Disentangling X-ray dichroism and birefringence in high-purity polarimetry

We report on the most precise simultaneous measurement of dichroism and birefringence in the X-ray range. The investigation of these optical anisotropies, which are rarely explored in this spectral range, shows that birefringence has a remarkable influence on polarization-resolved spectroscopy at X-ray absorption edges. We use precision X-ray polarizers providing a polarization purity of $10^{-8}$, which enabled an unprecedented dynamical range of six orders of magnitude. As benchmark systems for probing electronic anisotropies of correlated materials, we investigated orthorhombic La$_2$CuO$_4$ and monoclinic CuO. With an improved ab initio quantum code, we get very good agreement between experiment and theory, allowing a clear separation of dichroism and birefringence. A signature of non-diagonal tensor elements of the complex linear absorption coefficient in the anisotropy of $\sigma-\pi$ scattered photons was proven by experimental data and simulation.

Symmetry breaking interactions in condensed matter form the basis for the properties of modern functional materials like magnetism, superconductivity, multiferroicity and others. Therefore, the study of anisotropies in relevant degrees of freedom in these materials is the basis to understand the origin of these properties.
Electronic anisotropies are commonly studied by monitoring the polarization dependence of X-ray absorption spectra with techniques like X-ray magnetic circular dichroism (XMCD), X-ray magnetic linear dichroism (XMLD) or X-ray natural linear dichroism (XNLD) [1][2][3][4][5][6][7]. Additionally, anisotropies in condensed matter can also be studied by monitoring the X-ray optical activity induced upon transmission or scattering, which is a burgeoning technique in the X-ray regime. The corresponding changes in the X-ray polarization can be detected with very high sensitivity via two crossed linear polarizers between which the interaction of the X-rays with the material takes place [8][9][10][11][12].
The study of electronic anisotropies in correlated materials constitutes a novel field of application for this technique. In this method, the optical activity is monitored while tuning the X-ray energy across an absorption edge of a transition metal in the sample. Due to high polarization extinction ratio of up to 10 −10 [13], selected orbitals and their hybridization can be probed with high sensitivity.
Polarization changes of X-rays in the interaction with matter occur due to absorptive and dispersive effects. They are related to an anisotropy of the imaginary and the real part of the index of refraction, leading to dichroism and birefringence, respectively. In case of linearly polarized light, the first of these effects causes a rotation of the polarization vector due to an anisotropic absorption cross section in the sample. The second effect occurs due to the different propagation velocities of the two orthogonal polarization components, which leads to a phase shift between these polarization components and induces an ellipticity.
The spectral dependence of X-ray dichroism and birefringence is a key to understand the optical properties in the X-ray regime. They are of fundamental interest, because they are directly connected to the electronic properties of the material via the scattering amplitude. Local optical anisotropies of the absorbing atom can provide, for example, information about orbital orientation in molecules and solids or enables the determination of spin and orbital magnetic moments [1][2][3][4][5][6][7].
While the spectroscopic analysis of X-ray dichroism is widespread, that of X-ray birefringence is in its infancy [14][15][16][17]. More commonly, following the Kramers-Kronig relation, the real part of the refractive index is calculated by integration over the imaginary part, which is measured by X-ray absorption spectroscopy [18]. However, this approach works only if the integration is performed over a sufficiently large energy range, which is often not practical. Although the spectroscopic approach has advantages, the challenge is to disentangle both effectsdichroism and birefringence -for an accurate analysis of the X-ray polarization in the vicinity of an absorption edge.
