research papers
Optimization of a phasespace beam position and size monitor for lowemittance light sources
^{a}Physics and Engineering Physics, University of Saskatchewan, 116 Science Place, Saskatoon, SK S7N5E2, Canada, ^{b}Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA, and ^{c}Canadian Light Source, 44 Innovation Boulevard, Saskatoon, SK S7N2V3, Canada
^{*}Correspondence email: xshi@aps.anl.gov
The recently developed vertical phasespace beam position and size monitor (psBPM) system has proven to be able to measure the electronsource position, angle, size and divergence simultaneously in the vertical plane at a single location of a beamline. The optimization of the psBPM system is performed by raytracing simulation to maximize the instrument sensitivity and resolution. The contribution of each element is studied, including the monochromator, the Kedge filter, the detector and the sourcetodetector distance. An optimized system is proposed for diffractionlimited storage rings, such as the Advanced Photon Source Upgrade project. The simulation results show that the psBPM system can precisely monitor the source position and angle at high speed. Precise measurements of the source size and divergence will require adequate resolution with relatively longer integration time.
Keywords: beam position and size monitor; phase space; diagnostics and feedback; raytracing simulation; diffractionlimited storage rings.
1. Introduction
The newgeneration synchrotron facilities are being designed and built to achieve an ultrasmall emittance utilizing multibend achromat (MBA) lattices (Einfeld et al., 2014).
Measurements of electronbeam position and size are challenging and important for the operation of these new light sources (Eriksson et al., 2014; Tavares et al., 2014). The existing and planned diagnostics for measuring the source size for the MBA sources include pinhole imaging (Elleaume et al., 1995; Thomas et al., 2010), πpolarization imaging (Andersson et al., 2008; Breunlin & Andersson, 2016), doubleslit interferometry (Mitsuhashi, 1999; Naito & Mitsuhashi, 2006; Corbett et al., 2016) and Kirkpatrick–Baez (KB) mirrors (Renner et al., 1996; Zhu et al., 2018). Most of these systems use dedicated bending magnet (BM) beamlines. The larger size of the BM source, resulting from larger beta function, compared with other locations in the lattice, allows for more precise measurements.
The pinholecamera measurement with Xrays is the simplest system, but for source sizes of less than 10 µm it is impractical because diffraction by the pinhole complicates extracting information about the source size from the image. The doubleslit interferometry system has better resolution compared with pinhole imaging because the blurring caused by the source size reduces the contrast. In this case, the contrast is a measure of the source size, which does not rely on direct imaging. These measurements are photon hungry, and wavefront distortions caused by optical components can result in inaccurate sourcesize measurements.
The πpolarization technique, another interferencebased method, utilizes the outoforbital plane vertical (π) polarization of the BM beam and, similar to the doubleslit method, depends on source size to reduce the intensity `null' at the midplane.
KB mirror systems use two cylindrical mirrors, one focusing the photon beam horizontally and the other focusing it vertically onto a CCD to measure the source size. In order to beat the diffraction limit, the KB mirror system must use shortwavelength synchrotron radiation.
In all of these methods, knowledge of the pointspread function of the detection system is essential for the sourcesize deconvolution. The contribution from the detector resolution has to be minimized for small sourcesize measurements.
The vertical phasespace beam position and size monitor (psBPM) system (Samadi et al., 2015, 2019) developed at the Canadian Light Source (CLS) has demonstrated the ability to measure the source size and divergence, as well as the source position and angle in the vertical plane, at a single location and time. In this article, we will report on the process of optimizing the psBPM system for ultrasmall electronsourcesize measurements and provide an example for the Advanced Photon Source Upgrade (APSU) project (Borland et al., 2018).
1.1. psBPM system
A psBPM system, as shown in Fig. 1, contains a crystalbased monochromator, a Kedge filter and an area detector. The monochromator is tuned to the photon energy of the Kedge of the filter element. The system utilizes the large horizontal photon fan of the BM beamline to simultaneously measure the direct beam (unfiltered beam) and the part going through the Kedge filter (filtered beam). These beams include both σ and π polarizations, and the polarization effect is negligible in the analysis.
