In the recently developed technique of X-ray Birefringence Imaging, molecular orientational order in anisotropic materials is studied by exploiting the birefringence of linearly polarized X-rays with energy close to an absorption edge of an element in the material. In the experimental setup, a vertically deflecting high-resolution double-crystal monochromator is used upstream from the sample to select the appropriate photon energy, and a horizontally deflecting X-ray polarization analyzer, consisting of a perfect single crystal with a Bragg reflection at Bragg angle of approximately 45°, is placed downstream from the sample to measure the resulting rotation of the X-ray polarization. However, if the experiment is performed on a synchrotron bending-magnet beamline, then the elliptical polarization of the X-rays out of the electron orbit plane affects the shape of the output beam. Also, because the monochromator introduces a correlation between vertical position and photon energy to the X-ray beam, the polarization analyzer does not select the entire beam, but instead selects a diagonal stripe, the slope of which depends on the Bragg angles of the monochromator and the polarization analyzer. In the present work, the final background intensity distribution is calculated analytically because the phase space sampling methods normally used in ray traces are too inefficient for this setup. X-ray Birefringence Imaging data measured at the Diamond Light Source beamline B16 agree well with the theory developed here.

## I. INTRODUCTION

The technique of X-ray Birefringence Imaging, which we have reported very recently (Palmer *et al.*, 2014; 2015), represents in many respects the X-ray analogue of the polarizing optical microscope, which has been exploited across many scientific disciplines since the 19th Century as an instrument for exploring the structural anisotropy of materials. However, while the phenomena of optical birefringence (on which the operation of the polarizing optical microscope is based) and X-ray birefringence share several common characteristics, it is important to emphasize that optical birefringence relates to the anisotropy of the material as a whole (e.g., for a crystal, it depends on the symmetry of the crystal structure) whereas X-ray birefringence, when studied using an X-ray energy close to the absorption edge of a specific type of atom in the material, depends on the *local* anisotropy in the vicinity of the selected type of atom. As X-ray birefringence depends on the orientational properties of the bonding environment of the X-ray absorbing atom, studying the X-ray birefringence of a material has the potential to yield local structural information on the orientational properties of individual molecules and/or bonds. Initial studies of the phenomenon of X-ray birefringence (Palmer *et al.*, 2011; 2012), using a narrowly focused X-ray beam (and hence unable to provide spatially resolved mapping of the material), were carried out on anisotropic crystals containing brominated organic molecules. These studies showed that, with the incident X-ray energy tuned to the bromine K-edge, X-ray birefringence depends strongly on the orientation of the C–Br bonds in the material. The results demonstrated clearly that X-ray birefringence is a reliable strategy for determining the orientational properties of specific bonds in solids and for understanding changes in bond orientations that occur as a function of temperature (e.g., in materials that undergo order-disorder phase transitions).

Subsequently, we proposed and demonstrated (Palmer *et al.*, 2014; 2015) an experimental setup (described in more detail below) that allows X-ray birefringence measurements to be carried out in a *spatially resolved imaging mode*, representing the technique of X-ray Birefringence Imaging. With this setup, the X-ray birefringence of the material is mapped in a spatially resolved manner with resolution of the order of 10 *μ*m. Indeed, using a model material (the 1-bromoadamantane/thiourea inclusion compound in which all C–Br bonds are parallel to the crystal *c*-axis, which behaves as the “X-ray optic axis”), the observed intensity variation as a function of crystal orientation was found to be directly analogous to the behaviour of a uniaxial crystal in the polarizing optical microscope. X-ray Birefringence Imaging was also demonstrated to be a sensitive technique for characterizing changes in molecular orientational ordering associated with solid-state phase transitions, revealing *inter alia* the capability of the technique to identify and characterize “domain structures” (in which different regions of the material comprise orientationally distinct arrangements of the component molecules), leading to an understanding of the size, spatial distribution, and temperature dependence of the domain structures.

In the experimental setup for X-ray Birefringence Imaging (Figure 1), the sample is illuminated with linearly polarized X-rays tuned to an absorption edge of a selected element in the sample. Downstream from the sample, a polarization analyzer is oriented to select the orthogonal polarization state, and an imaging detector is in turn placed downstream from the analyzer. In principle, if no sample is present, no intensity should reach the detector, and thus X-ray Birefringence Imaging is a type of dark-field microscopy. When a birefringent sample is present, the sample can rotate the polarization of the incident linearly polarized X-rays, permitting a portion of the incident X-ray intensity to reach the detector. The angle of this rotation, and hence the intensity recorded at the detector, depends on the orientation of the X-ray optic axis of the sample (i.e., the C–Br bond orientation in the examples described above) relative to the incident polarization state. Therefore, a series of images recorded as a function of the orientation of the sample relative to the incident X-ray beam can yield detailed insights into the orientational properties of the molecules within the material.

