2.1.1 Beam divergence

As very simple geometric effect, a beam divergence leads to a deviation from a Bragg angle of exactly 45° for some parts of a beam, impinging on a perfect crystal, and thus a minor contribution in the $\pi$ -polarized component^[ Reference Schulze^52]. Assuming the X-ray beam is a Gaussian beam, the relationship between the divergence and the polarization purity is as follows^[ Reference Schulze^52, Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55]:

(7) $$\begin{align}{\mathcal{P}}_{\mathrm{Divergence}}^{\mathrm{Limit}}={\sigma}_{\mathrm{H}}^2+{\sigma}_{\mathrm{V}}^2,\end{align}$$

with ${\sigma}_{\mathrm{V}}$ and ${\sigma}_{\mathrm{H}}$ being the divergence in the vertical and horizontal direction, respectively.

In 2020, Bernhardt et al.^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55] experimentally verified Schulze’s^[ Reference Schulze^52] theoretical analysis by studying the effect of beam horizontal divergence on X-ray polarization purity at beamline ID18 of the European Synchrotron Radiation Facility (ESRF). The comparison of the X-ray polarization purity between the fitted data ( ${\mathcal{P}}_{\mathrm{exp}}$ ) and the calculated data ( ${\mathcal{P}}_{\mathrm{Divergence}}^{\mathrm{Limit}}$ , using Equation (7)) is presented in Table 1. The polarization purity from the fitted data points and the calculated limit match very well for all three horizontal beam divergences. When the horizontal divergence of X-ray was reduced from 17 to 8.4 μrad by a slit with variable gap and a V-shaped channel-cut (VCC), the X-ray polarization purity decreased to $1.4\times {10}^{-10}$ ^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55]. In addition, this paper and others^[ Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey^19, Reference Tschentscher, Bressler, Grünert, Madsen, Mancuso, Meyer, Scherz, Sinn and Zastrau^56] mention that 1 μrad divergence is available for the X-ray free electron laser (XFEL). Therefore, the X-ray polarization purity is limited to the order of ${10}^{-12}$ .

Table 1 Comparison of measured purity ${\mathcal{P}}_{\mathrm{exp}}$ against the calculated limit ${\mathcal{P}}_{\mathrm{Divergence}}^{\mathrm{Limit}}$ given by the beam divergence ${\sigma}_{\mathrm{H}}$ for ${\sigma}_{\mathrm{V}}=6.1\;\unicode{x3bc} \mathrm{rad}$ . Taken from Ref. [Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze55], licensed under CC BY 4.0.

2.1.2 Crystal quality

Crystal quality affects the polarization purity in two ways. First, for similar geometric reasons to the divergence, all parts of a (perfectly parallel) beam of finite size must experience the same 45° incidence angle to allow for the same polarization suppression^[ Reference Hart and Rodrigues^54]. Second, imperfect crystals have varying lattice constants that affect the reflectivity curves and thus the spectral/angular acceptance and integrated reflectivity. Thus, the properties of the crystal material must be taken into account to avoid the depolarization of X-rays.

Researchers^[ Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus^53, Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55] have used artificial diamonds containing a mass of crystalline defects produced by chemical vapor deposition (CVD) as a polarizer in X-ray polarimetry. Contrary to expectations and the prediction of Hart and Rodrigues^[ Reference Hart and Rodrigues^54], imperfections of artificial diamonds have no observable influence on the polarization purity of X-rays but lead to low peak reflectivity and low transmittance of polarizers^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55]. Furthermore, polarimetry at photon energies above 10 keV can benefit from imperfections because of the higher integrated reflectivity. For low photon energies, the nearly perfect crystals with high reflectivity are essential for the expected highly linearly polarized X-rays^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55].

2.1.3 Detour reflections (Umweganregungen)

Another limitation of the polarization purity is detour reflections (Umweganregung)^[ Reference Renninger^57]. These are the result of consecutive Bragg diffractions on different lattice planes and therefore different Bragg angles, yielding in sequence the same beam reflection angle as the primary reflection. This is similar to a cat’s-eye retro-reflector, where the rays bounce off several surfaces, in contrast to a mirror where only one reflection occurs. In fact, the detours are only possible in 3D crystals due to the abundance of lattice planes in directions off the main reflection.

As a result, every partial Bragg diffraction does not happen with a 45° Bragg angle such that the Brewster condition is not fulfilled, and no strong ratios of ${R}_{\pi }:{R}_{\sigma }$ are yielded, even in sequence. Yet, the overall intensity can be relatively weak compared to the beam from the 45° (main) Bragg diffraction. Still, these unpolarized contributions yield a limit for the polarization purity.

The Ewald sphere is a geometric construction to determine the diffraction direction of crystals, and diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere. The consecutive reflections case happens at nearby lattice planes in 3D crystals if there are more than two reciprocal lattice points that lie on the Ewald sphere. Under some azimuth angles, the incident beam excites not only the required intended reflection with a 45° Bragg angle but also secondary reflections – not with a 45° Bragg angle. As a result, the latter reflections will cause the depolarization of X-rays when the secondary, detoured reflections exit into the same exit direction of the principal reflection^[ Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus^51, Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus^53, Reference Renninger^57], and the polarization purity is suppressed. Marx et al.^[ Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus^51] provided the reflection system for a silicon crystal and an X-ray energy of 12.914 keV, as shown in Figure 4. The radius of the Ewald sphere is $1/\lambda$ , where $\lambda$ is the wavelength of the incident beam. The intersections of the colored lines are the multi-beam cases. The effect of multi-beam situations can be reduced by optimizing the crystal azimuth to avoid the excitation of secondary reflections^[ Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus^53]. In addition, using lower photon energies reduces the size of the Ewald sphere and therefore reduces the number of potential detour cases.

Figure 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 ${\overrightarrow{S}}_0$ and ${\overrightarrow{S}}_{\mathrm{h}}$ 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. [Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus51], with the permission of APS.

2.1.4 Material dependence

It is obvious from the previous sections that the material has a strong influence, mainly to provide crystals of the highest quality (cf. Section 2.1.2). At present, Si ( $Z=14$ )^[ Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus^51, Reference Marx, Uschmann, Höfer, Lötzsch, Wehrhan, Förster, Kaluza, Stöhlker, Gies, Detlefs, Roth, Härtwig and Paulus^58– Reference Marx-Glowna, Schulze, Uschmann, Kämpfer, Weber, Hahn, Wille, Schlage, Röhlsberger, Förster, Stöhlker and Paulus^60] and diamond ( $Z=6$ )^[ Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus^53, Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55] are widely available. Ge ( $Z=32$ )^[ Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze^61] exhibits a reflection for Cu X-ray tubes but is abandoned gradually. Silicon crystals with a perfect crystal structure, few impurities and a very mature preparation technology are adopted widely as polarizers.

For silicon and diamond, Bernhardt et al.^[ Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus^53] compared the reflectivity of those two materials for the Bragg reflection at a 45° angle, as illustrated in Figure 5. The solid line and dashed line are the reflectivity curves of diamond and silicon, respectively. The curve for diamond is higher but narrower than that for silicon. This is a quite general behavior^[ Reference Als-Nielsen and McMorrow^62] and the reason lies mainly in the number of electrons per atom, $Z$ . Silicon has more electrons, and thus scatters more intensity per lattice plane and fewer lattice planes are needed for Bragg reflection. This explains the wider spectral/angular width of the curve. On the other hand, the absorption per atom of silicon is higher, and thus the peak reflectivity is not as high as for diamond. The photon energy ${E}_{\mathrm{ph}}$ also plays a role here, as diamond has a smaller unit cell and thus the wavelength for the same (400) reflection is shorter. With higher photon energy, absorption reduces and penetration increases; hence, this also contributes to the narrower and higher curve of diamond.

