1.1

Motivation and current status

Diffracting perfect crystals are a mainstay of hard X-ray beamlines at synchrotrons and free electron lasers. The diffraction of X-rays through perfect crystals is governed by the dynamical theory, which has been reviewed by Zachariasen¹, Batterman and Cole², Authier³, and Shvyd’ko⁴. The ray tracing software packages McXTrace⁵, RAY⁶ and SHADOW⁷ have long included the calculation of X-ray reflectivity using the equations of dynamical diffraction for plane waves. However, if the X-ray beam is highly coherent, as on long synchrotron beamlines and free electron lasers, then ray tracing is no longer appropriate; one must use physical optics instead to describe the propagation of the wavefront. Despite the common use of diffracting perfect crystals and the well-known dynamical theory that describes them, software packages that can calculate X-ray wavefront propagation through diffracting perfect crystals remain scarce. The authors of PHASE⁸, a program written to calculate wavefront propagation through synchrotron beamlines, have proposed to include crystal optics⁹, but in SRW¹⁰ a module written to calculate diffraction from perfect crystals in the Bragg (reflection) geometry is now available¹¹. The latest Python version of SRW, which includes this module, can be downloaded from Github (https://github.com/ochubar/SRW). SRW can also be executed in the open-source cloud-based interface Sirepo¹² (https://www.sirepo.com/srw) and is one member of the bundle of software packages in OASYS¹³, which can also be downloaded from Github (https://github.com/oasys-kit).

The benchmarking of the SRW Bragg reflection propagator was performed on crystals whose thickness was many times larger than the Bragg reflection’s extinction length so that transmission through the crystal was negligible¹¹. This is usually the case for crystal monochromators in the Bragg geometry. However, if a diffracting crystal in the Bragg geometry is sufficiently thin or if its absorption is sufficiently low, it may transmit a significant portion of the incident wavefront. This “Bragg transmission” has not been as widely exploited as Bragg reflection, but in recent years, X-ray interferometers¹⁴ and an energy analyzer of ultra-high sub-meV resolution¹⁵ have relied on its special properties, and most of all, X-ray phase retarders have been constructed according to its principles. Therefore, this paper extends the SRW module for X-ray diffraction from perfect crystals in Bragg geometry to Bragg transmission, using phase retarders as an example of practical interest.

Phase retarders alter the polarization of the beam that passes through them. For example, a quarter-wave retarder switches the beam’s polarization between linear and circular, and a half-wave retarder rotates the beam’s plane of polarization. Because current sources of X-rays with energy > 5 keV at medium-energy synchrotrons produce only horizontally polarized beam, phase retarders are vital for measurements of electronic order, which require linear polarization of variable direction, and for X-ray circular dichroism. Therefore, they are widely used at many synchrotron beamlines, such as the Materials and Magnetism Beamline I16 of Diamond Light Source and the Integrated In Situ and Resonant Hard X-ray Studies (ISR) beamline at NSLS-II. Such practical utility is one good reason for demonstrating phase retarders in SRW, but there are two others. One is complexity: the design of most crystal optics is determined only by the intensity they diffract, but the design of a phase retarder demands accurate calculations of both intensity and phase of the diffracted wave. The other is the question of their compatibility with beam focusing, since X-ray phase retarders are sensitive to the angle between the diffracting net planes and the incident beam, and therefore are affected by the increased divergence that beam focusing introduces.

