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Abstract
The evolution from 3rd to 4th generation of storage rings significantly enhanced the coherence of synchrotron radiation sources, making coherent scattering techniques such as coherent X-ray diffraction imaging (CXDI) and X-ray photon correlation spectroscopy more accessible. In conformance with the design requirements of coherent beamlines at the High Energy Photon Source (HEPS), we have developed wave optics simulation software, the Coherence Analysis Toolbox, based on coherent modes decomposition and a wavefront propagation model. Simulations of beamline performance and a CXDI experiment on the hard X-ray coherent scattering beamline at HEPS were carried out. This software is open source and now available on GitHub.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Synchrotron radiation (SR) storage rings have advanced from the 3rd to 4th generation with development of multi-bend achromat lattices [1], which allows electron emittance to be reduced to enhance the brilliance and transverse coherence of the SR source. For example, the electron emittance of the European Synchrotron Radiation Facility – Extremely Brilliant Source (ESRF-EBS), Advanced Photon Source – Upgrade (APS-U) and Spring-8-II are 132 pm.rad, 42 pm.rad and 149 pm.rad, respectively [2]. The electron emittance of the High Energy Photon Source (HEPS), which is a new SR source under construction, reaches 34 pm·rad [3]. Coherent scattering techniques such as coherent X-ray diffraction imaging (CXDI) [4,5] and X-ray photon correlation spectroscopy (XPCS) [6] would greatly benefit from the enhanced coherence of SR from 4th generation sources. CXDI and XPCS strongly depend on the coherence of the X-ray beams at the sample, and it is therefore important to analyze the coherence of X-ray beamlines. Furthermore, coherent scattering experiments could be simulated more accurately by considering the coherence of the X-rays, which could also help to characterize the performance of 4th generation SR beamlines [7–9].
For example, the propagation of multiple wavefronts using Monte Carlo simulation considers the emittance and propagation of X-ray wavefronts for every sampled electron, and coherence is calculated statistically [10,11]. Based on this method, simulations of coherent scattering experiments, such as coherent scattering and XPCS, have been performed on 3rd generation SR sources [12,13]. Although accuracy could be ensured [14], the demands on computational resources reduced the flexibility of this method, leading to exploration of other approaches [15].
Taking advantage of the enhanced coherence of 4th generation SR sources, coherent modes decomposition (CMD) and wavefront propagation methods have become more efficient. Based on statistical optics, the spatial coherence of an X-ray beam can be fully described by cross-spectral density (CSD), the expansion of which results in orthonormal coherent modes [16]. For 4th generation SR, the number of required coherent modes for coherence analysis is about 102–103 [16], which is much more efficient than for wavefront analysis (∼103–105 for the Monte Carlo multi-electron method). Furthermore, an accurate estimate of the coherent fraction of the X-ray beam can be obtained using this method.
Several tools have been developed for coherent mode analysis of SR produced by storage rings. For example, prior to CMD, the CSD of the ESRF-EBS has been calculated with COMSYL (Coherent Modes for SYnchrotron Light) software using the brightness convolution theorem [16,17]. Also, Synchrotron Radiation Workshop (SRW) software has added an efficient CMD method by calculating the coherent modes at the beam waist [18]. And X-ray Tracing (XRT) software has incorporated two CMD methods base on the propagated wavefronts [19]. However, the CMD methods requires further development to reduce their high demand for computational resources.
In this study, we have developed CAT, a wave optics simulation software. First, a two-step singular-value decomposition (SVD) method was developed to achieve effective CMD of SR. Then, the coherent modes were propagated and re-diagonalized to analyze the coherence of SR along the beamline, and the performance of optical elements with different figure errors was also analyzed. Finally, X-ray beams of differing partial coherence were constructed for simulations of CXDI.
2. Model description
2.1 Coherent modes decomposition of SR X-rays
Coherent modes are usually decomposed from the CSD $W({{x_1},{x_2};{y_1},{y_2};\omega } )$ [20], and the CSD is in turn calculated from the wavefronts of the radiated electric fields:
Therefore, the eigenvalues can be calculated by decomposing the matrix A+A of size [n, n] instead of the matrix AA+ of size [Nx×Ny, Nx×Ny] [21]. If n << Nx×Ny, the SVD method is much more efficient than CSD decomposition. However, the number of electrons required by the Monte Carlo method is large. Thus, the SVD method does not always offer an advantage for CMD of SR from 4th generation SR source.
