusters $C_a$ and $C_{a'+1}$ do not belong to $Y_{(a,a')}$).]{\scalebox{0.15}{\includegraphics{figs/instance_for_level_2_1.jpg}}\label{fig: path_cluster_H}
}
\subfigure... |
1,116,691,499,391 | arxiv |
\section{Proof of Theorem~\ref{thm:stronger-mdc-qq3}}
Our proof in this section involves a refinement of modules, that is called {\em $p$-connected components}.
The notion of $p$-connectedness also generalizes connectivity in graphs.
A graph $G=(V,E)$ is $p$-connected if and only if, for every bipartition $(V_1,V_2)$... |
1,116,691,499,392 | arxiv |
\section*{Appendix}
\section{Proof of Theorem 1}\label{app:Th:compute_g}
\begin{proof}(Theorem 1)
Formally, denote $g_Z$ to estimate $\nabla_{\theta}\log Z_{\theta}$ as follows using importance sampling:
\begin{align}
g_Z &=\frac{1}{M_2} \sum_{\tau\in\mathcal{T}^{Q}_{M_2,1}}\frac{P(\tau|\theta,T)}{Q(\tau|\hat{\... |
1,116,691,499,393 | arxiv | \section{Introduction}
Dynamical spin systems have played a central role in non-equilibrium
statistical models. The Ising model is widely studied in
statistical mechanics, as it is simple and allows one to understand
many features of phase transitions. The non-equilibrium properties of
the Ising model follow from the s... |
1,116,691,499,394 | arxiv | \section{Introduction}
The discovery of the Higgs boson on July 2012 provided a well established
explanation of the mass origin of the fermions and gauge bosons of
the standard model (SM). However, despite this great breakthrough,
many questions are still open, such as the origin of neutrino mass
and its smallness, th... |
1,116,691,499,395 | arxiv | \section{I. Introduction}
\hspace*{3.35ex}Hole-doped perovskite manganites \textit{Re}$_{1-x}$\textit{A}$_{x}$MnO$_{3}$ (\textit{Re}: rare earth metal, \textit{A}: alkaline earth metal) have rich electronic and magnetic phase diagrams that depend on the hole concentration and the average ionic radius of the \textit{A}-... |
1,116,691,499,396 | arxiv | \section{#1}\setcounter{equation}{0}}
\def\ft#1#2{{\textstyle{{\scriptstyle #1}\over {\scriptstyle #2}}}}
\def\fft#1#2{{#1 \over #2}}
\renewcommand{\(}{\left(}
\renewcommand{\)}{\right)}
\newcommand{\partial}{\partial}
\newcommand{\begin{eqnarray}}{\begin{eqnarray}}
\newcommand{\end{eqnarray}}{\end{eqnarray}}
\newcomm... |
1,116,691,499,397 | arxiv | \section{I. introduction}
The observations of the cosmic microwave background (CMB) strongly
support inflation as the paradigm of early universe. To discover the
nature of inflation, intensive analysis of the CMB has been performed.
The latest results by the Planck
collaboration~\cite{Ade:2015lrj,Ade:2015xua} provide ... |
1,116,691,499,398 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
The Wide Field Infrared Survey Telescope (WFIRST) Coronagraph Instrument (CGI) is a high-contrast imager, polarimeter, and integral field spectrograph (IFS) that will enable the study of exoplanets and circumstellar disks at visible wavelengths ($\sim 550-850$~nm).\footnote{I... |
1,116,691,499,399 | arxiv | \section{Introduction}
The detection of single atoms with high fidelity is an important requirement for many modern atomic physics experiments, and in particular for quantum information science. Single particle imaging has been achieved using fluorescence imaging of ions~\cite{Neuhauser1980} and neutral atoms~\cite{Sch... |
1,116,691,499,400 | arxiv | \section{Introduction}
The GRism ACS Program for Extra-galactic Science (GRAPES)
project is a slitless spectroscopic survey of
the Hubble Ultra Deep Field (UDF) region (Beckwith 2006).