1.2

Development and operation of X-ray phase retarders

A phase retarder must be highly transparent and have a strongly anisotropic refractive index. For visible light, certain crystals such as quartz, calcite, and mica possess these useful properties. For X-rays, however, the refractive index of essentially all materials differs by only a few parts per million from that of vacuum when the beam is simply transmitted, and the absorption is often significant. The way around this lies in Molière’s realization that perfect crystals become birefringent close to a Bragg reflection, showing different refractive indices for polarization perpendicular to the “diffraction plane” spanned by the incident and reflected beams (s-polarization) and for polarization parallel to this diffraction plane (p-polarization)¹⁶. Baranova and Zel’dovich¹⁷ calculated the birefringence close to a single Bragg reflection, as well as both birefringence and gyrotropy (rotation of the polarization) when two Bragg reflections are simultaneously close to excitation. Because the degree and type of birefringence are highly sensitive to the beam’s incidence angle on the crystal, a user can rapidly vary the output polarization by a very small rotation of the crystal. Skalicky and Malgrange¹⁸ proved experimentally that perfect crystals near a Bragg reflection could indeed function as phase retarders. The effectiveness of X-ray phase retarders at synchrotron beamlines was demonstrated at the Cornell high-energy synchrotron source by Golovchenko et al¹⁹, and a quarter-wave retarder was first used to measure circular magnetic X-ray dichroism at the LURE synchrotron²⁰. The first two experiments were performed with crystals in Laue geometry, while the third was performed with a crystal in Bragg transmission. Lang and Srajer²¹ compared the practical advantages of Laue and Bragg phase retarders, and found that Bragg transmission was the best choice because it allows energy tunability and helicity switching without the need to collimate the incident beam with special optical components. X-ray phase retarders have been made from both silicon and diamond. Silicon can be manufactured in large ingots with extremely high purity and crystalline perfection, but diamond absorbs X-rays much less strongly. Diamond, therefore, is the material of choice and is sufficient if the incident beam can fit inside a single crystallographically perfect volume, as is usually the case at undulator beamlines. Bragg transmission phase retarders always operate outside the region of total reflection. When accepting a linearly polarized incident beam, they convert the polarization most efficiently when their diffracting net planes are rotated about the beam direction by 45° so that the s and p components are equal, as shown in Figure 1.

Figure 1.

Angles of rotation of phase retarder. θ sets the Bragg angle. χ determines the efficiency of the polarization conversion. The “diffraction plane” is the plane spanned by the incident beam and the reflected beam. When χ = 0, the normal to the atomic plane lies in the horizontal plane. Phase retarders are operated at χ = ±45°, where the polarization conversion is optimal.

1.3

Summary of benchmarking tests of SRW Bragg transmission module

The SRW Bragg transmission module is tested in three ways. First, the phase retarder is simulated by SRW in isolation when illuminated by a linearly horizontally polarized Gaussian beam of very large radius of curvature and hence low divergence, which may be treated as a quasi-plane wave. The total transmitted intensity, and its linearly vertically polarized and right circularly polarized components, are checked against analytical calculations derived from plane-wave dynamical diffraction theory. They are expected to agree well. Second, as a more realistic but still simple test, the phase retarder is included in an SRW simulation of the NSLS-II ISR beamline, which includes an undulator, a double-crystal monochromator, and a pair of focusing KB mirrors. Only the radiation from a single electron is used, and all optical components are ideal. These assumptions are made not only for simplicity, but also to see more clearly how and why the beam divergence affects the polarization conversion and the shape of the focal spot. This SRW simulation of the phase retarder in a beamline also opens the way to more complex simulations that take electron beam emittance and optical imperfections into account. Third, experimental measurements of the intensity converted to linear vertical polarization at various rotation angles χ are compared with both analytical formulas from plane-wave dynamical diffraction and with the SRW beamline simulation previously described. Discrepancies help to determine if the phase retarder is misaligned.

2.1

Material parameters

The phase retarders were made of diamond. The form factors were calculated from the fits of Waasmaier and Kirfel²². Anomalous dispersion corrections were determined by linear interpolation between NIST’s tabulated values at https://physics.nist.gov/PhysRefData/FFast/html/form.html²³. The temperature was 300 K. The Debye-Waller factor was calculated using the Debye model with Θ_M = 1880 K as measured by Schoening and Vermeulen²⁴. A lattice constant of 3.56712 Å and a thermal expansion coefficient of 1.06 × 10^–6 were provided for a temperature of 298 K by Stoupin and Shvyd’ko²⁵.