In the work published so far (Palmer *et al.*, 2014; 2015), X-ray Birefringence Imaging was performed using radiation produced by the bending magnet B16 of the Diamond Light Source synchrotron. Bending magnets produce X-ray beams of large cross-section (several millimeters) with a high degree of linear polarization in the horizontal plane, which is the plane of the electron orbit inside the storage ring. A vertically deflecting silicon double-crystal monochromator (DCM) set to the (1 1 1) Bragg reflection was used to select the photon energy, which in our previous experiments was 13 493 eV (Palmer *et al.*, 2014) or 13 495 eV (Palmer *et al.*, 2015). This energy is very close to the bromine K-edge, and was selected as the energy at which the X-ray birefringence is maximized while X-ray dichroism is relatively modest (thus, at this specific energy, the polarization-dependent contrast in the X-ray Birefringence Imaging data is maximized). For the theoretical calculations in the present work, we take the bromine K-edge energy as 13 474 eV (Bearden and Burr, 1967). The horizontally deflecting analyzer, which selects X-rays with vertical polarization, was either a silicon or a germanium single crystal oriented to the (5 5 5) Bragg reflection, for which the Bragg angle *θ _{B}* at a photon energy of 13 474 eV is 47.1933° or 44.7680°, respectively. An ideal analyzer would have $\theta B=45\xb0$, thus deflecting the X-ray beam by $2\theta B=90\xb0$; the (5 5 5) reflection is the one that most nearly approaches this ideal at the chosen photon energy. The overall setup of the X-ray birefringence imaging experiments at B16 is shown in Figure 1.

However, this apparatus differs from the ideal setup in several crucial ways that could lead to confusion or misinterpretation of the data if not thoroughly understood. These issues are now described.

### A. Angular dependence of the polarization of the bending-magnet beam

When the electrons circulating inside the storage ring pass between the poles of a bending magnet, their path is curved. Relativistic charged particles traveling on a circular arc produce well-collimated, wide-bandwidth electromagnetic radiation (synchrotron radiation). The high collimation is advantageous for use with diffracting crystal monochromators, the bandpass and efficiency of which are limited by their small angular acceptance. Furthermore, the wide bandwidth offers flexibility in choosing a desirable energy. However, only synchrotron radiation generated exactly in the plane of the electron orbit is fully horizontally polarized. The radiation outside this plane is elliptically polarized, with opposite helicities depending on whether the viewing point is above or below the horizontal plane. Jackson (1975) has shown that for a single electron the energy radiated with horizontal polarization per unit frequency interval *dω* per unit solid angle *d*Ω about a vertical angle of observation *β* is

where *R* is the radius of the arc travelled by the electron, *γ* is the total electron energy divided by *m _{e}c*

^{2}where

*m*is the electron mass, $\xi =\omega R3c(1\gamma 2+\beta 2)3/2$, and

_{e}*K*(

_{ν}*x*) is the modified Bessel function of order

*ν*. The corresponding quantity for vertical polarization is

The synchrotron radiation occurs within a narrow cone of angular width $\u223c1/\gamma $. Numerical calculations of the horizontal and vertical polarization components of the synchrotron radiation at B16 are provided in Sec. II A.

### B. Imperfect polarization suppression by the polarization analyzer

If the horizontally deflecting analyzer were set to a Bragg angle of exactly 45°, it would select only vertical polarization and would completely block all horizontally polarized X-rays. However, silicon and germanium crystals offer only a limited choice of Bragg reflections; therefore, in general it is not possible to find one with a Bragg angle of exactly 45° for the chosen X-ray energy. Because the analyzer is a perfect crystal, dynamical diffraction theory is used here to calculate the effectiveness of the analyzer for discriminating between horizontal and vertical polarization.

### C. Angle-energy dispersion introduced by Bragg reflections at the monochromator and polarization analyzer crystals

The wavelength *λ* of X-rays selected by Bragg diffraction from the $(hkl)$ lattice planes of a cubic crystal with lattice parameter *a* varies with the grazing angle of incidence *θ* of the X-rays according to Bragg's Law $2dhkl\u2009sin\u2009\theta =\lambda $, where $dhkl=a/h2+k2+l2$ is the perpendicular spacing between adjacent $(hkl)$ planes. This formula for $dhkl$ is valid for both silicon and germanium, which have cubic crystal lattices.

