3.1 Conventional two-beam scenario

The most extensively studied scenario in the present context is the conventional two-beam scenario^[ Reference Aleksandrov, Ansel’m and Moskalev ^{66 ]}, which envisions the head-on collision of the XFEL beam acting as probe and an NIR high-intensity pump^[ Reference Heinzl, Liesfeld, Amthor, Schwoerer, Sauerbrey and Wipf ^{67 ]}. It aims at measuring the birefringence phenomenon for probe light traversing a strong electromagnetic field that arises as a direct consequence of the effective nonlinear couplings of the electromagnetic field in Equation (6). If the strong field introduces a preferred direction, such as a unidirectional magnetic field or a linearly polarized laser field, the isotropy of the vacuum is broken. In turn, the vacuum polarized by the strong field effectively features two distinct transverse propagation eigenmodes for probe light associated with different refractive indices. Correspondingly, if initially linearly polarized light (photon number $N$ ) having overlap with both of these modes is sent through such a field, a small fraction ${N}_{\perp }/N\ll 1$ of its photons are scattered into a perpendicularly polarized mode, thereby effectively supplementing it with a small ellipticity. The large photon number and the high polarization degree of XFEL beams in conjunction with the available high-purity X-ray polarimetry, reaching polarization purities down to the $\mathcal{P}={10}^{-11}$ level for specific photon energies in the $\mathcal{O}(10)\kern0.22em \mathrm{keV}$ regime^[ Reference Schulze, Grabiger, Loetzsch, Marx-Glowna, Schmitt, Garcia, Hippler, Huang, Karbstein, Konôpková, Schlenvoigt, Schwinkendorf, Strohm, Toncian, Uschmann, Wille, Zastrau, Röhlsberger, Stöhlker, Cowan and Paulus ^{72 ]}, make XFEL beams the ideal probe for such an experiment^[ Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey ^{84 ]}. At the same time, NIR high-intensity lasers can reach the highest peak-field strengths on macroscopic scales extending over spatial distances of $\mathcal{O}(1)\kern0.22em \unicode{x3bc} \mathrm{m}$ and times of $\mathcal{O}(10)\kern0.22em \mathrm{fs}$ , which are well-compatible with the characteristic scales of XFEL pulses, making them the best choice for the pump. In the collision of linearly polarized laser beams the number of polarization-flipped signal photons ${N}_{\perp }$ is maximized for a relative polarization angle of $\pi /4$ . As the attainable signal photon number scales with ${\left(1-\cos {\vartheta}_{\mathrm{coll}}\right)}^4$ , where ${\vartheta}_{\mathrm{coll}}$ is the collision angle of the two beams, the counter-propagating geometry is favoured; see Figure 8 for a schematic illustration of the experimental setup.

The most advanced theoretical modelling of this specific scenario has used pulsed paraxial beams to describe the colliding laser fields. As the smallest XFEL beam waists that have been experimentally realized so far are of the order of 100 nm^[ Reference Mimura, Yumoto, Matsuyama, Koyama, Tono, Inubushi, Togashi, Sato, Kim, Fukui, Sano, Yabashi, Ohashi, Ishikawa and Yamauchi ^{99 ]}, the Rayleigh lengths that can be achieved for beams with a photon energy of ${\omega}_{\mathrm{X}}=\mathcal{O}(10)\kern0.22em \mathrm{keV}$ fulfil ${z}_{\mathrm{R},\mathrm{X}}\gtrsim \mathcal{O}(1)\kern0.22em \mathrm{mm}$ ; cf. the definitions in the context of Equation (23) above. This value is much larger than the Rayleigh range ${z}_{\mathrm{R},\mathrm{L}}=\mathcal{O}(10)\kern0.22em \unicode{x3bc} \mathrm{m}$ of a tightly focused NIR laser beam. It is therefore an excellent approximation to formally send ${z}_{\mathrm{R},\mathrm{X}}\to \infty$ when determining the induced quantum vacuum signal, that is, to adopt an infinite Rayleigh length approximation^[ Reference King, Di Piazza and Keitel ^{86 ,} Reference King and Keitel ^{87 ,} Reference King, Hu and Shen ^{90 ,} Reference Gies, Karbstein and Kohlfürst ^{100 ,} Reference Karbstein, Sundqvist, Schulze, Uschmann, Gies and Paulus ^{101 ]} for the probe beam. This significantly simplifies the calculation; cf. Ref. [Reference Karbstein and Oude Weernink102] for the limitations of this approximation. Modelling both laser fields as fundamental Gaussian beams at leading order in the paraxial approximation (multiplied by a Gaussian pulse envelope), the directional emission characteristics of the signal and the signal photon yield can be straightforwardly evaluated by standard computer algebra even for arbitrary collision angles and impact parameters^[ Reference King and Keitel ^{87 ,} Reference Dinu, Heinzl, Ilderton, Marklund and Torgrimsson ^{103 ,} Reference Karbstein and Sundqvist ^{104 ]}; for different measurement concepts see also Refs. [Reference Ataman105, Reference Formanek, Palastro, Ramsey, Weber and Di Piazza106]. Invoking further approximations, such as the inherently very small divergence of the signal in the X-ray domain and an effective waist approximation for the pump^[ Reference Karbstein ^{107 ]} (for the case of ${\vartheta}_{\mathrm{coll}}=\pi$ ), one obtains rather compact analytical scaling laws describing the dependence of the signal on the various parameters of the driving laser fields. Using these scaling laws, one can quickly assess the effects of varying the characteristic parameters and thus identify promising parameter regimes prior to performing large-scale numerical simulations; cf. also the discussion below.

For a relative polarization of $\pi /4$ between the colliding beams, the attainable number of polarization-flipped signal photons per shot can then be approximately expressed as follows^[ Reference Karbstein ^{107 ,} Reference Mosman and Karbstein ^{108 ]}:

(24) $$\begin{align} &{N}_{\perp} ={N}_{\mathrm{X}}\sqrt{\frac{3}{\pi }}\frac{{\left({c}_1-{c}_2\right)}^2{m}^8}{\pi }{\left(\frac{W_{\mathrm{L}}}{m}\frac{\omega_{\mathrm{X}}}{m}\right)}^2{\left(\frac{\lambda_{\mathrm{e}}}{w_{0,\mathrm{L}}}\right)}^4 \nonumber \\& \times \sqrt{g(0)g\left(\frac{w_{0,\mathrm{X}}}{w_{0,\mathrm{L}}}\right)}\exp \left(\! -2{\left(\frac{r_0}{w_{0,\mathrm{X}}}\right)}^2\left(\! 1-\sqrt{\frac{g\left(\!\frac{w_{0,\mathrm{X}}}{w_{0,\mathrm{L}}}\right)}{g(0)}}\right) \!\right)\! ,\end{align}$$

where ${c}_1$ and ${c}_2$ are the low-energy constants introduced in Equation (9), ${W}_{\mathrm{L}}$ is the laser pulse energy of the pump, ${w}_0$ denotes the beam waist, ${r}_0$ is the transverse impact parameter and we used the shorthand notation:

(25) $$\begin{align} & g(b):= \frac{1}{{\left(1+2{b}^2\right)}^2}\kern0.1em \nonumber \\ &\quad \times F\left(\frac{4{z}_{\mathrm{R},\mathrm{L}}\sqrt{1+2{b}^2}}{\sqrt{\tau_{\mathrm{X}}^2+{\tau}_{\mathrm{L}}^2/2}},\frac{2\left({z}_0-{t}_0\right)}{\sqrt{\tau_{\mathrm{X}}^2+{\tau}_{\mathrm{L}}^2/2}},\frac{\tau_{\mathrm{X}}}{\tau_{\mathrm{L}}}\right),\kern1em \mathrm{with} \nonumber\\ {} & F\left(\chi, {\chi}_0,\rho \right):= \sqrt{\frac{1+2{\rho}^2}{3}}\kern0.1em {\chi}^2\kern0.1em {\mathrm{e}}^{2\left({\chi}^2-{\chi}_0^2\right)}{\int}_{-\infty}^{\infty}\mathrm{d}K\kern0.1em {\mathrm{e}}^{-{K}^2}\kern0.1em \nonumber \\ &\quad \times {\left|\sum \limits_{s=\pm 1}{\mathrm{e}}^{2s\left(\rho K-i{\chi}_0\right)\chi}\kern0.1em \operatorname{erfc}\left(s\left(\rho K-i{\chi}_0\right)+\chi \right)\right|}^2.\end{align}$$

The entire dependence of Equation (24) on the longitudinal beam parameters, namely the Rayleigh length of the pump ${z}_{\mathrm{R},\mathrm{L}}$ , the $1/{\mathrm{e}}^2$ pulse durations ${\tau}_{\mathrm{X}}$ , ${\tau}_{\mathrm{L}}$ and the spatiotemporal offset ${z}_0-{t}_0$ between the probe and pump foci in longitudinal direction, is encoded in the function $g(b)$ defined in Equation (25). Note that in the limit of ${w}_{0,\mathrm{X}}\ll {w}_{0,\mathrm{L}}$ and $\left\{{\tau}_{\mathrm{X}},{\tau}_{\mathrm{L}}\right\}\gg {z}_{\mathrm{R},\mathrm{L}}$ we have $g(b)\sim {\left({z}_{\mathrm{R},\mathrm{L}}/{\tau}_{\mathrm{L}}\right)}^2$ ^[ Reference Karbstein ^{107 ]}, and the form of Equation (19) with characteristic distance $z\sim {z}_{\mathrm{R},\mathrm{L}}$ is recovered. The dependence of the signal on the impact parameter ${r}_0$ is particularly relevant, as experiments have shown that focusing leads to an inevitable beam jitter characterized by spatial fluctuations of the order of the beam waist, that is, ${r}_0\sim {w}_{0,\mathrm{L}}$ . Another important signal parameter is the $1/{\mathrm{e}}^2$ radial divergence:

(26) $$\begin{align}{\theta}_{\mathrm{signal}}={\theta}_{\mathrm{X}}\kern0.1em {\left(\frac{g\left(\frac{w_{0,\mathrm{X}}}{w_{0,\mathrm{L}}}\right)}{g(0)}\right)}^{-\frac{1}{4}},\end{align}$$

where ${\theta}_{\mathrm{X}}=2/\left({\omega}_{\mathrm{X}}{w}_{0,\mathrm{X}}\right)$ is the radial divergence of the probe beam, and thus also the divergence of the background ${N}_{\perp, \mathrm{bgr}}\sim \mathcal{P}{N}_{\mathrm{X}}$ to be registered at the detector.

