Fig. 1. Feynman diagram for the LbL scattering amplitude implying a four-photon self-interaction.

Fig. 2. Light-by-light scattering cross-section in microbarns () as a function of photon energy in the centre-of-mass frame, (calculated from Refs. [31,32,40]). The value of the cross-section probed by some past experiments is indicated by coloured dots ‘decorated’ by ‘laser lines’. This labelling of the processes (e.g., -to-) refers to the number of real photons in the in and out states (see insert). The -to- measurements (blue) were for a diphoton mass or more than 5 GeV (CMS^[9]) or more than 6 GeV (ATLAS^[7,8]). These are collectively represented on the plot at . The -to- process of Delbrück scattering off nuclei measured by Jarlskog et al.^[5] (see also Refs. [6,41]) is shown in green. An energy range is indicated (in cyan) for HED-HIBEF assuming the near head-on collision of an optical beam with central energy and XFEL beam with energies between 6 and . The red cross-dots represent the -to- laser experiments by Moulin et al.^[42], Bernard et al.^[43] (both all-optical), Inada et al.^[44], Yamaji et al.^[45] (both employing an XFEL) and Watt et al.^[46].

Fig. 3. Two variants of Delbrück scattering involving two virtual photons, . (a) Off a (static) Coulomb potential denoted by crosses. (b) Off a Lorentz boosted Coulomb potential in ultra-peripheral heavy-ion collisions.

Fig. 4. Two-step modification of the QED 4-photon vertex (left) to the Heisenberg–Euler four-point interaction (centre) and coupling to an external field scattering (right).

Fig. 5. Historical evolution of the results for vacuum birefringence experiments employing static magnetic fields (reproduced with permission from Ref. [78], where more details can be found). The green horizontal line represents .

Fig. 6. The STAR experiment. (a) Landau–Lifshitz process. (b) Modulation signalling vacuum birefringence by means of the optical theorem (reproduced with permission from Ref. [11]).

Fig. 7. Overview of different collision geometries: (a) the conventional head-on two-beam scenario; (b)–(e) three-beam scenarios where the XFEL is collided with two optical lasers. In (c) one of the optical beams is frequency doubled^[83].

Fig. 8. Schematic layout of the conventional scenario to measure vacuum birefringence. The XFEL beam is polarized with a channel-cut crystal, focused down to the interaction point with the counter-propagating high-intensity laser, recollimated and analysed with a second channel-cut crystal in a crossed position, such that only the -polarized component reaches the detector.

Fig. 9. Schematic layout of the dark-field scenario. The XFEL is focused with a beamstop creating a shadow in the converging (expanding) beam before (after) focus while retaining a central intensity peak in the focus where it collides with a counter-propagating high-intensity pump. X-ray optics image the beamstop to a matched aperture plane. The effective interaction with the pump is strongly localized and limited to the vicinity of the probe focus. Hence, given that the overlap factor is sufficiently similar to that in the conventional scenario in Figure 8, a scattering signal is induced in the shadow. A crystal polarizer directs its different polarization components to separate detectors.

Fig. 10. Planar three-beam configuration of a focused XFEL beam colliding in a plane with two focused optical beams at a collision angle .

Fig. 11. Example scattered photon signal for typical parameters at HED-HIBEF in the collision of the ReLaX optical photons with the EuXFEL photons. The coordinates are the scattering angles in and perpendicular to the collision plane, respectively.

Fig. 12. The number of scattered photons, parallel () and perpendicular () to the XFEL probe, as a function of the beam collision angle, plotted for different XFEL parameters. The probe propagates along the -axis, and the collision is in the – plane. The dashed curves () refer to the signal falling on the detector outside a central exclusion region of radius . The SASE and self-seeded options are taken from Table 2. Left: example results for total scattered photons. Right: photon scattering and birefringence (polarization flip).

Fig. 13. Feynman diagram for Coulomb-assisted birefringence.

Fig. 14. High-precision X-ray polarimetry. Left: the polarimeter consists of two channel-cut crystals acting as polarizer and analyser, respectively. Not shown is the telescope in between both and the optical laser responsible for polarizing the vacuum. Right: extinction curve around of the crossed-polarizer position reproduced from Ref. [72]. At a few data points, the corresponding detector signal is displayed: in the crossed-polarizer position not a single photon reaches the detector. To the level tested in this experiment, the polarizers are perfect.

Fig. 15. Experimental setup for the dark-field proof-of-principle experiment at the HED instrument of the EuXFEL. For completeness, here also the ReLaX beam path for the counter-propagating geometry is indicated.

Fig. 16. Result of the diffractive simulation. The sub-figures show the 2D intensity profiles of the beam along the beam path at various positions: (a) just behind the first obstacle, (b) at the pinhole position, close to beam focus, (c) before aperture A1, (d) behind aperture A1, (e) at an intermediate position and (f) at the detector, with a red square indicating the area into which the signal scattered at focus would be imaged. The axes are in units of m and the colour scale is logarithmic over three orders of magnitude.

Fig. 17. Example of optimization of the experimental parameters. The horizontal axis is the shadow factor, while the vertical one is the signal transmission factor. The diameter of the wire (obstacle O1) is encoded in the colour of the points.

Fig. 18. Design of microfabricated obstacles. The left-hand pane shows their shape (thickness as a function of position perpendicular to beam), while the other two panes show the transmission and phase shift induced to the XFEL beam, the combined effect of which is to deflect the beam on the edges rather than to scatter it.

Fig. 19. Diffraction simulation of various openings of the slits of apertures A1 and A2 while using the wires as obstacles. The first two figures show the simulated shadow and transmission factors, while the bottom figures show derived and factors, which are considered for optimization. Minimum values of the latter factors are desirable for our purpose.

Fig. 20. Diffraction simulation of various openings of the slits of apertures A1 and A2 while using the trumpet as obstacle O1 and the phase-corrected aperture as A1.

Fig. 21. Integrated reflectivity of the considered crystal cuts. For each crystal cut we in addition depict the value of in arbitrary units.