1]Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328
Dresden, Germany 2]Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85,
22607 Hamburg, Germany 3]Technische Universität Dresden, 01062 Dresden,
Germany 4]Centre for Mathematical Sciences, University of Plymouth, Plymouth,
PL4 8AA, UK 5]Department of Physics and Astronomy, Abbe Center of Photonics,
Friedrich-Schiller-Universität Jena, 07743 Jena, Germany 6]Helmholtz Institute
Jena, Fröbelstieg 3, 07743 Jena, Germany 7]European XFEL GmbH, Holzkoppel 4,
22869 Schenefeld, Germany 8]Center for Advanced Systems Understanding,
Untermarkt 20, 02826 Görlitz, Germany 9]GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany 10]ELI
Beamlines Facility, The Extreme Light Infrastructure ERIC, Dolní Břežany,
Czech Republic

# Letter of Intent: Towards a Vacuum Birefringence Experiment at the Helmholtz
International Beamline for Extreme Fields

(BIREF@HIBEF Collaboration)

N. Ahmadiniaz  C. Bähtz  A. Benediktovitch  C. Bömer  L. Bocklage  T. E. Cowan
J. Edwards  S. Evans  S. Franchino Viñas  H. Gies  S. Göde  J. Görs  J.
Grenzer  U. Hernandez Acosta  T. Heinzl  P. Hilz  W. Hippler  L. G. Huang  O.
Humphries  F. Karbstein  P. Khademi  B. King  T. Kluge  C. Kohlfürst  D. Krebs
A. Laso-García  R. Lötzsch  A. J. Macleod  B. Marx-Glowna  E. A. Mosman  M.
Nakatsutsumi  G. G. Paulus  S. V. Rahul  L. Randolph  R. Röhlsberger  N.
Rohringer  A. Sävert  S. Sadashivaiah  R. Sauerbrey  H.-P. Schlenvoigt  S. M.
Schmidt  U. Schramm  R. Schützhold  J.-P. Schwinkendorf  D. Seipt  M. S̆míd
T. Stöhlker  T. Toncian  M. Valialshchikov  A. Wipf  U. Zastrau  M. Zepf  [  [
[  [  [  [  [  [  [  [

###### Abstract

Quantum field theory predicts a nonlinear response of the vacuum to strong
electromagnetic fields of macroscopic extent. This fundamental tenet has
remained experimentally challenging and is yet to be tested in the laboratory.
A particularly distinct signature of the resulting optical activity of the
quantum vacuum is vacuum birefringence. This offers an excellent opportunity
for a precision test of nonlinear quantum electrodynamics in an uncharted
parameter regime. Recently, the operation of the high-intensity laser ReLaX
provided by the Helmholtz International Beamline for Extreme Fields (HIBEF)
has been inaugurated at the High Energy Density (HED) scientific instrument of
the European XFEL. We make the case that this worldwide unique combination of
an x-ray free-electron laser and an ultra-intense near-infrared laser together
with recent advances in high-precision x-ray polarimetry, refinements of
prospective discovery scenarios, and progress in their accurate theoretical
modelling have set the stage for performing an actual discovery experiment of
quantum vacuum nonlinearity.

###### Contents

  1. 1 Executive Summary
  2. 2 Introduction
    1. 2.1 History and Status
    2. 2.2 Light-by-light scattering with lasers
    3. 2.3 Vacuum Birefringence
    4. 2.4 Previous Experiments
  3. 3 Prospective Scenarios
    1. 3.1 Conventional two-beam scenario
    2. 3.2 Dark-field scenario
    3. 3.3 Planar three-beam set-up
    4. 3.4 Other opportunities: Coulomb-assisted birefringent scattering
  4. 4 Plans for experimental implementation
    1. 4.1 Conventional scenario
    2. 4.2 Dark-field scenario
      1. 4.2.1 Diffractive simulations
      2. 4.2.2 Obstacles
      3. 4.2.3 Performance evaluation
      4. 4.2.4 Ge analyser
  5. 5 Conclusions and Outlook

##  1 Executive Summary

This letter sets out the intention to perform a first measurement of vacuum
birefringence with real photons as a flagship experiment at the HED-HIBEF
instrument. Photons from the European X-ray Free Electron Laser (EuXFEL) will
be scattered at regions of the quantum vacuum polarised by the optical
Relativistic Laser at XFEL (ReLaX). Their polarisation will be measured and
compared to predictions from quantum electrodynamics (QED). Counting the
number of scattered photons that have flipped or not flipped their
polarisation allows experimental determination of the low-energy effective
field theory couplings of QED, first calculated over 80 years ago [1, 2, 3].
Measurement of vacuum birefringence with real photons can thus be viewed as a
first step on the way to harnessing the nonlinearity of the quantum vacuum.

QED predicts a self-interaction of the electromagnetic field that is mediated
by virtual electron-positron pairs (some of the constituents of the ‘quantum
vacuum’). This effect is purely quantum mechanical in nature: in classical
electromagnetism, the electromagnetic field obeys the superposition principle.
The self-interaction of the electromagnetic field has been observed in the
scattering of gamma rays in the Coulomb field of atomic nuclei (Delbrück
scattering) [4, 5] and more recently at the ATLAS and CMS detectors in the
generation of two real photons in the collision of two Coulomb fields [6, 7,
8]. So-called ‘vacuum polarisation’ has also been invoked to describe
anomalous polarisation of photons measured from strongly magnetised neutron
stars [9]. In addition, indirect evidence for vacuum birefringence in the
modulation of pairs created via the linear Breit-Wheeler process in the STAR
experiment has been noted [10]. However, the very small cross-section makes
direct probing of virtual electron-positron pairs by colliding and scattering
only real photons extremely challenging and has yet to be achieved. The high
number of photons available in laser beams, and the coherence of their
electromagnetic fields over spacetime scales much larger than that typically
probed by a virtual pair, suggest that colliding focussed laser pulses would
be a suitable way to measure this effect. Because the leading-order process is
a four-photon interaction, many signatures suggested to be probed in laser
experiments are similar to those from four-wave mixing. Among others, these
include: manipulation of the polarisation of intense laser pulses, frequency-
shifting effects [11, 12, 13, 14], vacuum diffraction [15, 16, 17] and vacuum
self-focussing [18, 19]. At the same time, any detection in laser beam
collisions is challenged by separating the signal of a modified polarisation,
momentum or energy in the scattered photons from the large background of the
laser fields.