Figure 5 Reflectivity of X-rays for the $\sigma$ –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.^[ Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus^51]. Note that ${1}^{"}\approx 5\;\unicode{x3bc} \mathrm{rad}$ . Reprinted from Ref. [Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus53], with the permission of AIP Publishing.

For applications, however, the integrated reflectivity can be of interest, for example, if the beam has a finite spectral bandwidth or divergence. The integrated reflectivity of diamond is much smaller than that of silicon. For example, a later work by Bernhardt et al.^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55] used diamonds with many crystalline defects and showed a peak reflectivity of only 50%–60%, while the rocking curve broadened by a factor of approximately 2.

There is also a material dependence of detour reflections. Tischler and Batterman^[ Reference Tischler and Batterman^63] provided that the resulting contribution of all reflections is dependent on the amplitude for each detour:

(8) $$\begin{align}{E}_{\mathrm{mult}}=\sum \limits_{i=1}^N{E}_i.\end{align}$$

Based on Equation (8), they calculated the $N$ -beam integrated intensities for the (622) reflection of germanium (Ge) and silicon (Si). The ratio of intensities is very close to the ratio of atomic numbers to the fourth power^[ Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus^51, Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55, Reference Marx, Uschmann, Höfer, Lötzsch, Wehrhan, Förster, Kaluza, Stöhlker, Gies, Detlefs, Roth, Härtwig and Paulus^58, Reference Schulze, Grabiger, Loetzsch, Marx-Glowna, Schmitt, Garcia, Hippler, Huang, Karbstein, Konôpková, Schlenvoigt, Schwinkendorf, Strohm, Toncian, Uschmann, Wille, Zastrau, Röhlsberger, Stöhlker, Cowan and Paulus^59]:

$$\begin{align*}{\left(\frac{E_{\mathrm{mult}}^{\mathrm{Ge}}}{E_{\mathrm{mult}}^{\mathrm{Si}}}\right)}^2={\left(\frac{0.038}{0.0075}\right)}^2\simeq 25.7\approx {\left(\frac{Z_{\mathrm{Ge}}}{Z\mathrm{Si}}\right)}^4=\left(\frac{32}{14}\right)\simeq 27.3.\end{align*}$$

Figure 6 Schematic of a channel-cut polarizer with $2\times 2$ reflections. Thin lines indicate the lattice planes for the 45° Bragg reflection, which are parallel to the surface in this case.

Consequently, polarizers made by materials with low Z values are favorable to further mitigate the impact of detour reflections, apart from choosing a good azimuth angle.

2.2.1 Consecutive reflections

It is well known that a polarized light beam can be produced by several transmissions through a number of glass plates, even though each plate is only a partial polarizer. Similarly, channel-cut crystals improve the polarization purity^[ Reference Hart and Rodrigues^54, Reference DuMond^65] since they stack a number of reflections into a single optical element.

Regarding multiple successive Bragg reflections between the walls of a channel cut in an ideal crystal to increase the polarization purity of X-rays, the ratio of intensities of two polarization states for X-rays polarized by $m$ consecutive Bragg reflections is given by the following^[ Reference Alp, Sturhahn and Toellner^66]:

(9) $$\begin{align}\frac{I_{\pi }}{I_{\sigma }}=\frac{\int {R}_{\pi}^m\left(\theta \right)\kern0.1em \mathrm{d}\theta }{\int {R}_{\sigma}^m\left(\theta \right)\kern0.1em \mathrm{d}\theta },\end{align}$$

with the notation of Equation (6). Hart and Rodrigues^[ Reference Hart and Rodrigues^54] calculated the ratio of ${I}_{\pi }:{I}_{\sigma }$ for multiple Bragg reflections in a grooved Ge polarizer using two-beam dynamical theory, as shown in Figure 7. The polarization purity decreases as the number of multiple Bragg reflections increases. In 1965, Bonse and Hart^[ Reference Bonse and Hart^64] found that multiple Bragg reflections between the walls of a channel-cut perfect crystal do not narrow the reflection curves considerably.

Recently, high polarization purity of X-rays was achieved by multiple reflections. In 2011, Marx et al.^[ Reference Marx, Uschmann, Höfer, Lötzsch, Wehrhan, Förster, Kaluza, Stöhlker, Gies, Detlefs, Roth, Härtwig and Paulus^58] reported that the highest purity of polarization of X-rays reaches $1.5\times {10}^{-9}$ based on $m=4$ reflections at Si (400) channel-cut crystals for 6 keV X-ray energy. Two years later^[ Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus^51], they obtained $2.4\times {10}^{-10}$ polarization purity of the X-ray using $m=6$ reflections. Here, the energy of the X-ray is 6 keV and the polarizer is Si (400) channel-cut crystals, the same as before.

2.2.2 Asymmetric cuts

The channel-cut crystals enhance the polarization purity of X-rays. However, the angular and spectral acceptance of channel-cut crystals tends to restrict the throughput of X-rays. To increase the acceptance of channel-cut crystals while maintaining the polarization filtering, researchers^[ Reference Alp, Sturhahn and Toellner^66– Reference Marx-Glowna, Uschmann, Schulze, Marschner, Wille, Schlage, Stöhlker, Röhlsberger and Paulus^68] came up with asymmetrically cut crystals with an asymmetry angle ${\alpha}_{\mathrm{c}}$ between the lattice planes and the surface. To quantify the asymmetry, the asymmetry parameter $b$ for a Bragg diffraction is defined^[ Reference Marx-Glowna, Uschmann, Schulze, Marschner, Wille, Schlage, Stöhlker, Röhlsberger and Paulus^68] by the following:

(10) $$\begin{align}b=\frac{\sin \left({\theta}_{\mathrm{B}}+{\alpha}_{\mathrm{c}}\right)}{\sin \left({\theta}_{\mathrm{B}}-{\alpha}_{\mathrm{c}}\right)}.\end{align}$$

Note that the asymmetry angle ${\alpha}_{\mathrm{c}}$ is negative if the incidence angle relative to the crystal surface is smaller than exit angle, as for the first reflection shown in Figure 8.

Figure 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 ${\alpha}_{\mathrm{c}}$ 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.

The angular acceptance of the crystal varies with the asymmetry angle as follows:

(11) $$\begin{align}{\left(\Delta \theta \right)}_{\mathrm{asymm}}=\sqrt{b}\cdot {\left(\Delta \theta \right)}_{\mathrm{symm}},\end{align}$$

with ${\left(\Delta \theta \right)}_{\mathrm{symm}}$ being the usual acceptance from a symmetric reflection.

Figure 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. [Reference Toellner, Alp, Sturhahn, Mooney, Zhang, Ando, Yoda and Kikuta67], with the permission of AIP Publishing.

Figure 9 displays the effects of the asymmetry angle on angular acceptance and polarization suppression^[ Reference Toellner, Alp, Sturhahn, Mooney, Zhang, Ando, Yoda and Kikuta^67]. The angular acceptance increases while the polarization suppression factor decreases when the asymmetry angle approaches 45°. However, a larger asymmetry angle requires larger crystals due to the beam footprint, imposing practical issues.