For a monochromator crystal (for which silicon has been used in our experimental work), if the central grazing incidence angle of the X-ray beam on the crystal is *θ _{B}*, then the wavelength

*λ*of the central ray selected by the crystal is $\lambda B=2dhkl\u2009sin\u2009\theta B$. Another ray with a slightly different incidence angle $\theta B+\beta $ fulfils Bragg's Law if it has a slightly different wavelength $\lambda =\lambda B+\Delta \lambda $, where $\Delta \lambda $ can be estimated by differentiation of Bragg's Law with respect to

_{B}*θ*

For the bending magnet radiation, the energy bandwidth is so broad and the vertical angular width is so small that its power spectrum may be considered constant across the DCM's bandpass of several eV over all significant values of *β*. However, the DCM selects slightly different wavelengths at different values of *β* and thus introduces an angle-dependent dispersion given by Eq. (3).

Similarly, for the horizontally deflecting polarization analyzer, which uses the $(h\u2032k\u2032l\u2032)$ Bragg reflection and for which the central incidence angle of the X-ray beam is $\theta B\u2032$, we may deduce

where *α* denotes a horizontal angle of deviation from the central ray.

## II. CALCULATIONS

### A. Bending magnet: Polarization distribution

Each electron circulating inside the Diamond Light Source storage ring carries a total energy close to 3.0 GeV. The usual current in the storage ring is 300 mA. The electrons pass through a bending magnet that exerts a vertical magnetic field of 1.44 T and therefore follow a curved path of 6.981 m radius in the horizontal plane. The emittance of the electron pulses is $\epsilon x=2.6\u2009nm\xb7rad$ horizontally and $\epsilon y=0.008\u2009nm\xb7rad$ vertically. If energy spread is neglected, the distribution of the electrons in the phase space $(x,x\u2032)$, where *x* is the horizontal position and $x\u2032$ is the horizontal angle, at any point along the electron beam path is related to the Twiss parameters $\alpha x$, $\beta x$, and $\gamma x$ by

where $\beta x\gamma x\u2212\alpha x2=1$. Although the Twiss parameters vary along the electron beam path, the emittance remains constant. In the same way, in the phase space $(y,y\u2032)$

The waist of the electron beam in the *x* or *y* direction is the position along the electron beam path where $\alpha x=0$ or $\alpha y=0$, respectively. If, at some initial position, the Twiss parameters of the electron beam are $\alpha j0$, $\beta j0$, and $\gamma j0$ ($j=x,y$), then after travelling a distance *s* these parameters change to

Therefore, the distance from the initial position to the waist is $swj=\alpha j0/\gamma j0$.

The source of the radiation entering B16 is neither at a waist of the electron beam nor at the center of the bending magnet but is 107 mm along the electron beam path from the entrance to the bending magnet. At this position, approximate values of the Twiss parameters are $\alpha x=0.573$, $\beta x=0.574\u2009m$, $\alpha y=1.678$, $\beta y=23.346\u2009m$, yielding $swx=0.248\u2009m$ and $swy=10.267\u2009m$. A more sophisticated calculation of the waist position would include the nonzero energy spread of the electron beam and the associated dispersion $\eta j(s)$, defined by

where *δ _{x}* and

*δ*are the displacements along

_{y}*x*and

*y*, respectively, of an electron with energy

*E*slightly different from the central energy

_{e}*E*of 3.0 GeV. The dispersion in each direction propagates along the electron beam path according to the relations $\eta j(s)=\eta j0+\eta j0\u2032s$, $\eta j\u2032(s)=\eta j0\u2032$. At the source point, $\eta x=0.0328\u2009m$ and $\eta x\u2032=\u22120.0835$, while the vertical dispersion is negligible throughout the ring. The position of the horizontal waist is therefore adjusted to 0.325 m downstream from the source point, thus placing it close to the center of the 0.933 m long bending magnet. At the horizontal waist, the root mean square (rms) horizontal beam size is $\sigma x=34.5\mu m$, and at the

_{e0}*virtual*vertical waist the rms vertical beam size is $\sigma y=7.0\mu m$.