The first direct measurement of vacuum birefringence has been envisioned as a
flagship experiment for HED-HIBEF from its initial inception in 2011. HED-
HIBEF combines x-rays with an optical pump beam thereby considerably
increasing the cross-section, which scales with the centre-of-mass energy to
the sixth power, compared to all-optical set-ups. High-precision x-ray
polarimetry has been developed over the last decade so that it is now possible
to generate a beam of x-rays that are polarised in the same state to a degree
of better than one in one hundred billion [20]. This allows to substantially
reduce the background for those x-ray photons that scatter into the ‘flipped’
polarisation mode as a signal of vacuum birefringence. At the same time the
theoretical tools to make quantitatively accurate predictions of quantum
vacuum signals in experimentally realistic laser fields have been advancing;
see the recent review [21] and references therein. In light of these
developments, we detail several experimental scenarios that can be realised by
combining the EuXFEL with the optical laser ReLaX to measure the birefringence
of the vacuum with real photons. One of these scenarios features the ‘dark
field’ method of blocking part of the XFEL before it is focussed and collides
with the optical beam so that in the detector plane in the shadow, there is a
region of very few background XFEL photons, which is suitable for detecting a
signal [22]. The actual suppression that can be achieved with the EuXFEL beam
will be determined when HED-HIBEF uses its priority access to measure
background rates in 2024. In this letter we give details for the experimental
implementation of this scenario. Finally, testing QED in an uncharted
parameter regime can also constrain the parameter space beyond the Standard
Model physics, for instance, that of weakly interacting particles with a small
mass [23]. In the Summary, the potential of HED-HIBEF experiments to search
for new degrees of freedom is discussed.

The fundamental physics prediction of vacuum birefringence represents a prime
example of a fascinating and a priori counter-intuitive phenomenon that can
happen when a seemingly trivial state (“the vacuum”) is subjected to extreme
conditions (“ultra-intense electromagnetic fields”). Its detection with state-
of-the-art technology constitutes a formidable challenge and is at the edge of
what is currently possible. The BIREF@HIBEF collaboration brings together
experts in strong-field QED, x-ray optics, high-intensity lasers, and laser-
plasma physics and thus adopts an interdisciplinary approach to meeting this
challenge and performing a discovery experiment.

##  2 Introduction

The quantum vacuum, i.e. the ground state of a quantum field theory such as
QED, behaves as a nonlinear, polarisable medium in reaction to strong
electromagnetic fields. An electromagnetic wave probing the polarised vacuum
may in turn change its polarisation state as a result of _vacuum
birefringence_. Microscopically, this is caused by the possibility of _light-
by-light scattering_ in QED.

###  2.1 History and Status

The idea of light-by-light (LbL) scattering by now has a venerable history.
According to Scharnhorst’s thorough account [24], its absence in the classical
theory has already been stated by Kepler. In modern terms, this is explained
by the linearity of Maxwell’s equations of electrodynamics entailing the
superposition principle: electromagnetic field solutions can be added at will
and remain solutions. In _quantum_ electrodynamics, the situation is
different. The presence of vacuum polarisation due to virtual pair
fluctuations implies the existence of four-photon scattering amplitudes as
first pointed out by Halpern [25] and Debye (according to Heisenberg [26]).
The presence of this amplitude, represented by the Feynman diagram of Fig. 1,
implies that the quantum theory becomes nonlinear and photons (self) interact
through an effective four-photon vertex. This fact becomes manifest in terms
of the celebrated Heisenberg-Euler (HE) Lagrangian, a low-energy effective
field theory for QED [3] discussed in more detail below.

Figure 1: $O(\alpha^{2})$ Feynman diagram for the LbL scattering amplitude
implying a 4-photon self-interaction.

Heisenberg’s students Euler and Kockel did the first calculation of the LbL
cross section at low energy [1], while Landau’s students Akhiezer and
Pomeranchuk obtained the high-energy result [27]. Due to the four Lorentz
indices of the amplitude, the full cross section is somewhat tedious to work
out, and it took until 1950 for a full answer to materialise [28, 29] (see
also [30, 31]). Since then, the cross section has become (advanced) textbook
material [32, 33, 34, 35, 36].

As usual in QED, the cross section for LbL scattering may be constrained via
dimensional analysis. The basic QED parameters are Planck’s constant, $\hbar$,
and the speed of light, $c$, which signal the unification of quantum mechanics
with special relativity. Henceforth, though, we will choose natural units,
$\hbar=c=1$, unless stated otherwise. The remaining parameters are thus the
electron mass and charge, $m$ and $e$, respectively. The latter defines the
QED coupling strength, $\alpha=e^{2}/4\pi=1/137$, which tells us that QED is
(normally) perturbative. We may also form the typical QED length scale, given
by the electron Compton wavelength, ${\mathchoice{\hbox{\raise 5.06941pt\hbox
to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\displaystyle\lambda$}}}{\hbox{\raise
5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\textstyle\lambda$}}}{\hbox{\raise
3.54857pt\hbox to0.0pt{\kern 0.81665pt\vrule
width=2.85832pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptstyle\lambda$}}}{\hbox{\raise
2.5347pt\hbox to0.0pt{\kern 0.58331pt\vrule
width=2.04166pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptscriptstyle\lambda$}}}}_{e}:=1/m\simeq
3.8\times 10^{-13}\,{\rm m}$, and the typical QED electric field magnitude,
$E_{S}:=m^{2}/e\simeq 1.3\times 10^{18}\,{\rm V}/{\rm m}$, also known as the
Sauter-Schwinger limit [37, 38]. Any electric field, when localised within a
Compton wavelength, will contain modes above threshold ($q^{2}>4m^{2}$) and
thus produce pairs with a probability given by a perturbative amplitude
(squared) [35]. If the field magnitude exceeds the Sauter-Schwinger limit,
non-perturbative, sub-threshold pair production becomes possible.

With the parameters defined, we return to the LbL cross section. At low
energies, the dominant energy scale is the electron mass, so the cross section
(an area) must be proportional to $1/m^{2}$. At high energies, masses are
irrelevant, and the cross section is inversely proportional to the square of
the total energy $\omega_{*}$ in the centre-of-mass (CM) frame, $\sigma\sim
1/\omega_{*}^{2}$. The precise results for the total unpolarised cross section
are [34]

$\displaystyle\sigma$
$\displaystyle=\frac{973\,\alpha^{4}}{10125\,\pi}\left(\frac{\omega_{*}}{m}\right)^{6}\frac{1}{m^{2}}\qquad$
$\displaystyle(\omega_{*}\ll m)\;,$ (1) $\displaystyle\sigma$
$\displaystyle=4.7\,\alpha^{4}/\omega_{*}^{2}\qquad$
$\displaystyle(\omega_{*}\gg m)\;.$ (2)

The energy dependence of the cross section is depicted in Fig. 2 together with
the outcome of some past experiments to be discussed below.