The natural vertical opening angle of the photon beam (Schwinger, 1949) provides a Gaussiantype profile for the unfiltered side of the beam at photon energies well above the of the BM source. It is the central location and width of this unfiltered beam that is used in the data analysis. The photonbeam opening angle also provides a range of incident angles onto the monochromator crystal. This range of angles can give an energy range about the central Kedge energy. The Kedge will introduce a steptype function through this energy range. The location and width of the Kedge are used in the analysis of the filtered data.
The vertical profiles of the filtered and unfiltered beams contain the information of the electronsource position, angle, size and divergence (Samadi et al., 2015). The position of the Kedge location in the filtered beam, y_{edge}, is a direct measure of the electronsource position, y_{eSource}. In other words,
The electronsource size, , can be extracted from the spatial width of the measured Kedge on the detector, σ_{edge}, by (Samadi et al., 2019)
where D is the sourcetodetector distance, is the natural width of the Kedge of the filter element translated from an energy width to an angular width (see Section 2) and is the angular acceptance of the monochromator (Warren, 1969; Zachariasen, 1945). The electronsource emission angle, , and divergence, , can be obtained from the simultaneously measured unfiltered beam position, y_{beam}, and width, σ_{beam}, by
and
respectively, where is the natural opening angle of the photon beam (Schwinger, 1949). In the following sections, each term in equations (1)–(4) will be quantitatively analyzed with numerical simulation.
1.2. Simulation tools and method
The system measures the beam along the direction perpendicular to the orbital plane, which is also the diffraction plane of the monochromator that is typically vertical. Taking this direction, the system can be described by the propagation of the photon beam through phase space, which minimally includes three dimensions, the energy, E, the vertical spatial coordinate, y, and the vertical angular coordinate, y′. To describe the system in sufficient resolution each dimension needs at least a grid size of 10^{3}, which gives a total matrix size of 10^{9}. To reduce the computation effort, Monte Carlo based geometrical ray tracing is used for this work.
All simulations are performed using the ShadowOui program (Rebuffi & Sánchez del Río, 2016) in the OASYS (Rebuffi & Sanchez del Rio, 2017) environment. In ShadowOui, each type of source and optical element is defined as an individual `widget'. The BM source is simulated using the `Bending Magnet' widget, which requires input of electronsource size, electron emittance () and magnetic field of the BM. A BM point source (zero emittance) (PS) can be created by setting both and to zero, which is used to generate the photonbeam distribution representing the singleelectron emission or singleelectron pointspread function from the BM. The BM source includes both horizontal (σ) and vertical (π) polarization components; however, only about 4% of the total intensity is contained in the π polarization.
The monochromators considered in this work are single crystals in the Bragg and Laue geometry. All crystals are simulated using the `Plane Crystal' widgets in ShadowOui, where the crystal is set to be autotuned to the Kedge energy, E_{K}, of the selected filter element.
The Kedge filter is the next optical element downstream of the monochromator. The builtin module in ShadowOui for filter absorption does not contain lifetime broadening for the Kedge spectrum, which is the main contributor to the edge width, σ_{edge}, in real measurements. Therefore, a dedicated Python script was made inside the OASYS environment to simulate the filter absorption by assigning each ray an intensity scaling factor based on its photon energy and the transmission curve. The transmission through the filter is calculated by
where is the energydependent massattenuation coefficient around the Kedge of the filter, ρ is the concentration and t is the effective filter thickness. The Kedge spectrum depends on the corelevel width, which is normally described by a Lorenz function (Babanov et al., 1998). In this work, to be consistent with the experimental results (Samadi et al., 2015), a Gaussian function is used.
A typical simulation to achieve sufficient statistics requires 5 × 10^{7} to 5 × 10^{8} rays, which is challenging to run and store as a single simulation. Therefore, a recursive loop is implemented to accumulate results of multiple runs (typically 100 to 3200), each of which contains 5 × 10^{5} rays. The vertical photonbeam profiles are recorded as histograms that collect rays at the detector position. The histograms are weighted by the ray intensity which contains information on the crystal reflectivity and the filter transmission. The bin size of the histograms is a representation of the pixel size of the detector. The vertical profiles of the filtered beam, I_{filtered}(y), and the unfiltered beam, I_{0}(y), are collected and stored for post analysis [see Fig. 2(a)].