An overview of the calculated polarization purities of X-rays for different asymmetry angles ${\alpha}_{\mathrm{c}}$ and different numbers of channel-cut reflections $m$ is listed in Table 2 with an explanation of each parameter by annotations. For the same number of reflections, the polarization purity at an asymmetry angle of $-{28}^{\circ }$ is better than that at an asymmetry angle of $-{43}^{\circ }$ . In their paper, Marx-Glowna et al. ^[ Reference Marx-Glowna, Uschmann, Schulze, Marschner, Wille, Schlage, Stöhlker, Röhlsberger and Paulus^68] chose a $-{28}^{\circ }$ asymmetry angle and four reflections because of the practical limitation of the crystal size and preparation (high-quality surface), and obtained experimentally $\left(2.2\pm 2.0\right)\,{\times}\, {10}^{-9}$ polarization purity finally.

Table 2 Calculated polarization purity $\mathcal{P}$ for asymmetry angle ${\alpha}_{\mathrm{c}}$ and number of reflections $n$ . Here, ${D}_{-}$ is the accepted beam divergence, ${S}_{+}$ is the beam footprint on the crystal surface and $I/{I}_0$ is the peak reflectivity. Taken from Ref. [Reference Marx-Glowna, Uschmann, Schulze, Marschner, Wille, Schlage, Stöhlker, Röhlsberger and Paulus68], licensed under CC BY 4.0.

2.2.3 Quasi-channel-cuts

It may be necessary to realize the two opposing surfaces by two separate crystals. This is called a quasi-channel-cut. It may help to tune the Bragg reflections separately by an angle offset^[ Reference Hart and Rodrigues^54] since the reflectivity curves for both polarizations ${R}_{\sigma }$ and ${R}_{\pi }$ can have different widths and positions in an angle.

Furthermore, not all materials can be grown as large bulk as is done for silicon. For example, diamond is quite attractive because of its high thermal conductivity and low absorption in the X-ray region^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55], but it is very challenging to produce at large sizes and to obtain high-quality diamond with few dislocations and stacking faults. A reflectivity as high as 99% of hard X-rays from nearly defect-free diamond crystals at near-normal incidence has been reported^[ Reference Shvyd’ko, Stoupin, Blank and Terentyev^69]. Nevertheless, low crystal quality and complicated production processes constrain the development of diamond in polarizers. Polarization purities of $8.9\times {10}^{-10}$ ^[ Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus^53] and ${3\times {10}^{-10}}$ ^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze^55] are reported for using artificial diamond crystals manufactured by CVD.

The setup of a quasi-channel-cut is technically challenging in providing sufficient angular stability of both surfaces.

2.2.4 Temporal effects

Another feature, relevant for applications at XFELs in particular, is the inherent pulse stretching effect for Bragg crystals^[ Reference Shvyd’ko and Lindberg^70, Reference Hrdý^71]. Due to the scattering at the lattice planes happening over many lattice planes (leading to the finite spectral bandwidth), a short X-ray pulse will become temporally stretched. The ray will enter the crystal at a certain depth where it is effectively diffracted out, being the Bragg-case extinction depth $e$ . For a symmetric reflection, the projection of penetration will add to the pulse envelope^[ Reference Hrdý^71] as follows:

(12) $$\begin{align}\Delta \tau =2e\sin {\theta}_{\mathrm{B}}/c.\end{align}$$

This effect increases obviously with the Bragg angle and the number of consecutive reflections, and depends as well on the photon energy and material. The latter dependency is not straightforward. Higher photon energy usually leads to deeper penetration, but higher $Z$ of the material leads to stronger diffraction per lattice plane and hence reduced penetration.

3.2.1 PW lasers and XFELs

In 2006, Heinzl et al.^[ Reference Heinzl, Liesfeld, Amthor, Schwoerer, Sauerbrey and Wipf^20] considered a petawatt laser system with 140 fs pulse duration, 150 J pulse energy and ${10}^{22}\;\mathrm{W}/{\mathrm{cm}}^2$ intensity in the focal region to induce vacuum birefringence. A schematic diagram of the experimental setup is depicted in Figure 10. The high-power optical laser pulse is focused by an off-axis parabolic mirror (OAP), and the linearly polarized X-ray laser pulse collides with the optical laser pulse at the interaction area. Then, a small ellipticity of the X-ray pulse caused by vacuum birefringence will be detected. The whole process happens in a vacuum chamber.

Schlenvoigt et al.^[ Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey^19] proposed an experimental scheme (Figure 11) based on the European XFEL and HED instrument in conjunction with the Relativistic Laser at Xfel (ReLaX) laser system being developed by the Helmholtz International Beamline for Extreme Fields (HIBEF). In this figure, the main part is the setup for vacuum birefringence detection. The PW laser is also focused by an OAP into the interaction area. The propagation of the XFEL is worth introducing in detail. A well-collimated XFEL is measured by an intensity monitor (IM) to record the number of X-ray photons. Then, the XFEL becomes a linearly polarized beam with $\mathcal{P}\sim {10}^{-11}$ polarization purity after the polarizer (Pol). The first compound refractive lenses (CRLs) are used to focus the XFEL to the interaction point to overlap with the PW laser focus. The second CRLs are for re-collimation of the X-rays. The analyzer (Ana) is the same as the Pol in material and geometry but crossed to the Pol and only allows photons of flipped polarization to pass, which will be detected by the detector (Det). Comparing the photon numbers of the initial XFEL pulse and the polarization-flipped pulse, vacuum birefringence can be detected.

Moreover, the authors have studied the effect of plasma from residual gas particles on the signal of vacuum birefringence and proposed the method of vacuum cleaning. They plan to introduce another laser called a cleaning laser to ionize the gas particles, shown in yellow and named the cleaned volume, illustrated in the circle at the bottom left of Figure 11. The cleaned volume is much larger than the interaction volume, shown in pink. A static electric field is applied to remove charged particles from the cleaned volume. At the same time, the surrounding gas will repopulate the volume by diffusion, which can be mitigated by correct timing of the cleaning laser pulse. The bottom right shows the fundamental idea of probing QED vacuum birefringence by combining an XFEL and PW laser.

Subsequently, Shen et al.^[ Reference Shen, Bu, Xu, Xu, Ji, Li and Xu^35] and Xu et al. ^[ Reference Xu, Shen, Xu and Liang^77] presented an experimental design revolving around a 100 PW laser and a 12.914 keV XFEL beam with the Station of Extreme Light (SEL) at the Shanghai HIgh repetition rate XFEL aNd Extreme light (SHINE) facility. According to the parameters of the 100 PW laser and adopting the analysis of Schlenvoigt et al.^[ Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey^19], the ellipticity is about $2\times {10}^{-10}$ and about 170 photons with flipped polarization should be produced by vacuum birefringence if the total photon number at the interaction is ${10}^{12}$ .

There are further works presenting estimates for laser–XFEL studies, concentrating more on modeling and refined beam geometries^[ Reference Karbstein and Sundqvist^33, Reference Karbstein, Gies, Reuter and Zepf^84]. They also consider 30 J, 30 fs, 1 PW laser systems in conjunction with ${10}^{12}$ probe photons. Recently, Mosman and Karbstein^[ Reference Mosman and Karbstein^85] discussed in detail that modeling for ReLaX and the European XFEL as Schlenvoigt et al.^[ Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey^19] did. However, they used more realistic laser and XFEL parameters, for example, the accelerator setpoint and bunch charge dependency on the number of probe photons, yielding $N\sim {10}^{11}$ . This number is valid for the SASE mode of the European XFEL; the spectral matching aspect was published later^[ Reference Schulze, Grabiger, Loetzsch, Marx-Glowna, Schmitt, Garcia, Hippler, Huang, Karbstein, Konôpková, Schlenvoigt, Schwinkendorf, Strohm, Toncian, Uschmann, Wille, Zastrau, Röhlsberger, Stöhlker, Cowan and Paulus^59], which effectively reduces the available number of photons. They also discussed XFEL pulse lengthening for channel-cut polarizers (cf. Section 2.2.4). Effectively, the polarizer before the interaction will stretch the X-ray pulses to approximately 100 fs. This can help in experiments to reduce the temporal jitter effect.