Figure 2: Light-by-light scattering cross section in microbarn ($1\,\upmu{\rm
b}=10^{-30}\,{\rm cm}^{2}$) as a function of photon energy in the centre-of-
mass frame, $\omega_{\ast}$ (calculated from [30, 31, 39]). 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.
$2$-to-$2$) refers to the number of real photons in the in and out state (see
insert). The $0$-to-$2$ measurements (blue) were for a diphoton mass
$>5\,\textrm{GeV}$ (CMS, [8]) or $>6\,\textrm{GeV}$ (ATLAS, [6, 7]). These are
collectively represented on the plot at
$\omega_{\ast}/m=\sqrt{s}/2m=5\,\textrm{GeV}$. The $1$-to-$1$ process of
Delbrück scattering off nuclei measured by Jarlskog et al. ([4], see also [40,
5]) is shown in green. An energy range is indicated (in cyan) for HED-HIBEF
assuming a near head-on collision of an optical beam with central energy
$1.55\,\textrm{eV}$ and XFEL beam with energies between $6\,\textrm{keV}$ and
$12.9\,\textrm{keV}$. The red cross-dots represent the $2$-to-$2$ laser
experiments by Moulin et al. [41], Bernard et al. [42] (both all-optical),
Inada et al. [43], Yamaji et al. [44] (both employing an XFEL) and Watt et al.
[45].

We note that the cross section (1) is universal in the sense that it is
entirely given in terms of QED parameters ($\alpha$ and $m$) and the Lorentz
invariant kinematic factor $s=4\omega_{*}^{2}$, the total energy (squared) in
the CM frame. Thus, measuring the cross section is indeed another experimental
test of QED. The actual observable is the number $N^{\prime}$ of scattered
photons given by

$\displaystyle N^{\prime}=Nn_{\rm L}\Delta z\,\sigma\;,$ (3)

where $N$ is the number of incoming (probe) photons, $n_{\rm L}=N_{\rm L}/V$
the target photon density and $\Delta z$ the target thickness, i.e. the
spatial extent of the probed photon distribution.

At CM energies of order $\omega_{*}\sim m\sim 0.5\,{\rm MeV}$, the cross
section is of the order of $10^{-30}\,{\rm cm}^{2}$ which, although five
orders of magnitude smaller than the Thomson cross section,
$\sigma_{\mathrm{Th}}=(8\pi/3)\alpha^{2}/m^{2}$, is not particularly small.
However, in this energy regime, available photon fluxes (i.e., $N$ and $N_{\rm
L}$) are too small to lead to a measurable number $N^{\prime}$ of events. At
low energies, the situation is reversed. For example, in the optical regime,
one has large photon fluxes, but the cross section is exceedingly small,
$\sigma\sim 10^{-64}\,{\rm cm}^{2}$ for $\omega_{*}=1.5\,{\rm eV}$.
Nevertheless, it has been suggested early on that (intense) lasers may be
employed to measure this process [46, 11, 47]. The current bound on the cross
section from an all-optical scattering experiment is $\sigma<1.5\times
10^{-48}\,{\rm cm}^{2}$ at $\omega_{*}=0.8\,{\rm eV}$ [42], a considerable
improvement of the earlier result [41]. With the advent of XFELs, the CM
energy can be increased by about four orders of magnitude (from $1\,{\rm eV}$
to $10\,{\rm keV}$). Experiments at SACLA (Japan) have produced a bound of
$\sigma<1.9\times 10^{-23}$ cm2 at $\omega_{*}=6.5\,{\rm keV}$ [44]. At this
energy, the QED prediction is $2.5\times 10^{-43}\,{\rm cm}^{2}$. So both
experiments are off the QED value by about $20$ orders of magnitude, see Table
1. For comparison, we note that the cross section for neutrino electron
scattering (for neutrino energies of $10^{2}\,{\rm eV}$) is about
$10^{-58}\,{\rm cm}^{2}$ and has not been measured yet [48].

Facility | Experiment | $\omega_{*}$ | Bound | QED value
---|---|---|---|---
LULI | all-optical (two beam) [41] | $1.7\;{\rm eV}$ | $\sigma<9.9\times 10^{-40}\,{\rm cm}^{2}$ | $1.6\times 10^{-64}\;{\rm cm}^{2}$
LULI | all-optical (three beam) [42] | $0.8\;{\rm eV}$ | $\sigma<1.5\times 10^{-48}\;{\rm cm}^{2}$ | $10^{-66}\;{\rm cm}^{2}$
SACLA | XFEL + XFEL [44] | $6.5\,{\rm keV}$ | $\sigma<1.9\times 10^{-23}\;{\rm cm}^{2}$ | $2.5\times 10^{-43}\;{\rm cm}^{2}$
HED-HIBEF | XFEL ($8766$ eV) + optical | $116\;{\rm eV}$ |  | $1.81\times 10^{-53}\;{\rm cm}^{2}$
Table 1:  Experimental bounds obtained for the LbL scattering cross section
and QED predictions (in historical order). The last row refers to the
experiment proposed in this LoI which aims to reach the sensitivity for the
QED value stated.

Fig. 2 also shows some experiments at high energy, $\omega_{*}\gtrsim m$.
These have actually observed variants of Delbrück scattering [49] where
photons couple to nuclear Coulomb fields. This may be realised by scattering
photons off nuclei (charge $Ze$) [Fig. 3 (a)] or via ultra-peripheral heavy-
ion collisions [Fig. 3 (b)]. In either case, the two intermediate photons are
virtual, with their four-momentum $q$ off-shell, $q^{2}\neq 0$. Counting
vertices and dimensions, the cross section for $\omega_{*}<m$ can be estimated
as [34]

$\sigma\sim
Z^{4}\alpha^{6}\left(\frac{\omega_{*}}{m}\right)^{4}\frac{1}{m^{2}}\;.$ (4)

This yields an increase compared to LbL, recall (1), if
$Z^{2}\alpha>\omega_{*}/m$ which is easily achieved for $\omega_{*}<m$. In
particular, the nuclear charge can be viewed as a _coherent_ enhancement
factor which is also present at larger energies, $\omega_{*}>m$ [34]. As a
result, the Delbrück processes of Fig. 3 have indeed been measured, both for
nuclei at rest (Fig. 2, green dots, [4, 40, 5]) and in heavy-ion collisions
(Fig. 2, blue dots, [6, 8, 7]). Fig. 2 clearly shows that the Delbrück cross
section exceeds the LbL one – by at least five orders of magnitude for
$\omega_{*}\gg m$. For additional details we refer to the overview presented
in the introduction of [50].