Equation (24) predicts the maximum value of ${N}_{\perp }$ for optimal collisions ( ${r}_0={z}_0={t}_0=0$ ) and other fixed parameters to be reached for small probe waists, ${w}_{0,\mathrm{X}}\ll {w}_{0,\mathrm{L}}$ . In this case, one obtains the polarization-flip counts

(27) $$\begin{align}{\displaystyle \begin{array}{c||c|c}f/\#& f/1& f/2\\\hline {}{N}_{\perp }& 0.044& 0.0045\end{array}},\end{align}$$

for an XFEL energy of ${\omega}_{\mathrm{X}}=9835\kern0.1em \mathrm{eV}$ and the standard self-seeding parameters in Table 2 with $f/1$ ( $f/2$ ) focusing. Self-seeding is the option of choice for experiments aiming at the detection of polarization-flip signals, as the probe bandwidth $\Delta \omega$ is of the same order as the quite narrow acceptance bandwidth of crystal polarimeters. In contrast, the self-amplified spontaneous emission (SASE) bandwidth is typically much larger; cf. Table 2. In the determination of ${N}_{\perp }$ we also took into account that the quasi-channel-cut (QCC) crystal acting as polarizer increases the probe pulse duration from its original value given in Table 2 to ${\tau}_{\mathrm{FWHM}}=50\kern0.22em \mathrm{fs}$ ^[ Reference Karbstein, Sundqvist, Schulze, Uschmann, Gies and Paulus ^{101 ]}. Using $\mathcal{P}={10}^{-11}$ ^[ Reference Schulze, Grabiger, Loetzsch, Marx-Glowna, Schmitt, Garcia, Hippler, Huang, Karbstein, Konôpková, Schlenvoigt, Schwinkendorf, Strohm, Toncian, Uschmann, Wille, Zastrau, Röhlsberger, Stöhlker, Cowan and Paulus ^{72 ]} and neglecting real-world imperfections (such as inevitable losses coming with the lenses used for focusing), the values in Equation (27) imply the associated signal-to-background ratios to be given by the following:

(28) $$\begin{align}{\displaystyle \begin{array}{c||c|c}f/\#& f/1& f/2\\\hline {}{N}_{\perp }/\left(\mathcal{P}{N}_{\mathrm{X}}\right)& 0.022& 0.0023\end{array}}.\end{align}$$

These values are less than unity so the signal is background dominated. Moreover, in the limit considered, we have ${\theta}_{\mathrm{signal}}\approx {\theta}_{\mathrm{X}}$ , and the dependence of the signal on the impact parameter ${r}_0$ simplifies to ${N}_{\perp}\sim \exp (-2{({r}_0/{w}_{0,\mathrm{L}})}^2)$ .

On the other hand, for matching waists, ${w}_{0,\mathrm{X}}={w}_{0,\mathrm{L}}$ , the signal is reduced by roughly a factor of three to ${N}_{\perp}\simeq 0.017$ ( $0.0015$ ) for $f/1$ ( $f/2$ ) focusing. At the same time the divergence of the signal, ${\theta}_{\mathrm{signal}}\approx 1.6\kern0.1em {\theta}_{\mathrm{X}}$ , becomes wider than that of the probe. This can be intuitively understood as follows: for a counter-propagating geometry the transverse extent of the strong-field interaction region is effectively determined by the transverse profile of the product ${\mathrm{\mathcal{E}}}_{\mathrm{X}}(x)\kern0.1em {\mathrm{\mathcal{E}}}_{\mathrm{L}}^2(x)$ of the focus field profiles of the probe and pump, the Fourier transform of which governs the far-field distribution of the signal. When ${w}_{0,\mathrm{X}}\gtrsim {w}_{0,\mathrm{L}}/\sqrt{2}$ , the transverse extent of the signal emission region is smaller than the focal spot of the probe, and in turn ${\theta}_{\mathrm{signal}}>{\theta}_{\mathrm{X}}$ ; the factor of $\sqrt{2}$ is due to the linear (quadratic) dependency on the probe (pump) profile. Furthermore, as ${N}_{\perp}\sim \exp \left(-1.2{({r}_0/{w}_{0,\mathrm{L}})}^2\right)$ for $f/1$ focusing, the dependence on ${r}_0$ – and thus the sensitivity to beam jitter in the experiment – becomes milder. Clearly, when ${\theta}_{\mathrm{sig}}>{\theta}_{\mathrm{X}}$ , a measurement of the full signal requires collecting photons scattered outside the forward cone of the probe on the detector. However, as ${\theta}_{\mathrm{X}}\sim 1/{w}_{0,\mathrm{X}}$ , the most tightly focused probes make the highest demands on the diameter of the collection optics.

The above simple comparison of just two different focusing options for the probe clearly illustrates that assessing the best choice of parameters for a feasible vacuum birefringence experiment is a nontrivial task. We emphasize that improved modelling accounting for the details of the laser fields actually available in experiments will become computationally demanding and challenging.

The possibility of achieving ${\theta}_{\mathrm{sig}}>{\theta}_{\mathrm{X}}$ suggests that angular cuts can be used to improve the signal-to-background ratio in experiments: collecting only those photons scattered outside a minimum diffraction angle ${\vartheta}_{\mathrm{min}}$ , the signal-to-background ratio can be enhanced to ${\left.{N}_{\perp }/\left(\mathcal{P}{N}_{\mathrm{X}}\right)\right|}_{\vartheta >{\vartheta}_{\mathrm{min}}}>1$ at the expense of reduced absolute photon numbers. Following this route, it is even possible to achieve ${\left.{N}_{\parallel }/{N}_{\mathrm{X}}\right|}_{\vartheta >{\vartheta}_{\mathrm{min}}}>1$ , such that one can hope to also measure the $\parallel$ -polarized signal component^[ Reference Inada, Yamazaki, Yamaji, Seino, Fan, Kamioka, Namba and Asai ^{44 ,} Reference Karbstein and Mosman ^{109 ]}. However, an experimental implementation of such a diffraction-assisted measurement concept^[ Reference King, Di Piazza and Keitel ^{17 ,} Reference King and Keitel ^{87 ]} requires that any unwanted background, for example, stray light from the lenses used for focusing, can be controlled and inhibited from reaching the detector; cf. Ref. [Reference Doyle, Khademi, Hilz, Sävert, Schäfer, Schreiber and Zepf110] for a related background study at optical frequencies. This would necessitate setting up a dedicated filtering and imaging system along the lines outlined in Section 4 for the dark-field scenario.

Finally, with regard to an actual experimental implementation we need to account for several non-ideal effects associated with the optical system: (i) in experiments each reflection at a diamond surface of the QCC polarizer and analyser comes with a reduction of the number of photons by about $2\%$ ^[ Reference Karbstein, Sundqvist, Schulze, Uschmann, Gies and Paulus ^{101 ]} (the setup in Figure 8 employs eight reflections in total); (ii) the transmission of lenses preserving the polarization purity can be estimated as $25\%$ ; (iii) the acceptance bandwidth of the polarizer, $\Delta {\omega}_{\mathrm{diamond}}^{400}=80\kern0.22em \mathrm{meV}$ , entails a reduction of its throughput by $\Delta {\omega}_{\mathrm{diamond}}^{400}/\Delta \omega =80/300$ for self-seeding. In consequence, both the signal and background photon numbers, ${N}_{\perp }$ and $\mathcal{P}{N}_{\mathrm{X}}$ , respectively, are reduced by a factor of ${(0.98)}^8\times {(0.25)}^2\times 80/300\simeq 0.014$ from their ideal theoretical values stated in Equation (27) above.

While this keeps the signal-to-background ratios ${N}_{\perp }/$ $(\mathcal{P}{N}_{\mathrm{X}})$ given in Equation (28) unaltered, it results in quite a drastic reduction of the signal photon numbers attainable per shot. In particular, the best value attainable for ${w}_{0,\mathrm{X}}\ll {w}_{0,\mathrm{L}}$ then gives rise to just ${N}_{\perp}\simeq 6.2\times {10}^{-4}$ ( $6.4\times {10}^{-5}$ ) polarization-flipped signal photons per shot reaching the detector for $f/1$ ( $f/2$ ) focusing. Assuming Poissonian statistics,

(29) $$\begin{align} n & {>\#}^2\kern0.1em \frac{1}{2}{\left(\left({N}_{\mathrm{sig}}+{N}_{\mathrm{bgr}}\right)\ln \left(1+\frac{N_{\mathrm{sig}}}{N_{\mathrm{bgr}}}\right)-{N}_{\mathrm{sig}}\right)}^{-1} \nonumber \\& {=\#}^2\kern0.1em \frac{N_{\mathrm{bgr}}}{N_{\mathrm{sig}}^2}\left(1+\mathcal{O}\left(\frac{N_{\mathrm{sig}}}{N_{\mathrm{bgr}}}\right)\right)\end{align}$$

shots are required for a measurement with a significance of $\#\sigma$ ^[ Reference Cowan ^{111 ]}, where ${N}_{\mathrm{sig}}$ and ${N}_{\mathrm{bgr}}$ are the number of signal and background events, respectively. This implies that $n{>\#}^2\times 7.5\times {10}^4$ ( ${\#}^2\times 6.9\times {10}^6$ ) optimal shots are required for a $\#\sigma$ confirmation of the vacuum birefringence signal with the presently available parameters at HED-HIBEF and $f/1$ ( $f/2$ ) focusing. Given a repetition rate of 1 Hz, this translates into the requirement of more than ${\#}^2\times 21\kern0.22em \mathrm{h}$ ( ${\#}^2\times 80\kern0.22em \mathrm{days}$ ) optimal shots with zero spatiotemporal offset. Lasers such as ReLaX can operate for long periods continuously. The total number of full power shots on the gratings and optics easily allow for continuous shooting on a month scale. We also note that if pointing and timing stabilization is implemented, the jitter of ReLaX can be reduced to less than the temporal and spatial sigma. In these circumstances, jitter does not materially affect the overall estimate relative to other uncertainties, such as shot-to-shot XFEL performance in seeded mode.

3.2 Dark-field scenario

A prospective variation of the conventional two-beam scenario detailed in Section 3.1 envisions modifying the probe beam so as to feature a shadow, or equivalently a dark field, in the converging and expanding beam while retaining a central intensity peak in the focus where it is collided with the counter-propagating high-intensity pump^[ Reference Karbstein, Ullmann, Mosman and Zepf ^{23 ,} Reference Karbstein and Mosman ^{112 ]}. In the experiment the shadow in the probe beam is generated by a well-defined beamstop inserted into the incident beam; see Figure 9 for an illustration. To demonstrate the underlying principle, we focus here on the case of a circularly symmetric shadow imprinted in the incident probe beam. Experimentally, this annular beam approach was pioneered in Refs. [Reference Peatross, Chaloupka and Meyerhofer113, Reference Zepf, Tsakiris, Pretzler, Watts, Chambers, Norreys, Andiel, Dangor, Eidmann, Gahn, Machacek, Wark and Witte114] for the detection of weak nonlinear optics signals driven by high-intensity lasers. By construction, the background is substantially reduced in the dark field, so that it may even be possible to access both polarization components of the nonlinear vacuum response. To this end, the shadow in the probe beam is to be imaged onto a polarization-sensitive detector employing an advanced filtering and imaging system. This will consist of appropriately designed and placed beamstops and apertures so as to prevent any unwanted background from reaching the detector together with the signal. We quantify the background scattered into the dark field by the shadow quality

(30) $$\begin{align}\mathcal{S}=\frac{1}{N_{\mathrm{X}}}{\int}_{A_{\mathrm{d}\mathrm{et}}}{\mathrm{d}}^2x\kern0.1em \frac{{\mathrm{d}}^2{N}_{\mathrm{X}}\left(x,y\right)}{\mathrm{d}x\kern0.1em \mathrm{d}y}\end{align}$$

Figure 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 ${\mathrm{\mathcal{E}}}_{\mathrm{X}}(x){\mathrm{\mathcal{E}}}_{\mathrm{L}}^2(x)$ 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.

that measures the fraction of the total number of input probe photons ${N}_{\mathrm{X}}$ registered by a detector of acceptance area ${A}_{\mathrm{det}}$ in the shadow in the absence of a quantum vacuum signal, that is, for a vanishing pump field. In a theoretically idealized calculation using ideal flat-top far-field profiles and neglecting diffraction at the obstacles and apertures put into the beam path, we have $\mathcal{S}=0$ . However, in any real experiment, we inevitably have $\mathcal{S}>0$ , that is, a finite background in the shadow, due to diffraction and scattering off the optical elements (cf. Section 4.2 below). The discussion of this setup follows Ref. [Reference Karbstein, Ullmann, Mosman and Zepf23] and more detail can be found therein.

In the focus, the information about the far-field profile of the beam is encoded in a characteristic Airy pattern around the main peak. The peak-field driven quantum vacuum nonlinearities producing the signal are sizable only in the vicinity of the overlapping beam foci. Thus, by ensuring that the transverse profile of the overlap factor ${\mathrm{\mathcal{E}}}_{\mathrm{X}}(x)\kern0.1em {\mathrm{\mathcal{E}}}_{\mathrm{L}}^2(x)$ effectively matches the focus profile of a beam without shadow^[ Reference Karbstein ^{115 ]}, one can induce a signal component scattered into the shadow located in the expanding probe beam (cf. Section 3.1 above).