Ray-tracing simulations of the bending magnet radiation were performed with SHADOW (Sánchez del Río *et al.*, 2011) using the emittances, waist sizes, and waist positions given above. The horizontally and vertically polarized components were calculated with 500 000 rays at 13 474 eV. The total intensity of each ray is normalised to 1; however, the rays are distributed pseudo-randomly in angle and position according to the appropriate probability distribution. The angular width of the horizontal aperture was 0.4 mrad. No aperture was used to limit the vertical spread of the rays. The vertical angles of all the rays were organized in a 200-bin histogram, and in each bin the sum of the intensities of all the rays was calculated in each polarization component. The results are shown in Figure 2. Because the positions and angles of the rays have been calculated stochastically, some random noise is left in the computation of the histogram; therefore, a Fourier transform low-pass filter was applied to obtain a smoothed distribution of intensity as a function of vertical angle *β*.

### B. Polarization dependence of crystal reflectivities

Because both the monochromator and the analyzer are composed of large, essentially perfect crystals, X-ray diffraction by these crystals must be treated using 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. Many reviews of dynamical diffraction are available; that of Zachariasen (1945) is used here. The crystal is treated as a medium for which the dielectric constant *ε* is scalar but has the same spatial periodicity as the crystal lattice. For X-rays, $\epsilon =1+\psi $, where $\psi \u223c10\u22125$. A Fourier series may be written for *ψ*

where **r** is the real-space position vector and **G** is a reciprocal lattice vector of the crystal. Each Fourier component $\psi G$ is related to the corresponding structure factor $FG$ by

where $qe$ is the electron charge, $me$ is the electron mass, $\omega 0$ is the angular frequency of the incident X-rays, and $V$ is the volume of the unit cell of the crystal. Now, let $k0$ be the wave vector of a monochromatic plane wave incident on the crystal ($|k0|=1/\lambda $), let $n\u0302$ be the unit inward normal to the crystal surface, and let the crystal be oriented so that only the single reciprocal lattice vector $H$ fulfills the Bragg condition. (Note that $H$ is perpendicular to a set of lattice planes in real space with interplanar spacing $d$ equal to $1/|H|$.) Then, we can define an asymmetry parameter $b$

and a deviation parameter $A$

By solving Maxwell's equations for the electric displacement inside the crystal and connecting it with the electric displacement in the vacuum outside the crystal with appropriate boundary conditions, we obtain the following formula for the X-ray reflectivity $RH$ from a thick crystal:

where

and $K$ is a polarization factor equal to 1 if the incident X-rays are polarized normal to the diffraction plane that contains $k0$ and $H$(*s*-polarization) and $|\u2009cos\u20092\theta B|$ if the X-rays are polarized parallel to the diffraction plane (*p*-polarization). If the wavelength is held constant while the incidence angle $\theta $ of the X-ray beam is varied near the Bragg angle $\theta B$, and if the Bragg angle is not close to 90°, then

and thus we can obtain the rocking curve $RH(\theta )$ from Eq. (16).

The lattice parameters for silicon and germanium at 22.5 °C are *a*_{Si} = 5.4310 Å (Mohr *et al.*, 2012) and *a*_{Ge} = 5.6578 Å (Baker and Hart, 1975), respectively. For the atomic form factor calculations, the fits of Waasmaier and Kirfel (1995) have been applied. Anomalous dispersion corrections to the form factors are linearly interpolated from the tables of Henke *et al.* (1993). Debye-Waller factors are determined at 22.5 °C using a Debye model with characteristic temperature $\Theta M$ of 543 K for silicon and 290 K for germanium (Batterman and Chipman, 1962). Silicon and germanium both have the diamond crystal structure (space group $Fd3\xafm$); for the structure factor calculations, the origin of the real-space crystal lattice is placed at a midpoint of a nearest-neighbour bond, which is a centre of inversion. Then, $\psi \u2212H=\psi H$. The resulting values of $\psi H$ are shown in Table I. Because all the diffracting crystals are cut so that their incident and diffracted beams make the same angle with the crystal surface (the “symmetric” geometry), all the rocking curves are calculated with $b=\u22121$ as displayed in Figure 3. For the Si (1 1 1) DCM, there is very little difference between the reflectivities for *s*- and *p*-polarization. This is because of the small Bragg angle of 8.4375°, which would make $K=0.9569$ for *p*-polarization. However, for the Si (5 5 5) analyzer, the Bragg angle is 47.1933° and thus $K=0.07649$ for *p*-polarization, resulting in the strong suppression of this component. For the Ge (5 5 5) analyzer, the Bragg angle is 44.7680° and thus $K=0.008098$ for *p*-polarization, so that the suppression of this component is almost complete. However, note that the reflectivity of the *s*-polarized component is drastically reduced as well because germanium absorbs the X-rays much more strongly than silicon.