(a)

(b)

Figure 3: Two variants of Delbrück scattering involving two _virtual_ photons,
$\gamma^{*}$. (a) Off a (static) Coulomb potential denoted by crosses. (b) Off
a Lorentz boosted Coulomb potential in ultra-peripheral heavy-ion collisions.

###  2.2 Light-by-light scattering with lasers

At BIREF@HIBEF an intense optical laser ($\omega_{\rm L}=1.55\;{\rm eV}$) will
be combined with the European XFEL (which can run at different frequencies
from about $5$ to $24\;{\rm keV}$, e.g. at $\omega_{\rm X}=8766\;{\rm eV}$).
This implies a CM energy given by the geometric mean of
$\omega_{*}=(\omega_{\rm L}\omega_{\rm X})^{1/2}=116\;{\rm eV}$. According to
(1), the QED cross section will then be $\sigma=1.81\times 10^{-53}\;{\rm
cm}^{2}$. This is still fairly small, but the coherence of the optical laser
background drastically enhances the amplitude by a factor proportional to the
photon number.

The calculation of the relevant cross section thus proceeds in two steps.
First, one employs a low-energy approximation by adopting the Heisenberg-Euler
effective Lagrangian with a point-like four-photon vertex. Intuitively, the
vacuum polarisation loop can no longer be resolved as the effective theory is
only valid for distances much larger than the Compton wave length. In a second
step, one replaces two of the photon legs in Fig. 1 by an external
electromagnetic field representing the intense laser focus. The two steps are
depicted in terms of Feynman diagrams in Fig. 4.

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

Scattering off an optical laser background $F$ yields a coherent enhancement
factor $F^{2}$ in the amplitude. This factor is proportional to the photon
number $N_{\rm L}$ of the optical laser, hence analogous to the squared
nuclear charge, $Z^{2}$, in Delbrück scattering. Thus, replacing $Z^{2}$ by
$F^{2}\sim N_{\rm L}$ in (4), the cross section is expected to behave as

$\sigma\sim\alpha^{4}N_{\rm
L}^{2}\left(\frac{\omega_{*}}{m}\right)^{6}\frac{1}{m^{2}}\;.$ (5)

The dimensionless constant of proportionality can be calculated from the low-
energy effective field theory of QED,

$\mathcal{L}_{\mathrm{eff}}=\mathcal{S}+\mathcal{L}_{\textrm{HE}}+c_{22}F_{\mu\nu}\Box
F^{\mu\nu}+\ldots\;\qquad\mathcal{L}_{\textrm{HE}}=c_{1}\mathcal{S}^{2}+c_{2}\mathcal{P}^{2}\;,$
(6)

where $\mathcal{L}_{\textrm{HE}}$ is the celebrated Heisenberg-Euler (HE)
Lagrangian [3] to lowest order in field strengths. Its crucial ingredients are
the fundamental Lorentz and gauge invariants,

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}(\mathbf{E}^{2}-\mathbf{B}^{2})\;,$
(7) $\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}=\mathbf{E}\cdot\mathbf{B}\;,$
(8)

with electromagnetic field strength tensor
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ and its dual,
$\tilde{F}^{\mu}=(1/2)\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$, gauge
potential $A^{\mu}$ and low-energy constants $c_{1}$, $c_{2}$ and $c_{22}$.
The invariant $\mathcal{S}$ is just the Maxwell Lagrangian while
$\mathcal{S}^{2}$ and $\mathcal{P}^{2}$ are the leading-order nonlinear
corrections first employed in [1]. The ellipsis represents corrections of
higher order in both field strength [3] and derivatives [51, 52, 53, 54, 55].
The low-energy constants are

$c_{1}=\frac{8\alpha^{2}}{45m^{4}}\;,\quad
c_{2}=\frac{14\alpha^{2}}{45m^{4}}\;,\quad c_{22}=\frac{\alpha}{60\pi
m^{2}}\;,$ (9)

where the powers in $\alpha$ and $m$ follow from vertex counting and
dimensional analysis. These coefficients receive higher loop corrections that
are parametrically suppressed with additional powers of $\alpha\ll 1$. They
account for higher-order vacuum polarisation effects arising from the
interaction of charges in the vacuum polarisation loop cf., e.g., [56].

In this context, we also emphasise that the structure of Eq. (6) is generic
for any nonlinear extension of classical electrodynamics that respects Lorentz
covariance, U(1) gauge invariance, and a charge conjugation parity symmetry.
Hence, a measurement of the QED predictions in Eq. (9) inherently also implies
a restriction of the parameter space of other potential non-linear extensions
of classical electrodynamics in and beyond the Standard Model of particle
physics.

In QED, the last constant, $c_{22}$, is the weight of the leading-order
_linear_ vacuum polarisation term and was already calculated by Dirac and
Heisenberg in 1934 [57, 26]. Its QED analog, valid for all energies, implies a
scale dependence, hence ‘running’, of the electric charge which decreases with
distance. Intuitively, this corresponds to charge screening caused by the
virtual pair dipoles in the vacuum. This implies in turn that the vacuum can
be viewed as a polarisable medium with both linear and nonlinear response to
an external field. The associated response functions are given by the second
derivatives of the effective Lagrangian with respect to $F$ (or $\mathbf{E}$
and $\mathbf{B}$). Focussing on the nonlinear case we can define macroscopic
fields $\mathbf{D}$ and $\mathbf{H}$ as the derivatives of the Lagrangian
$\mathcal{L}_{\mathrm{eff}}=\mathcal{S}+\mathcal{L}_{\mathrm{HE}}$,

$\displaystyle D_{i}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}_{\mathrm{eff}}}{\partial
E_{i}}\equiv\varepsilon_{ij}E_{j}\;,$ (10) $\displaystyle H_{i}$
$\displaystyle=$
$\displaystyle-\frac{\partial\mathcal{L}_{\mathrm{eff}}}{\partial
B_{i}}\equiv\mu_{ij}^{-1}B_{j}\;,$ (11)

where the (static) nonlinear response functions are the tensors [58]

$\displaystyle\varepsilon_{ij}$ $\displaystyle=$
$\displaystyle(1+2c_{1}\mathcal{S})\,\delta_{ij}+2c_{2}\,B_{i}B_{j}\;,$ (12)
$\displaystyle\mu_{ij}^{-1}$ $\displaystyle=$
$\displaystyle(1+2c_{1}\mathcal{S})\,\delta_{ij}-2c_{2}\,E_{i}E_{j}\;.$ (13)

These correspond to the dielectric and permeability tensors, respectively, and
thus explicitly show the electric and magnetic response of the vacuum to
external fields. In the words of Weisskopf [59]: ‘When passing through
electromagnetic fields, light will behave as if the vacuum had acquired a
dielectric constant different from unity due to the influence of the fields.’