A two-step SVD method was developed here. First, A is divided into k parts [A1, A2, …, Ai, …, Ak] along the direction of column, and the size of Ai is [Nx×Ny, n/k]. Second, SVD is performed on Ai to generate coherent modes:
A schematic diagram of this method is shown in Fig. 1. The efficiency of the two-step SVD method was compared with the CSD decomposition method. Both methods adopt the sparse eigen-solvers of the SciPy library [22]. Figure 2 shows the advantage of two-step SVD over CSD decomposition for CMD of SR from 4th generation sources.
Fig. 2. The efficiencies of CSD decomposition and two-step SVD. Using the electron beam and undulator parameters (Supplement 1,Table S1, column HXCS), all the calculations were performed on a workstation with an Intel Xeon Gold 6226 2.70 GHz and 64 GB of memory, running the Ubuntu 18.04 LTS operating system. 20 CPU cores were used for parallel computations. The truncation number, t, of the CMD was 200, and the size of the wavefront [Nx, Ny] was [200, 200].
The validity of the method used by CAT was evaluated by comparison with SRW and COMSYL software. As comparison of CSDx and CSDy calculated by SRW and CAT is shown in Fig. 3(a). The coherent modes of ESRF-EBS were calculated by CAT (Fig. 3(d)) and compared with the reported results (see Supplement 1 and Figs. S1–S5 for more details and comparisons).
Fig. 3. Comparison of SRW/COMSYL with CAT software. (a) CSDx and CSDy calculated by SRW and CAT. (b) The coherent modes of ESRF-EBS calculated by CAT, m is the index of mode. Details and more comparisons are shown in Supplement 1, Fig. S1.
2.2 Propagation of coherent modes
Different propagators were applied in CAT to achieve high propagation efficiency for various optical layouts. In the following description, the coherent mode $\varphi $(x, y) was propagated from the input optic plane (x1, y1) to the output optic plane (x2, y2) over a distance d.
First, the Fresnel diffraction integral was used for long-distance propagation [23],
Second, the angular spectrum method (ASM) was used for propagation over short-distance [23],
Third, the Chirp-Z transform (CZT) method [26–29] was used for sub-micrometer focusing. For the coherent mode $\varphi ({x,y} )$, the discrete representation of one dimension is $x = [{{x_1},{x_2}, \ldots ,{x_n}, \ldots ,{x_N}} ]$, where $n \in [{0,N - 1} ]$, N is the size of the discrete representation of the coherent mode, ${x_n} = n\varDelta r$, and $\varDelta r$ is the pixel size of the coherent mode. The transformed discrete dataset using the CZT is $z = [{{z_1},{z_2}, \ldots ,{z_m}, \ldots ,{z_M}} ]$, where $m \in [{0,M - 1} ]$, M is the size of the transformed dataset, ${z_1} = {k_0}$ defines the starting point of the z, ${z_m} = {k_0} + m\varDelta k$, and $\varDelta k = 2\pi /({M\varDelta r} )$ defines the pixel size of z. The relationship between the x and z datasets is
where $a = {e^{i{k_0}}}$ and $w = {e^{i\varDelta k}}$. The CZT method can be speed up by Bluestein method, which ensures the efficiency of this propagator.The propagators in CAT were verified by comparison with SRW [30] . The results are shown in Fig. 4. An optical layout of hard X-ray coherent scattering (HXCS) beamlines with an advanced-KB mirror was used as an example of the performance of the propagators (see Supplement 1, Section 3 and Fig. S7).
Fig. 4. (a) The wavefront was focused by an ideal lens. The distance between the source and lens was 10 m, and the focal length of the lens was 5 m (1:1 focus). Here, the SRW standard propagator was the Fresnel propagator used by SRW, which was calculated by using the convolution theorem. (b) The wavefront was diffracted by a slit. The distance between the source and slit was 10 m, and the distance between the slit and final screen was 10 m. The size of the slit was 50 µm.