Note that this dependency immediately implies a signal in the dark field for ${w}_{0,\mathrm{X}}\gtrsim {w}_{0,\mathrm{L}}/\sqrt{2}$ . In this case, the Gaussian pump intensity profile ensures that the Airy pattern of the probe in the focus is effectively damped out such that the transverse profile of the overlap factor ${\mathrm{\mathcal{E}}}_{\mathrm{X}}(x)\kern0.1em {\mathrm{\mathcal{E}}}_{\mathrm{L}}^2(x)$ is characterized by a single Gaussian peak. This results in a Gaussian far-field distribution of the signal. At the same time, the fraction of the total induced signal that is scattered into the dark field decreases with ${w}_{0,\mathrm{L}}$ , because the far-field divergence scales inversely with the source size. On the other hand, it is clear that for ${w}_{0,\mathrm{X}}\ll {w}_{0,\mathrm{L}}$ the pump field appears as effectively constant on the transverse scales of variation of the probe. In this case, the dependence of the overlap factor ${\mathrm{\mathcal{E}}}_{\mathrm{X}}(x)\kern0.1em {\mathrm{\mathcal{E}}}_{\mathrm{L}}^2(x)$ on the transverse coordinate $r$ is governed by the $r$ dependence of ${\mathrm{\mathcal{E}}}_{\mathrm{X}}(x)$ . Hence, in this limit, the dark field is also present in the far-field angular distribution of the signal. These considerations indicate that pump and probe waists of the same order maximize the signal scattered into the dark field.

In what follows, we discuss the dark-field scenario based on a few well-justified approximations. This allows us to provide simplified analytical expressions for the relevant quantities. (The full calculations accounting for the details of the probe beam can be found in Refs. [Reference Karbstein and Mosman112, Reference Karbstein115].) To be specific, we limit ourselves to ${w}_{0,\mathrm{X}}\gtrsim {w}_{0,\mathrm{L}}/\sqrt{2}$ and make use of the fact that in this parameter regime, the focus profiles of both colliding laser fields can be well-approximated as Gaussian^[ Reference Karbstein ^{115 ]}. In addition, we assume a relative polarization of $\pi /4$ between the colliding beams and a detector of radial opening angle ${\theta}_{\mathrm{det}}$ placed centrally in the dark zone of the outgoing far field. The number ${N}_{\perp}^{\bullet }$ of $\perp$ polarized signal photons registered (per shot) by this detector is then well-approximated by the following^[ Reference Karbstein ^{115 ]}:

(31) $$\begin{align}{N}_{\perp}^{\bullet}={N}_{\perp}\kern0.1em {\left(1-\nu \right)}^2\kern0.1em \frac{1-{\mathrm{e}}^{-1}}{1+\nu}\left(1-{\mathrm{e}}^{-2{\left({\theta}_{\mathrm{det}}/{\theta}_{\mathrm{signal}}\right)}^2}\right),\end{align}$$

with ${N}_{\perp }$ and ${\theta}_{\mathrm{det}}\le {\theta}_{\mathrm{in}}$ as given in Equations (24) and (26). For the same scenario, the complementary signal is somewhat larger, namely:

(32) $$\begin{align}{N}_{\parallel}^{\bullet}={\left(\frac{c_1+{c}_2}{c_1-{c}_2}\right)}^2\kern0.1em {N}_{\perp}^{\bullet}\simeq 13.4\kern0.1em {N}_{\perp}^{\bullet }.\end{align}$$

Here, $\nu ={\left({\theta}_{\mathrm{in}}/{\theta}_{\mathrm{out}}\right)}^2$ measures the fraction of the transverse area of the incident XFEL beam that is blocked by the beamstop. This blocking fraction can be parameterized in terms of the inner and outer radial divergences, ${\theta}_{\mathrm{in}}$ and ${\theta}_{\mathrm{out}}$ , of the ‘hollow’ probe beam featuring the central shadow^[ Reference Karbstein, Ullmann, Mosman and Zepf ^{23 ]}:

(33) $$\begin{align}{\theta}_{\mathrm{out}}={\theta}_{\mathrm{X}}\kern0.1em \sqrt{\frac{2\left(1-{\mathrm{e}}^{-1}\right)}{1+\nu }}\kern1em \mathrm{and}\kern1em {\theta}_{\mathrm{in}}=\sqrt{\nu}\kern0.1em {\theta}_{\mathrm{out}}.\end{align}$$

Equation (31) follows upon integrating the differential number of signal photons derived in Refs [Reference Karbstein107, Reference Mosman and Karbstein108] over solid angle. In addition, one has to account for the reduction of the peak field of the probe^[ Reference Karbstein and Mosman ^{112 ]} to match the central focus peak of a flat-top beam with a perfect central shadow in its far field.

Upon identifying ${\theta}_{\mathrm{det}}={\theta}_{\mathrm{in}}$ in Equation (31) and using that in the considered parameter regime, the factor $2{\left({\theta}_{\mathrm{in}}/{\theta}_{\mathrm{sig}}\right)}^2$ is sufficiently small so as to justify a limitation to the leading non-vanishing contribution; we find ${N}_{\perp}^{\bullet}\sim \nu {\left(1-\nu \right)}^2/{\left(1+\nu \right)}^2$ , from which we infer that the choice $\nu =\sqrt{5}-2\simeq 0.24$ maximizes the signal in the shadow for optimal collisions. We also choose ${w}_{0,\mathrm{X}}={w}_{0,\mathrm{L}}/\sqrt{2}\simeq 0.7\kern0.1em {w}_{0,\mathrm{L}}$ (which should be close to the optimal waist spot ratio^[ Reference Karbstein, Ullmann, Mosman and Zepf ^{23 ]}) and an XFEL energy of ${\omega}_{\mathrm{X}}=8766\kern0.22em \mathrm{eV}$ compatible with the use of a germanium crystal to separate the polarization components of the signal; cf. also Section 4.2.4 below for the details. Adopting ${\theta}_{\mathrm{det}}={\theta}_{\mathrm{in}}$ to collect the maximum number of signal photons scattered into the dark-field on the detector (for the parameters in Table 2 attainable with the self-seeding option), the second identity in Equation (33) then results in ${\theta}_{\mathrm{in}}\simeq 28.3\kern0.22em \unicode{x3bc} \mathrm{rad}/\#$ , for $f/\#$ focusing. For these parameters, Equation (31) predicts a signal yield, which can be tabulated as follows:

(34) $$\begin{align}{\displaystyle \begin{array}{c||c|c}f/\#& f/1& f/2\\\hline {}{N}_{\perp}^{\bullet }& 0.0016& 1.2\times {10}^{-4}\end{array}}.\end{align}$$

We also note that for a detector with a half opening angle, that is, for ${\theta}_{\mathrm{det}}={\theta}_{\mathrm{in}}/2$ , these photon numbers are reduced by a factor of approximately/almost equal to 0.3. As in Section 3.1 we can also assess the dependence of the signal photon number on the impact parameter using Equation (31). For the present parameters and $f/1$ focusing we infer ${N}_{\perp}^{\bullet}\sim \exp \left(-1.8{({r}_0/{w}_{0,\mathrm{L}})}^2\right)$ . Note, however, that the approximation invoked here accounts only for the central peak in the focus profile of the probe. It is therefore insensitive to the oscillatory behaviour of the signal induced by the Airy ring structure for ${r}_0\gtrsim {w}_{0,\mathrm{X}}$ and, in turn, is expected to somewhat overestimate the drop of the signal with ${r}_0$ .

In the dark-field scenario, the polarization purity of the probe photons does not need to be improved beyond the one supplied by the EuXFEL itself, which is of the order of ${10}^{-6}$ ^[ Reference Geloni, Kocharyan and Saldin ^{116 ]}. Therefore, the analyser, which separates the differently polarized signal components scattered into the dark field, can work with a low number of reflections. This allows for a transmission bandwidth larger than that of the self-seeded XFEL and, at the same time, comes with a negligible influence on the probe pulse duration and a negligible loss at each reflection. Using a single reflection off germanium, a polarization purity of the order of ${10}^{-3}$ can be readily achieved. Employing several such reflections, the polarizer can be further enhanced, thus achieving a purity on par with the polarization purity of the XFEL. In consequence, the losses associated with the setup in Figure 9 are dominated by the lenses and not the analyser. Due to the relaxed polarization purity requirement, standard beryllium lenses can now be used, each of which introduces a loss in the number of photons of about $50\%$ . This reduces the signal and background photon numbers registered at the detector by a factor of $0.25$ and thus implies a reduction of the full signal scattered into the dark field to ${N}_{\perp}^{\bullet}\simeq 4.0\times {10}^{-4}$ ( $3.0\times {10}^{-5}$ ) for $f/1$ ( $f/2$ ) focusing. Assuming the germanium polarizer in Figure 9 to achieve a polarization purity of ${10}^{-6}$ , the associated background in the dark field then consists of ${N}_{\mathrm{bgr}}=\mathcal{S}{N}_{\mathrm{X}}\times 0.25\times {10}^{-6}$ ( $\mathcal{S}{N}_{\mathrm{X}}\times 0.25$ ) XFEL photons for the $\perp$ ( $\parallel$ ) polarized mode. Identifying ${N}_{\mathrm{sig}}={N}_{\perp}^{\bullet }$ , and given that ${N}_{\mathrm{sig}}\ll {N}_{\mathrm{bgr}}$ , Equation (29) then implies the need for $n{>\#}^2\times \mathcal{S}\times 3.2\times {10}^{11}$ ( ${\#}^2\times \mathcal{S}\times 5.5\times {10}^{16}$ ) optimal shots to achieve a $\#\sigma$ measurement of the $\perp$ polarized component of the nonlinear vacuum response with $f/1$ ( $f/2$ ) focusing. For ${N}_{\mathrm{sig}}={N}_{\parallel}^{\bullet }$ , the analogous values for the $\parallel$ polarized component are $n{>\#}^2\times \mathcal{S}\times 1.8\times {10}^{15}$ ( ${\#}^2\times \mathcal{S}\times 3.0\times {10}^{17}$ ).

To compare with the number of optimal shots needed for a measurement of the $\perp$ polarized signal in the conventional scenario of Section 3.1, we refer back to the discussion below Equation (29). From this we conclude that one needs a shadow quality of $\mathcal{S}\lesssim {10}^{-7}$ for the dark-field approach to match the precision of the conventional scenario.

Finally, we note that the measurement of both polarization components of the nonlinear vacuum response allows one to determine the low-energy constants ${c}_1$ and ${c}_2$ in Equation (6) individually, and not just their difference as in the conventional scenario. Moreover, we emphasize that a simultaneous measurement of both components gives direct access to the ratio

(35) $$\begin{align}\frac{N_{\perp }}{N_{\parallel }}={\left(\frac{c_1-{c}_2}{c_1+{c}_2}\right)}^2=\frac{9}{121}\left(1+\frac{260}{99}\frac{\alpha }{\pi }+\mathcal{O}\left({\alpha}^2\right)\right),\end{align}$$

where we have also included higher-order (two-loop) contributions^[ Reference Ritus ^{117 ,} Reference Dittrich and Reuter ^{118 ]}. Importantly, the ratio in Equation (35) is independent of the parameters of the colliding beams and thus insensitive to fluctuations in experimental parameters, such as spatiotemporal jitter or intensity fluctuations, for sufficiently large photon numbers^[ Reference Karbstein, Ullmann, Mosman and Zepf ^{23 ]}.