3.2.2 XFEL only

The common method of detecting vacuum birefringence is combining the XFEL with a PW-class optical laser. A novel way to detect vacuum birefringence by the collision of two consecutive XFEL pulses under a finite angle has been put forward by Karbstein et al.^[ Reference Karbstein, Sundqvist, Schulze, Uschmann, Gies and Paulus^86]. This idea takes the scaling of background field intensity (cf. Section 3.3) with wavelength, ${I}_{\mathrm{BG}}\propto {\lambda}^{-2}$ , into account and complements it with the higher repetition rate of XFELs compared to PW-class optical lasers.

Recently, the pulse duration of an XFEL was measured directly^[ Reference Osaka, Inoue, Yamada, Inubushi, Matsumura, Sano, Tono, Yamauchi, Tamasaku and Yabashi^87] to approximately 10 fs. This experiment demonstrates that the nonlinear regime of optics may be accessed in the X-ray domain, that is, sufficiently high photon densities can be produced. With a typical pulse energy of 1 mJ and the aforementioned pulse duration, the XFEL pulse power is about 100 GW. With $\lambda \sim 0.1\;\mathrm{nm}$ and further typical beam and focusing parameters ( ${F}_{\#}\sim 100$ ), spot sizes below 100 nm are reasonable^[ Reference Schropp, Boye, Feldkamp, Hoppe, Patommel, Samberg, Stephan, Giewekemeyer, Wilke, Salditt, Gulden, Mancuso, Vartanyants, Weckert, Schöder, Burghammer and Schroer^88, Reference Schropp, Hoppe, Meier, Patommel, Seiboth, Lee, Nagler, Galtier, Arnold, Zastrau, Hastings, Nilsson, Uhlén, Vogt, Hertz and Schroer^89], and thus focus areas of approximately ${10}^{-10}\ {\mathrm{cm}}^2$ . As a result, intensities of ${I}_{\mathrm{BG}}\sim {10}^{20}\ \mathrm{W}/{\mathrm{cm}}^2$ can be obtained with XFEL pulses for the background field. That is approximately 10% of optical laser peak intensities. While the ellipticity – being the photon polarization flip probability – scales as ${\delta}^2\propto {\left({I}_{\mathrm{BG}}\right)}^2$ , the number of flipped photons per unit time (e.g., operating hour) scales with the repetition rate, being easily ${10}^2\ \mathrm{to}\ {10}^4$ Hz and 1 MHz in future facilities (cf. Table 3). As a consequence, the number of flipped photons per unit time can compete with or even exceed the numbers of XFEL–laser combined schemes. Technically, a PW-class laser installation alongside an XFEL is not necessary, but an even more complex X-ray beam path must be realized.

Table 3 Overview of XFEL facilities. Bold facility names indicate facilities with an ultra-intense laser in operation. Italic represents planned facilities. Adapted from Ref. [Reference Nakamiya and Homma90], licensed under CC BY-NC-ND 4.0.

From the installation diagram (Figure 12) we can see that the XFEL beam is focused twice to the interaction region from different directions. This employs the pulse train, such that pulse $n$ is the probe for pulse $n+1$ . Along with the beam path, the first set of CRLs generates the pump pulse ( $b=2$ ), where no high polarization purity is required. Further downstream follows a CRL to re-collimate the beam for a delay path, matching the pulse repetition time. Then the XFEL passes through the polarizer made by a channel-cut diamond to become the probe pulse ( $b=1$ ). Karbstein et al.^[ Reference Karbstein, Sundqvist, Schulze, Uschmann, Gies and Paulus^86] also mentioned the losses and the pulse deformations of the XFEL pulse caused by optical elements. In the meantime, each XFEL pulse train should be controlled well to achieve the best possible spatio-temporal overlap of the focused pump and probe beams. Furthermore, each optical element must be of sufficient perfection, for example, the high reflectivity of diamond crystals and the high perfect focus of CRLs.

Figure 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 $ \vartheta_{\mathrm{coll}}$ . 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. [Reference Karbstein, Sundqvist, Schulze, Uschmann, Gies and Paulus86], licensed under CC BY 4.0.

Apart from X-ray polarimetry, scientists have presented a different approach for measuring vacuum birefringence using multi-MeV to GeV photons^[ Reference King and Elkina^37, Reference Nakamiya and Homma^91]. King and Elkina^[ Reference King and Elkina^37] carried out the analytical calculations and numerical simulations for the measurement of vacuum birefringence by multi-MeV photons, instead of X-ray or optical photons. Nakamiya and Homma^[ Reference Nakamiya and Homma^91] proposed combining a 10 PW laser system with a 1 GeV gamma-ray photon source to probe the vacuum birefringence effect and designed the gamma-ray polarimeter to measure the polarization flip of the probe gamma-rays. They derived theoretically the phase retardation of GeV probe photons via pairwise topology of the Bethe–Heitler process in a polarimeter, and concluded it would be possible to observe the vacuum birefringence effect with the accuracy of 4.7% for the averaged phase retardation $\left\langle G\right\rangle$ of 0.72 if ${10}^4$ conversion pairs are available.

