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1.INTRODUCTIONMany synchrotron beamlines that produce hard X-rays (photon energy ≥ 3 keV) select a suitable wavelength by using diffraction from perfect silicon crystal monochromators. To select photon energies up to about 20 keV, diffraction in the Bragg case is used. In this case the diffracted beam emerges from the surface upon which the incident beam falls. By using perfect crystals one can obtain a resolving power δE/E ≈ 10-4 in the photon energy E, which is suitable for many types of experiments. In the following, practical examples of the application of the Bragg case will be discussed. X-ray diffraction from large perfect crystals must be treated with the theory of dynamical diffraction, as the simpler kinematic theory does not account for extinction of the incident wave or for the coherent coupling of the incident and diffracted waves inside the crystal. Dynamical diffraction of plane waves from perfect crystals was treated at length in the early twentieth century by Darwin,1 Prins,2 Ewald3 and von Laue4; reviews of their theory have been provided by Zachariasen5 and by Batterman and Cole6 among others. Dynamical diffraction of spherical waves from absorbing perfect crystals was subsequently treated by Kato7,8 in the Laue case and by Saka, Katagawa and Kato9 in the Bragg case. Although spherical-wave dynamical diffraction has not been widely applied to synchrotron beamline modeling, the formulas for plane-wave reflectivity are routinely used for calculations of rocking curves by XOP10 and in ray traces by SHADOW.11 However, despite this long history, there is no widely available software to calculate the complete complex amplitude distribution of a synchrotron X-ray beam diffracted by a crystal. Not only the widespread use of perfect crystal monochromators, but also the increasing need for diffraction-limited focusing, make it worthwhile to fill this gap. The software package “Synchrotron Radiation Workshop,” or SRW, was written by Chubar and Elleaume. SRW propagates X-ray wavefields through any combination of drift spaces, apertures, lenses or short mirrors. However, SRW could not include perfect crystals until now. Here the integration of a new module for Bragg-case dynamical diffraction from a perfect crystal is presented. This module has been tested in the extreme cases of approximately plane-wave and spherical-wave incident beam diffracted either vertically or horizontally by Bragg reflections of arbitrary asymmetry. 2.METHOD2.1User operationThe perfect crystal propagator is implemented in the SRW C/C++ library. SRW-Python, the latest version of SRW, operates under Python 2.7 and higher. Python 3.3.3 was used to run the simulations of this article. The perfect crystal is defined in SRW-Python by the class SRWLOptCryst. The user supplies the following parameters as input:
For the tests of this article, the electric susceptibilities were calculated by the Web application code X0h.13 Only cases in which the incident and diffracted beams lie in a common plane perpendicular to the crystal surface are calculated here. Also, only horizontal and vertical beam deflection are considered as these cases permit the use of standard Fast Fourier Transform algorithms without interpolation (see §2.2). Nonetheless, the great majority of crystal monochromators installed at synchrotron beamlines can now be simulated. 2.2PrinciplesA Fourier transform is performed on the wavefront. As each Fourier component is a plane wave, plane-wave dynamical diffraction theory is used to generate the corresponding component of the diffracted wavefront. The formulas used here for evaluating the complex amplitude DH and the wave vector kH of a diffracted plane wave from an incident plane wave of complex amplitude D0 and wave vector k0 are given in Zachariasen.5 An inverse Fourier transform is then applied over all the diffracted plane waves to produce the final wavefront. The application of the discrete Fourier transform to the sampled points of the incident and diffracted wave-fronts requires some modifications for asymmetric reflections in order to convert correctly between the reciprocal and the real representation. The following set of orthogonal unit vectors is defined as fixed to the crystal:
The reciprocal lattice vector that is normal to the diffracting atomic planes is denoted by h. In the following, Another set of orthogonal unit vectors Because diffraction from crystals deflects X-ray beams through a large angle, it is convenient to choose a new laboratory coordinate system for the diffracted beam. The orthogonal unit vectors x’ and y’ and z’ are defined starting from a form of Snell’s Law, kHT = koT + hT, where are the components of ko, h and kH parallel to the crystal surface. Because |kH| = 1/λ0, Let can be used to calculate their components in the diffracted beam coordinate system. Now, without loss of generality, each of the two independent orthogonally polarized components of the frequency-domain transverse electric field of the incident wavefront may be treated separately as a scalar. Let Uo(r) be one of these components given in the (x, y, z) coordinate system. The wavefront will then be represented in reciprocal space by the Fourier transform: where Rcryst(K0) is the crystal’s complex reflectivity function as given by Zachariasen5 for a plane wave of wave vector K0 having the polarization of U0(r). KH is the diffracted beam wave vector determined from K0 by