In this letter, enabled by a newly developed high-purity X-ray polarimeter, a full investigation of the polarization changes at an atomic absorption edge is presented. As a benchmark system for correlated materials and a arXiv:2003.00849v1 [physics.ins-det] 27 Feb 2020 Figure 1. a) The sample was mounted on an Eulerian cradle with the b-axis anti-parallel to the beam. An avalanche photodiode behind the X-ray analyzer in crossed position to the polarizer detected the σ − π scattered photons. Simultaneously ionization chambers (IC) measured the transmitted intensity through the sample. b) Enlarged view of the sample along the beam. parent compound for cuprate superconductors, CuO and La 2 CuO 4 were investigated at the Cu K-absorption edge.
For the investigation of polarization changes, the following experimental setup at the synchrotron source PETRA III (beamline P01) was used, which is also shown in Fig. 1a). A Si (111) monochromator was used for energy adjustment. The sample was mounted on a Eulerian cradle to facilitate high-precision angular adjustments, ϕ and χ. It was located between two monolithic Si(440) channel-cut crystals, which act as polarizer and analyzer [13,19]. The polarizer generates a highly pure linear σ − polarization that strikes the sample in the horizontal plane. Polarization changes due to X-ray dichroism and X-ray birefringence lead to σ−π scattering of the sample. This is measured as linear π −polarization in the vertical plane behind the analyzer in crossed position to the polarizer with an avalanche photodiode. The transmitted intensity through the sample was also controlled by ionization chambers (IC) before and behind the sample.
The polarizers work on the principle that reflections at a Bragg angle close to θ B = 45 • suppress the polarization component along the diffraction plane, π, following a cos 2θ dependence [20]. Thus, each Bragg reflection on the channel-cut crystals with a Bragg angle close to 45 • acts like a polarization filter. Therefore, a high degree of polarization sensitivity can be achieved by multiple Bragg reflections. In this study six reflections per channel-cut were used.
Previous measurements with this type of highprecision X-ray polarizers were performed at a fixed energy [9][10][11][12]. In this work, this highly sensitive measuring method was further developed into a spectroscopy technique following Siddons et al. [21]. In our study, the near-edge region of the Cu K-absorption edge is scanned to probe the symmetry of the Cu atom and its hybridizations with the surrounding atoms.
To investigate the entire energy range of the Cu K-absorption edge with a high polarization purity with 1 eV energy steps, the Bragg angle of the polarizer crystals have to be varied simultaneously with the Bragg angle of the Si(111) monochromator. According to the Bragg condition, the Bragg angle on the Si(440) plane have to be varied in an angular range from θ B = 45.78 • to θ B = 46.06 • to cover the energy range from 8970 eV to 9010 eV. Calculations by dynamical theory of X-ray diffraction show that a polarization purity of 1 · 10 −7 at 8970 eV up to 5 · 10 −9 at 9010 eV can be achieved. The experimental polarization purity at 8970 eV was nevertheless 1.3 · 10 −8 . The deviation to the theoretical value indicates small strain in the channel cut crystal. For the detection of σ − π scattered photons, the rocking curve of the analyzer channel-cut in crossed position to the polarizer has to be scanned for each energy. The energy resolution of the X-ray polarimeter is determined by the full width half maximum of the Si(440) polarizer and analyzer rocking curves of 62 meV. This value guarantees that the spectral features due to X-ray dichroism and X-ray birefringence are not widened.
The angular dependence of X-ray dichroism and X-ray birefringence is ruled by the point group of the crystal [22]. We investigated two different point groups: monoclinic CuO with point group C 2h (2/m) and space group C2/c (a = 4.6837Å, b = 3.4226Å, and c = 5.1288Å, β = 99.54 • [23]) and orthorhombic La 2 CuO 4 with point group D 2h (mmm) and space group Bmab (a = 5.352Å, b = 5.400Å and c = 13.157Å [24]). Both materials show trichroism, which is described by the angular dependence of the dipole absorption cross section: with complex tensor components σ D (l, m), spherical harmonics Y m l and quantum numbers m and l [22]. The polarization vectorˆ = (cos χ cos ϕ, cos χ sin ϕ, − sin χ) is chosen according to the experimental setup where χ represents the angle between the a − axis of the crystal and the electric field vector of the synchrotron beam as shown in Fig. 1b). The electric field vector is parallel to the aaxis at is maximal for ϕ = 0. For CuO with point group C 2h (2/m), where σ D (2, 1) = σ D (2, −1) = 0, the dipole absorption cross section is We chose the sample orientation ϕ = 0 for both sample materials, which corresponds to the electric field vector lying in the a-c-plane. This allows to detect the full anisotropy of σ D (2, 0) and the real part σ Dr (2, 2) of the dipol absorption cross section σ D (2, 2). Therefore, the crystals were polished in (010) orientation with an accuracy of ≤ 0.1 • to 33 µm (CuO) and 23.5 µm (La 2 CuO 4 ) thin disks. The thickness of the samples was determined by transmission measurement and comparison with Henke data [25]. The adjustment of the crystal axes within the a − c plane was confirmed by Laue method.