3.4.1 XFEL facilities

XFELs are indispensable sources for structural analysis and have contributed to the development of ultra-fast processes. XFEL facilities have blossomed all over the world^[ Reference Huang, Deng, Liu, Wang and Zhao^90]. In Europe, Deutsches Elektronen-Synchrotron (DESY)^[ Reference Ackermann, Asova, Ayvazyan, Azima, Baboi, Bähr, Balandin, Beutner, Brandt, Bolzmann, Brinkmann, Brovko, Castellano, Castro, Catani, Chiadroni, Choroba, Cianchi, Costello, Cubaynes, Dardis, Decking, Delsim-Hashemi, Delserieys, Di Pirro, Dohlus, Düsterer, Eckhardt, Edwards, Faatz, Feldhaus, Flöttmann, Frisch, Fröhlich, Garvey, Gensch, Gerth, Görler, Golubeva, Grabosch, Grecki, Grimm, Hacker, Hahn, Han, Honkavaara, Hott, Hüning, Ivanisenko, Jaeschke, Jalmuzna, Jezynski, Kammering, Katalev, Kavanagh, Kennedy, Khodyachykh, Klose, Kocharyan, Körfer, Kollewe, Koprek, Korepanov, Kostin, Krassilnikov, Kube, Kuhlmann, Lewis, Lilje, Limberg, Lipka, Löhl, Luna, Luong, Martins, Meyer, Michelato, Miltchev, Möller, Monaco, Müller, Napieralski, Napoly, Nicolosi, Nölle, Nuñez, Oppelt, Pagani, Paparella, Pchalek, Pedregosa-Gutierrez, Petersen, Petrosyan, Petrosyan, Petrosyan, Pflüger, Plönjes, Poletto, Pozniak, Prat, Proch, Pucyk, Radcliffe, Redlin, Rehlich, Richter, Roehrs, Roensch, Romaniuk, Ross, Rossbach, Rybnikov, Sachwitz, Saldin, Sandner, Schlarb, Schmidt, Schmitz, Schmüser, Schneider, Schneidmiller, Schnepp, Schreiber, Seidel, Sertore, Shabunov, Simon, Simrock, Sombrowski, Sorokin, Spanknebel, Spesyvtsev, Staykov, Steffen, Stephan, Stulle, Thom, Tiedtke, Tischer, Toleikis, Treusch, Trines, Tsakov, Vogel, Weiland, Weise, Wellhöfer, Wendt, Will, Winter, Wittenburg, Wurth, Yeates, Yurkov, Zagorodnov and Zapfe^{92, 93]}, one of the accelerator centers, contains three large accelerators: PETRA III, FLASH and European XFEL. FLASH supplies soft X-rays and PETRA III and EuXFEL supply hard X-rays. Italian Elettra Sincrotrone Trieste^{[ 94]} has two advanced light sources: Elettra and FERMI. The third-generation synchrotron radiation facility Elettra produces synchrotron radiation in a wavelength range from infrared to hard X-rays, while FERMI is a seeded free electron laser working in the ultraviolet and soft X-ray range. SwissFEL^[ Reference Milne, Schietinger, Aiba, Alarcon, Alex, Anghel, Arsov, Beard, Beaud, Bettoni, Bopp, Brands, Brönnimann, Brunnenkant, Calvi, Citterio, Craievich, Divall, Dällenbach, D’Amico, Dax, Deng, Dietrich, Dinapoli, Divall, Dordevic, Ebner, Erny, Fitze, Flechsig, Follath, Frei, Gärtner, Ganter, Garvey, Geng, Gorgisyan, Gough, Hauff, Hauri, Hiller, Humar, Hunziker, Ingold, Ischebeck, Janousch, Juranić, Jurcevic, Kaiser, Kalantari, Kalt, Keil, Kittel, Knopp, Koprek, Lemke, Lippuner, Sancho, Löhl, Lopez-Cuenca, Märki, Marcellini, Marinkovic, Martiel, Menzel, Mozzanica, Nass, Orlandi, Loch, Panepucci, Paraliev, Patterson, Pedrini, Pedrozzi, Pollet, Pradervand, Prat, Radi, Raguin, Redford, Rehanek, Réhault, Reiche, Ringele, Rittmann, Rivkin, Romann, Ruat, Ruder, Sala, Schebacher, Schilcher, Schlott, Schmidt, Schmitt, Shi, Stadler, Stingelin, Sturzenegger, Szlachetko, Thattil, Treyer, Trisorio, Tron, Vetter, Vicario, Voulot, Wang, Zamofing, Zellweger, Zennaro, Zimoch, Abela, Patthey and Braun^{95, 96]} is Switzerland’s X-ray free electron laser with a hard X-ray free electron laser with 0.1 nm wavelength and 20 fs pulse duration at the Paul Scherrer Institute (PSI).

In the USA, the Linac Coherent Light Source (LCLS)^[ Reference Emma, Akre, Arthur, Bionta, Bostedt, Bozek, Brachmann, Bucksbaum, Coffee, Decker, Ding, Dowell, Edstrom, Fisher, Frisch, Gilevich, Hastings, Hays, Ph, Huang, Iverson, Loos, Messerschmidt, Miahnahri, Moeller, Nuhn, Pile, Ratner, Rzepiela, Schultz, Smith, Stefan, Tompkins, Turner, Welch, White, Wu, Yocky and Galayda^{97, 98]} at SLAC achieved the first lasing and FEL saturation at 0.15 nm in 2009. Its upgrade LCLS-II^[ Reference Galayda^99] is designed to produce high-energy X-rays covering the energy range from 200 eV to 1.5 keV for soft X-rays and from 1 to 5 keV for hard X-rays. A further upgrade^[ Reference Raubenheimer^100] from 4 to 8 GeV beam energy will extend the photon energy range to at least 12.8 keV.

In Asia, the first FEL facility was the SPring-8 Ångström Compact Free-Electron Laser (SACLA)^[ Reference Pile^101] in Japan, with a peak X-ray laser power of 1 GW and wavelength of 0.1 nm. It has matured to multi-beamline, soft and hard X-ray operation^[ Reference Yabashi, Tanaka and Ishikawa^102, Reference Tono, Hara, Yabashi and Tanaka^103] and extreme intensities^[ Reference Osaka, Inoue, Yamada, Inubushi, Matsumura, Sano, Tono, Yamauchi, Tamasaku and Yabashi^87]. The Pohang Accelerator Laboratory X-ray Free Electron Laser (PAL-XFEL)^[ Reference Ko, Kang, Heo, Kim, Kim, Min, Yang, Baek, Choi, Mun, Park, Suh, Shin, Hu, Hong, Jung, Kim, Kim, Na, Park, Park, Jung, Jeong, Lee, Lee, Lee, Oh, Suh, Han, Kim, Jung, Kim, Lee, Lee, Sung, Mok, Yang, Parc, Lee, Lee, Shin, Kim, Kim, Lee, Park, Kim, Park, Eom, Rah, Kim, Nam, Park, Park, Kim, Kwon, An, Park, Kim, Hyun, Kim, Kim, Yu, Kim, Kang, Kim, Kim, Lee, Lee, Park, Koo, Kim and Lee^76] produces wavelengths of 0.1 and 1 nm for hard and soft X-rays, respectively. In China there are two facilities^[ Reference Liu, Feng, Gu, Gao, Deng, Zhang, Sun, Chen, Zhang, Fang, Wang, Zhou, Leng, Gu, Yin, Gu, Fang, Wang and Zhao^104, Reference Shen, Ji, Zhang, Bu and Xu^105]: the Soft X-ray Free Electron Laser (SXFEL) performs the shortest wavelength of 2 nm, and the hard XFEL SHINE with a 0.05 nm wavelength is being constructed. All of these facilities with brilliant X-rays enable scientists to gain insights into the properties, ultrafast processes and essences of matter.

We are not discussing synchrotron sources for two reasons. They are currently limited in beam divergence and thus in polarization purity. Furthermore, their pulses are much longer than background field pulses and contain far fewer photons than XFEL pulses.

In the context of X-ray polarimetry of vacuum birefringence, it must be noted that probe beam photon numbers are often over-estimated in experiment proposals. Typical XFEL pulse energies are of the order of 1 mJ, which yields at 12 keV about ${10}^{11}$ photons^[ Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey^19]. In addition, the usual spectral bandwidth of XFELs is of the order of 1% in SASE mode. With self-seeding schemes^[ Reference Saldin, Schneidmiller, Shvyd’ko and Yurkov^106– Reference Amann, Berg, Blank, Decker, Ding, Emma, Feng, Frisch, Fritz, Hastings, Huang, Krzywinski, Lindberg, Loos, Lutman, Nuhn, Ratner, Rzepiela, Shu, Shvyd’ko, Spampinati, Stoupin, Terentyev, Trakhtenberg, Walz, Welch, Wu, Zholents and Zhu^108], a reduction to 10^–4 is possible at the cost of reduced pulse energy. This sets requirements on the spectral/angular acceptance of crystal optics, such as channel-cut crystal polarizers, particularly the widths of Bragg reflectivity curves, as shown in Figure 5.