2.3 Beamline analysis method
The X-ray beam at the optical element was simulated by propagating the coherent modes to the optical plane of the element. For an ideal lens, the phase transformation is
where f is the focal length. For a KB mirror, the phase transformation in one direction wasCoherence analysis of the propagated coherent modes has two steps. First, the CSD of the X-ray beam is calculated by Eq. (2). Second, a re-diagonalization of coherent modes is required when the modes are spatially filtered by the optical elements. SVD was deemed to be an appropriate and highly efficient method here since the number of the propagated coherent modes in SR from a 4th generation source is quite small.
CAT provides an efficient way of simulating the performance of optical elements with different figure errors. Take reflective X-ray optics for example, first, the figure error was transformed into a phase error,
where errors is the figure error, θ is the incident angle, errorp is the phase error. Second, the phase error was added with the phase of the coherent mode,3. Simulations
3.1 SR source
Based on the parameters of the electron beam and undulator (Table S1), CMD of the HXCS beamline was performed at 12.4 keV to analyze beam coherence. The emitted wavefronts were calculated at a distance of 20 m. Then two-step SVD was performed to calculate the coherent modes ([Nx, Ny] = [150, 150]). As shown in Fig. 5, the calculated CF of the SR source for the HXCS beamline was 20.7%, and the occupation of coherent modes with m = 0 to m = 29 already exceeded 90%, which demonstrated the high coherence of the HEPS (see Supplement 1, Section 2 and Fig. S6 for the occupations).
3.2 Coherence analysis of the beamline
An optical layout of HXCS is shown in Fig. 6(a). The sample was placed at 2.7 m from the focus and coherence was adjusted by a secondary source slit (SSS). The secondary source was focused by a 4:3 focus system (object distance = 40 m, imaging distance = 29.3 m). Based on the data in Table S1 (Supplement 1, Section 1), the SR photon beam divergence was calculated to be 4.3 µrad (horizontal direction) and 3.8 µrad (vertical direction). Thus, the size of the propagated X-ray beam incident on the lens was 172 µm (4.3 µrad×40 m, RMS). The diameter of the lens was set to 600 µm to receive most of the incident X-ray beam, and its focal length was 16.91 m.
Fig. 6. (a) HXCS optical layout designed for XPCS and traditional plane wave CXDI. (b) Intensity, (b) CSDx and (c) CSDy of the secondary source.
The secondary source was focused by an ideal lens. As shown in Fig. 6(b)–6(d), the intensity, CSDx ($W({{x_1},{x_2};{y_1} = 0,{y_2} = 0;\omega } )$), and CSDy ($W({{x_1} = 0,{x_2} = 0;{y_1},{y_2};\omega } )$) of the secondary source were calculated using propagated coherent modes based on Eq. (2) ([Nx, Ny] = [120, 120]). Based on Fig. 6(b), the size of the secondary source (horizontal×vertical) was 8.81 µm×4.14 µm (RMS), consistent with the theoretical value (7.69 µm×4.10 µm, RMS). The coherence lengths were calculated [31] to be 1.68 µm×2.59 µm (RMS, Fig. 6(c), 6(d)). The SSS was used to modify the partial coherence of the X-ray beam. The size of the slit was adjustable from 21.0 µm×21.0 µm to 12.6 µm×12.6 µm. Finally, the modified coherent modes were propagated from the secondary source to the sample (10 µm×10 µm).
3.3 Simulation of CXDI
Simulation of CXDI was based on the coherent modes and the sample. Using the projection approximation, a sample can be described by a transmission function T(x, y) [12]. When a coherent mode $\varphi ({x,y} )$ illuminates the sample, the exit wave can be represented by $T({x,y} )\varphi ({x,y} )$, which then propagates to the detector. For CXDI, the distance d between the sample and the detector is large enough to allow the propagation to be treated as Fraunhofer diffraction. Thus, the diffraction signal $\varphi ({{q_x},{q_y}} )$ is proportional to a scaled FT of $\varphi ({x,y} )$,
Therefore, the diffraction pattern at the detector can be calculated by summing the intensities of all the propagated coherent modes:Simple CXDI simulations were carried out with different values of partial coherence represented by the CF. The coherent modes propagated at the sample were re-diagonalized within 10 µm×10 µm ([Nx, Ny] = [1000, 1000]), and the results are shown in Fig. 7 (SSS = 21 µm). CMD showed that the CF of the X-ray beams at the sample increased from 89.0% to 95.6% as the width of the SSS decreased from 21.0 µm to 12.6 µm.