The simulated photonbeam profiles are then analyzed based on the same dataanalysis process developed for experimental results (Samadi et al., 2019). The edge profile, f_{edge}(y), shown in Fig. 2(b), is obtained by
The edge profile and the unfiltered beam profile are both fitted to a Gaussian function with widths of σ_{edge} and σ_{beam}, and center positions, y_{edge} and y_{beam}, respectively, given by
and
The position and angle at the electronbeam source are extracted from the fitted y_{beam} and y_{edge} values using equations (1) and (3), respectively. The electronbeam source size is obtained by deconvolving the edge width, σ_{edge,PS}, of a zeroemittance point source from that of the BM source, σ_{edge,BM}, with a finite electronbeam size, given by
Comparing equation (9) with equation (2), the simulated σ_{edge,PS} term represents the total contribution of and . The electronbeam divergence is then obtained from the photonbeam widths for the BM source, σ_{beam,BM}, and for the zeroemittance point source, σ_{beam,PS}, by
The simulated σ_{beam,PS} term represents in equation (4). The simulation error is calculated as the standard deviation of results from 100 separate raytracing calculations unless otherwise specified.
2. Optimization process
The optimization process involves aspects of the system that determine its ability to best measure source properties. These factors include the monochromator, the Kedge filter, detector characteristics, and arrangement of these components (measurement geometry).
Two cases are considered: a BM at the CLS and a BM for the APSU.
For the CLS, the simulation study is for a 1.354 T BM and an electron beam with = 52.7 µm and = 6.35 µrad (Bergstrom & Vogt, 2008). Unless specified, all simulations were performed with a monochromator tuned to the barium Kedge energy (37.441 keV), a 35 mg cm^{−2} barium Kedge filter and a sourcetodetector distance of D = 10 m.
2.1. Monochromator
The monochromator is one of the most critical components of a psBPM system. The effect of the monochromator and the choice of the Kedge filter are closely related to each other through the angle–energy dispersion from Bragg's law. The dispersion effect of the monochromator crystal projects the absorptionedge energy width, , into an angular width, (measured by the spatial width on the detector at distance D) through the relationship
where θ_{K} is the of the monochromator crystal at the filter Kedge energy, E_{K}. In general, to achieve small requires a filter with small , a high Kedge energy and a small This section and Section 2.2 below show in detail how these terms contribute to the measurement.
There are several choices for the crystal material, reflection geometry and lattice planes that will now be considered.
2.1.1. Crystal material and geometry
Singlecrystal materials are considered for the monochromator. Highquality semiconductor crystals are commonly available as a consequence of the semiconductor industry drive to improve device performance. ) can be used to describe the diffraction properties of such crystals. Silicon is the most common monochromator crystal used for Xray beamlines because of its availability, degree of perfection and ability to handle synchrotronradiation heat loading.
(Zachariasen, 1945The diffraction geometry from crystals falls into two broad categories. The reflection or Bragg geometry has lattice planes mostly parallel to the crystal's surface; Xrays impinge upon and diffract out of the same surface. In transmission or Laue geometry the lattice planes are mostly perpendicular to the crystal surface; Xrays impinge upon one surface and exit through another by diffracting through the crystal.
Laue geometry has two practical advantages over Bragg geometry because it allows a smaller footprint (a smaller crystal) and reduced thermal deformation from the photonbeam heat load. Nevertheless, based on the diffraction profiles [see Fig. 3(a)] calculated using the XCRYSTAL module (Sanchez del Rio et al., 2015) in XOP (Sánchez del Río & Dejus, 2011), Bragg geometry is preferred owing to the higher reflectivity and narrower bandwidth compared with Laue geometry.