A relative spectral width of ${10}^{-4}$ at ${\theta}_{\mathrm{B}}=45{}^{\circ}$ , matching the self-seeding bandwidth, requires a reflectivity curve width of ${10}^{-4}\ \mathrm{rad}={20}^{"}$ . Thus, the dashed curve in Figure 5, having a width of approximately ${6}^{"}$ , would have a spectral transmission of approximately 0.3 for self-seeded FEL pulses. Using an asymmetric cut (see Section 2.2.2), can increase the reflectivity curve width. As an example, in order to widen the curve by the required factor of approximately 3, an asymmetry angle ${\alpha}_{\mathrm{c}}\sim -40{}^{\circ}$ would be necessary. This would increase the beam footprint on the channel-cut crystal surface by approximately 10.

For higher photon energies, for example, the solid curve in Figure 5 for ${E}_{\mathrm{ph}}\sim 10\;\mathrm{keV}$ , the reflectivity curves become narrower due to deeper penetration. As a consequence, improving the bandwidth by the asymmetry angle ${\alpha}_{\mathrm{c}}$ becomes increasingly impractical.

3.4.2 PW-class laser facilities

Ultra-intense pulsed lasers are known to produce the highest light intensities on Earth and are thus favorable tools to generate the background field in their focus, polarizing the vacuum helped by a counter-propagating probe light. Table 4 has already provided estimated ellipticities for relevant peak intensities. Those intensities are in reach of currently available laser technology. Recently, laser scientists at the Center for Relativistic Laser Science (CoReLS) in Korea reported a peak laser intensity exceeding ${10}^{23}\ \mathrm{W}/{\mathrm{cm}}^2$ ^[ Reference Yoon, Kim, Choi, Sung, Lee, Lee and Nam^109], generated from a 4 PW laser pulse.

Ultra-intense laser systems are nowadays commercially available and are widely employed. They are in general based on the chirped pulse amplification scheme (CPA)^[ Reference Strickland and Mourou^110] and exhibit pulse durations of from approximately 10 fs to approximately 1 ps, and typical wavelengths are 800 nm or 1.05 μm. An overview can be found at The International Committee on Ultrahigh Intensity Lasers (ICUIL)^{[ 111]}, where they provide an interactive map^{[ 112]}.

PW-class lasers are currently the top class of existing facilities, but there are several dozen and hence too many to list them all. Here we concentrate on facilities and projects significantly exceeding 1 PW.

The Extreme Light Infrastructure (ELI)^[ Reference Gales, Tanaka, Balabanski, Negoita, Stutman, Tesileanu, Ur, Ursescu, Andrei, Ataman, Cernaianu, D’Alessi, Dancus, Diaconescu, Djourelov, Filipescu, Ghenuche, Ghita, Matei, Seto, Zeng and Zamfir^{113– 116]} is an advanced laser-based research infrastructure with multiple sites. One of the sites is Extreme Light Infrastructure Nuclear Physics (ELI-NP), which succeeded in delivering the 10 PW @ 1 shot per minute in 2019.

In the UK, the Central Laser Facility (CLF)^[ Reference Chekhlov, Divall, Ertel, Hawkes, Hooker, Ross, Matousek, Hernandez-Gomez, Musgrave, Tang, Winstone, Neely, Clarke, Foster, Hancock, Wyborn and Collier^{117, 118]}, part of the Rutherford Appleton Laboratory, is dedicated to high-energy laser systems. There are five laser facilities: ULTRA, Artemis, OCTOPUS, Gemini and Vulcan. Here, Gemini is a dual beam laser system with 2 × 15 J, 30 fs laser pulses. Vulcan has two kinds of laser modes. In its long pulse mode, the laser energy is up to 2.6 kJ with nanosecond pulse duration. In the short pulse mode, it has up to 1 PW peak power with 500 fs pulse duration and the focal intensity is about ${10}^{21}\ \mathrm{W}/{\mathrm{cm}}^2$ . Soon, the peak power will be increased from 1 PW (500 J in 500 fs) to 20 PW (400 J in 20 fs). It will be a unique beamline to examine matter under extreme conditions.

In France, the Apollon laser system is a multi-beam, multi-petawatt facility to generate 10 PW pulses of 150 J energy and 15 fs (full width at half maximum (FWHM)) duration at a repetition rate of 1 shot per minute. The first available laser beam delivered on-target pulses of 10 J average energy, 24 fs duration and 1 PW nominal power in 2021^[ Reference Burdonov, Fazzini, Lelasseux, Albrecht, Antici, Ayoul, Beluze, Cavanna, Ceccotti, Chabanis, Chaleil, Chen, Chen, Consoli, Cuciuc, Davoine, Delaneau, d’Humières, Dubois, Evrard, Filippov, Freneaux, Forestier-Colleoni, Gremillet, Horny, Lancia, Lecherbourg, Lebas, Leblanc, Ma, Martin, Negoita, Paillard, Papadopoulos, Perez, Pikuz, Qi, Quéré, Ranc, Söderström, Scisciò, Sun, Vallières, Wang, Yao, Mathieu, Audebert and Fuchs^119].

The Institute of Applied Physics of the Russian Academy of Sciences has established a large infrastructure project, the Exawatt Center for Extreme Light Studies (XCELS)^[ Reference Mukhin, Soloviev, Perevezentsev, Shaykin, Ginzburg, Kuzmin, Mart’yanov, Shaikin, Kuzmin, Mironov, Yakovlev and Khazanov^{120– 122]}. The aim of the project is to build high-power lasers with 200 PW power and 25 fs pulse duration by assembling 12 identical laser channels with 15 PW power for each. The intensity in the focus is expected to be approximately ${10}^{25}\ \mathrm{W}/{\mathrm{cm}}^2$ . This provides an opportunity to peer into the fundamental processes and unknown phenomena of high-energy physics.

Furthermore, the project of Shanghai HIgh repetition rate XFEL aNd Extreme light facility (SHINE) was founded^[ Reference Gan, Yu, Wang, Liu, Xu, Li, Li, Yu, Wang, Liu, Chen, Peng, Xu, Yao, Zhang, Chen, Tang, Wang, Yin, Liang, Leng, Li and Xu^123] in Shanghai, China. In the future, the SEL of SHINE will provide a laser system with 100 PW and an expected focused laser intensity of $2\times {10}^{23}\ \mathrm{W}/{\mathrm{cm}}^2$ ^[ Reference Wang, Liu, Lu, Chen, Long, Li, Chen, Chen, Bai, Li, Peng, Liu, Wu, Wang, Li, Xu, Liang, Leng and Li^124]. This is the basis for the proposed experiments^[ Reference Shen, Bu, Xu, Xu, Ji, Li and Xu^35, Reference Xu, Shen, Xu and Liang^77] at SHINE.

In view of vacuum birefringence, the gain of ellipticity ${\delta}^2$ with laser peak intensity is obviously quite strong, ${\delta}^2\propto {\left({I}_{\mathrm{BG}}\right)}^2$ (see Equation (16a)). This is of particular importance for the trade-off of ellipticity against purity $\mathcal{P}$ . Table 4 indicates that an intensity of ${I}_{\mathrm{BG}}\sim 2\times {10}^{23}\ \mathrm{W}/{\mathrm{cm}}^2$ can yield an ellipticity ${\delta}^2\sim 2\times {10}^{-10}$ , exceeding the best purities so far measured, $\mathcal{P}\sim {10}^{-11}$ (cf. Table 5). In this way, the signal would always exceed the background from the finite extinction. In addition, it should be noted that such intensity has already been demonstrated with a 4 PW laser system^[ Reference Yoon, Kim, Choi, Sung, Lee, Lee and Nam^109].