Material . | $H$ $(h\u2009k\u2009l)$ . | $\psi 0r$ . | $\psi 0i$ . | $\psi Hr$ . | $\psi Hi$ . |
---|---|---|---|---|---|

Si | (1 1 1) | $\u22125.35702\xd710\u22126$ | $\u22124.50233\xd710\u22128$ | $+2.82736\xd710\u22126$ | $+3.14769\xd710\u22128$ |

Si | (5 5 5) | $\u22125.35702\xd710\u22126$ | $\u22124.50233\xd710\u22128$ | $\u22127.86237\xd710\u22127$ | $\u22122.39693\xd710\u22128$ |

Ge | (5 5 5) | $\u22121.05332\xd710\u22125$ | $\u22129.66306\xd710\u22127$ | $\u22121.70992\xd710\u22126$ | $\u22124.88988\xd710\u22127$ |

Material . | $H$ $(h\u2009k\u2009l)$ . | $\psi 0r$ . | $\psi 0i$ . | $\psi Hr$ . | $\psi Hi$ . |
---|---|---|---|---|---|

Si | (1 1 1) | $\u22125.35702\xd710\u22126$ | $\u22124.50233\xd710\u22128$ | $+2.82736\xd710\u22126$ | $+3.14769\xd710\u22128$ |

Si | (5 5 5) | $\u22125.35702\xd710\u22126$ | $\u22124.50233\xd710\u22128$ | $\u22127.86237\xd710\u22127$ | $\u22122.39693\xd710\u22128$ |

Ge | (5 5 5) | $\u22121.05332\xd710\u22125$ | $\u22129.66306\xd710\u22127$ | $\u22121.70992\xd710\u22126$ | $\u22124.88988\xd710\u22127$ |

As a final note for Sec. II C, the rocking curve $RH$ may be written in terms of a variable $\eta $ defined as

where $z$ is defined in Eq. (18). The peak of $RH(\eta )$ lies in the region $|\eta |\u22641$.

### C. Selection of X-rays through the monochromator and polarization analyzer in crossed geometry

The crossed X-ray monochromator and polarization analyzer are shown schematically in Figure 4. A right-handed Cartesian set of axes $x\u0302$, $y\u0302$, and $z\u0302$ is defined so that $+x\u0302$ points along the direction of the central X-ray and $+y\u0302$ points upward. A general wave vector $k$ can be defined by its magnitude $k$, horizontal deviation angle $\alpha $ and vertical deviation angle $\beta $ by the relation

$k0$ is defined as $k(k=k0,\alpha =0,\beta =0)$. The intensity of each ray as it is created by the source is taken as 1. All crystals are symmetrically cut and are assumed to be perfectly aligned relative to the X-ray beam and to each other. Thermal distortion of the DCM crystals is neglected.

Now, let $H1$ be the reciprocal lattice vector that is normal to the diffracting planes of the first crystal of the DCM. The interplanar spacing is then $d1=1/|H1|$. Let $\theta 1$ be the grazing incidence angle of the central ray of the beam on these diffracting atomic planes. Then

where $\theta i1$ is the grazing incidence angle of the wave vector $k$ on the diffracting atomic planes of the first DCM crystal. Now assume that $k0$ fulfils Bragg's Law on the first DCM crystal; that is, $\lambda 0=k0\u22121=2d1\u2009sin\u2009\theta 1$. It is shown in Shvyd'ko *et al* (1998) that the deviation parameter $A1$ defined in Eq. (15) is

and, using the relation of $k0$ to $d1$ and $\theta 1$, we obtain a second expression for $\theta i1$

Now, Eq. (26) can be expanded in a Taylor series about $\alpha =0$, $\beta =0$, and $k=k0$ because $\alpha $ and $\beta $ are much smaller than 1 (see Figure 2) and $\delta k=k\u2212k0$ is much smaller than $k$ for all rays that are transmitted through the DCM. Furthermore, $A1$ is also very small (typically $\u223c10\u22125$) for those rays that are diffracted by the DCM with significant efficiency. Taking the lowest-order terms in each quantity $\alpha $, $\beta $, and $\delta k/k0$ and ignoring the product of these terms with $A1$ or with each other, we obtain