Snell’s Law. r’ = x’ Note that then to a good approximation the elements of RH0 are By substituting these into Eq. 10 and the resulting expression for KH into the Fourier transform of Eq. 9, one finds Thus the diffracted wavefront in real space is calculated by an inverse Fourier transform of the product of the plane-wave components of the incident wavefront with the respective complex reflectivities of the crystal. However, although the usual rectangular mesh of sampled points r of the incident wavefront will be transformed by Eq. 8 into a rectangular mesh of points (K0x, K0y) in reciprocal space, Eq. 13 will in general create an oblique mesh of points (x’,y’) that is not suitable for standard 2-D Fast Fourier Transform algorithms. The real-space mesh of the diffracted wavefront will be rectangular if a12 = a21 = 0 while a11 ≠ 0 and a22 ≠ 0; that is, if the deflection of the beam is either vertical (in the yz-plane) or horizontal (in the xz-plane). For vertical deflection, a11 = 1 and a22 = −|b|, where b is the asymmetry factor. For horizontal deflection, a11 = −|b| and a22 = 1. We note that the diffracted wavefront will also be rectangular if a11 = a22 = 0 while a12 ≠ 0 and a21 ≠ 0, corresponding to a 90° coordinate rotation that exchanges x’ and y’. Otherwise, the oblique mesh must be interpolated onto a rectangular mesh. 3.RESULTS3.1Crystal typeAll crystals are silicon, the most common material for X-ray monochromators. The thickness t of the crystal was set to 10 mm. Because this is many times larger than the reflections’ extinction lengths, there is no significant energy flow from the bottom surface of the crystal. 3.2Collimated Gaussian beam: Rocking curve comparison with XOP3.2.1Incident wave propertiesFirst, a linearly horizontally polarized Gaussian source of 600 µm FWHM width in both horizontal (x) and vertical (y) directions is modeled. The source produces monochromatic X-rays of 9131 eV. The Rayleigh length zR of a Gaussian beam is given by where λ is the wavelength of the radiation and w0 is the radius of the Gaussian wave at its waist, here assumed to be located at the source. The radius w(z) at any distance z along the beam axis from the waist is defined here as the distance from the axis at which the intensity drops to 1/e2 its central value. w0 is thus the FWHM at the waist divided by and the beam radius at arbitrary distances by If the crystal is placed 19 m downstream from the source, then R = 1.900 x 106 m and w is essentially the same as w0. Therefore, this source produces a highly collimated beam that can be used to test the crystal’s calculations for the plane-wave case. 3.2.2SRW rocking curves: calculation methodThe incident Gaussian wavefront at the crystal 19 m downstream from the source was calculated on a grid of points over a region twice as large as the FWHM of the Gaussian beam, which is large enough to ensure that the incident beam intensity is negligible at the edges and therefore does not cause artefacts under Fourier transformation. The number Nx x Ny of points on the grid was determined automatically by SRW. The intensities at all points on the grid were summed to obtain the total incident flux Tinc. Then the crystal propagator was applied to this wavefront, and the resulting diffracted wavefront was propagated through a 5 m drift space without any semi-analytical treatment of the leading quadratic terms of the phase. (Attempts to treat these phase terms semi-analytically created artefacts in the phase of the diffracted wavefront.) The number of points on the diffracted wave mesh remained the same as for the incident wave mesh, but the range of x or y was rescaled appropriately by the factor 1/|b|, where b is the asymmetry factor of the Bragg reflection. The intensities at all points on the grid of the final wavefront were summed to obtain the total diffracted flux Tdiff, from which SRWs value for the reflectivity, RSRW, was determined: By stepping through a set of deviations δθ from the center of the rocking curve, it was possible to have SRW generate a rocking curve that could be compared to the results of XOP’s module for flat perfect crystals, XINPRO.10 However, while XOP assumes that the origin of δθ is the angle θB that strictly fulfills Bragg’s Law, SRW sets the origin of _dTh at the true center of the rocking curve, which is shifted from θB because the diffracting crystal’s index of refraction is slightly different from the vacuum’s value of 1. The angular shift is given by Zachariasen:5 Therefore, in the figures of this section, Δθrefr was added to all values of _dTh in the SRW trials in order to shift the angular values to those of XOP. This serves as a check on SRWs ability to calculate the refraction shift correctly. 3.2.3Comparison of rocking curves of SRW and XOPInformation on the diffraction cases for which rocking curves were calculated by SRW is displayed in Table 1. For X-rays of energy 9131 eV, silicon has Ψ0 = -1.173103 × 10-5 + 2.138459 × 10-7i and, because its crystal lattice is centrosymmetric, Ψ-H = ΨH. Table 1.Diffraction cases of silicon crystals for which rocking curves calculated by SRW (see § 3.2.2) and XOP10 were compared. The SRW rocking curves were calculated using a large (600 µm FWHM) collimated Gaussian beam that approximates a plane wave (see § 3.2.1). The XOP rocking curves were calculated using standard plane-wave dynamical diffraction theory. The source is linearly horizontally polarized and produces monochromatic radiation at 9131 eV.