Table 5 Timeline of precision X-ray polarimetry. Here, $m$ denotes the number of reflections per channel-cut crystal, ${\sigma}_{\mathrm{H}}$ and ${\sigma}_{\mathrm{V}}$ represent the beam divergence, $\mathcal{P}$ is the obtained polarization purity and ${\mathcal{P}}_{\mathrm{Divergence}}^{\mathrm{Limit}}$ is calculated from the divergence according to Equation (7). For the current record^[ Reference Marx-Glowna, Grabiger, Lötzsch, Uschmann, Schmitt, Schulze, Last, Roth, Antipov, Schlenvoigt, Sergueev, Leupold, Röhlsberger and Paulus^72], the nominal instrument’s beam divergence was reduced by slits at the polarimeter.

3.4.3 Combined facilities

For the purpose of vacuum birefringence experiments where the background field is generated by an ultra-intense laser and X-ray polarimetry is employed for detection, it is mandatory to combine XFELs with such lasers. In most cases, XFEL facilities host several beamlines and/or several instruments per beamline. Thanks to the wide range of applications and thus an existing market, it is relatively straightforward to equip an XFEL instrument with an ultra-intense laser. Even for systems below the PW level there are enough use cases to use XFELs as probes in laser–matter interactions.

Referring to Table 3, there are three out of four existing hard X-ray facilities equipped with an ultra-intense laser. We can exclude soft X-ray FELs since a key for detection is a short probe wavelength. The facilities and instruments are: MEC at LCLS/SLAC, SACLA EH6 and HED/HIBEF at the European XFEL. Their respective parameters^[ Reference Shen, Bu, Xu, Xu, Ji, Li and Xu^35, Reference Huang, Deng, Liu, Wang and Zhao^{91, 98, 111,} Reference Nagler, Arnold, Bouchard, Boyce, Boyce, Callen, Campell, Curiel, Galtier, Garofoli, Granados, Hastings, Hays, Heimann, Lee, Milathianaki, Plummer, Schropp, Wallace, Welch, White, Xing, Yin, Young, Zastrau and Lee^125– Reference Yabuuchi, Kon, Inubushi, Togahi, Sueda, Itoga, Nakajima, Habara, Kodama, Tomizawa and Yabashi^127] are listed in Table 6.

Table 6 Overview of facilities combining XFEL beams with PW-class lasers. Planned facilities are shown in italic. Please note that there is no common factorial relation between laser power and peak intensity. Focusing F-numbers very among the facilities, adapted to their overall mission. Furthermore, beam quality can reduce the encircled energy in the focal spot and therefore reduce the peak intensity^[ Reference Yabuuchi, Kon, Inubushi, Togahi, Sueda, Itoga, Nakajima, Habara, Kodama, Tomizawa and Yabashi^127]. The provided laser pulse wavefront control for the final focusing and reasonably tight focusing, ${10}^{22}\ \mathrm{W}/{\mathrm{cm}}^2$ per 1 PW, is realistic.

The Matter in Extreme Conditions (MEC) instrument is the facility at LCLS that produces extreme matter states with an intense laser radiation, where LCLS provides complete imaging and optical diagnostics methods. Nagler et al.^[ Reference Nagler, Arnold, Bouchard, Boyce, Boyce, Callen, Campell, Curiel, Galtier, Garofoli, Granados, Hastings, Hays, Heimann, Lee, Milathianaki, Plummer, Schropp, Wallace, Welch, White, Xing, Yin, Young, Zastrau and Lee^125] presented an overview of the beamline, the capabilities of the instrumentation and highlights of experiments. Glenzer et al.^[ Reference Glenzer, Fletcher, Galtier, Nagler, Alonso-Mori, Barbrel, Brown, Chapman, Chen, Curry, Fiuza, Gamboa, Gauthier, Gericke, Gleason, Goede, Granados, Heimann, Kim, Kraus, MacDonald, Mackinnon, Mishra, Ravasio, Roedel, Sperling, Schumaker, Tsui, Vorberger, Zastrau, Fry, White, Hasting and Lee^130] summarized the first experiment of laser-compressed solids and the measurements of highly accurate X-ray diffraction and X-ray Thomson scattering on the MEC instrument at LCLS. Fletcher et al.^[ Reference Fletcher, Curry, Gauthier, Glenn, Chen, Cunningham, Descamps, Frost, Galtier, Heimann, Kim, Mo, Ofori-Okai, Peebles, Seiboth, Treffert, Dyer, McBride and Glenzer^131] investigated bremsstrahlung from relativistic electrons generated by the interaction of a high-intensity femtosecond laser with solid μm-thick aluminum and polypropylene targets, and measured the energy spectrum and temperature of hot electrons via differential X-ray energy filtering.

Similar to the LCLS, the SACLA XFEL facility opened after the completion of commissioning^[ Reference Yabuuchi, Kon, Inubushi, Togahi, Sueda, Itoga, Nakajima, Habara, Kodama, Tomizawa and Yabashi^127, Reference Tono, Togashi, Inubushi, Sato, Katayama, Ogawa, Ohashi, Kimura, Takahashi, Takeshita, Tomizawa, Goto, Ishikawa and Yabashi^132]. This experimental platform is equipped with two beams of 800 nm wavelength, 1 Hz repetition rate and 12.5 J maximum energy in a 25 fs pulse duration and a 500 TW peak power after pulse compression. Yabuuchi et al.^[ Reference Yabuuchi, Kon, Inubushi, Togahi, Sueda, Itoga, Nakajima, Habara, Kodama, Tomizawa and Yabashi^127] characterized the light source performance during the commissioning of the experimental platform and confirmed that the XFEL and the high-intensity laser can operate normally with dedicated diagnostics.

In Europe, the HED scientific instrument at the European XFEL is a unique platform for experiments in extreme conditions of pressure, temperature or electromagnetic field^[ Reference Zastrau, Appel, Baehtz, Baehr, Batchelor, Berghäuser, Banjafar, Brambrink, Cerantola, Cowan, Damker, Dietrich, Di Dio Cafiso, Dreyer, Engel, Feldmann, Findeisen, Foese, Fulla-Marsa, Göde, Hassan, Hauser, Herrmannsdörfer, Höppner, Kaa, Kaever, Knöfel, Konôpková, García, Liermann, Mainberger, Makita, Martens, McBride, Möller, Nakatsutsumi, Pelka, Plueckthun, Prescher, Preston, Röper, Schmidt, Seidel, Schwinkendorf, Schoelmerich, Schramm, Schropp, Strohm, Sukharnikov, Talkovski, Thorpe, Toncian, Toncian, Wollenweber, Yamamoto and Tschentscher^128, Reference Nakatsutsumi, Appel, Baehtz, Chen, Cowan, Göde, Konopkova, Pelka, Priebe, Schmidt, Sukharnikov, Thorpe, Tschentscher and Zastrau^133]. Zastrau et al.^[ Reference Zastrau, Appel, Baehtz, Baehr, Batchelor, Berghäuser, Banjafar, Brambrink, Cerantola, Cowan, Damker, Dietrich, Di Dio Cafiso, Dreyer, Engel, Feldmann, Findeisen, Foese, Fulla-Marsa, Göde, Hassan, Hauser, Herrmannsdörfer, Höppner, Kaa, Kaever, Knöfel, Konôpková, García, Liermann, Mainberger, Makita, Martens, McBride, Möller, Nakatsutsumi, Pelka, Plueckthun, Prescher, Preston, Röper, Schmidt, Seidel, Schwinkendorf, Schoelmerich, Schramm, Schropp, Strohm, Sukharnikov, Talkovski, Thorpe, Toncian, Toncian, Wollenweber, Yamamoto and Tschentscher^128] presented the scientific scope, technical infrastructure, diagnostics and experimental platforms. The HED scientific instrument supports a variety of X-ray methods, including X-ray polarimetry. The HIBEF user consortium contributes the high-intensity and high-energy laser systems^[ Reference Garcia, Höppner, Pelka, Bähtz, Brambrink, Di Dio Cafiso, Dreyer, Göde, Hassan, Kluge, Liu, Makita, Möller, Nakatsutsumi, Preston, Priebe, Schlenvoigt, Schwinkendorf, Šmíd, Talposi, Toncian, Zastrau, Schramm, Cowan and Toncian^129, Reference Ahmadiniaz, Bussmann, Cowan, Debus, Kluge and Schützhold^134] and their operation for users.