Intensive efforts are being dedicated to the studies of crystal quality and thermal mechanical design of monochromators, which are not in the scope of this work. We limit the following discussions to single Bragg silicon crystals and focus on the optical optimization of the psBPM system.
2.1.2. planes
The intrinsic angular bandwidth of a monochromator, , can be modeled using standard ). Reflection from highindices planes [e.g. Si(440)] has smaller angular bandwidth, as shown in Fig. 3(b). This effect can be clearly seen in the DuMond diagrams (DuMond, 1937) shown in Fig. 4, where a zeroemittance BM source is monochromated by a Si(111) crystal [Fig. 4(a)] and a Si(440) crystal [Fig. 4(b)] to the barium Kedge energy. Assuming the photon beam is absorbed by a barium filter with a sharp edge ( = 0), the intrinsic bandwidth of the crystal is spatially projected onto the detector plane (the y axis in Fig. 4). The Si(440) reflection contributes to a much smaller edge width, yet gives a larger [steeper y versus energy slope in Fig. 4(b)]. This has the effect of limiting the energy range that the monochromator will cover with the photonbeam divergence from the source. A limited energy range is not ideal when the filteredge width, , is nonzero based on equation (11). Table 1 shows the angular width of the barium Kedge in angle, , with different crystal reflections. For the same energy edge width ( = 5.6 eV, or a FWHM of 13.2 eV assuming a Gaussian distribution) (Babanov et al., 1998), a crystal with lower reflection indices is preferred.
(Zachariasen, 1945

2.2. Kedge filter
The choice of the Kedge filter determines the energy to be selected by the monochromator. Also, the Kedge width affects the ability to determine the source size [see equation (2)] as it needs to be accounted for in the overall edgewidth measurement. Since the monochromator and energy will determine the from the source, the ability to accurately determine the center and width of the distribution will rely on the statistical fitting of the vertical profile of the beam. The same applies to the Kedge filter where a statistical fit to the edge location and width is performed. The for those measurements depends upon the Kedge filter element attenuation (), concentration (ρ) and thickness (t) [see equation (5)]. The product of concentration and thickness is commonly referred to as the projected concentration (mass per area) of the filter.
2.2.1. Kedge choice
The natural energy width of the Kedge of an element is dominated by the lifetime of the electron–hole in the Kshell. Both the Kedge energy and the edge width increase with the (KeskiRahkonen & Krause, 1974). As described in Section 2.1, the Kedge selection must be considered along with the selection of the monochromator crystal. Fig. 5 shows the simulated vertical photonbeam profiles indicating the edge widths that contain contributions from both the monochromator and the filter Kedge. Even though the crystal bandwidth is smaller for the Si(440) crystal (Fig. 4), the total edge width is spatially larger in y (see Fig. 5) because of the increased energy dispersion of the (440) reflection compared with the (111) reflection.
Quantitatively, the total contribution from both the monochromator and the Kedge filter add in quadrature as
The calculated , and values for different filter elements and crystal reflections are summarized in Table 2, where is the fitted Gaussian width of the diffraction profile calculated using XOP (Sánchez del Río & Dejus, 2011), and is calculated using equation (11) with extracted from Fig. 1 in the work by Babanov et al. (1998). As the element goes up, the total contribution from the Kedge width and monochromator width becomes smaller, which implies a better sensitivity for detecting the electronsource size based on equation (2). Since the total contribution is mostly dominated by the Kedge width, the bandwidth of the monochromator has relatively less effect. Therefore, crystals with lower reflection indices [i.e. Si(111)] are preferred because of the smaller Table 2 also shows that the reduction of the total width is not that dramatic when going to a higher than iodine. Considering that most of the BM sources have critical energies much less than 30 keV, going to a higher energy leads to a rapid reduction in as well. One should therefore choose as high an energy as possible while maintaining sufficient flux.