3.3 Planar three-beam setup

From a scattering point of view, vacuum birefringence is explained by polarization flips without momentum transfer and hence corresponds to an LbL forward-scattering process. One can, however, relax this restrictive assumption and ‘open up’ phase space, by considering genuine four-photon scattering processes with non-vanishing transfer of both energy and momentum. These are the fundamental processes that HED-HIBEF will measure. If two optical beams are employed instead of one, then each beam can couple one optical photon to the incoming XFEL photon, giving more control over the momentum and energy of the scattered signal photon^[ Reference Ahmadiniaz, Cowan, Grenzer, Franchino-Viñas, Garcia, Šmid, Toncian, Trejo and Schützhold ^{83 ,} Reference King, Hu and Shen ^{90 ]}. In particular, the scattered photon’s momentum and energy can be shifted with respect to the XFEL probe photon so that the signal can be measured in a region of phase space with lower noise than the standard two-beam configuration. A disadvantage of this is that the number of scattered photons in this region is smaller than the total number of photons scattered in the two-beam configuration.

Let the field $F$ be given as a sum of the XFEL, ${F}_x$ , two optical beams, ${F}_1$ and ${F}_2$ , and the scattered photon ${F}_{x'}$ , such that $F={F}_x+{F}_1+{F}_2+{F}_{x'}$ . Then, for the low-energy HE Lagrangian given in Equation (6), the scattering matrix element, $S=-i\int \mathrm{d}^4x\kern0.1em {\mathrm{\mathcal{L}}}_{\mathrm{int}}$ contains many channels:

(36) $$\begin{align}S & =\underset{\mathrm{X}\hbox{-} \mathrm{ray}+\mathrm{one}\;\mathrm{optical}\kern0.17em \mathrm{beam}}{\underbrace{{S}_{\mathrm{x}11{\mathrm{x}}^{\prime }}+{S}_{\mathrm{x}22{\mathrm{x}}^{\prime }}}}+\underset{\mathrm{X}\hbox{-} \mathrm{ray}+\mathrm{two}\;\mathrm{optical}\kern0.17em \mathrm{beam}\mathrm{s}}{\underbrace{{S}_{\mathrm{x}12{\mathrm{x}}^{\prime }}}}+\cdots \nonumber \\&\quad +\underset{\mathrm{X}\hbox{-} \mathrm{ray}\;\mathrm{merging}}{\underbrace{{S}_{\mathrm{x}\mathrm{x}1{\mathrm{x}}^{\prime }}}}+\cdots +\underset{\mathrm{X}\hbox{-} \mathrm{ray}\;\mathrm{s}\mathrm{plitting}}{\underbrace{{S}_{\mathrm{x}1{\mathrm{x}}^{\prime }{\mathrm{x}}^{\prime }}}}+\cdots,\end{align}$$

where the subscripts on each of the amplitudes count the number of fields included (e.g., ${S}_{\mathrm{x}11{\mathrm{x}}^{\prime }}$ involves one photon from the XFEL, two from optical beam $1$ and one scattered photon). If the optical beam is such that the field’s tensor structure is constant, as is the case for plane-wave-like fields (e.g., that of the paraxial Gaussian beam), then the kinematics of each amplitude can be illustrated by separating out the fields’ space–time dependency from their tensor structure, and writing kinematic factors $\Gamma (q)$ defined in terms of the momentum transfer $q$ . For example, ${S}_{\mathrm{x}11{\mathrm{x}}^{\prime }}=\mathfrak{S}{\Gamma}_{\mathrm{x}11{\mathrm{x}}^{\prime }}(q)$ , where $\mathfrak{S}$ is a geometrical factor depending on the polarization of the photons and ${\Gamma}_{\mathrm{x}11{\mathrm{x}}^{\prime }}(q)=\int {F}_1^2(x)\exp \left(- iq\cdot x\right)\mathrm{d}^4x$ . We note this has a form analogous to a far-field diffraction integral, where the X-ray photon ‘diffracts’ on regions of the vacuum polarized by the intense optical beams^[ Reference Di Piazza, Hatsagortsyan and Keitel ^{16 ,} Reference King, Di Piazza and Keitel ^{86 ]}. These kinematic factors can be used to demonstrate the channels in the three-beam scenario; here it will suffice to consider monochromatic beams, for example, ${F}_j(x)={F}_j\cos \left({k}_j\cdot x\right)$ for $j\in \left\{1,2\right\}$ . Then we see the following:

(37) $$\begin{align}{\Gamma}_{\mathrm{x}11{\mathrm{x}}^{\prime }}(q)\propto \underset{\mathrm{frequency}\kern0.17em \mathrm{upshift}}{\underbrace{\delta \left(2{k}_1-q\right)}}+\underset{\mathrm{frequency}\kern0.17em \mathrm{downshift}}{\underbrace{\delta \left(2{k}_1+q\right)}}+\underset{\mathrm{elastic}}{\underbrace{2\delta (q)}},\end{align}$$

that is, the ‘X-ray+one optical beam’ channel in principle includes frequency-shifting parts. However, since the scattered photons with momentum ${\mathrm{\ell}}^{\prime }$ must obey the vacuum dispersion relation ${\mathrm{\ell}}^{\prime}\cdot {\mathrm{\ell}}^{\prime}=0$ , the frequency-shifting channels are effectively inaccessible^[ Reference King, Hu and Shen ^{90 ]}. Compare this to the ‘X-ray+two optical beams’ channel, for which the corresponding kinematic factor is as follows:

(38) $$\begin{align}{\Gamma}_{\mathrm{x}12{\mathrm{x}}^{\prime }}(q) &\propto \underset{\mathrm{frequency}\kern0.17em \mathrm{upshift}}{\underbrace{\delta \left({k}_1+{k}_2-q\right)}}+\underset{\mathrm{frequency}\kern0.17em \mathrm{downshift}}{\underbrace{\delta \left({k}_1+{k}_2+q\right)}}+\underset{\mathrm{quasi}\hbox{-} \mathrm{elastic}}{\underbrace{\delta \left({k}_1-{k}_2+q\right)}}\nonumber \\ &\quad +\underset{\mathrm{quasi}\hbox{-} \mathrm{elastic}}{\underbrace{\delta \left(-{k}_1+{k}_2+q\right)}}.\end{align}$$

We see that by tuning ${k}_1$ and ${k}_2$ there is more control to scatter a photon with a momentum slightly different from the X-ray one, whilst simultaneously satisfying the on-shell condition ${\mathrm{\ell}}^{\prime}\cdot {\mathrm{\ell}}^{\prime}=0.$ It is indeed possible to pick three photons with not too different momenta and specific collision angles to access the frequency up- and down-shifting channels^[ Reference Lundstrom, Brodin, Lundin, Marklund, Bingham, Collier, Mendonca and Norreys ^{14 ,} Reference Lundin, Marklund, Lundstrom, Brodin, Collier, Bingham, Mendonca and Norreys ^{15 ,} Reference Bernard, Moulin, Amiranoff, Braun, Chambaret, Darpentigny, Grillon, Ranc and Perrone ^{43 ,} Reference Ahmadiniaz, Cowan, Grenzer, Franchino-Viñas, Garcia, Šmid, Toncian, Trejo and Schützhold ^{83 ,} Reference Gies, Karbstein, Kohlfürst and Seegert ^{89 –} Reference Berezin and Fedotov ^{92 ]}. However, for the X-ray + two optical photons collision at HED-HIBEF, only the ‘quasi-elastic’ channels (of absorbing one photon from an optical beam and emitting one photon to an optical beam) are kinematically accessible. Furthermore if the absorbed and emitted photons have different momenta, the reaction is only accessible if one allows for the pulses having a finite bandwidth $\Delta$ , that is, for non-monochromatic beams, leading to a momentum transfer in the scattering process of $q=\pm \left({k}_1-{k}_2\right)+\Delta$ . Although labelled ‘quasi-elastic’, these channels can support a measurable transverse momentum shift and, if one of the optical lasers is frequency doubled, also an energy shift in the scattered photons; see Figure 7. This ‘three-beam’ kinematic picture can also be used to model frequency shifting in collisions of two beams with wide bandwidths, where the two photon momenta, ${k}_1$ and ${k}_2$ , take different values supported by the bandwidth^[ Reference King and Keitel ^{87 ,} Reference Sundqvist and Karbstein ^{119 ]}.

Here we focus on a planar collision geometry for the three beams, depicted in Figure 10. Using the kinematic factors in Equations (37) and (38) we can reinterpret the ‘X-ray+one optical beam’ and ‘X-ray+two optical beams’ scattering matrix channels in terms of positions in the detector plane:

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

(39) $$\begin{align}S\approx \underset{\mathrm{central}\kern0.17em \mathrm{peak}}{\underbrace{{S}_{\mathrm{x}11\mathrm{x}'}+{S}_{\mathrm{x}22\mathrm{x}'}}}+\underset{\mathrm{sidepeaks}}{\underbrace{{S}_{\mathrm{x}12\mathrm{x}'}}}.\end{align}$$

Here we neglect all the channels that are nonlinear in the X-ray photons as they are accompanied by a suppression factor, the ratio of optical to X-ray field strengths. The centre of the side peaks can be easily found from Equation (38), at a scattering angle approximately twice the ratio of the transverse optical photon momentum to the X-ray momentum: ${\theta}_x=\pm \arctan \left(2{\omega}_{\mathrm{optical}}\sin \psi /{\omega}_{\mathrm{x}\hbox{-} \mathrm{ray}}\right)$ , where ${\theta}_x$ is the scattering angle in the plane of the collision. The widths of the peaks can be ascertained by considering the bandwidth of the lasers. The number of scattered photons $N$ is calculated using $N=V\int \frac{\mathrm{d}^3{\mathrm{\ell}}^{\prime }}{{\left(2\pi \right)}^3}{\left|S\right|}^2$ , where ${\mathrm{\ell}}^{\prime }$ is the scattered photon momentum, and only the terms in Equation (39) are included in $S$ . The beams are modelled using the leading-order paraxial Gaussian solution from Equation (23), and parameters for the optical and EuXFEL beams are taken from Table 2. An example of the distribution of scattered photons is given in Figure 11, which assumes perfect alignment of the three beams, where the grid lines indicate the predicted central positions of the side peaks. Figure 11 demonstrates the key utility of the three-beam setup: the transverse momentum kick given to the signal photons can potentially be used to direct the signal into a region with low noise, albeit at the cost of a reduced signal. The larger the collision angle, the larger the momentum kick, but the lower the signal. This reduction is plotted in Figure 12. At small collision angles the three peaks constructively interfere since they are in the same region of phase space. As the collision angle is increased, the peaks separate and the signal reduces, while constructive interference becomes less pronounced.

Figure 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 ${\theta}_{x,y}$ are the scattering angles in and perpendicular to the collision plane, respectively.

Figure 12 The number of scattered photons, parallel ( ${N}_{\parallel }$ ) and perpendicular ( ${N}_{\perp }$ ) to the XFEL probe, as a function of the beam collision angle, plotted for different XFEL parameters. The probe propagates along the $z$ -axis, and the collision is in the $x$ – $z$ plane. The dashed curves ( ${N}_{\parallel, \perp}^{>200\;\unicode{x3bc} \mathrm{rad}}$ ) refer to the signal falling on the detector outside a central exclusion region of radius $200\kern0.22em \unicode{x3bc} \mathrm{rad}$ . 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).