Another combined XFEL–laser facility will be the SEL at SHINE, which is designed to achieve laser intensities sufficient to explore the vacuum birefringence effect by colliding an XFEL^[ Reference Shen, Bu, Xu, Xu, Ji, Li and Xu^35, Reference King, Hu and Shen^135].

Figure 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. [Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze61], with the permission of AIP Publishing.

In view of vacuum birefringence, currently operating facilities provide laser intensities of 10¹⁹– ${10}^{21}\ \mathrm{W}/{\mathrm{cm}}^2$ ^[ Reference Nagler, Arnold, Bouchard, Boyce, Boyce, Callen, Campell, Curiel, Galtier, Garofoli, Granados, Hastings, Hays, Heimann, Lee, Milathianaki, Plummer, Schropp, Wallace, Welch, White, Xing, Yin, Young, Zastrau and Lee^125, Reference Yabuuchi, Kon, Inubushi, Togahi, Sueda, Itoga, Nakajima, Habara, Kodama, Tomizawa and Yabashi^127, Reference Garcia, Höppner, Pelka, Bähtz, Brambrink, Di Dio Cafiso, Dreyer, Göde, Hassan, Kluge, Liu, Makita, Möller, Nakatsutsumi, Preston, Priebe, Schlenvoigt, Schwinkendorf, Šmíd, Talposi, Toncian, Zastrau, Schramm, Cowan and Toncian^129]. Thus, ellipticities should be expected to be approximately ${10}^{-14}$ , much smaller than the current best polarization purities. Reasons for those comparably low intensities could be (i) operating at lower energy and power levels due to longer lifetime and cost efficiency, (ii) focusing that is not too tight for target debris management (for laser–matter interactions) and (iii) wavefront distortions re-distributing energy out of the focus peak into a halo.

These restrictions can be lifted for experiments dedicated to vacuum birefringence, for example, by providing a dedicated focusing element. We repeat here that a peak intensity of approximately ${10}^{23}\ \mathrm{W}/{\mathrm{cm}}^2$ has already been demonstrated with a 4 PW laser system^[ Reference Yoon, Kim, Choi, Sung, Lee, Lee and Nam^109]. That said, the laser at the HED instrument of the European XFEL could reach, with slightly more energy (12.5 J) and shorter pulse duration (25 fs), a peak power of 400 TW and thus approximately ${10}^{22}\ \mathrm{W}/{\mathrm{cm}}^2$ peak intensity, resulting in an ellipticity of $6\times {10}^{-12}$ instead of $4\times {10}^{-13}$ , as estimated by Mosman and Karbstein^[ Reference Mosman and Karbstein^85].

3.4.4 X-ray optics

Besides precision X-ray polarizers, CRLs are indispensable optical elements in the vacuum birefringence experimental setup with two purposes. Primarily, the XFEL beam must be focused into the interaction volume with the tightly focused PW laser. This is the purpose of the first CRL. On the other hand, the polarizers require a low divergence to provide a high extinction ratio. Therefore, the first CRLs must be located after the polarizer. In addition, the XFEL must be re-collimated by the second CRL, after the interaction but before the analyzer. In essence, two sets of CRLs are already inside the polarimeter setup. Therefore, the effects of the CRL material on the polarization must be studied and a suitable material must be found. Grabiger et al.^[ Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze^61] studied how the lens material itself influences the X-ray polarization. The setup is shown in Figure 13^[ Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze^61].

Grabiger et al.^[ Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze^61] analyzed three different grades of beryllium samples: high purity (PF-60), optical grade (O-30-H) and ultra-high purity grade (IF-1) beryllium. The results in the upper part of Table 7 clearly indicate that the beryllium samples greatly affect polarization purity. In this regard, the explanation given by the authors is the polycrystalline state of beryllium and they suggested two better options to focus X-rays. One alternative way is to employ reflective optical components, such as Kirkpatrick–Baez mirrors. Another option is to manufacture X-ray lenses from either single-crystal materials such as diamond, or from materials with an amorphous structure, such as glassy carbon and polymers.

Table 7 Deterioration of polarization purity by CRL materials. Upper part for flat Be samples at approximately 8 keV; lower part for CRL telescopes with $2\times \sim 6\;\mathrm{m}$ focal length at approximately 13 keV. Data taken from Ref. [Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze61], with the permission of AIP Publishing, and from Ref. [Reference Marx-Glowna, Grabiger, Lötzsch, Uschmann, Schmitt, Schulze, Last, Roth, Antipov, Schlenvoigt, Sergueev, Leupold, Röhlsberger and Paulus72], licensed under CC BY 4.0.

Those materials were studied later by the same group^[ Reference Marx-Glowna, Grabiger, Lötzsch, Uschmann, Schmitt, Schulze, Last, Roth, Antipov, Schlenvoigt, Sergueev, Leupold, Röhlsberger and Paulus^72]. They now used a synchrotron source for better sensitivity and investigated the impact of CRLs of different materials on the polarization purity, mimicking the general scheme of focusing and re-collimation proposed for vacuum birefringence X-ray polarimetry experiments^[ Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey^19, Reference Karbstein and Sundqvist^33, Reference Shen, Bu, Xu, Xu, Ji, Li and Xu^35, Reference Karbstein, Gies, Reuter and Zepf^84, Reference Karbstein, Sundqvist, Schulze, Uschmann, Gies and Paulus^86]. However, they employed rather long focal lengths of approximately 6 m. For shorter focal lengths, more CRL materials would be exposed to the beam and probably deteriorate the purity more strongly than currently measured. The results are listed in the lower part of Table 7. CRLs were fabricated out of Be, SU-8 photo-polymer, diamond and glassy carbon. From all of those materials, the CRLs fabricated from SU-8 showed the least depolarization of X-rays.

It should be noted that data for Be cannot be easily compared across both parts of Table 7. The effective thickness of the Be CRLs is not provided, and both experiments employed different photon energies.

In regard to X-ray polarimetry for vacuum birefringence, the results show another limit. Even though the deteriorated purity $\left(3.3\pm 1.5\right)\times {10}^{-11}$ is close to the instrument purity $\left(1.4\pm 0.9\right)\times {10}^{-11}$ , the deterioration was measurable. Without speculating about scalings, a birefringence experiment using CRLs as currently known would have a limit of at least the measured level of $\mathcal{P}\sim {10}^{-11}$ . Hence, the prospects for lowering the background signal $NT\mathcal{P}$ by better X-ray optics are dismal.