2.2.2. Filter concentration
The choice of filter concentration and thickness (projected concentration) will affect the sensitivity and accuracy of the sourcesize measurement. Fig. 6 shows the extracted (a) edge jumps, (b) edge profiles and (c) source sizes calculated with different Ba filter concentrations. When the filter projected concentration is low (e.g. 7 mg cm^{−2}), the absorptionedge contrast is low, which gives a lower intensity edge profile and higher noise level. Therefore, the extracted source sizes have larger uncertainties, shown by the error bars in Fig. 6(c). On the other hand, when the filter projected concentration is too high (e.g. 140 mg cm^{−2}), the filter absorbs most of the light on the highenergy side (negative yvalue side) of the spectrum, which tends to broaden the fitted edge width and thus gives a larger source size. The relative fitting error is also large in high filterprojectedconcentration cases. As a result, the best filter projected concentration for Ba is around 35 mg cm^{−2}. In practice, it is easy to optimize the filter projected concentration experimentally by analyzing the measurement error and accuracy as demonstrated by the simulation.
2.3. Geometry
The basic geometry of the psBPM system is shown in Fig. 1. Other than the obvious arrangement where the system elements must intercept the incident and diffracted beams, the only relevant distance is the sourcetodetector distance, D, as indicated in equations (1)–(4).
The sourcetodetector distance, D, must be optimized to maximize the sensitivity of the psBPM system. Simulation was carried out using the parameters described in Section 2 with variable distances, D. The standard deviation (RMS error) of the simulated electronsource size and position, which is a good measure of the sensitivity of the psBPM system, is plotted as a function of D in Figs. 7(a) and 7(b), respectively. The sensitivity for detecting the source position, y, is linearly related to the choice of D [see Fig. 7(a)]. More importantly, the sensitivity for measuring the source size is inversely proportional to D^{2} [see Fig. 7(b)].
It is therefore beneficial to reduce D to optimize the sensitivity of the psBPM system. Because of the physical space limitation in a typical beamline, a distance of 10 m would be a reasonable choice for existing or planned BM beamlines that are dedicated to source diagnostics. Another concern of having a short distance is that the quadratic increase of the incidentpower density will increase the thermal deformation on the monochromator crystals, which will degrade the accuracy of the size and angle measurements. In that case, an aggressive cooling scheme will be required.
2.4. Detector
The determination of the unfiltered beam location and width as well as the filtered beam Kedge location and width relies on curve fitting to the measured I_{0}(y) and f_{edge}(y) profiles using equations (7) and (8), respectively. The edge width is normally in the range of a few tens of microradians as shown in Table 2. There must also be enough spatial resolution across the edge width to ensure an accurate fitting. Fig. 8 shows the simulated source size and divergence as a function of the pixel size (bin size of the histograms) with the total (number of rays) kept constant. A pixel size of a few tens of micrometres is adequate to ensure the accuracy of the source size and divergence measurements. Previous experiments (Samadi et al., 2019) show that a detector with 100 µm pixel size is sufficient to measure thirdgeneration synchrotron source sizes. Overall, the accuracy of the curvefitting procedure is more sensitive to the integrated than to the pixel size of the detector.
The nextgeneration synchrotrons have the source size and divergence one order of magnitude smaller. A similar study shows that a pixel size of 10 µm is expected to be sufficient for the APSU source, assuming perfect detectors. However, the noise level (dark noise and others) on the detector will affect the curvefitting results and reduce the measurement sensitivity. The smallest measurable size of the psBPM system will be limited by the
the detector resolution and the noise level, which needs further study.3. Example of a psBPM system for APSU
Based on all of the above studies, an optimized configuration is proposed for the lowemittance APSU project. The APSU will have 42 pm rad natural emittance (Borland et al., 2018). Simulations were performed with the source parameters at the M3 BM with = 4.9 µm and = 2.8 µrad, a singleBragg Si(111) monochromator tuned to the barium Kedge energy (37.441 keV), a 35 mg cm^{−2} barium filter, a sourcetodetector distance of D = 10 m, and a detector pixel size of 10 µm.
Using the simulation procedure described in Section 1.2, the ability of the psBPM to measure the source properties was analyzed. Fig. 9 shows the predicted output source properties as a function of the input values that varied relative to their nominal values by as low as 5%. The source position and angular position were simulated with 5 × 10^{7} rays and obtained from equations (1) and (8), and equations (3) and (7), respectively. The source size and divergence were studied with 5 × 10^{8} rays and extracted using equations (9) and (10), respectively.