To estimate the number of detectable photons, we exclude a central portion of the detector, which will be dominated by the probe XFEL photons. From knowledge of the XFEL beam and numerical simulations (such as those detailed in Section 4.2.1) we determine that a $200\kern0.22em \unicode{x3bc} \mathrm{m}$ radius for the central exclusion region on the detector would be sufficient. The entailed reduction in signal photon count is demonstrated in Figure 12. On the one hand, we would like a larger transverse momentum kick of the scattered photons so that the optical beam opening angle does not need to be made so large, and the detectable signal remains sufficient. This would suggest using lower energy XFEL photons, for example, the $6\kern0.22em \mathrm{keV}$ mode. On the other hand, the cross-section, $\sigma$ , has a strong scaling with CM energy, ${\omega}_{\ast }$ as $\sigma \propto {\omega}_{\ast}^6$ , and so to maximize the total number of photons scattered, we would use the higher energy XFEL photons, for example, the $12.9\kern0.22em \mathrm{keV}$ mode. From our analyses using the HED-HIBEF parameters, we find that the $9.8\kern0.22em \mathrm{keV}$ mode is a good compromise in this planar three-beam scenario.

Some example results are given in Table 3. Due to the bandwidth of the analyser being much narrower than the bandwidth of the XFEL beam in SASE operation, for observing birefringence we use the self-seeded parameters for which the XFEL has a narrow enough bandwidth such that scattered photons are accepted by the analyser.

Table 3 Example number of signal photons scattered in collision of the XFEL probe with two optical beams.

The planar three-beam scenario is a setup of interest for BIREF@HIBEF. As the dark-field scenario has been selected for priority access beamtime at the EuXFEL, we omit here a full discussion of the effect of optical components on the measurability of the numbers of photons detailed in Table 3, and how the HE low-energy constants ${c}_1$ and ${c}_2$ can be determined in this setup. See Ref. [Reference Macleod and King120] for further details.

3.4 Other opportunities: Coulomb-assisted birefringent scattering

In the preceding sections, we have rigorously examined a variety of scenarios for detecting vacuum birefringence, each presenting unique challenges and insights. These explorations spanned across diverse experimental setups, employing a range of pump and probe fields to elucidate the nature of vacuum birefringence. These varying approaches, each with its own merits and limitations, collectively contribute to a better understanding of the phenomenon. These options are concisely summarized in Table 4. In what follows, we first provide an overview of the available choices for both pump and probe fields. Then, we venture into a new direction by combining two options for the pump field to explore a birefringent signal.

Table 4 Summary of choices for pump and probe fields in vacuum birefringence detection. Increasing the strength of the pump fields results in a smaller interaction volume, while a higher wave number in the probe fields leads to fewer photons per shot.

Figure 13 Feynman diagram for Coulomb-assisted birefringence.

Table 4 presents three distinct choices each for the pump and probe fields. In the case of the pump field, one can opt for a static magnetic field, laser focus or the electric field of a nucleus, each with its own set of pros and cons. For example, the first column details the characteristics of a static magnetic field, where we achieve a field strength of several teslas. This leads to an extremely small change in the vacuum refractive index ( $\delta n$ ), and the other columns provide similar metrics for alternative pump field options. Regarding the probe field, there are also multiple choices. Take the optical laser, for instance: it generates a large number of photons $N$ but with relatively low energy $\mathcal{O}\left(\mathrm{eV}\right)$ . Table 4 illustrates various experiments that have utilized different combinations of pump and probe fields. For instance, as detailed in Section 2, experiments such as PVLAS^[ Reference Ejlli, Valle, Gastaldi, Messineo, Pengo, Ruoso and Zavattini ^{78 ]} and BMV^[ Reference Battesti, Da Souza, Batut, Robilliard, Bailly, Michel, Nardone, Pinard, Portugall, Trénec, Mackowski, Rikken, Vigué and Rizzo ^{121 ]} employed the options detailed in the first column: a static field as the pump and an optical laser as the probe. HED-HIBEF is designed to follow the approach outlined in the second column, using a focused array of several optical lasers for the pump field and X-rays for the probe. In addition, for the measurement of Delbrück scattering^[ Reference Jarlskog, Joensson, Prünster, Schulz, Willutzki and Winter ^{5 ,} Reference Schumacher, Borchert, Smend and Rullhusen ^{41 ]}, the electric field of a nucleus as pump field and high-energy gamma photons as the probe have been utilized; for more details see Ref. [Reference Ahmadiniaz, Cowan, Sauerbrey, Schramm, Schlenvoigt and Schützhold122]. In this section, we utilize the combination of a nuclear Coulomb field, denoted as ${\mathbf{E}}^{\mathrm{ext}}$ , and the strong magnetic field ${\mathbf{B}}^{\mathrm{ext}}$ provided by a focused high-intensity laser pulse as the pump field. This is coupled with an XFEL beam, which serves as the probe field; see Refs. [Reference Ahmadiniaz, Cowan, Sauerbrey, Schramm, Schlenvoigt and Schützhold122, Reference Ahmadiniaz, Bussmann, Cowan, Debus, Kluge and Schützhold123]. An illustrative Feynman diagram for this process resembles the one shown in Figure 3(a), with the key distinction being that one of the Coulomb fields is replaced by an interaction with an external magnetic field ${\mathbf{B}}^{\mathrm{ext}}$ ; see Figure 13. This scenario may offer several advantages: the nuclear Coulomb field adds a significant momentum transfer $\Delta \mathbf{k}$ to the flip of the XFEL polarization^[ Reference Ahmadiniaz, Cowan, Sauerbrey, Schramm, Schlenvoigt and Schützhold ^{122 ,} Reference Ahmadiniaz, Bussmann, Cowan, Debus, Kluge and Schützhold ^{123 ]}. Compared to the past experiments measuring Delbrück scattering at photon energies $O\left(100\kern0.1em \mathrm{MeV}\right)$ ^[ Reference Jarlskog, Joensson, Prünster, Schulz, Willutzki and Winter ^{5 ,} Reference Akhmadaliev, Kezerashvili, Klimenko, Malyshev, Maslennikov, Milov, Milstein, Muchnoi, Naumenkov, Panin, Peleganchuk, Popov, Pospelov, Protopopov, Romanov, Shamov, Shatilov, Simonov and Tikhonov ^{6 ,} Reference Schumacher, Borchert, Smend and Rullhusen ^{41 ]}, we obtain a large interaction volume whose space–time scale is set by the momentum transfer and thus exceeds the Compton wavelength. Since all involved field strengths are sub-critical, hence well below the Schwinger limit, we can use the low-field expansion of the HE Lagrangian in Equation (6) with the low-energy constants ${c}_1$ and ${c}_2$ given in Equation (9). In the next step, we decompose the electric and magnetic fields in Equations (6) and (8) as $\mathbf{E}\to \mathbf{E}+{\mathbf{E}}^{\mathrm{ext}}$ and $\mathbf{B}\to \mathbf{B}+{\mathbf{B}}^{\mathrm{ext}}$ and assume that the magnetic field ${\mathbf{B}}^{\mathrm{ext}}$ is approximately constant. In addition to the static Coulomb field ${\mathbf{E}}^{\mathrm{ext}}$ of the nucleus, we have space–time-dependent XFEL fields $\mathbf{E}$ and $\mathbf{B}$ . After inserting the field decomposition into Equation (6) we obtain the effective Lagrangian for the XFEL field:

(40) $$\begin{align}{\mathrm{\mathcal{L}}}_{\mathrm{XFEL}}=\frac{1}{2}\left(\mathbf{E}\cdot {\epsilon}^{\mathrm{ext}}\cdot \mathbf{E}+\mathbf{B}\cdot {\left({\mu}^{-1}\right)}^{\mathrm{ext}}\cdot \mathbf{B}+\mathbf{E}\cdot {\Psi}^{\mathrm{ext}}\cdot \mathbf{B}\right),\end{align}$$

where we have introduced the notation $\mathbf{E}\cdot {\epsilon}^{\mathrm{ext}}\cdot \mathbf{E}\equiv {E}_i{\epsilon}_{ij}^{\mathrm{ext}}{E}_j$ , etc., for the quadratic forms in the Lagrangian. Its Hessian defines the dynamical vacuum response functions^[ Reference Baier and Breitenlohner ^{59 ]}, that is, the permittivity tensor, ${\epsilon}_{ij}^{\mathrm{ext}}$ , the permeability tensor, ${({\mu}_{ij}^{-1})}^{\mathrm{ext}}$ , and the magneto-electric coupling tensor, ${\Psi}_{ij}^{\mathrm{ext}}$ , all of which are functions of the pump fields, ${\mathbf{E}}_{\mathrm{ext}}$ and ${\mathbf{B}}_{\mathrm{ext}}$ :

(41) $$\begin{align} & {\epsilon}_{ij}^{\mathrm{ext}} =\frac{2{\alpha}^2}{45{m}^4}\left(8{E}_i^{\mathrm{ext}}{E}_j^{\mathrm{ext}}+14{B}_i^{\mathrm{ext}}{B}_j^{\mathrm{ext}} +{\delta}_{ij}\left({({\mathbf{E}}^{\mathrm{ext}})}^2-{({\mathbf{B}}^{\mathrm{ext}})}^2\right)\right),\nonumber\\ & {}{\left({\mu}_{ij}^{-1}\right)}^{\mathrm{ext}}=\frac{2{\alpha}^2}{45{m}^4}\left(8{B}_i^{\mathrm{ext}}{B}_j^{\mathrm{ext}}+14{E}_i^{\mathrm{ext}}{E}_j^{\mathrm{ext}}\right.\nonumber \\&\left.\qquad\qquad +{\delta}_{ij}\left({({\mathbf{B}}^{\mathrm{ext}})}^2-{({\mathbf{E}}^{\mathrm{ext}})}^2\right)\right),\nonumber\\ & {}{\Psi}_{ij}^{\mathrm{ext}}=\frac{2{\alpha}^2}{45{m}^4}\left(-8{E}_i^{\mathrm{ext}}{B}_j^{\mathrm{ext}}+14{B}_i^{\mathrm{ext}}{E}_j^{\mathrm{ext}} +14{\delta}_{ij}({\mathbf{E}}^{\mathrm{ext}}\cdot {\mathbf{B}}^{\mathrm{ext}})\right).\end{align}$$

These tensors characterize the vacuum polarizability, which varies with the type of pump laser used, as detailed in Table 4. The equations of motion derived from Equation (40) turn into macroscopic Maxwell equations in a medium upon introducing the electric displacement field $\mathbf{D}={\epsilon}^{\mathrm{ext}}\cdot \mathbf{E}+{\Psi}^{\mathrm{ext}}\cdot \mathbf{B}$ and the magnetic displacement field $\mathbf{H}=-\left({\left({\mu}^{-1}\right)}^{\mathrm{ext}}\cdot \mathbf{B}+{\left({\Psi}^{\mathrm{ext}}\right)}^{\mathrm{T}}\cdot \mathbf{E}\right)$ . In the usual manner, a suitable combination yields the inhomogeneous wave equation:

(42) $$\begin{align}\square \mathbf{D}=\nabla \times \left[\nabla \times \mathbf{D}\right]+{\partial}_t\left[\nabla \times \mathbf{H}\right]={\mathbf{J}}^{\mathrm{eff}}.\end{align}$$

The resulting wave equation can be analysed using standard Green function methods (see e.g., Ref. [Reference Marklund and Shukla19]). The source term ${\mathbf{J}}^{\mathrm{eff}}$ , owing to its relatively small magnitude, permits the use of the Born approximation to solve the equation. To proceed we decompose the XFEL electric displacement field $\mathbf{D}$ into an incoming plane wave ${\mathbf{D}}^{\mathrm{in}}$ and a small scattering component ${\mathbf{D}}^{\mathrm{out}}$ . Assuming a stationary time-dependence of ${\mathrm{e}}^{- i\omega t}$ for the XFEL, we arrive at a Helmholtz equation^[ Reference Ahmadiniaz, Bussmann, Cowan, Debus, Kluge and Schützhold ^{123 ]}. This equation can be solved using Green function methods. Focusing on the long-distance behaviour, we find the following scattering amplitude^[ Reference Jackson ^{124 ]}:

(43) $$\begin{align}\mathfrak{U}=\frac{1}{4\pi \mid {\mathbf{D}}_{\omega}^{\mathrm{in}}\mid}\kern0.1em {\mathbf{e}}_{\mathrm{out}}\cdot \int \mathrm{d}^3r\kern0.1em \exp \left(-i\kern0.1em {\mathbf{k}}_{\mathrm{out}}\cdot \mathbf{r}\right)\kern0.1em {\mathbf{J}}_{\omega}^{\mathrm{eff}},\end{align}$$

where ${\mathbf{k}}_{\mathrm{out}}$ is the wave vector and ${\mathbf{e}}_{\mathrm{out}}$ is the polarization unit vector of the scattered XFEL radiation. The scattering amplitude straightforwardly yields the differential cross-section, $\mathrm{d}\sigma /\mathrm{d}\Omega ={\left|\mathfrak{U}\right|}^2$ . For Coulomb-assisted Delbrück scattering, cf. Figure 13, the amplitude can be expressed as follows:

(44) $$\begin{align}\mathfrak{U}&=\frac{\omega^2}{4\pi}\int \mathrm{d}^3r\kern0.1em {\mathrm{e}}^{i\Delta \mathbf{k}\cdot \mathbf{r}}\Big({\mathbf{e}}_{\mathrm{out}}\cdot {\epsilon}^{\mathrm{e}\mathrm{xt}}\cdot {\mathbf{e}}_{\mathrm{in}}+{\mathbf{e}}_{\mathrm{out}}\cdot {\Psi}^{\mathrm{e}\mathrm{xt}}\cdot \left({\mathbf{n}}_{\mathrm{in}}\times {\mathbf{e}}_{\mathrm{in}}\right) \nonumber \\&\quad +{\mathbf{e}}_{\mathrm{in}}\cdot {\Psi}^{\mathrm{e}\mathrm{xt}}\cdot \left({\mathbf{n}}_{\mathrm{out}}\times {\mathbf{e}}_{\mathrm{out}}\right)\nonumber\\ &{}\quad +\left({\mathbf{n}}_{\mathrm{out}}\times {\mathbf{e}}_{\mathrm{out}}\right)\cdot {\left({\mu}^{-1}\right)}^{\mathrm{e}\mathrm{xt}}\cdot \left({\mathbf{n}}_{\mathrm{in}}\times {\mathbf{e}}_{\mathrm{in}}\right)\Big).\end{align}$$

Here ${\mathbf{n}}_{\mathrm{in}}={\mathbf{k}}_{\mathrm{in}}/\omega$ and ${\mathbf{n}}_{\mathrm{out}}={\mathbf{k}}_{\mathrm{out}}/\omega$ are the initial and final propagation directions, respectively, while ${\mathbf{e}}_{\mathrm{in}}$ and ${\mathbf{e}}_{\mathrm{out}}$ are the corresponding polarization vectors; $\Delta \mathbf{k}={\mathbf{k}}_{\mathrm{in}}-{\mathbf{k}}_{\mathrm{out}}$ is the momentum transfer or recoil. Figure 13 suggests a non-vanishing contribution of the magneto-electric tensor ${\Psi}^{\mathrm{ext}}$ , that is, the second and the third terms of Equation (44). Since the external magnetic field ${\mathbf{B}}^{\mathrm{ext}}$ provided by an intense laser focus is approximately constant, the spatial integral in Equation (44) yields the Fourier transform of the nuclear Coulomb field, ${\mathbf{E}}^{\mathrm{ext}}={\mathbf{e}}_{\mathrm{r}} Ze/\left(4\pi {r}^2\right)$ ( $Z$ being the atomic number), that is, $\int \mathrm{d}^3r\kern0.1em {\mathrm{e}}^{i\Delta \mathbf{k}\cdot \mathbf{r}}{\mathbf{e}}_{\mathrm{r}} Ze/\left(4\pi {r}^2\right)= iZe\Delta \mathbf{k}/{\left|\Delta \mathbf{k}\right|}^2$ .

Note that the spatial integration is effectively cut off by the momentum transfer $\Delta \mathbf{k}$ , implying a large interaction volume extending over many XFEL wavelengths for small scattering angles (near-forward direction), $\mid {\mathbf{n}}_{\mathrm{in}}-{\mathbf{n}}_{\mathrm{out}}\mid \ll 1$ . However, when the effective interaction volume becomes of the order of the spatiotemporal scales of variation of the optical laser pulse, the approximation of a constant magnetic field can no longer be justified; see also Ref. [Reference Lee125].

Let us consider forward scattering and focus on the leading-order contribution $\sim 1/\mid \Delta \mathbf{k}\mid$ . We thus approximate ${\mathbf{n}}_{\mathrm{in}}\approx {\mathbf{n}}_{\mathrm{out}}\equiv \mathbf{n}$ and consider the birefringent signal where ${\mathbf{e}}_{\mathrm{in}}$ and $\mathbf{n}$ are (nearly) orthogonal, that is, ${\mathbf{e}}_{\mathrm{in}}\approx \mp \mathbf{n}\times {\mathbf{e}}_{\mathrm{out}}$ and ${\mathbf{e}}_{\mathrm{out}}\approx \pm \mathbf{n}\times {\mathbf{e}}_{\mathrm{in}}$ . (The $\mp$ and $\pm$ symbols refer to two possible orientations of ${\mathbf{e}}_{\mathrm{in}}$ and ${\mathbf{e}}_{\mathrm{out}}$ : either parallel or anti-parallel to the cross-product of $\mathbf{n}$ with the other polarization vector.) This simplifies the integrand in Equation (44), and the birefringent amplitude is finally given by the following:

(45) $$\begin{align}{\mathfrak{U}}_{\perp}^{\Psi^{\mathrm{ext}}} &=\pm i\kern0.1em \frac{12{\alpha}^2}{45{m}^4}\kern0.1em \frac{Ze}{{\left(\Delta \mathbf{k}\right)}^2}\frac{\omega^2}{4\pi}\left(\left({\mathbf{e}}_{\mathrm{out}}\cdot {\mathbf{B}}^{\mathrm{ext}}\right)\left({\mathbf{e}}_{\mathrm{out}}\cdot \Delta \mathbf{k}\right) \right. \nonumber \\&\left. \quad -\left({\mathbf{e}}_{\mathrm{in}}\cdot {\mathbf{B}}^{\mathrm{ext}}\right)\left({\mathbf{e}}_{\mathrm{in}}\cdot \Delta \mathbf{k}\right)\right).\end{align}$$

We maximize the birefringence signal by aligning $\Delta \mathbf{k}$ with ${\mathbf{B}}^{\mathrm{ext}}$ and ${\mathbf{e}}_{\mathrm{in}}$ (or ${\mathbf{e}}_{\mathrm{out}}$ ). Assuming a strong magnetic field of the order of ${B}^{\mathrm{ext}}={10}^6\;\mathrm{T}$ (which can be achieved through laser focusing) and a large XFEL frequency of $\omega =24\;\mathrm{keV}$ (which involves additional experimental challenges), the birefringence signal is encoded in the differential cross-section in the forward direction:

(46) $$\begin{align}\frac{\mathrm{d}{\sigma}_{\perp}^{\Psi^{\mathrm{ext}}}}{\mathrm{d}\Omega}={\left|{\mathfrak{U}}_{\perp}^{\Psi^{\mathrm{ext}}}\right|}^2\approx {10}^{-25}\frac{Z^2}{{\left(\Delta \mathbf{k}\right)}^2}. \end{align}$$

The magnitude of the signal in Equation (46) is comparable to the other proposals for the vacuum birefringence experiments discussed above. In the Coulomb-assisted scenario, there are two main enhancement factors: firstly, the large number, $\mathcal{O}\left({10}^{12}\right)$ , of polarized XFEL photons; secondly, the large number $N$ of nuclei. In one possible scenario, a cubic carbon cluster with an edge length of $100\kern0.1em \mathrm{nm}$ is ionized completely using the pre-pulse of a high-intensity laser of the order of $\mathcal{O}\left({10}^{22}\kern0.22em \mathrm{W}/{\mathrm{cm}}^2\right)$ . For small momentum transfer (of the order of eV), particle-in-cell (PIC) simulations show that the amplitudes of $N={10}^8$ nuclei would have the same phase and add up coherently, thus acting as one giant nucleus of charge ${Z}_{\mathrm{eff}}= NZ$ . The birefringent signal would be of the order of $\mathcal{O}\left({10}^{-5}\right)$ photons per shot and thus could be detected in complete analogy with the scenarios discussed above; see also Refs. [Reference Inada, Yamazaki, Yamaji, Seino, Fan, Kamioka, Namba and Asai44, Reference Yamaji, Inada, Yamazaki, Namba, Asai, Kobayashi, Tamasaku, Tanaka, Inubushi, Sawada, Yabashi and Ishikawa45, Reference Heinzl, Liesfeld, Amthor, Schwoerer, Sauerbrey and Wipf67, Reference Battesti, Beard, Böser, Bruyant, Budker, Crooker, Daw, Flambaum, Inada, Irastorza, Karbstein, Kim, Kozlov, Melhem, Phipps, Pugnat, Rikken, Rizzo, Schott, Semertzidis and Zavattini79, Reference Schlenvoigt, Heinzl, Schramm, Cowan and Sauerbrey84]. For a more detailed discussion, we refer to the recent publications in Refs. [Reference Ahmadiniaz, Bussmann, Cowan, Debus, Kluge and Schützhold123, Reference Ahmadiniaz, Cowan, Ding, Lopez, Sauerbrey, Shaisultanov and Schützhold126], which also address a number of potential background processes, including nuclear Thomson scattering, nuclear resonances, Delbrück scattering and electronic Compton scattering. Among these processes, electronic Compton scattering seems the most important.

Another background atomic effect that could flip the polarization of X-ray photons is birefringence and dichroism of atomic X-ray susceptibility due to the alignment of electrons in ions generated by strong-field ionization by the polarized optical laser pulse. The X-ray dichroism of ionized krypton gas was experimentally demonstrated in Refs. [Reference Young, Arms, Dufresne, Dunford, Ederer, Höhr, Kanter, Krässig, Landahl, Peterson, Rudati, Santra and Southworth127–Reference Goulielmakis, Loh, Wirth, Santra, Rohringer, Yakovlev, Zherebtsov, Pfeifer, Azzeer, Kling, Leone and Krausz129] and theoretically studied in Refs. [Reference Santra, Dunford and Young130,Reference Rohringer and Santra131] for X-rays resonant to transition between the 1s core atomic orbital and 4p_3/2 valence atomic orbital where an aligned electron hole was produced. From these studies, the probability of polarization flipping can be estimated as $\sim {\left(\sin \left(2\theta \right)n{\lambda}^2L\Delta {\rho}_{\mathrm{aniso}}\right)}^2{A}^2/\left(\Delta {\omega}^2+{\gamma}^2/4\right)$ , where $\theta$ is an angle between the polarizations of the optical and X-ray pulses; $n$ and $L$ are the concentration and the length of the gas sample, respectively; $\lambda$ is the X-ray wavelength; $\Delta {\rho}_{\mathrm{aniso}}$ is the difference between probabilities of finding a valence orbital hole with distinct projection quantum numbers – this quantity describes the alignment of the valence hole; $A$ , $\Delta \omega$ and $\gamma$ are the spontaneous emission rate, the detuning and the linewidth, respectively, for the corresponding valence-to-core transition. For the case of strong-field ionized ${\mathrm{Kr}}^{+}4{\mathrm{p}}^{-1}$ , the alignment of the valence hole $\Delta {\rho}_{\mathrm{aniso}}$ can reach the value of $64\%$ , and for an atomic concentration 10 ${}^{18}\kern0.1em {\mathrm{cm}}^{-3}$ , an interaction length of a few mm and an X-ray photon energy resonant to the valence-to-core transition, the polarization flipping probability could reach $\mathcal{O}$ (10 ${}^{-5}$ ). With decreasing gas concentration and increasing detuning, this probability rapidly drops off; thus avoiding the valence-to-core transition is necessary to suppress the atomic birefringence background. From the atomic, molecular and optical (AMO) and plasma physics point of view, the high-purity X-ray polarimetry measurements described in this letter could provide unique insights into the anisotropy properties of atomic transitions.