For a theoretical description of polarization changes by the sample, we use the complex linear absorption coefficient which is connected to the complex refractive index. The real part µ is related to the absorption cross section and therefore proportional to the imaginary part of the resonant scattering amplitude f , which links the electronic properties of the crystal lattice with the optical properties of the sample. The imaginary part µ is correspondingly proportional to the real part of f [26].
To obtain the matrix of complex linear absorption coefficient, ab initio simulations were performed with the FDMNES code following the density functional theory [27]. The relativistic full potential approach was used, including the spin-orbit interaction. The self-consistent electronic structures around the absorbing atom were calculated in a cluster with radius up to 6Å. The code includes all the steps of the calculation of the polarization change in a material up to the final transmitted intensity after the analyzer.
For the calculation of polarization changes due to anisotropies in the real or imaginary part of the complex linear absorption coefficient, the Jones matrix formalism is used. The derivation is given in detail in the supplemental material [URL will be inserted by publisher]. Accordingly, the π − polarized X-ray intensity after the sample normalized to the impinging σ − polarized X-ray intensity on the sample, I σπ , is given by where µ σπ = µ πσ (centrosymmetric crystal), τ = (µ ππ − µ σσ ) 2 + 4µ σπ µ πσ , l is the thickness of the sample, and χ is the angle of the sample around the beam propagation direction [26,28,29].
After introducing the theoretical background, we will now discuss the influence of the symmetry of the complex linear absorption coefficient µ on the σ − π scattered intensity I σπ . According to equation (5), the following behavior is expected: If the non-diagonal tensor elements µ πσ are zero, I σπ is maximal at χ = ±45 • for all energies, since I σπ is then proportional to sin 2 2χ. In contrast, the maxima of I σπ as function of χ are energy dependent, if non-diagonal tensor elements µ πσ are present.
Both symmetry cases of the complex linear absorption coefficient µ were experimentally proven by investigating La 2 CuO 4 (µ πσ = 0) and CuO (µ πσ = 0). Fig. 2 shows the measured spectra of the σ − π scattered photons for both crystals as a function of the angle χ. In agreement with the theory, all spectral features of La 2 CuO 4 are in phase, whereas for CuO a phase shift of I σπ in χ is induced energy dependently. Furthermore I σπ has a symmetry of π/2 in χ for both crystals, which can easily be explained by their centrosymmetry. Interestingly, we found that the measured σ − π scattered intensity for CuO is constant for all angles χ at 8988 eV, which is in disagreement with the simulation.
After discussing the experimental and theoretical σ −π scattered intensity qualitatively, they are now compared quantitatively. We found experimentally that the maximum percentage of scattered σ − π photons is higher for CuO with 2.2 % (Fig. 2 a)) than for La 2 CuO 4 with 0.1 % (Fig. 2 f)). The intensity of simulated σ − π scattered photons depends very sensitively on the thickness of the sample. This could be the reason for the fact that the calculated intensity of the σ−π scattered photons of 2.3% for CuO (Fig. 2 b)) and 0.16 % for La 2 CuO 4 (Fig. 2 g)) is a bit higher than the experimental data. Nevertheless, the simulation is in very good qualitative agreement with the experiment.