The psBPM system has an excellent ability to measure the source position and angular position as shown in Figs. 9(a) and 9(b). The measurement of source position and angular position is fast and considered real time. The source size and divergence can be extracted at the same time, which is one of the main features of the psBPM system. The sensitivity to the sourcesize variation is about 10% of the nominal source size in these calculations because of the limited statistics. The sourcesize measurement is the most photonhungry component of the system. In real measurements, increasing the acquisition time will improve the sensitivity, but with limited measurement speed.
The sensitivity of the psBPM system is SHADOW raytracing is that rays can have fractional intensities to account for the crystal reflectivity and filter absorption. Therefore, a single ray can represent a large number of photons. From the previous studies (Samadi et al., 2019), simulation with 1 × 10^{7} rays gives the same sensitivity as the measurement of source size performed with a level of 1.2 × 10^{10} photons Hmrad^{−1} (where H means horizontal) To achieve the sensitivity shown in Fig. 9(c), simulation with 5 × 10^{8} rays indicates that a level of 5.9 × 10^{11} photons Hmrad^{−1} is needed for measuring the APSU source size. Considering an Si(111) Bragg crystal monochromator with no filter, this requires a minimum of 1.5 s exposure time.
driven, but nonlinearly. The required level can only be estimated with the comparison of experimental and simulation results. One feature of4. Conclusion
The psBPM system can precisely measure electronbeam source position and angle, which are relative to the Kedge location in the filtered side of the photon beam as well as the central location of the unfiltered beam. The system can also provide accurate measurements of the electronsource size and divergence from knowledge of the Kedge width and the full photonbeam width. The simultaneous measurement of all four source properties in the vertical plane is a unique feature of the psBPM system. In principle, the system can also be used to measure the source position and size in the horizontal plane but a separate horizontally deflecting monochromator will be required.
Factors that affect the sensitivity and resolution of the system include the choice of monochromator, Kedge filter, geometry of the system and detector. The optimized configuration contains lowindex crystal reflections, a highenergy Kedge filter and a relatively small sourcetodetector distance. The filterelement concentration must be selected to ensure enough absorption contrast while maintaining a reasonable transmission on the highenergy side of the Kedge. Compared with other systems, the psBPM monitor has lessdemanding requirements on detector resolution, which makes it capable of highspeed measurements.
It is also worth pointing out that the psBPM system can measure a wide size range. The larger the source size, the easier (or faster) it can be measured, as long as the sourcesize contribution () is smaller than the natural opening angle of the photon beam.
A singlecrystal monochromator may generate Compton scatter at the detector location which reduces signal contrast. To achieve a higher sensitivity, the use of a twocrystal monochromator should be considered. Another concern is fluorescence from the Kedge filter, some of which may also provide background in the detector. Other considerations for a practical system include mechanical stability and thermal management of the monochromator.
Simulations validated by measurement show that the psBPM system is suitable for nextgeneration light sources. An optimized system for the APSU source was presented as an example to demonstrate the performance. The source position and angular motion can be monitored with high precision and at high speed; while the sourcesize measurement is photon hungry, which creates a tradeoff between measurement speed and resolution. Because of the relatively simple configuration of the psBPM monitor, it can coexist and operate in parallel with other systems at the same beamline.
Acknowledgements
The authors would like to thank Dr Luca Rebuffi (Argonne National Laboratory) for the OASYS support and Dr Les Dallin (Canadian Light Source) for the invaluable discussions.
Funding information
This work was supported by the US Department of Energy, Office of Basic Energy Sciences, under Contract No. DEAC0206CH11357, Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant, Canadian Institutes of Health Research (CIHR) Team Grant – Synchrotron Medical Imaging, CIHR Training Grant – Training in Health Research Using Synchrotron Techniques, the Canada Research Chair Program, Saskatchewan Health Research Foundation Team Grant, and the University of Saskatchewan.
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