Figures 1 and 2 show the comparisons made between SRW and XOP for Si (1 1 1), which is used in most double crystal monochromators at X-ray synchrotron beamlines, and for Si (6 4 2), which is to be applied to a high-energy-resolution X-ray monochromator on the upcoming NSLS-II inelastic X-ray scattering beamline 10-ID. Note the good agreement between SRW and XOP in all four cases. That for the (6 4 2) reflection the angular width and peak reflectivity are lower in horizontal deflection than in vertical deflection is a result of the beam polarization relative to the “diffraction plane” that contains k0 and kH. Because the source is linearly horizontally polarized, vertical deflection (Fig. 2 (a)) results in p-polarization, in which the electric field points perpendicular to the diffraction plane, while horizontal deflection (Fig. 2(b)) results in p-polarization, in which the electric field lies within the diffraction plane. In the latter case, the strength of the Bragg reflection, which is given by ΨH, is effectively reduced by cos 2θB. Therefore the incident wave penetrates the crystal more deeply and suffers increased absorption. 3.3Divergent Gaussian beam: real-space distributions of intensity and phase at various distances from crystal3.3.1Incident wave propertiesEqs. 14-16 in §3.2.1 are now applied to a Gaussian source of 2 µm FWHM width in both horizontal and vertical directions. The diffracting silicon crystal is still placed 19 m downstream from the source. The quantities that define the Gaussian beam are as follows: This beam is strongly divergent and hence approximates a spherical wave. The linear horizontal polarization and the monochromatic spectrum at 9131 eV are as before. To check SRW’s calculations, both the intensity and the phase of the Gaussian wavefront at the crystal position are analyzed. A contour plot of the wavefront’s intensity is shown in Fig. 3, and the profiles through the center of the wavefront along x and y are shown in Fig. 4. The FWHM of the best-fit Gaussian is 568.1 µm in both directions, matching the theoretical value very well. Figure 5 shows that the phase distribution through the center of the wavefront is quadratic in both x and y, as expected for a Gaussian beam. If the phase ϕ(x) is fit to a quadratic polynomial Aphx2 + Bphx + Cph, then the radius of the wavefront is π/(λAph) = 19.0005 m, again in good agreement with theory. The same quadratic fit applied in y gives the same curvature. Figure 3.Contour plot of wavefront intensity (arbitrary units) produced by the 2 µm FWHM Gaussian source at the crystal position 19 m downstream. The horizontal and vertical axes are x in millimeters and y in millimeters, respectively.  3.3.2Bragg reflections examined with divergent Gaussian beamThe silicon crystal’s Bragg reflection is set to (6 4 2), whose spacing d and susceptibility coefficient ΨH have already been listed in Table 1. The examples are shown in Table 2. δθ = 0 to set the crystal orientation at the true center of the rocking curve. Table 2.Diffraction cases of silicon crystals for which diffraction of a divergent Gaussian beam was calculated. The source has a FWHM width of 2 µm in both horizontal and vertical directions, is linearly horizontally polarized, and produces monochromatic radiation at 9131 eV.