The standard sample used for simulations was a part of the Siemens star (Fig. 8(a), binary sample, created by PyNX [32], 10 µm×10 µm, 256×256 pixels). The spatial dimension of the diffraction pattern was 800×800 pixels (FT of the sample, Fig. 8(b)), and the oversampling ratio was 9.8, which satisfies the requirements of CXDI for accurate sample reconstruction [33]. The sample was reconstructed by the hybrid input-output (HIO) algorithm and error reduction (ER) method [34]. And 5000 HIO iterations and 500 ER iterations were applied.
The results of the reconstruction are shown in Fig. 9. It was observed that the quality of the reconstruction increased with coherence. Although the coherence length (10.3 µm×18.0 µm) covered the size of the sample when the CF was 89.0%, an acceptable reconstruction could not be achieved until the CF increased to 94.7%. Therefore, an increase in the quality of the CXDI reconstruction would require an X-ray beam with a higher CF, which could be achieved by decreasing the width of the SSS. However, the flux of the X-ray beam was reduced (see Supplement 1, Section 6 and Fig. S10).
4. Conclusion
In summary, we have developed CAT, a wave optics simulation software. A two-step SVD method was developed to perform effective coherent modes decomposition of the highly coherent X-rays from a 4th generation SR source. Taking advantage of coherent mode propagation and re-diagonalization, the coherence at different optic planes can be easily calculated, and the influence of surface errors in the optics can be simulated. The effect of partial coherence on CXDI experiments was analyzed. Applications of the method and the related code developed here (see Supplement 1 Section 7) were also presented. CAT is open source software available on GitHub, and further development will be pursued.
Acknowledgments
This work was supported by High Energy Photon Source (HEPS), a major national science and technology infrastructure in China.
Disclosures
The authors declare no conflicts of interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. R. Hettel, “DLSR design and plans: an international overview,” J. Synchrotron Radiat. 21(5), 843–855 (2014). [CrossRef]
2. B. Oh, J. Ko, J. Lee, G. Jang, and S. Shin, “Characteristics of Photon Beam and Preservation of Coherence in Fourth-Generation Light Sources,” Appl. Sci. 11(24), 11896 (2021). [CrossRef]
3. Y. Jiao, G. Xu, X. H. Cui, Z. Duan, Y. Y. Guo, P. He, D. Ji, J. Li, X. Li, C. Meng, Y. Peng, S. Tian, J. Wang, N. Wang, Y. Wei, H. Xu, F. Yan, C. Yu, Y. Zhao, and Q. Qin, “The HEPS project,” J. Synchrotron Radiat. 25(6), 1611–1618 (2018). [CrossRef]
4. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015). [CrossRef]
5. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006). [CrossRef]
6. F. Giovanni, E. B. Knudsen, T. Ramos, D. Carbone, and J. W. Andreasen, “A Monte Carlo ray-tracing simulation of coherent X-ray diffractive imaging,” J. Synchrotron Radiat. 27(1), 134–145 (2020). [CrossRef]
7. X. Shi, R. Reininger, R. Harder, and D. Haeffner, “X-ray optics simulation and beamline design for the APS upgrade,” Proc. SPIE 10388, 103880C (2017). [CrossRef]
8. R. Luca and X. Shi, “Advanced simulation tools in the OASYS suite and their applications to the APS-U optical design,” Proc. SPIE 11493, 1149303 (2020). [CrossRef]
9. A. Madsen, A. Fluerasu, and B. Ruta, “Structural dynamics of materials probed by X-ray photon correlation spectroscopy,” Synchrotron light sources and free-electron lasers 2, 1–21 (2015). [CrossRef]
10. O. Chubar and P. Elleaume, “Accurate and efficient computation of synchrotron radiation in the near field region,” Proc. of the EPAC98 Conference 11493, 1177–1179 (1998). [CrossRef]
11. O. Chubar, L. Berman, Y. S. Chu, A. Fluerasu, S. Hulbert, M. Idir, K. Kaznatcheev, D. Shapiro, Q. Shen, and J. Baltser, “Development of partially-coherent wavefront propagation simulation methods for 3rd and 4th generation synchrotron radiation sources,” Proc. SPIE 8141, 814107 (2011). [CrossRef]