2.2

First benchmark: quasi-plane wave

If the incident wave is planar and the crystal is unbounded, the results for intensity and output polarization can be computed from the analytical equations derived from dynamical diffraction theory as in Giles et at²⁰. In an SRW simulation, the incident wave must be spatially bounded, but an approximation to a plane wave can be obtained by using a monochromatic, horizontally linearly polarized Gaussian of large cross section. In this simulation, the Gaussian source has a root mean square source size of 600 μm in both the horizontal and the vertical directions. One of the two symmetric (111) diamond phase retarders used at the ISR beamline of NSLS-II is placed 36.2 m downstream from the source. These phase retarders, along with the photon energy and the radius of the incident wavefront, are given in Table 1. Because the radius of the wavefront at the entrance of each phase retarder is very large, the wavefront can indeed be considered essentially flat. Figure 2 compares the transmission of a quasi-plane wave of 8708 eV through the 510 μm thick phase retarder as calculated by SRW with that of a true plane wave as predicted by dynamical diffraction theory and also with measurements taken at NSLS-II ISR. Figure 3 makes the same comparisons for a quasi-plane wave of 6844 eV and the 240 μm thick phase retarder. The agreement between the SRW simulation and plane-wave dynamical diffraction theory is excellent, as it should be. The measurements of the linearly vertically polarized intensity in part (d) of both figures are similar to the calculations in part (a), although the maxima do not appear at quite the expected angles. This already indicates that there are errors in the alignment of the phase retarder in the beamline, as will be discussed later.

Table 1.

Thickness of the diamond symmetric (111) phase retarders used at NSLS-II ISR, photon energies, and radii of quasi-planar wavefront entering each phase retarder in the first benchmark.

Figure 2.

First benchmark: 510 μm thick symmetric (111) diamond phase retarder in 8708 eV X-rays. (a)-(c): Check of SRW simulation of phase retarder in quasi-plane wave (red squares) against plane-wave dynamical diffraction theory (black line). LV = linear vertical polarization, CR = right circular polarization. (d) Experimental measurements collected at the NSLS-II ISR beamline for the linearly vertically polarized beam downstream from the phase retarder. The vertical dashed lines mark the positions of the maxima in (a). The polarization analyzer used to separate out the linearly vertically polarized component was a LiF crystal oriented to the (400) Bragg reflection. The plots were recorded in steps of 5° in χ.

Figure 3.

First benchmark: 240 μm thick symmetric (111) diamond phase retarder in 6844 eV X-rays. (a)-(c): Check of SRW simulation of phase retarder in quasi-plane wave (red squares) against plane-wave dynamical diffraction theory (black line). LV = linear vertical polarization, CR = right circular polarization. (d) Experimental measurements collected at the NSLS-II ISR beamline for the linearly vertically polarized beam downstream from the phase retarder.

The vertical dashed lines mark the positions of the maxima in (a). The polarization analyzer used to separate out the linearly vertically polarized component was a Cu crystal oriented to the (220) Bragg reflection. The plots were recorded in steps of 5° in χ.

2.3

Second benchmark: idealized version of NSLS-II ISR beamline

Here only the 240 μm thick phase retarder at 6844 eV is examined. The optical layout of the NSLS-II ISR beamline is shown in Figure 4. The source is an ideal U23 undulator of 2.8 m length. Its gap is adjusted so that the desired photon energy falls on its third harmonic. The undulator radiation is linearly polarized in the horizontal direction. Only singleelectron radiation is considered here. The horizontally focusing mirror (HFM) is horizontally deflecting and tangentially curved. The grazing angle of incidence at its center is 2.618 mrad. Its dimensions are 1 m long and 0.1 m wide. The vertically focusing mirror (VFM) is vertically deflecting and tangentially curved. Its grazing angle of incidence and its dimensions are the same as those of the HFM. The double-crystal monochromator (DCM) is composed of two non-dispersively arranged, vertically deflecting symmetric Si (111) crystals. The phase retarder is set to χ = 45°. Thermal distortions and figure errors on the optical components are neglected.

Figure 4.

Optical layout of NSLS-II beamline. Copied from Sirepo interface¹². HFM = horizontally focusing mirror. DCM = double-crystal monochromator. VFM = vertically focusing mirror.