4.1 Conventional scenario

The decisive bottlenecks in the conventional scenario are the quality of the X-ray polarizers and the X-ray lenses. For the former, early works suggested that an extinction ratio of the order of ${10}^{-10}$ would be necessary to prove vacuum birefringence induced by a petawatt-class laser. The extinction ratio is also known as polarization purity. More detailed analyses, together with the fact that only $300\kern0.22em \mathrm{TW}$ is available at the EuXFEL, led to the insight that an extinction ratio of ${10}^{-12}$ is actually required for the X-ray polarimeter. Of equal importance is that the X-ray lenses do not compromise the polarization purity.

At the University of Jena and the Helmholtz Institute Jena, a research and development programme to eliminate these two bottlenecks was started about 15 years ago. The polarizers are perfect crystals into which a trench is cut – which is why they are referred to as channel-cut crystals or simply channel cuts (CCs). The wavelength is chosen such that the X-rays are Bragg-reflected at the (inner) walls of the channel at a Bragg angle of 45°, the Brewster angle for X-rays. At the Brewster angle, the polarization component parallel to the diffraction plane (p-component) of the X-rays field is suppressed. Zig-zagging multiple successive reflections increases the suppression of the p-component. For some crystals attractive for X-ray polarimetry – namely diamond – cutting a channel is not an option. In this case, one would precision-mount two separate crystals perfectly in parallel, an arrangement known as artificial or QCC.

Figure 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 ${0.01}^{\circ }$ of the crossed-polarizer position reproduced from Ref. [Reference Schulze, Grabiger, Loetzsch, Marx-Glowna, Schmitt, Garcia, Hippler, Huang, Karbstein, Konôpková, Schlenvoigt, Schwinkendorf, Strohm, Toncian, Uschmann, Wille, Zastrau, Röhlsberger, Stöhlker, Cowan and Paulus72]. 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.

As early as 2010, an extinction ratio of $1.5\times {10}^{-9}$ at $6.457\kern0.22em \mathrm{keV}$ photon energy and of $9.0\times {10}^{-9}$ at $12.914\kern0.22em \mathrm{keV}$ was demonstrated. The (400) and (800) Bragg reflexes of Si were used^[ Reference Marx, Uschmann, Höfer, Lötzsch, Wehrhan, Förster, Kaluza, Stöhlker, Gies, Detlefs, Roth, Härtwig and Paulus ^{132 ]}. Advances in the production and processing of the crystals and improvements in the brilliance of X-ray sources led relatively quickly to a further improvement in polarization purity to $2.4\times {10}^{-10}$ and $5.7\times {10}^{-10}$ for the same X-ray wavelengths as above^[ Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus ^{85 ]}. The originally communicated requirements on X-ray polarimetry to detect vacuum birefringence were thus largely fulfilled. While these requirements were to become more and more demanding, some of the opportunities arising from the advent of polarizers providing polarization purities several orders of magnitude better than the previous state-of-the-art were quickly exploited (e.g., Refs. [Reference Marx-Glowna, Grabiger, Lötzsch, Uschmann, Schmitt, Schulze, Last, Roth, Antipov, Schlenvoigt, Sergueev, Leupold, Röhlsberger and Paulus21, Reference Marx, Schulze, Uschmann, Kämpfer, Lötzsch, Wehrhan, Wagner, Detlefs, Roth, Härtwig, Förster, Stöhlker and Paulus85, Reference Schulze, Loetzsch, Rüffer, Uschmann, Röhlsberger and Paulus133, Reference Schmitt, Joly, Schulze, Marx-Glowna, Uschmann, Grabiger, Bernhardt, Loetzsch, Juhin, Debray, Wille, Yavaş, Paulus and Röhlsberger134]). Prominent applications were the detection of quantum optics and QED effects in the X-ray range^[ Reference Heeg, Wille, Schlage, Guryeva, Schumacher, Uschmann, Schulze, Marx, Kämpfer, Paulus, Röhlsberger and Evers ^{135 –} Reference Haber, Schulze, Schlage, Loetzsch, Bocklage, Gurieva, Bernhardt, Wille, Rüffer, Uschmann, Paulus and Röhlsberger ^{137 ]}.

In order to achieve further improvements in polarization purity, the limiting factors had to be identified^[ Reference Schulze ^{138 ,} Reference Marx-Glowna, Schulze, Uschmann, Kämpfer, Weber, Hahn, Wille, Schlage, Röhlsberger, Förster, Stöhlker and Paulus ^{139 ]} and eliminated. These are the so-called Umweganregungen and the finite divergence of the X-rays. Strictly speaking, the latter is not a real problem, as it only occurs at the synchrotron and laboratory sources used for polarimeter development, whereas the divergence of the XFEL beam is negligible. With regard to the Umweganregungen, it turned out that these cannot be avoided in principle. However, there are certain highly symmetrical reflexes for which all of them interfere destructively. What is required, though, is a highly precise adjustment not only of the Bragg angle, but also of the azimuthal orientation of the polarimeter crystals, which is the degree of freedom that can be exploited by a rotation around the normal of the diffracting lattice planes. Novel methods have been developed for this task that enable quick adjustment of the polarimeter crystals and thus efficient use of the precious XFEL beamtime.

In parallel, another branch of the research and development of X-ray polarimeters was the use of diamond crystals. These are advantageous for several reasons. Notable among these are the small lattice constant, which allows the use of shorter wavelengths, the low atomic number, which is advantageous with regard to Umweganregungen, the high reflectivity of almost 100% and, finally, the excellent thermal conductivity. First experiments were performed with relatively cheap CVD crystals in a QCC setup with only two reflections^[ Reference Bernhardt, Marx-Glowna, Schulze, Grabiger, Haber, Detlefs, Loetzsch, Kämpfer, Röhlsberger, Förster, Stöhlker, Uschmann and Paulus ^{140 ]}. Nevertheless, a polarization purity of $8.9\times {10}^{-10}$ at $9.837\kern0.22em \mathrm{keV}$ ((400) Bragg reflection) was obtained, limited by the divergence of the synchrotron radiation. With four crystals for both the polarizer and the analyser, and some control on the beam divergence, another record polarization purity of $1.4\times {10}^{-10}$ was achieved^[ Reference Bernhardt, Schmitt, Grabiger, Marx-Glowna, Loetzsch, Wille, Bessas, Chumakov, Rüffer, Röhlsberger, Stöhlker, Uschmann, Paulus and Schulze ^{141 ]}.

The first experiment at a free-electron X-ray laser was performed at the HED instrument of the EuXFEL. The silicon 400 reflection and six reflections in the CC were used. A polarization purity of $8\times {10}^{-11}$ was achieved; see Figure 14 for an illustration. It should be emphasized that this number is not determined by the quality of the polarizers but rather by the photon flux of the XFEL. In other words, $8\times {10}^{-11}$ is the upper limit for the actual extinction ratio; the extinction curve is in fact compatible with perfect polarization^[ Reference Schulze, Grabiger, Loetzsch, Marx-Glowna, Schmitt, Garcia, Hippler, Huang, Karbstein, Konôpková, Schlenvoigt, Schwinkendorf, Strohm, Toncian, Uschmann, Wille, Zastrau, Röhlsberger, Stöhlker, Cowan and Paulus ^{72 ]}. It should also be emphasized that the limited photon flux was due to limited beamtime, the XFEL running in SASE mode and other factors, that is, they do not constitute principal road blocks. This is highlighted by the fact that the present record polarization purity of $\left(1.4\pm 0.9\right)\times {10}^{-11}$ was achieved at a third generation synchrotron facility^[ Reference Marx-Glowna, Grabiger, Lötzsch, Uschmann, Schmitt, Schulze, Last, Roth, Antipov, Schlenvoigt, Sergueev, Leupold, Röhlsberger and Paulus ^{21 ]}. The theoretical limit for the polarization purity as determined by the finite laser beam divergence is of the order of ${10}^{-14}$ . It has yet to be shown whether this limit can be reached or whether other effects kick in.

Another bottleneck is the lenses of the X-ray telescope. Due to a refractive index slightly smaller than unity in the X-ray regime, stacks of concave lenses must be used, so-called compound refractive lenses (CRLs). Irrespective of that, it is known that X-ray diffraction in crystals can lead to birefringence. It is also known that metals, including beryllium, the preferred material for X-ray lenses, have a micro-crystalline structure. Accordingly, the question has been whether this is indeed a limiting factor and, if so, how to bypass it. The Jena group has addressed both. In short, the answer is that there is little if any hope that beryllium, which is attractive because of high transmission, can be used. Already $500\kern0.22em \unicode{x3bc} \mathrm{m}$ of beryllium degrades the polarization purity to the level of ${10}^{-6}$ ^[ Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze ^{142 ]}.

One alternative is the use of amorphous lens materials. Two options have been tested so far: CRLs made from glassy carbon and lithographically produced CRLs made from the photoresist SU-8. The disadvantage of the latter is that the production method only allows one-dimensional focusing. Therefore, two SU-8 CRLs, rotated by 90° with respect to each other, have to be used for focusing and recollimation. This reduces the transmission of the total setup. On the other hand, these lenses are of very good optical quality and allow close to diffraction-limited performance^[ Reference Grabiger, Marx-Glowna, Uschmann, Loetzsch, Paulus and Schulze ^{142 ]}. For glassy carbon, the situation was exactly the opposite. Meanwhile, better components have become available and tests will be carried out soon.

Figure 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.

Still another alternative, at first glance maybe paradoxically, is lenses made from perfect crystals, specifically diamond. They can be rotated around the optical axis such that they exhibit no birefringence. Also this option will be tested soon.

The conclusion is that the conventional scenario is still very promising and could indeed be the fastest way to prove vacuum birefringence. Essentially, it must be shown that the polarimeters can achieve an extinction ratio of the order of ${10}^{-12}$ . So far, there is no evidence whatsoever that this might not be the case.

4.2 Dark-field scenario

The setup described in Section 3.2 and schematically shown in Figure 9 was adapted to fit the experimental conditions of the HED instrument of the EuXFEL for the proof-of-principle experiment, as can be seen in Figure 15. The experiment is designed to be used with XFEL photon energy of $8766\kern0.22em \mathrm{eV}$ , so that a 440 plane of a germanium crystal can be used as a polarization analyser. The XFEL beam is coming from the left-hand side of the picture, entering the IC1 experimental chamber about $-1.2\kern0.22em \mathrm{m}$ upstream from the interaction point. The whole setup is in vacuum chambers up until the Kapton window located before the detectors, at about $5\kern0.1em \mathrm{m}$ downstream from the target chamber center (TCC). At the entrance point ( $-1.1\kern0.22em \mathrm{m}$ ), obstacle O1 is located, which is made of a vertical wire-like structure with a diameter of 125–150 $\unicode{x3bc} \mathrm{m}$ to block the central part of the beam. The focusing lens (L1) is located 47 cm upstream from focus, and consists of 12 beryllium lenses with a $50\kern0.22em \unicode{x3bc} \mathrm{m}$ central radius of curvature. The beryllium lenses have a $400\kern0.22em \unicode{x3bc} \mathrm{m}$ diameter, which is the limiting factor for the incoming beam size. Just after the lens stack, obstacle O2 is located, made of simple wire with a size matching O1. A pinhole of about $50\kern0.22em \unicode{x3bc} \mathrm{m}$ diameter is placed exactly in the focus, which prevents radiation scattered on the first lens from propagating further. The image of O1 projected by lens L1 is at the position 85 cm downstream from the focus, where aperture A1 is located. The scattering on the edges of these slits is a critical factor for the success of the setup, and therefore several possibilities for how to limit the beam are being pursued, as described further. The second lens stack (L2), consisting of six lenses of the same type as lens 1, is located 114 cm downstream from focus, to image the focal plane onto the detector plane. An intermediate aperture (A2) is located in a separate chamber at the image of obstacle O2, that is, about $4\kern0.22em \mathrm{m}$ downstream of focus. Six to seven metres downstream, at the end of the experimental hutch, is where the detector bench is located. This can either host detectors in the direct beam, which is useful for initial alignment and testing, or can be used for measurement of the polarization distinction. Alternatively, an analyser setup hosting two detectors and Ge crystal can be put in the hutch, as described below.