From conventional optics, it is known that media with dielectric constant
$\varepsilon$ and permeability $\mu$ have optical properties characterised by
an index of refraction, $n\equiv\sqrt{\varepsilon\mu}$. In view of (12) and
(13) it must thus be possible to ascribe a nontrivial refractive index to the
vacuum. This can indeed be done by studying the eigenvalues of the dielectric
and permeability tensors, $\varepsilon_{ij}$ and $\mu_{ij}$ [60, 36, 61]. One
finds that the vacuum is characterised by _two_ nontrivial refractive indices,
$n_{1}$ and $n_{2}$, so that the vacuum acts like a _birefringent_ medium.

###  2.3 Vacuum Birefringence

A web search for the term ‘vacuum birefringence’ dates its first appearance to
the year 1964 when it appeared in the title of a paper by Klein and Nigam [60]
who employed the tensors (12) and (13) stemming from the low-energy HE
Lagrangian. However, the vacuum refractive indices were first calculated (for
arbitrary photon energies!) by Toll in his unpublished PhD thesis [62]
supervised by Wheeler. Other early and important contributions have been made
by Erber [63] (preceding Klein and Nigam!), Baier and Breitenlohner [58] and
Narozhny [64]. The utilisation of lasers has first been suggested and analysed
in [65]. The HED-HIBEF experiment is based on the optimal scenario of
combining an XFEL probe beam with an intense optical laser background. This
was first suggested in [66].

To avoid confusion we recall that the term ‘vacuum birefringence’ refers to
the refractive or dispersive properties of the vacuum (encoded in the real
part of the refractive indices), while the term ‘vacuum dichroism’ describes
its absorptive properties (encoded in the imaginary part of the refractive
indices). In other words, dichroism amounts to a direction dependent
absorption of probe photons, hence a reduction of probe intensity [67, 68].

A quick way to derive the magnitude of the birefringence effect is to employ
the differential cross section in the CM frame which for a $2\to 2$ process
(with all masses equal) has the universal form

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^{2}}\frac{1}{s}\,|\mathfrak{M}|^{2}\equiv|f(\mathbf{l},\mathbf{l}^{\prime},\theta)|^{2}\;.$
(14)

Here $\mathfrak{M}$ is the invariant amplitude depending on photon
polarisations and momenta while $f$ is the scattering amplitude for momentum
transfer, $\mathbf{l}\to\mathbf{l}^{\prime}$, and scattering angle $\theta$.
Microscopically, vacuum birefringence corresponds to forward scattering
($\mathbf{l}=\mathbf{l}^{\prime}$, $\theta=0$) with a polarisation flip of a
probe photon passing through an intense laser background. We thus assume the
involvement of two laser photons with the same polarisations and four vectors,
while the probe polarisation flips, $\varepsilon\to\varepsilon^{\prime}$ with
$\varepsilon\cdot\varepsilon^{\prime}=0$. Choosing the optimal scenario of a
$45^{\circ}$ angle between probe and laser polarisations, one may use the
textbook formulae in [32] to find the forward-flip amplitude $\mathfrak{M}(0)$
and hence the forward scattering amplitude,

$f(\mathbf{l},\mathbf{l},0)\equiv
f(0)=\frac{\mathfrak{M}(0)}{8\pi\sqrt{s}}=\frac{4\alpha^{2}}{15\pi}\left(\frac{\omega_{*}}{m}\right)^{3}\frac{1}{m}\;.$
(15)

Another textbook formula (see e.g. [69], Ch. 1.5) relates the index of
refraction to the forward scattering amplitude. The flip amplitude, in
particular, defines the _difference_ of the vacuum refractive indices,

$\Delta n=\frac{2\pi n_{\rm
L}}{\omega_{*}^{2}}f(0)=\frac{8\alpha^{2}}{15}\frac{I_{\rm L}}{m^{4}}\;.$ (16)

Here we have used that the laser photon density, $n_{\rm L}=N_{\rm L}/V=I_{\rm
L}/\omega_{*}$, is proportional to intensity, $I_{\rm L}$, measured in the CM
system. This result is consistent with the intensity scaling that led to (5)
and the calculation [64] of the refractive indices in terms of the vacuum
polarisation tensor (the covariant unification of dielectric and permeability
tensors),

$n_{i}=1+4c_{i}I_{\rm L}\quad\mbox{whence}\quad\Delta n=4(c_{2}-c_{1})I_{\rm
L}\;.$ (17)

This may be written in a more covariant way as follows. One introduces the
probe 4-vector $k=\omega\ell$ with frequency $\omega$ and a dimensionless
null-vector $\ell$, $\ell^{2}=0$. One then forms the ‘null-energy projection’
[70], $T_{\ell\ell}\equiv\ell_{\mu}T^{\mu\nu}\ell_{\nu}$, of the Maxwell
energy-momentum tensor,
$T^{\mu\nu}=F^{\mu\alpha}F_{\alpha}^{\;\;\nu}-g^{\mu\nu}\mathcal{S}$, the
traceless part of the squared field strength. For any field configuration,
$F^{\mu\nu}$, one can thus write

$\displaystyle\frac{\Delta
n}{T_{\ell\ell}}=c_{2}-c_{1}=\frac{2\alpha^{2}}{15m^{4}}\;,$ (18)

which obviously just measures the difference of the leading-order low-energy
constants in the HE Lagrangian. The difference $\Delta n$ in refractive
indices induces an ellipticity $\delta\equiv k_{\rm X}z\Delta n/2$ for a
linearly polarised probe beam of wave number $k_{\rm X}$ traversing the
polarised vacuum across a distance $z$. The experimental signature is then the
flip probability,