Figure 5 compares the SRW simulations of the phase retarder’s total and polarization-resolved transmission on the idealized NSLS-II ISR beamline with the analytical calculations of plane-wave dynamical diffraction theory. The total transmission calculated by SRW as a function of deviation from the Bragg angle tracks the plane-wave calculations very well. However, the minima and maxima of the linearly horizontally (LH) polarized and the linearly vertically (LV) polarized components are less prominent in the SRW simulations. The sharper these extrema are in the plane-wave calculations, the more severely they are weakened when the undulator beam is used. Also, at the edges of the total reflection region, where the plane-wave calculations show a rapid oscillation of the transmitted beam between LH and LV polarization, the SRW calculations with undulator beam show an averaged value for both polarizations. Figure 6 shows that this smearing of the angular dependence of the LH and LV polarization components results from the nonzero divergence of the undulator beam. Figure 6(a) displays the LV component of the wavefront 1 m downstream from the phase retarder on the idealized NSLS-II ISR beamline as simulated by SRW when θ is –75 μrad from the center of the total reflection region. At this point, Figure 5(c) shows that the LV transmission through the phase retarder reaches a minimum. The incomplete suppression of the LV transmission in the beamline simulation results from the leakage of the outer sections of the undulator beam. One may similarly see the leakage of the LH component of the outer parts of the undulator beam when θ is –50 μrad from the center of the total reflection region, where Figure 5(b) shows that the LH transmission reaches a minimum. Figure 6(c) and (d) show the spatial distribution of, respectively, the LH and LV components of the wavefront 1 m downstream from the phase retarder on the idealized NSLS-II ISR beamline when θ is –20 μrad from the center of the total reflection region. Here, the rapid oscillation of polarization with θ seen in Figure 5(b) and (c) is mirrored in the fringes of the spatial distribution.

Figure 5.

Second benchmark: SRW simulations of idealized NSLS-II ISR beamline (red squares) compared with analytical calculations from dynamical diffraction theory (black line). (a) Total transmission through phase retarder, including all polarization components. (b) Linearly horizontally (LH) polarized component of transmission through phase retarder. (c) Linearly vertically (LV) polarized component of transmission through phase retarder.

Figure 6.

SRW simulations of spatial distribution of LH and LV components of single-electron wavefront at NSLS-II ISR beamline, 1 m downstream from the phase retarder. The field of view in each image is 1.38 mm horizontal × 1.66 mm vertical. (a) and (b) are taken, respectively, at a minimum for LV transmission and at a minimum for LH transmission. (c) and (d) are the LH and LV components when the phase retarder is oriented to the left edge of the total reflection region, where the polarization of the transmitted beam oscillates rapidly with incidence angle.

The focusing of the beam onto a sample may bring forth more subtle effects. Kato²⁶ showed that the direction of the time- and space-averaged energy flow of the wavefields propagating in a perfect crystal near a Bragg reflection depends on the incidence angle of the X-ray beam. The energy flow will lie between the direction of the incident beam and that of the diffracted beam, and a μrad rotation of the crystal relative to the incident beam may change the direction of the energy flow by several degrees. This would in principle cause a small displacement of the beam transmitted through the phase retarder as θ is varied to change the beam polarization. Figure 7 compares the position of the LV component of the beam at the focal spot for Δθ = –150 μrad, which is an LV maximum, and for Δθ = –100 μrad, which is close to the next LV minimum. The beam shifts from the upper left to the lower right by about 1 μm. This does not seem very large, but is still a significant fraction of the idealized FWHM focal spot size of 13.3 μm horizontal × 8.7 μm vertical.

Figure 7.

Spatial distribution of LV component of single-electron wavefront of idealized NSLS-II ISR at focal point (the sample) as calculated by SRW. (a) Δθ = –150 μrad. (b) Δθ = –100 μrad. (c) Image (a) – Image (b), showing the shift of about 1 μm in the beam’s position. The FWHM of the focal spot is 13.3 μm horizontal × 8.7 μm vertical. The field of view in each image is 38.2 μm horizontal × 18.9 μm vertical.