The influence of birefringence and dichroism on the σ − π scattered intensity I σπ was investigated by simulating both effects separately. This is achieved by neglecting the anisotropic part of the real or imaginary part of the complex linear absorption coefficient µ in order to disregard the influence of dichroism or birefringence, respectively. Fig. 2 c) and d) show the impact of dichroism and birefringence, respectively, to the σ − π scattered intensity for CuO and h) and i) for La 2 CuO 4 . It turns out that the total σ − π scattered intensity I σπ is the sum of the σ − π scattered intensity due to dichroism and birefringence. Our results show that X-ray birefringence greatly broadens the σ − π scattered photon spectra energetically and thus has a huge impact on polarization changes at an X-ray absorption edge.
The analysis of I σπ has considerable advantages over conventional X-ray absorption measurements such as XNLD: It is background free and thus exhibits a high dynamic response to polarization changes, which is illustrated in Fig. 3. The intensity of the σ − π scattered photons behind the analyzer in cross-polarization setting was measured for CuO for different angles χ. In order to determine the background, an energy scan was performed without a sample in the beam, which turned out to be several orders of magnitude below the measured signal. Six orders of magnitude dynamics and nearly background free measurements enable unprecedented sensitivity to polarization changes in the energy range from 8970 eV to 9010 eV. In conclusion, this letter reports the first comprehensive experimental and theoretical investigation of birefringence and dichroism at the Cu K-edge for two different crystal systems. By measuring the X-ray dichroism via the orientation dependent transmission of the sample, the imaginary part of the complex linear absorption coefficient can be identified. The real part which corresponds to X-ray birefringence can be calculated by comparing the measurement and simulation of the σ − π scattered photon intensity. This is especially interesting for the determination of optical constants of materials, which are not accessible via ab initio calculations such as influences of ion implantations, doping of materials or strongly correlated systems that can not be modeled with theoretical approaches at present. In order to avoid the problem of integration via a finitely measured absorption cross section using the Kramers-Kronig relation, this method is excellently suited and can answer questions of fundamental importance. High polarization sensitivity is particularly suitable for observing small anisotropies as early indicators of phase transitions during or even long before reaching critical parameters. Especially interesting is this new approach to investigate very weak anisotropies of quadrupolar or octopolar transitions in the pre-edge region like recently detected in Gd 3 Ga 5 O 12 at the Gd L1-absorption edge [30]. In contrast to XNLD, this method does not require spectra of orthogonal orientations to be subtracted from each other. Instead, the measurement of anisotropies with high angular resolu-tion is directly and quickly accessible. Futhermore, in analogy to an optical polarization microscope, it is also possible to image polarization anisotropies with spatial resolution.
We would like to thank Jèrôme Debray from Institut Néel for preparation of thin sample slides, Heike Marschner and Orthrud Wehrhan for support by their expertise in crystallography and Claudia Rödl, Martin von Zimmermann and Paul Schenk for enlightening discussions. This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant We use the (2x2) Jones Matrix defined in a () basis,  horizontal,  vertical, and the wave vector perpendicular to this basis.

Transmittance matrix
We use the transmittance matrix under the  basis where  and  are 2 directions perpendicular to the wave vector. We note  the complex linear absorption coefficient. Following Lovesey and Collins9 F 1 , after a distance l, the transmittance matrix is given by: with :

Polarization matrix
In the following we also use the Jones matrix for the polarization: When rotating the sample by an angle  the linear polarization gets an angle - versus its  We use the (2x2) Jones Matrix defined in a () basis,  horizontal,  vertical, and the wave vector perpendicular to this basis.

Transmittance matrix
We use the transmittance matrix under the  basis where  and  are 2 directions perpendicular to the wave vector. We note  the complex linear absorption coefficient. Following Lovesey and Collins9 F 1 , after a distance l, the transmittance matrix is given by:

Polarization matrix
In the following we also use the Jones matrix for the polarization:

Analyzer matrix
The analyzer matrix (or polarizer after the sample) is given versus its rotation angle, , and its Bragg angle, , by: = ( cos − sin cos 2 sin cos 2 cos ) For a perfect analyzer crystal cos 2 = 0.