If the second DCM crystal is aligned exactly parallel to the first, it diffracts all rays from the first DCM crystal back into their original direction. Hence, the wave vector $k$ defined in Eq. (21) is also valid for the X-rays incident on the polarization analyzer. However, now the reciprocal lattice vector that is normal to the diffracting atomic planes of the analyzer is

where $d2$ is the spacing between consecutive diffracting atomic planes of the analyzer and $\theta 2$ is the grazing angle of incidence of the central ray of the X-ray beam on these planes. Proceeding just as for the first DCM crystal, we find

where $\theta i2$ is the grazing incidence angle of the wave vector $k$ on the diffracting atomic planes of the analyzer. When the analyzer is perfectly aligned with respect to the DCM, then the wave vector $k0$ of the central ray fulfils the Bragg condition $\lambda 0=k0\u22121=2d2\u2009sin\u2009\theta 2$. The deviation parameter $A2$ from the Bragg condition is then

Expanding this in a Taylor series up to the lowest-order terms in $\alpha $, $\beta $, and $\delta k$ yields

Note that this is exactly like Eq. (27) except that the subscript 1 has been replaced by 2 and that the angles $\alpha $ and $\beta $ have been switched.

Before continuing, it is relevant to give a brief explanation of the meaning of Eqs. (27) and (32). Equation (27) shows the relationship between $\alpha $, $\beta $, $\delta k$, and $A1$ for all rays that are selected by the DCM. Consider, for example, only the rays in the *xy*-plane ($\alpha =0$) that have the same deviation $A1$ from the Bragg condition. Equation (27) then reduces to $\delta kk0=\u2212(\beta +A12\u2009sin\u20092\theta 1)cot\theta 1$, which yields in quantitative form the angle-wavelength dispersion described in Sec. I C. Similarly, if we consider only rays with equal $\delta k$ and equal $A1$, the resulting quadratic relationship between $\alpha $ and $\beta $ shows the correction that must be made to the Bragg condition for nonzero horizontal angles. Equation (32) shows the relationship between $\alpha $, $\beta $, $\delta k$, and $A2$ for all rays that are selected by the analyzer.

If the quadratic terms in Eqs. (27) and (32) are neglected, there remains a system of two linear equations in the five variables $\alpha $, $\beta $, $\delta k$, $A1$, and $A2$. Three of these variables determine the other two. Choosing $\alpha $, $A1$, and $A2$ as the independent variables, we find

Note that Eq. (34) is simply the angle-wavelength dispersion introduced to the beam by the first DCM crystal.

If $A1$ and $A2$ are neglected, then the relationships among the remaining three variables $\alpha $, $\beta $, and $\delta k$ for X-rays transmitted through the crossed monochromator/analyzer setup in this simple linear approximation may be understood graphically by an extension of the DuMond diagram (DuMond, 1937) to three dimensions. The ordinary DuMond diagram is simply a graph of Bragg's Law (wavelength or photon energy versus incidence angle) over a very small angular range in which the wavelength (photon energy) variation may be treated as essentially linear. For the crossed monochromator/analyzer setup, the monochromator is represented by drawing a line in the $(\beta ,\delta k)$ plane with a slope given by the right-hand side of Eq. (34), then defining a plane that is parallel to this line and to the $\alpha $-axis. The analyzer is represented by drawing a line in the $(\alpha ,\delta k)$ plane with a slope given by the middle expression of Eq. (34), then defining a plane parallel to this line and to the $\beta $-axis. The intersection of these two planes yields a line in $(\alpha ,\beta ,\delta k)$-space. Only X-rays with parameters lying within a thin volume close to this line will be diffracted efficiently by both the monochromator and the analyzer. Figure 5 shows this procedure. Note that the line of intersection is not parallel to any of the coordinate axes. The thin volume around this line therefore occupies only a very small portion of any rectangular region in $(\alpha ,\beta ,\delta k)$-space for which the edges are parallel to the coordinate axes. Because most ray-tracing programs sample rays only within such rectangular regions, the probability that a ray meets the condition for efficient diffraction by both the monochromator and the analyzer is very small, and calculations of the instrumental background of the crossed monochromator/analyzer setup require an unreasonably large number of rays. These considerations justify the numerical calculations of the instrumental background carried out below.