Finally, one can examine whether the LV component of the focal spot changes in shape if Δθ is changed. Figure 8 shows the LV component of the idealized NSLS-II ISR beamline’s focus at two LV maxima: θ = –150 μrad and Δθ = –50 μrad. Figure 5(c) shows that the first maximum has a very gradual dependence on Δθ, while the second is very sharp and thus is smeared out by the nonzero divergence of the focused undulator beam. The plots do in fact reveal that the profile for the second, sharper maximum is slightly narrower than the profile for the broader first maximum. The oscillations in the second maximum are also less prominent.

Figure 8.

SRW simulation of LV component of focal spot at sample position of idealized NSLS-II ISR beamline at two LV maxima. (a) Δθ = –150 μrad (b) Δθ = –50 μrad. (c) Profiles of both focal spots along dotted lines labelled “Profile” in (a) and (b).

2.4

Comparison of SRW with experimental data

The experimental apparatus used to vary the beam polarization and measure its LV component on the NSLS-II ISR beamline is shown in Figure 9. In order to minimize air absorption at lower photon energies, the beam from the phase retarder is sent through an evacuated flight path on its way to the analyzer crystal, which deflects the beam horizontally with a Bragg angle of 45°. At 8708 eV, the analyzer is a LiF crystal oriented to the symmetric (400) Bragg reflection. At 6844 eV, the analyzer is a Cu crystal oriented to the symmetric (220) Bragg reflection. This analyzer suppresses the LH component of the beam from the phase retarders, leaving only the LV component to be passed onward to the avalanche photodiode detector. The intensity recorded by the avalanche photodiode is normalized to the total transmitted intensity recorded by an ion chamber between the phase retarder and the polarization analyzer.

Figure 9.

Photographs of polarization-modifying apparatus on the NSLS-II ISR beamline. (a) Diamond phase retarders mounted on a Huber chi circle. (b) Assembly of analyzer and detector mounted on 2θ arm of Huber 6-circle diffractometer.

As before, for this paper, the 240 μm thick phase retarder at 6844 eV is examined. Figure 10(a) displays a series of experimental scans in θ of the LV component transmitted by the phase retarder at values of χ from –50° to +5°. Figure 10(b) shows a series of theoretical LV scans in θ at various values of χ as predicted by plane-wave dynamical diffraction. The experimental scans look similar to the theoretical calculations, but close examination reveals a subtle discrepancy: while the positions of the LV maxima are independent of χ in the theoretical calculations, the left-hand LV maximum in the experimental data shifts progressively toward the center of the rocking curve as χ increases beyond –25°.

Figure 10.

Scans in θ at various values of χ. (a) Experimental measurements of normalized LV intensity downstream from 240 μm thick phase retarder (same as in Figure 3(d)). (b) Plane-wave analytical calculations of the ideal LV transmissivity.

Furthermore, one would naively expect the LV intensity at the maxima to have a sin²(2χ) dependence. This is borne out by Figure 11(a), which shows the plane-wave dynamical diffraction calculation, and Figure 11(b), which provides the SRW simulation on the idealized NSLS-II ISR beamline. However, the dependence of the experimental LV maxima on sin²(2χ), plotted in Figure 11(c), is imperfect. One possible cause, which remains to be investigated, is that the length of the beam path through the phase retarder does not remain constant as χ is varied because the phase retarder is misaligned in the beam.

Figure 11.

(a) Fits to sin²(2χ) of theoretical LV transmissivity of phase retarder calculated by plane-wave dynamical diffraction at the left-hand (black squares) and right-hand (red circles) LV maxima. Solid curves are the best fits. (b) Fits to sin²(2χ) of theoretical LV intensity downstream from phase retarder calculated by SRW on the idealized NSLS-II ISR beamline at the left-hand (black squares) and right-hand (red circles) LV maxima. Solid curves are the best fits. (c) Fits to sin²(2χ + φ) of experimental measurements of LV intensity from phase retarder at the left-hand (black squares) and right-hand (red circles) LV maxima. Solid curves are the best fits.