12. O. Chubar, M. Rakitin, Y. C. K. Chen-Wiegart, A. Fluerasu, and L. Wiegart, “Simulation of experiments with partially coherent X-rays using synchrotron radiation workshop,” Proc. SPIE 10388, 1038811 (2017). [CrossRef]
13. O. Chubar, L. Wiegart, A. Fluerasu, M. Rakitin, J. Condie, P. Moeller, and R. Nagler, “Simulations of coherent scattering experiments at storage ring synchrotron radiation sources in the hard x-ray range,” Proc. SPIE 11493, 1149310 (2020). [CrossRef]
14. A. He, O. Chubar, M. Rakitin, L. Samoylova, C. Fortmann-Grote, S. Yakubov, and A. Buzmakov, “Parallel performance of “Synchrotron Radiation Workshop” code: partially coherent calculations for storage rings and time-dependent calculations for XFELs,” Proc. SPIE 11493, 114930H (2020). [CrossRef]
15. Z. Zhu, H. Xu, L. Hu, M. Li, P. Liu, Y. Dong, and L. Zhou, “A wave optics model for the effect of partial coherence on coherent diffractive imaging,” J. Synchrotron Radiat. 28(2), 499–504 (2021). [CrossRef]
16. M. Glass, M. Sanchez, and del Rio, “Coherent modes of X-ray beams emitted by undulators in new storage rings,” Europhys. Lett. 119(3), 34004 (2017). [CrossRef]
17. M. Sanchez del Rio, R. Celestre, M. Glass, G. Pirro, J. R. Herrera, R. Barrett, J. Cesa da Silva, P. Cloetens, X. Shi, and L. Rebuffi, “A hierarchical approach for modeling X-ray beamlines: application to a coherent beamline,” J. Synchrotron Radiat. 26(6), 1887–1901 (2019). [CrossRef]
18. R. Li and O. Chubar, “CPU and memory efficient coherent mode decomposition for the partially coherent x-ray simulations,” Proc. SPIE 11493, 114930G (2020). [CrossRef]
19. R. Khubbutdinov, A. P. Menushenkov, and I. A. Vartanyants, “Coherence properties of the high-energy fourth-generation X-ray synchrotron sources,” J. Synchrotron Radiat. 26(6), 1851–1862 (2019). [CrossRef]
20. A. S. Ostrovsky, Coherent-mode representations in optics (SPIE Press, 2006).
21. B. Anders., Noise and vibration analysis: signal analysis and experimental procedures (John Wiley & Sons, 2011).
22. P. Peterson, “F2PY: a tool for connecting Fortran and Python programs,” International Journal of Computational Science and Engineering 4(4), 296–305 (2009). [CrossRef]
23. J. Li and Y. Wu, “An indirect algorithm of Fresnel diffraction,” Opt. Commun. 282(4), 455–458 (2009). [CrossRef]
24. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44(2), 407–414 (1997). [CrossRef]
25. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006). [CrossRef]
26. L. Rabiner, R. W. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969). [CrossRef]
27. Y. Hu, Z. Wang, X. Wang, S. Ji, C. Zhang, J. Li, W. Zhu, D. Wu, and J. Chu, “Efficient full-path optical calculation of scalar and vector diffraction using the Bluestein method,” Light: Sci. Appl. 9(1), 1–11 (2020). [CrossRef]
28. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006). [CrossRef]
29. L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18(4), 451–455 (1970). [CrossRef]
30. O. Chubar and C. Rafael, “Memory and CPU efficient computation of the Fresnel free-space propagator in Fourier optics simulations,” Opt. Express 27(20), 28750–28759 (2019). [CrossRef]
31. T. Takashi and H. Kitamura, “SPECTRA: a synchrotron radiation calculation code,” J. Synchrotron Radiat. 8(6), 1221–1228 (2001). [CrossRef]
32. V. Favre-Nicolin, G. Girard, S. Leake, J. Carnis, Y. Chushkin, J. Kieffer, P. Paleo, and M. I. Richard, “PyNX: high-performance computing toolkit for coherent X-ray imaging based on operators,” J. Appl. Crystallogr. 53(5), 1404–1413 (2020). [CrossRef]
33. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE signal processing magazine 32(3), 87–109 (2015). [CrossRef]
34. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef]