The final step is to calculate the final intensity of a ray with given $\alpha $, $A1$, and $A2$. Here, a distinction must be made between horizontal and vertical polarization. Taking Eq. (16), we define $RH1s(A1)$ and $RH1p(A1)$ as the reflectivity of the first DCM crystal with $K=1$ and $K=|\u2009cos\u20092\theta 1|$, respectively. Because the second DCM crystal is identical to the first and is assumed to be ideally aligned, its reflectivity for each polarization is the same. Similarly, we define $RH2s(A2)$ and $RH2p(A2)$ as the reflectivity of the polarization analyzer with $K=1$ and $K=|\u2009cos\u20092\theta 2|$, respectively. Finally, we define $Ih[\beta (\alpha ,A1,A2)]$ and $Iv[\beta (\alpha ,A1,A2)]$ as, respectively, the components of the initial intensity of the ray polarized in the horizontal or vertical direction. The value of $\beta $ is of course given by Eq. (33). The final horizontally polarized intensity $Jh$ and vertically polarized intensity $Jv$ are then

The initial intensities $Ih$ and $Iv$ are shown in Figure 2, and the crystal reflectivities $RH1s$, $RH1p$, $RH2s$, and $RH2p$ are shown in Figure 3. The intensity distribution that is finally observed by a camera at given values of $\alpha $ and $\beta $ is the sum of the intensities of all rays for which $A2(\u2009sin\u2009\theta 1\u2009sin\u2009\theta 2)\u2212A1(\u2009sin\u2009\theta 2\u2009sin\u2009\theta 1)=C$, where $C$ is a constant equal to $(4\u2009cos\u2009\theta 1\u2009sin\u2009\theta 2)[\beta \u2212(\u2009tan\u2009\theta 1\u2009tan\u2009\theta 2)\alpha ]$ as shown in Eq. (33). Here, we choose $A2$ as the independent variable for the numerical calculations because the sensitivity of the analyzer reflectivity to $A2$ is much greater than the sensitivity of the monochromator reflectivity to $A1$. By integrating with respect to $A2$, we find the diffracted intensity distribution in the camera

Because the *s*- and *p*-polarized reflectivities for a Si (1 1 1) DCM are almost identical, the degree of horizontal polarization suppression is determined almost entirely by the analyzer.

Using all this information, we can determine how the beam from the analyzer appears if the source size is much smaller than the distance from the source to the analyzer; that is, if the source may be approximated as a point. In that case, the horizontal and vertical coordinates within the beam are simply the corresponding angles $\alpha $ and $\beta $ multiplied by the distance from the source to the analyzer. Equation (33) shows that the section of the incident beam selected by the monochromator and the analyzer is a stripe that is tilted from the horizontal direction by an angle $arctan(tan\u2009\theta 1/\u2009tan\u2009\theta 2)$. This tilt angle is 7.82° if the Si (5 5 5) analyzer is used and 8.51° if the Ge (5 5 5) analyzer is used. The vertical width of the stripe may be estimated from the reflectivity terms in Eqs. (35) and (36). For the horizontally polarized intensity, the definition of $\eta $ in Eq. (20) and the knowledge that the efficiency of a Bragg reflection is significant only for $|\eta |\u22641$ sets upper and lower limits on the values of $A1$ and $A2$ at which the reflectivities are significantly different from zero

where the subscripts 1 and 2 refer to the monochromator and the analyzer, respectively. Similar bounds on $A1$ and $A2$ can be determined for the vertically polarized intensity. The vertical width of the stripe for horizontal polarization ($\Delta \beta h$) and for vertical polarization ($\Delta \beta v$) are therefore given by Eq. (33)

For the Si (1 1 1) DCM and the crossed Si (5 5 5) analyzer, $\Delta \beta h=28.3\mu rad$ and $\Delta \beta v=27.4\mu rad$. For the Si (1 1 1) DCM and the crossed Ge (5 5 5) analyzer, $\Delta \beta h=27.1\mu rad$ and $\Delta \beta v=26.7\mu rad$.

Although both the horizontally and the vertically polarized intensities are selected along a tilted stripe of the same angle and almost the same width, their intensity distributions are very different as shown by the theoretical calculations of Figure 6. The horizontally polarized component of the intensity reaching the detector is uniform over the area of the selected tilted stripe and is determined by the nearness to 45° of the Bragg angle of the analyzer. The Ge (5 5 5) analyzer performs better, as expected, yielding a maximum $Kh$ that is smaller by a factor of 6000 than the maximum $Kh$ passed through the Si (5 5 5) analyzer. On the other hand, the vertically polarized component of the intensity reaching the detector is determined chiefly by the vertically polarized component of the X-ray beam emerging from the bending magnet. Because this falls to zero at $\beta =0$, as shown in Figure 2(b), a dark horizontal band cuts across the vertically polarized tilted stripe. The image that appears in the detector is the sum of the horizontally and vertically polarized components of the final beam. If the Si (5 5 5) analyzer is used, its imperfect discrimination allows the horizontal polarization to dominate the final image. On the other hand, if the Ge (5 5 5) analyzer is used, the almost complete suppression of the horizontal polarization allows the vertical polarization to dominate. Therefore, a dark horizontal band is predicted to appear in the final image if the Ge (5 5 5) analyzer is used, but not if the Si (5 5 5) analyzer is used.