$\frac{N^{\prime}}{N}=\delta^{2}=\frac{16\alpha^{4}}{225}\frac{I_{\rm
L}^{2}}{m^{8}}\left(\frac{z}{{\mathchoice{\hbox{\raise 5.06941pt\hbox
to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\displaystyle\lambda$}}}{\hbox{\raise
5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\textstyle\lambda$}}}{\hbox{\raise
3.54857pt\hbox to0.0pt{\kern 0.81665pt\vrule
width=2.85832pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptstyle\lambda$}}}{\hbox{\raise
2.5347pt\hbox to0.0pt{\kern 0.58331pt\vrule
width=2.04166pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptscriptstyle\lambda$}}}}_{\rm
X}}\right)^{2}=\frac{4\alpha^{2}}{225}\left(\frac{I_{\rm L}}{I_{\rm
S}}\right)^{2}\left(\frac{z}{\lambda_{\rm X}}\right)^{2}\;,$ (19)

with the Sauter-Schwinger intensity $I_{\rm S}=E_{\rm
S}^{2}=m^{4}/4\pi\alpha=4.7\times 10^{29}\,{\rm W}/{\rm cm}^{2}$. Eq. (19)
shows explicitly that an optimal scenario will maximise the target intensity
$I_{L}$ and its spatial extent, $z$, while simultaneously minimising the
(reduced) probe wave length, ${\mathchoice{\hbox{\raise 5.06941pt\hbox
to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\displaystyle\lambda$}}}{\hbox{\raise
5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\textstyle\lambda$}}}{\hbox{\raise
3.54857pt\hbox to0.0pt{\kern 0.81665pt\vrule
width=2.85832pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptstyle\lambda$}}}{\hbox{\raise
2.5347pt\hbox to0.0pt{\kern 0.58331pt\vrule
width=2.04166pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptscriptstyle\lambda$}}}}_{\rm
X}\equiv\lambda_{\rm X}/2\pi$ [66]. At HED-HIBEF this is realised by combining
a high-intensity optical laser with an XFEL. A rough estimate with $I_{\rm
L}=10^{21}\,{\rm W}/{\rm cm}^{2}$, $z=10\;\upmu\textrm{m}$ and
${\mathchoice{\hbox{\raise 5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\displaystyle\lambda$}}}{\hbox{\raise
5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\textstyle\lambda$}}}{\hbox{\raise
3.54857pt\hbox to0.0pt{\kern 0.81665pt\vrule
width=2.85832pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptstyle\lambda$}}}{\hbox{\raise
2.5347pt\hbox to0.0pt{\kern 0.58331pt\vrule
width=2.04166pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptscriptstyle\lambda$}}}}_{\rm
X}=0.02\,{\rm nm}$ (for $\omega_{\rm X}=10\,{\rm keV}$) yields
$N^{\prime}/N\approx 10^{-12}$. Obviously, such a minuscule flip probability
demands a very high accuracy of the required polarisation measurements.
Indeed, the original suggestion to employ XFEL beams as a probe [66] has been
an incentive to improve the polarisation purity of x-rays by several orders of
magnitudes to a current record of $10^{-11}$ [71]. A topical review of the
technical difficulties involved and how they are being addressed is given in
[72].

###  2.4 Previous Experiments

As early as 1929, Watson [73] tried to measure a vacuum refractive index $n$
depending linearly on a static magnetic field, $B$. He used an interferometer
to measure a small induced frequency shift, but did not find any effect and
thus produced the upper bound of $(n-1)/B<4\times 10^{-7}\,{\rm T}^{-1}$ (in
modern notation). In 1960 Jones [74] measured the velocity of light in a
magnetic field to high precision and found no deviation from its vacuum value.
A year later, Erber [63] discussed a number of settings (including Watson’s)
that might allow measurement of a _quadratic_ dependence of the vacuum
refractive indices, hence a Cotton-Mouton effect [75, 76], as predicted by
QED. Klein and Nigam [60] suggested using a static electric field inside a
plane capacitor but ruled the birefringence effect to be way too small. In
1979 Iacopini and Zavattini pointed out that one may use strong static
magnetic fields and optimise the geometric factor
$(z/{\mathchoice{\hbox{\raise 5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\displaystyle\lambda$}}}{\hbox{\raise
5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\textstyle\lambda$}}}{\hbox{\raise
3.54857pt\hbox to0.0pt{\kern 0.81665pt\vrule
width=2.85832pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptstyle\lambda$}}}{\hbox{\raise
2.5347pt\hbox to0.0pt{\kern 0.58331pt\vrule
width=2.04166pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptscriptstyle\lambda$}}}})^{2}$.
To this end, one should propagate a stable optical laser beam through a
magnetic cavity and increase the optical path length $z$ through multiple
reflections. This idea has been realised in the PVLAS experiment, which
recently celebrated its 20th anniversary [77]. There are also a number of
competing or complementary experiments for which an overview may be found in
[78] together with an extensive list of references. Fig. 5, reproduced from
[77], presents the historical evolution of the experimental results, the most
recent of which are still about an order of magnitude above the QED
prediction. (The error bars represent an uncertainty of one sigma.) The
current best value has been reported by the PVLAS-FE experiment as

$\frac{\Delta n^{(\mathrm{PVLAS-FE})}}{B^{2}}=(+19\pm 27)\times 10^{-24}\,{\rm
T}^{-2}\;,$ (20)

where $B$ denotes the external _static_ magnetic field being probed. To
compare with the QED result, we note that for a static magnetic field probed
at the right angle, $T_{\ell\ell}=B^{2}$. Introducing the magnetic Sauter-
Schwinger field strength, $B_{\rm S}=m^{2}/e=4.41\times 10^{9}\,{\rm T}$, the
general result (18) leads to

$k_{\mathrm{CMV}}\equiv\frac{\Delta
n^{\mathrm{stat}}}{B^{2}}=c_{2}-c_{1}=\frac{2\alpha^{2}}{15m^{4}}=\frac{\alpha}{30\pi}\frac{1}{B_{\rm
S}^{2}}=4\times 10^{-24}\,{\rm T}^{-2}\;.$ (21)

This constant, containing only the basic QED parameters, has been called the
Cotton-Mouton constant of the vacuum (hence the acronym CMV) in [78]. The same
value is obtained for the HED-HIBEF scenario of probing an optical laser
background, cf. (16) and (18), as $T_{\ell\ell}=4F^{2}=4I_{\rm L}$ (assuming a
head-on collision). Hence,

$\frac{\Delta n^{\mathrm{CF}}}{4F^{2}}=c_{2}-c_{1}\;,$ (22)

where CF stands for ‘crossed field’ with $F=E=B$ and
$\mathbf{E}\cdot\mathbf{B}=0$.