In order to align and monitor the X-ray beam, the diode screen with the photodiode, as well as several fluorescent screens, is located along the beam propagation axis. For the ultimate discovery experiment the ReLaX optical beam will counter-propagate with the XFEL beam, focused down by an $f/1$ off-axis parabolic mirror (OAP). In this experiment the ReLaX beam is expected to deliver 300 TW in 25 fs, and be focused down to a focal spot FWHM diameter of $1.3\kern0.22em \unicode{x3bc} \mathrm{m}$ . This focusing scheme, beam routing and corresponding diagnostics are yet to be developed.

Figure 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 $\unicode{x3bc}$ m and the colour scale is logarithmic over three orders of magnitude.

There are two key parameters to evaluate the quality of the dark-field setup. The shadow factor $\mathcal{S}$ was already defined in the theory in Section 3.2. The definition from the experimental point of view is that it is the ratio of X-ray photons in the active area of the final detector to the number of X-ray photons entering the chamber if there is no scattering at the focal spot involved.

The aim of the design of a good setup is then to minimize the $\mathcal{S}$ factor while keeping the overall transmission sufficient. Therefore a transmission factor $T$ is defined to indicate a chance of producing and detecting vacuum birefringence scattering processes. As the angular distribution of vacuum birefringence scattered photons is not known at this point, we define $T$ simply as $T={T}_0\times {T}_1$ , where ${T}_0$ is the transmission from entrance to the chamber towards the focal position (therefore indicating how many photons will be interacting with the field), and ${T}_1$ is a transmission from the focal point towards the active area of the detector when both obstacles O1 and O2 would be removed – therefore indicating chance that a scattered photon will be detected. By the active area of the final detector, we mean an area containing the image of the focal spot, therefore an area where vacuum birefringence scattering can be detected. Distinguishing this area, which might be smaller than $10\kern0.22em \unicode{x3bc} \mathrm{m}$ , from the remaining area of the detector is an important part of the setup.

Therefore, it can be said that the factor $T$ represents the signal in the experiment, while the shadow factor $\mathcal{S}$ represents the noise. The standard method of optimization would be to maximize the signal-to-noise ratio, represented by the fraction $T/\mathcal{S}$ . However, as will be shown later, this would lead to prohibitively low signal values (effectively closing up the apertures), providing zero measured photons. Therefore a measure of ${T}^6/\mathcal{S}$ was identified as an optimal quantity to be maximized, or less exactly, minimizing the $\mathcal{S}$ factor while keeping the $T$ at a reasonable $10\%$ level.

4.2.1 Diffractive simulations

In order to design and optimize the X-ray beam path and all its elements, the whole setup was simulated by a complex simulation using the LightPipes framework^[ Reference Vdovin and van Goor ^{143 ]}, which is designed to propagate a coherent beam where diffraction is essential. Most of the simulations presented here have a $700\kern0.22em \unicode{x3bc} \mathrm{m}$ box size, resolution of 47 nm and therefore $15{,\!}000\times 15{,\!}000$ simulated points. This was tested to be sufficient to avoid numerical imprecision for given cases. All obstacles and apertures are modelled as two-dimensional (2D) maps of thickness, that is, their exact three-dimensional (3D) shape is neglected. The thickness is converted into transmission and phase shifts by using the online Henke tables^[ Reference Henke, Gullikson and Davis ^{144 ]}. The full result of a typical simulation is shown and described in Figure 16.

An example of parameter optimization is shown in Figure 17. Here each point represents one set of parameters defining the experimental geometry, and its colour corresponds to the diameter of the obstacle O2 in $\unicode{x3bc} \mathrm{m}$ . It can be seen that optimal obstacles are rather large with a diameter of around 140 $\unicode{x3bc}$ m. It is very useful to see that the typical optimal simulations are following the trend $n\approx {s}^6$ , where $n$ corresponds to the noise of the measurement (which is proportional to the shadow factor), and signal $s$ is representing the possibility to detect a signal photon. That means that a simple optimization of the signal-to-noise ratio would lead to prohibitively small signals. Therefore, in further considerations, an approach to minimize the noise while keeping the signal on the $10\%$ level was used.

Figure 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.

4.2.2 Obstacles

As basic types of obstacles will be round wires, the diffraction and scattering on the edges of the wires are critical. Thus, their surfaces shall be polished to high surface quality, either by focused ion beam (FIB) or electrochemical etching. Commercially available diameters of 200 and $175\kern0.22em \unicode{x3bc} \mathrm{m}$ will be used and simulations have confirmed that the diameter of O2 shall be smaller than that of O1.

Alternatively, a custom-made microfabricated shape was developed to minimize the diffraction and was produced with electroplating. This method uses a focused laser or electron beam to create the desired shape from bulk silicon. Afterwards, nickel is deposited through electroplating. The so-called trumpet shape obtained in this way exhibits a quadratic increase of thickness as a function of decreasing distance from the axis, which transfers to an exponential increase of its opacity. Therefore, as the tip of such a shape is very thin, given by the manufacturing possibilities of the order of a single $\unicode{x3bc} \mathrm{m}$ , the diffraction on such a tip is very limited. On the other hand, the close-to-transparent edge might cause a significant refraction effect, which diverts the photons away from the axis, and therefore such photons do not propagate further downstream and do not contribute to the noise on the detector.

A variant of such an object is depicted in Figure 18. The left-hand pane shows the thickness profile as a function of radial distance. The profile is symmetrical, only positive values are shown and are in a wire-like geometry, that is, extended over the not shown dimension. The second and third subplots show the calculated absorption and phase shift caused by the obstacle.

Figure 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.

Figure 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 $\mathcal{S}/{T}^2$ and $\mathcal{S}/{T}^6$ factors, which are considered for optimization. Minimum values of the latter factors are desirable for our purpose.

4.2.3 Performance evaluation

Once a set of all beamline elements is given, it is still not trivial to find the optimal settings for the opening of both apertures A1 and A2. Opening of each aperture has its optimal value, which could not be independent of the other one. In general, if an aperture is open too wide, a significant amount of scattered radiation can go through, while if it is closed too much, the signal transmission is decreased and, eventually, additional scattering on the slit edge can be produced. Therefore, a set of simulations for various openings is performed for a given set of components. Figure 19 shows the simulated $\mathcal{S}$ and $T$ factors in the top row, and the derived quantities $\mathcal{S}/{T}^2$ and $\mathcal{S}/{T}^6$ in the second row, for the case in which both O1 and O2 are tungsten wires with respective diameters of 200 and $175\kern0.22em \unicode{x3bc} \mathrm{m}$ . The best performance, that is, the minimum value, of $\mathcal{S}/{T}^6=2\times {10}^{-4}$ is achieved for openings of 150 and $80\kern0.22em \unicode{x3bc} \mathrm{m}$ , respectively. A different size of A1 increases this factor significantly, while A2 does not have that strong an influence, showing that the majority of signal reduction is done on the O1–A1 pair, while the O2–A2 pair plays rather a minor role.

This effect is seen even more strongly in simulations with optimized O1–A1 components, as seen in Figure 20. Here, the trumpet shape for O1 is used, and A1 is used with the soft edges of W slits with a plastic phase corrector. By this approach, we improve the best-simulated value for $\mathcal{S}$ by a factor of approximately/almost equal to 30, and the value for $\mathcal{S}/{T}^6$ by a factor of approximately/almost equal to 20. What is interesting is to see that this value is for an A2 size of $300\kern0.22em \unicode{x3bc} \mathrm{m}$ , and is essentially not changing if the aperture is opened more, showing that the cleaning on the first set of components is so superior, that the O2–A2 set does not produce any improvement.

Figure 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.

Figure 21 Integrated reflectivity of the considered crystal cuts. For each crystal cut we in addition depict the value of $I{\omega}^2$ in arbitrary units.

4.2.4 Ge analyser

As mentioned above, the advantage of the dark-field scheme relies on combining two methods to distinguish the LbL scattered photons from the non-scattered (direct) XFEL photons, the angular scattering and the change of polarization. The set of obstacles and apertures is performing the angular selection, while the second one is yet to be done by a polarization analyser. The main requirements when designing an analyser for these setups were identified as follows:

1. (i) bandwidth covering the majority of the XFEL spectrum;

2. (ii) possibility to measure both polarization states.

As we are relying on crystal-based polarizers, where the setting of the Bragg angle to 45° ensures reflection of only the s-polarization, we have to admit that, to our knowledge, all considered crystal reflections in their default configuration have insufficient bandwidth, that is, they are below or at $0.2\kern0.22em \mathrm{eV}$ , while the bandwidth of the self-seeded XFEL is of the order of $1\kern0.22em \mathrm{eV}$ .

This is pushing the design to use asymmetrical cut crystals, where the angle of incidence is not equal to the Bragg angle. In this configuration, the bandwidth can be significantly improved without losing much of the peak reflectivity, and therefore also the integrated reflectivity is increased. A summary of the performance of considered or common crystals is shown in Figure 21, where the integrated reflectivity is plotted as a function of photon energy, where the given plane can serve as a polarizer. However, it is important to realize that for the success of the vacuum birefringence experiment, the quantity $I{\omega}^2$ , where $I$ is the integrated reflectivity (approximated as peak reflectivity times Darwin width) and $\omega$ is the probe photon energy, is a good indicator for a suitable crystal, as the number of scattered photons scales as ${\omega}^2$ . The data points in Figure 21 labelled as assym are those used with asymmetrical cuts with various degrees of asymmetry. That clearly shows that either the Ge 440 or Ge 335 in asymmetrical cuts is the crystals of choice. From those the Ge 440 was chosen, which fixes the experimental constraint to $8766\kern0.22em \mathrm{eV}$ . Various cuts and their integrated reflectivities are summarized in Table 5. Theoretically, it is clearly better to employ shallower incidence, which however requires a larger crystal surface with perfect quality. Therefore, various crystals with different cuts are being used for the upcoming experimental campaign, to find which will perform best in given experimental conditions.

Table 5 Parameters of different cuts of Ge 440 crystal reflection.

The second condition – the possibility to measure both polarization states – could be easily fulfilled by employing a thin transmissive crystal, as is done in other spectrometers already used, for example, at the HED instrument^[ 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 ^{145 ]}. However, two issues arise if that were to be used in the asymmetric (shallow Bragg angle) geometry: the crystal would have to be extremely thin, and its surface would have to be very precise along a large area. A combination of these two constraints leads us to the conclusion that such precision is not feasible. Therefore, the so-called Baronova configuration^[ Reference Baronova, Stepanenko and Stepanenko ^{146 ]} is employed, which utilizes a thick crystal in reflective geometry, but in such a geometry that both polarization states are reflected on different planes, and therefore in different directions. This setup will be adopted at the XFEL such that the native XFEL polarization (horizontal) will be reflected above the beam, while the polarization-flipped photons will be reflected horizontally to another camera. The reflectivities for both beams shall be in the predicted value of about $50\kern0.22em \unicode{x3bc} \mathrm{rad}$ .