These predictions are borne out by the experimentally measured images in Figure 7, which were taken with a MiniFDI CCD camera from Photonic Sciences Ltd. This camera has 1392 × 1040 pixels of 6.4 *μ*m size, and the images were taken with 2 × 2 binning. The intensity distributions and the tilt angle of the instrumental background are in good agreement with the theoretical calculations, both for the Si (5 5 5) analyzer and for the Ge (5 5 5) analyzer. In order to show the effectiveness of the X-ray Birefringence Imaging method, the bright trace left by a birefringent sample is also labelled in Figure 7. The specific samples are a single crystal of the 1-bromoadamantane/thiourea inclusion compound (Chao *et al.,* 2003) at 280 K in the case of the Si (5 5 5) analyzer (Figure 7(a); data from Palmer *et al.* (2014)) and a single crystal of the 1,8-dibromooctane/urea inclusion compound (Harris *et al.*, 1991; Guillaume *et al.*, 1994) at 100 K in the case of the Ge (5 5 5) analyzer (Figure 7(b); data from Palmer *et al.* (2015)). The X-ray optic axis of the sample is oriented by the diffractometer to an angle of +45° (Figure 7(a)) and +44° (Figure 7(b)) with respect to the horizontal $z\u0302$-axis of Figure 4. As the X-ray polarization is rotated by the birefringent sample, the sample effectively increases the transmission of X-rays to the detector. The theoretical estimate of the width of the instrumental background stripe from top to bottom, calculated for the specific distance of the CCD camera from the bending-magnet source (approximately 44 m), is $(44m)\Delta \beta v$, which is 1.21 mm if the Si (5 5 5) analyzer is used and 1.17 mm if the Ge (5 5 5) analyzer is used. The experimental images show that the width of the stripe [0.871 mm using Si (5 5 5) and 0.846 mm using Ge (5 5 5)] is somewhat narrower than the calculated values, which can be ascribed to the absorption of X-rays in the crystals, which causes the diffracted intensity in the rocking curves of Figure 3 to drop off within the range $|\eta |\u22641$.

## III. CONCLUSIONS

The technique of X-ray Birefringence Imaging has been developed at the synchrotron bending-magnet beamline B16 of Diamond Light Source and has been shown to be a powerful technique for mapping, in a spatially resolved manner, the local orientational properties of specific molecules and/or bonds in anisotropic materials. The experimental setup for X-ray Birefringence Imaging consists of a double-crystal monochromator to select an appropriate wavelength and a crystal polarization analyzer in crossed geometry. Extended DuMond diagrams indicate that this crossed setup is difficult to model using standard ray-tracing programs; therefore, calculations of its instrumental background have been performed from first principles. The bending-magnet beam covers a large area and hence is preferable to undulator beam for X-ray Birefringence Imaging experiments, and it delivers sufficient power densities to image the sample without causing beam damage to the sample. However, with the combination of a bending-magnet source and a crossed monochromator/analyzer geometry, a significant amount of background is passed through to the camera. The background comes largely from the elliptical polarization of the bending-magnet beam off the plane of the electron orbit, and also partly from the fact that the suppression of the horizontal polarization by the analyzer is imperfect. Calculation and experiment both show that the background is a bright tilted stripe with a horizontal dark band in the plane of the electron orbit. It should be stressed that the tilt angle of the stripe is not caused by any geometrical distortion of the beam cross-section; therefore, no spatial transformation of the image of the sample is necessary. Rather, the tilt angle is determined by the crossed angle-wavelength dispersion of the monochromator and the analyzer.

## ACKNOWLEDGMENTS

The authors wish to thank Ian Martin and Riccardo Bartolini of the Diamond Light Source Accelerator Physics group for providing the electron beam parameters for B16. We also thank Manuel Sánchez del Río of the ESRF for his assistance in setting up the simulation of the B16 X-ray source in SHADOW. We thank EPSRC (studentship to G.R.E.-G) and Cardiff University for financial support. This work was undertaken on B16 at Diamond Light Source under proposal MT8323-1.