Figure 5: Historical evolution of the results for vacuum birefringence
experiments employing static magnetic fields (reproduced from [77] where more
details can be found.) The green horizontal line represents
$k_{\mathrm{CMV}}\equiv c_{2}-c_{1}$.

Strong-field vacuum polarisation effects also play an important role in
astrophysics and cosmology. For instance, extreme field magnitudes can be
found in the magnetosphere of neutron stars and magnetars. Several years ago
it was claimed that measurements of the polarisation degree of such neutron
stars provide evidence for an active role of vacuum birefringence [9]. The
subsequent scientific debate [79] highlights the necessity of a controlled
laboratory verification of the effect which in turn should provide input for
improved models of neutron star environments.

Very recently the STAR collaboration has reported the observation of (linear)
Breit-Wheeler pair production [80], i.e. the process $\gamma+\gamma\to
e^{+}+e^{-}$. As this was realised in ultra-peripheral heavy-ion collisions,
the pair production is actually proceeding via the Landau-Lifshitz process
[81] at low photon virtualities, see Fig. 6 (a). Via the optical theorem, this
process is related to Delbrück scattering as observed by the ATLAS
collaboration – upon ’cutting’ the fermion loop in Fig. 3 (b). It has been
shown that polarisation flips induced by this diagram lead to modulations in
the angular pair spectra as shown in Fig. 6 (b) reproduced from the recent
review [10]. The authors of [10] have interpreted this as an indirect signal
for vacuum birefringence.

(a)

(b)

Figure 6: The STAR experiment. (a) Landau-Lifshitz process. (b) Modulation
signalling vacuum birefringence by means of the optical theorem (reproduced
from [10]).

##  3 Prospective Scenarios

As detailed in the previous section, real light-by-light scattering has a very
small cross-section. Any prospective scenario must be able to deliver a signal
that can be distinguished from the noise of the background photons. At the
moment, three different proposals – and viable combinations thereof – are
available that constitute prospective routes toward a first measurement of the
effect at HED-HIBEF. All these proposals are intimately related and look very
similar at first glance. However, they differ decisively in the details and
thus come with different experimental challenges. Two of these are two-beam
scenarios using the collision of the XFEL with the intense ReLaX beam, and the
third one summarises three-beam scenarios requiring the XFEL to collide with
two intense laser beams, generated by splitting the original ReLaX beam into
two beams. (See Fig. 7 for an overview of different collision geometries.)

Figure 7: Overview of different collision geometries: (a) is the conventional
head-on two-beam scenario; (b)-(e) are three-beam scenarios where the XFEL is
collided with two optical lasers. In (c) one of the optical beams is
frequency-doubled [82].

Focussing only on the essentials (see Secs. 3.1-3.3 below for a detailed
discussion of each of these scenarios), the two-beam scenarios are:

  * (1)

The conventional vacuum birefringence scenario [66, 83] that envisions the
collision of the two beams in a counter propagation geometry. It uses
exclusively the polarisation flip of x-ray probe photons. This small quantum
vacuum signal needs to be separated from the large background of probe photons
traversing the high-intensity pump focus without interaction. A key parameter
for the measurement of this effect is the quality of the employed polarimeter
which is typically quantified in terms of its polarisation purity $\cal P$.
This represents the fraction of background photons registered in the ideally
empty, perpendicularly polarised mode [84]. For the XFEL and intense laser
parameters attainable at HED-HIBEF the signal is found to be background
dominated [71]. However, by an appropriate choice of the beam waists, the
divergence of the signal can be made wider than that of the probe, such that
angular cuts can be used to improve the signal-to-background ratio at the
expense of reduced absolute photon numbers [16, 85, 86, 87].

  * (2)

A dark-field approach [22] based on modifying the probe beam with a well-
defined beam stop such as to exhibit a shadow in both the converging and
expanding beam while retaining a peaked focus profile. This should allow
access to both the parallel and perpendicular polarised components of the
nonlinear vacuum response scattered into the shadow where the background is
significantly reduced. In this scenario, the key parameter is the quality of
the shadow which can be quantified in terms of the unwanted background
measured within the shadow. It remains to be shown that a sufficiently good
shadow quality $\cal S$ ensuring a signal-to-background ratio above unity can
be realised in experiments for the XFEL and intense laser parameters available
at HED-HIBEF. However, the outcome of an elementary proof-of-concept
experiment at an x-ray tube [22] and the results of numerical diffraction
simulations performed by our collaboration look sufficiently promising that
this scenario is the one we have selected to first explore experimentally; cf.
also Sec. 4 below.

Finally, the three-beam scenarios aim at verifying quantum vacuum nonlinearity
in

  * (3)

four-wave-mixing processes [13, 14, 88, 89, 90, 82, 91] where the XFEL is
collided with two intense optical laser beams that are derived from the same
source and are focussed to the same spot. Since the optical beams have the
same frequency and propagate in different directions, this enables a new kind
of quasi-elastic scattering signal at the XFEL photon frequency, the
generation of which involves the absorption of a field quantum from one of the
intense beams and emission into the other. Due to the associated finite
momentum transfer, this signal is scattered into a well-defined direction away
from the forward direction of the XFEL. Moreover, three-beam setups can also
induce sizeable signals characterised by a frequency shift. While an enlarged
phase space seems beneficial, it remains to be shown that the additional
challenges coming with the experimental control of three laser beams can be
mastered in a way such as to benefit from the additional signal photon
channels facilitated by three-beam scenarios.