GRAPES exploits the high spatial
resolution and low background of the G800L grism on the
Advanced Camera for Surveys (ACS). The resul... |
1,116,691,499,401 | arxiv | \section{Introduction}
\setcounter{equation}{0}
Space-time dependent or `local' couplings $\lambda^i = \lambda^i(x)$ are a
standard tool in quantum field theory. They may be viewed as sources for
composite operators. Well-defined operator
insertions are obtained by functionally varying the generating
functional with... |
1,116,691,499,402 | arxiv | \section{Introduction}
A \emph{hole} is a chordless (or an induced) cycle in a graph. The \emph{chordality} of a graph $G$, denoted by $\mathcal{C}(G)$, is defined to be the size of a largest hole in $G$, if there exists a cycle in $G$. If $G$ is acyclic, then its chordality is taken as $0$. A graph $G$ is \emph{$k$-ch... |
1,116,691,499,403 | arxiv | \section{Introduction}
\label{S:1}
\subsection*{Federated Learning Overview}
The ever-growing use of big data systems, industrial-scale IoT platforms, and smart devices contribute to the exponential growth in data dimensions~\cite{s19204354}. This exponential growth of data has accelerated the adoption of machine lea... |
1,116,691,499,404 | arxiv | \section{Introduction}\label{intro}
About thirty ultra-compact X-ray binaries (UCXBs) candidates have been identified to date \citep{Zand2007}. These systems are a subclass of low mass X-ray binaries (LMXBs) with very short orbital period, typically $\lesssim $ 80 min. The compact nature of UCXBs implies that the comp... |
1,116,691,499,405 | arxiv | \section{Introduction}\label{sec:intro}
Let $t$ be a rooted tree and $n_i(t)$ the number of nodes in $t$ having $i$ children. The sequence $(n_i(t),i\geq 0)$ is called the degree sequence of $t$, and satisfies $ \sum_{i\ge 0} n_i(t)=1+\sum_{i\ge 0} in_i(t)=|t|$, the number of nodes in $t$.
The aim of this paper is... |
1,116,691,499,406 | arxiv | \section{Introduction}
Higher-spin (HS) gauge theory is a
theory of an infinite set of gauge fields of all spins. Since
gauge invariant HS interaction vertices contain higher derivatives
of degrees increasing with spin
\cite{Bengtsson:1983pd,Berends:1984wp,Fradkin:1987ks,Fradkin:1991iy},
HS gauge theory is n... |
1,116,691,499,407 | arxiv | \section{Introduction}
Consider the optimal stopping problem
\begin{equation}\label{eq.intro.optimalstopping}
\sup_{\tau\in \mathcal{T}} \mathbb{E}_x[\delta(\tau)f(X_\tau)],
\end{equation}
where $X=(X_t)_{t\in[0,\infty)}$ is a time-homogeneous Markov process taking values in some state space $\mathbb{X}$, $\delta$ i... |
1,116,691,499,408 | arxiv | \section{Introduction}
The derivation of four-dimensional
low-energy effective actions arising from string theory requires
a detailed understanding of the geometries used as compactification
spaces. Since the early days of string theory much research has focused on the study of
Calabi-Yau manifolds of complex dime... |
1,116,691,499,409 | arxiv |
\section{Introduction}
Quarks and gluons, the fundamental degrees of freedom of Quantum Chromodynamics (QCD)
never appear in Nature as asymptotic one-particle states. This phenomena is called confinement.
Instead, only their bound states, i.e., mesons and baryons, are observed in experiment
and these hadronic state... |
1,116,691,499,410 | arxiv | \section*{Abstract}
{\bf
Following the many contributions KLOE-2 has done to Dark Matter (DM) searches, an alternative model, where the Dark Force mediator is an hypothetical leptophobic B boson, in contra-position to the U boson or "dark photon", is investigated. The B boson couples mainly to quarks and it can be sear... |
1,116,691,499,411 | arxiv | \section{Introduction}
Understanding the chemical diversity of materials in star- and planet-forming regions is one of the key topics of present-day astronomy.
Formation of stars and planets can occur in various kinds of galaxies, which differ in size, shape, age, and environment.
It is thus important to understand h... |
1,116,691,499,412 | arxiv | \section{Introduction}\label{sec: intro}
Laser cavitation is widely used for medical applications~\cite{Peng2008}, including ablation of biological tissue~\cite{Vogel2003} and lithotripsy~\cite{Fried2018a,Marks2007}. Lately, lasers have also been investigated for their use in laser-actuated needle-free jet injectors (N... |
1,116,691,499,413 | arxiv | \section{Introduction} \label{sec:Introduction}
Only the most violent objects in the universe are capable of producing
high-energy particles in non-thermal acceleration processes and emit
photons with energies in the X-ray band and above. Spectral and
morphological studies in the X-ray and gamma-ray bands have beco... |
1,116,691,499,414 | arxiv | \section{Introduction}
A region $\Omega \subset \mathbb C$, the boundary of which is a polygonal curve with angles multiple of $\pi/2$, is considered in this article. This region models the shape of a channel under a dam. The calculation of fluid flow in such channel boils down to a conformal mapping problem of $\Omeg... |
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