This case was chosen because its small Darwin width acts as an angular filter on the incident wavefront. The height of the beam after diffraction should be roughly (9.89 µrad)(19 m) = 0.188 mm, which is noticeably smaller than the incident wavefront’s height. The filtering is clear in Fig. 6, which shows the diffracted wavefront after propagation through drift spaces of various lengths. Figure 6.Contour plots of wavefront intensity (arbitrary units) produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = 0° followed by propagation through a drift space. The horizontal and vertical axes are x’ in millimeters and y’ in millimeters, respectively. The lengths of the drift spaces are (a) 1 mm, (b) 2 m, (c) 4 m, (d) 6 m, (e) 8 m and (f) 10 m.  Cross sections taken through the center of the diffracted wavefront provide further detail:
Figure 7.(a) Horizontal profile of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = 0° followed by propagation through drift spaces of lengths shown in the legend. (b) Size wx’ of best-fit Gaussian to the profiles in (a) as a function of drift space length compared to theoretical values calculated by Eq. 16.  Figure 8.(a) Horizontal phase variation of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = 0° followed by propagation through drift spaces of lengths shown in the legend. (b) Radius of the wavefront calculated from the highest term of the best-fit quadratic polynomials of the plots in (a) as a function of drift space length compared to theoretical values calculated by Eq. 15.  Figure 9.(a) Vertical profile of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = 0° followed by propagation through drift spaces of lengths shown in the legend. (b) Size wyi of best-fit Gaussian to the profiles in (a) as a function of drift space length.  Figure 10.(a) Vertical phase variation of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = 0° followed by propagation through drift spaces of lengths shown in the legend. (b) Radius of the wavefront calculated from the highest term of the best-fit quadratic polynomials of the plots in (a) as a function of drift space length compared to theoretical values calculated by Eq. 15.  Figure 11.Vertical profile of intensity and of phase residual after subtraction of best-fit quadratic polynomial for diffracted wavefront of Si (6 4 2), α = 0 after a 1 mm drift space.  This example will show how a wavefront is collimated by a highly asymmetric Bragg reflection. Because of the large asymmetry of this reflection, the vertical width of the beam that will be accepted by this reflection will be roughly (34.5 /xrad)(19 m) = 0.656 mm, which is greater than the vertical size of the incident wavefront. Figure 12 shows SRW’s calculation of the diffracted wavefront after propagation through drift spaces of various length. Note that SRW does indeed predict, in accordance with experiment, that a much larger portion of the beam will be diffracted. Also note that SRW’s diffracted wavefront spreads horizontally but not vertically with increasing distance from the crystal, indicating already that SRW predicts the expected collimation. Figure 12.Contour plots of wavefront intensity (arbitrary units) produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = +66° followed by propagation through a drift space. The horizontal and vertical axes are x’ in millimeters and y’ in millimeters, respectively. The lengths of the drift spaces are (a) 1 mm, (b) 2 m, (c) 4 m, (d) 6 m, (e) 8 m and (f) 10 m.  Cross sections taken through the center of the diffracted wavefront provide the following information:
Figure 13.(a) Horizontal profile of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = +66° followed by propagation through drift spaces of lengths shown in the legend. (b) Size wxi of best-fit Gaussian to the profiles in (a) as a function of drift space length compared to theoretical values calculated by Eq. 16.  Figure 14.(a) Horizontal phase variation of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = +66° followed by propagation through drift spaces of lengths shown in the legend. (b) Radius of the wavefront calculated from the highest term of the best-fit quadratic polynomials of the plots in (a) as a function of drift space length compared to theoretical values calculated by Eq. 15.  Figure 15.(a) Vertical profile of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = +66° followed by propagation through drift spaces of lengths shown in the legend. (b) Size wy’ of best-fit Gaussian to the profiles in (a) as a function of drift space length.  Figure 16.(a) Vertical phase variation of wavefront produced by vertically deflecting crystal diffraction of the divergent Gaussian beam by Si (6 4 2) at α = +66° followed by propagation through drift spaces of lengths shown in the legend. (b) Radius of the wavefront calculated from the quadratic term of the best-fit quadratic polynomials of the plots in (a) as a function of drift space length.  4.CONCLUSIONSA new wavefront propagator for crystal diffraction has been integrated into SRW and tested under several different Bragg reflections, crystal orientations and asymmetry angles in conjunction with drift spaces. The incident wavefront was simulated as either a highly collimated Gaussian for comparisons with plane-wave dynamical diffraction or as a highly divergent Gaussian for comparisons with spherical-wave dynamical diffraction. The tests of this report have yielded results in line with theory. Beyond the results reported here, the short-term priority is to examine further X-ray diffraction cases with fully and partially coherent radiation beams of XFEL and storage ring sources. In the medium term, more economical crystal propagators requiring lower sampling densities and less CPU time will be investigated. Future additions to the SRW package could include, first of all, a new crystal propagator module for the Laue case, which could be written with only slight modifications to the present propagator for the Bragg case. This would extend the functionality of SRW to X-ray phase plates and to crystal monochromators designed for high photon energies. In the longer term, support for diffraction from bent crystals could be added using already developed models such as Penning-Polder14 or Takagi-Taupin.15, 16 ACKNOWLEDGMENTSThe authors thank Kawal Sawhney of Diamond Light Source for his constant support and David Laundy of Diamond Light Source for stimulating discussions. The present work was supported by the US Department of Energy, Contract No. DE-AC02-98CH10886. REFERENCESDarwin, C. G.,
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