Fig. 1. Basic scheme of polarimetry. Essential is the pair of polarizers with different and variable orientations to each other to study the effect of a sample in between on the polarization.

Fig. 2. Basic diffraction geometry for anomalous transmission of X-rays (Borrmann effect). Reprinted from Ref. [45], with the permission of AIP Publishing.

Fig. 3. Geometry of the Bragg diffraction at 45°. Unpolarized radiation is polarized because the -component, being in the plane of incidence, is not allowed for reflection (Brewster’s law). Used with the permission of SPIE, from Ref. [50]; permission conveyed through Copyright Clearance Center, Inc.

Fig. 4. Kossel pattern of silicon at 12.914 keV. The bold black circle represents the exploited Si (800) reflection used for suppression of the component. All other possible reflections are depicted by thin colored circles. The vectors and describe the direction of the incident and diffracted wave, respectively. In order to avoid degradation of the polarization purity due to multiple-beam cases, the azimuth has to be chosen such that the ‘distance’ to the closest undesired reflections is as large as possible. Reprinted from Ref. [51], with the permission of APS.

Fig. 5. Reflectivity of X-rays for the –polarization in 45° symmetric Bragg scattering geometry as a function of the angle of incidence, according to dynamical theory calculations. Solid line: the (400) Bragg reflection in diamond for 9.831 keV. Dashed line: the (400) Bragg reflection in silicon for 6.457 keV, as used by Marx et al.^[51]. Note that . Reprinted from Ref. [53], with the permission of AIP Publishing.

Fig. 6. Schematic of a channel-cut polarizer with reflections. Thin lines indicate the lattice planes for the 45° Bragg reflection, which are parallel to the surface in this case.

Fig. 7. Polarization ratios for m-fold multiple Bragg reflection polarizers using the Ge (440) Bragg reflection. Reprinted from Ref. [54], with the permission of Taylor & Francis.

Fig. 8. The geometry for an asymmetrically cut channel-cut crystal with a Bragg angle near 45°. The lattice planes, as indicated in Figure 6, are oriented 45° to the beam, yet the crystal surface is slanted. The asymmetry angle is the angle between surface and lattice planes. It is negative for the case shown at the first surface where the incident beam is shallow and leaves with a larger diameter.

Fig. 9. The effect of an asymmetric cutting angle on both the angular acceptance and the resulting polarization suppression for a silicon (840) channel-cut crystal. Reprinted from Ref. [67], with the permission of AIP Publishing.

Fig. 10. Proposed experimental setup for the demonstration of vacuum birefringence: a high-intensity laser pulse is focused by an F/2.5 off-axis parabolic mirror. A hole is drilled into the parabolic mirror in alignment with the z-axis (axes as indicated) in such a way that an X-ray pulse can propagate along the z-axis through the focal region of the high-intensity laser pulse. Using a polarizer–analyzer pair, the ellipticity of the X-ray pulse may be detected. Shown in grey: extension of the setup for the generation of counter-propagating laser pulses and a high-intensity standing wave, which may be used for pair creation. Reprinted from Ref. [20], with the permission of Elsevier.

Fig. 11. Schematic views of the experimental setup. Top: several meter-long parts of the X-ray beamline centered around the interaction point with the optical components inside a vacuum chamber. Left: zoom into a cm-sized neighborhood of the focus where the cleaning electrodes will be placed. Bottom left: another zoom into the cleaned region. The focus of the cleaning laser is about 10 μm wide. However, only a fraction (pink) of the cleaned region will be employed as the interaction region, where the PW optical laser () and the XFEL beam () are focused and superimposed. Bottom right: fundamental idea of probing QED vacuum birefringence caused by an intense optical laser with the XFEL beam. Beams are counter-propagating with their foci overlapping in space and time. To maximize the effect, the polarization directions must differ by 45°. A slight ellipticity in the polarization of the out-going probe pulse will occur. Used with the permission of IOP Publishing, from Ref. [19]; permission conveyed through Copyright Clearance Center, Inc.

Fig. 12. Illustration of the experimental setup utilizing compound refractive lenses (CRLs) to focus and re-collimate the XFEL beam. Reflections at diamond crystals change the propagation direction, and a pair of diamond quasi-channel-cuts serve as the polarizer and analyzer, respectively. The original XFEL beam is focused with a CRL to constitute the pump field; the beam focus defines the interaction point. Subsequently, it is defocused with a CRL and by reflection at two diamond crystals directed back to the interaction point under an angle of . Before reaching the interaction point, it is polarized with a diamond polarizer and the resulting probe beam focused to the interaction point with a CRL. Finally, it is defocused with another CRL, analyzed with a diamond analyzer and the signal registered with a charge-coupled device. Taken from Ref. [86], licensed under CC BY 4.0.

Fig. 13. Sketch of the experimental setup investigating CRL material properties. The multilayer mirrors collimate the X-rays from the rotating anode X-ray source. The combination of the polarizer, analyzer and charge-coupled device camera allows for polarization sensitive imaging. Reprinted from Ref. [61], with the permission of AIP Publishing.

Fig. 14. Schematic setup for nuclear resonant scattering with the polarization filtering method. The incoming radiation from the left is polarized by the first channel-cut crystal. Subsequently, the beam impinges on the magnetically anisotropic sample under investigation. The green arrow indicates the direction of the external magnetic field that induces optical activity via X-ray magnetic linear dichroism. The analyzer crystal in the crossed setting transmits only the photons that have undergone nuclear resonant - to –scattering. Taken from Ref. [68], licensed under CC BY 4.0.

Fig. 15. An illustrated experimental setup of strong magnetic field generation by interaction of an ultra-short relativistic optical laser pulse with solid matter, probed by an XFEL via Faraday rotation. Taken from Ref. [140], licensed under CC BY 4.0.

Fig. 16. Exploded view of the OSO-8 polarimeter assemblies. The crystal reflector employs approximately Bragg angle and is thereby polarization-filtering. Reprinted from Ref. [151], with the permission of Springer Nature.