For the experimental implementation of a given scenario, the central interest
is in the number of x-ray signal photons that can be discerned from the
typically large background of the EuXFEL beam. Only these constitute a
signature of quantum vacuum nonlinearity that is detectable in an experiment.
To address this theoretically, we model the near-infrared (NIR) high-intensity
and XFEL beams driving the nonlinear quantum vacuum signals as paraxial
solutions of the wave equation, supplemented with a Gaussian pulse envelope.
In general, the electric field of a paraxial beam can be expanded as
$\mathbf{E}=\mathbf{E}^{(0)}(\varsigma)+\varsigma\,\mathbf{E}^{(1)}(\varsigma)+\cdots$,
i.e., in powers of the small parameter $\varsigma=w_{0}/z_{\rm R}$, where
$w_{0}$ is the beam waist and $z_{\rm R}=\omega w_{0}^{2}/2$ is the Rayleigh
length of the fundamental Gaussian beam solution; $\omega=2\pi/\lambda$ is the
oscillation frequency of a beam with central wavelength $\lambda$. Throughout
this work, we truncate the paraxial expansion at leading order and use
$\mathbf{E}\approx\mathbf{E}^{(0)}(\varsigma)$. Apart from the probe beam
featuring a central shadow in the dark-field scenario, the description of
which requires the superposition of several Laguerre-Gaussian or Hermite-
Gaussian modes, we model all laser fields as linearly polarised fundamental
Gaussian beams. For a beam propagating in positive $\rm z$ direction, this
implies the following electric field profile,

$E=E_{0}\,\exp\left\\{-\left(\frac{t-z}{\tau/2}\right)^{2}-\left(\frac{r}{w}\right)^{2}\right\\}\cos\left\\{\omega(t-z)-\frac{z}{z_{\rm
R}}\left(\frac{r}{w}\right)^{2}+\arctan\frac{z}{z_{\rm R}}\right\\}\,,$ (23)

where $E_{0}$ is the field amplitude, $\tau$ is the phase pulse duration,
$r=\sqrt{x^{2}+y^{2}}$ and $w=w_{0}\sqrt{1+(z/z_{\rm R})^{2}}$. This, in
particular, implies that the peak intensity $I_{\textrm{peak}}$ of the optical
laser pulse is related to the pulse energy $W$ via $I_{\rm
peak}=8\sqrt{2/\pi}\,W/(\pi w_{0}^{2}\tau)=2\times 10^{23}\,[{\rm
Wcm}^{-2}]W[{\rm J}](w_{0}[\upmu{\rm m}])^{-2}(\tau[{\rm fs}])^{-1}$. The
focal width and pulse duration are related to the full-width-at-half-maximum
(FWHM) parameters via: $w_{0}=0.85\,w_{\rm FWHM}$ and $\tau=1.7\,\tau_{\rm
FWHM}$. Standard parameters to be available at HED-HIBEF are given in Tab. 2.

European XFEL $N$ $2\times 10^{11}$ @ $\omega=8\ldots 10\,{\rm keV}$ (self-
seeded) $1\times 10^{11}$ @ $\omega=12\ldots 13\,{\rm keV}$ (self-seeded)
$1\times 10^{12}$ @ $\omega=8\ldots 10\,{\rm keV}$ (SASE) $5\times 10^{11}$ @
$\omega=12\ldots 13\,{\rm keV}$ (SASE) $\Delta\omega$ $300\,{\rm meV}$ (self-
seeded) $10^{-3}\times\omega$ (SASE) $\tau_{\rm FWHM}$ $25\,{\rm fs}$ ReLaX
$\lambda$ $800\,{\rm nm}$ $W$ $4.8\,{\rm J}$ $\tau_{\rm FWHM}$ $30\,{\rm fs}$
$w_{\rm FWHM}$ $\\#\times 1.3\,\upmu{\rm m}$

Table 2: European XFEL and ReLaX parameters. The ReLaX focal width given here
is for $f/\\#$ focusing.

###  3.1 Conventional two-beam scenario

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

The most extensively studied scenario in the present context is the
conventional two-beam scenario [65], which envisions the head-on collision of
the XFEL beam acting as probe and a NIR high-intensity pump [66]. 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 Eq. (6): If the strong
field introduces a preferred direction, such as, e.g., a unidirectional
magnetic field or a linearly polarised laser field, the isotropy of the vacuum
is broken. In turn, the vacuum polarised by the strong field effectively
features two distinct transverse propagation eigenmodes for probe light
associated with different refractive indices. Correspondingly, if initially
linearly polarised 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 is scattered into a perpendicularly polarised mode, thereby
effectively supplementing it with a small ellipticity. The large photon number
and the high polarisation degree of XFEL beams in conjunction with the
available high-purity x-ray polarimetry, reaching polarisation purities down
to the ${\cal P}=10^{-11}$ level for specific photon energies in the ${\cal
O}(10)\,{\rm keV}$ regime [71], makes XFEL beams the ideal probe for such an
experiment [83]. At the same time, NIR high-intensity lasers can reach the
highest peak field strengths on macroscopic scales extending over spatial
distances of ${\cal O}(1)\,\upmu{\rm m}$ and times of ${\cal O}(10)\,{\rm fs}$
which are well-compatible with the characteristic scales of XFEL pulses,
making the best choice for the pump. In the collision of linearly polarised
laser beams the number of polarisation-flipped signal photons $N_{\perp}$ is
maximised for a relative polarisation angle of $\pi/4$. As the attainable
signal photon number scales with $(1-\cos\vartheta_{\rm coll})^{4}$, where
$\vartheta_{\rm coll}$ is the collision angle of the two beams, the counter-
propagating geometry is favoured; see Fig. 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 realised so far are of the
order of $100\,{\rm nm}$ [92], the Rayleigh lengths that can be achieved for
beams with a photon energy of $\omega_{\rm X}={\cal O}(10)\,{\rm keV}$ fulfil
$z_{\rm R,X}\gtrsim{\cal O}(1)\,{\rm mm}$; cf. the definitions in the context
of Eq. (23) above. This value is much larger than the Rayleigh range $z_{\rm
R,L}={\cal O}(10)\,\upmu{\rm m}$ of a tightly focussed NIR laser beam. It is
therefore an excellent approximation to formally send $z_{\rm R,X}\to\infty$
when determining the induced quantum vacuum signal, i.e., to adopt an infinite
Rayleigh length approximation [85, 86, 93, 89, 94] for the probe beam. This
significantly simplifies the calculation; cf. [95] 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 [86, 96,
97]; for different measurement concepts see also [98, 99]. 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 [100] (for
the case of $\vartheta_{\rm coll}=\pi$), 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 polarisation of $\pi/4$ between the colliding beams, the
attainable number of polarisation-flipped signal photons per shot can then be
approximately expressed as [100, 101]

$N_{\perp}=N_{\rm
X}\,\sqrt{\frac{3}{\pi}}\frac{(c_{1}-c_{2})^{2}m^{8}}{\pi}\left(\frac{W_{\rm
L}}{m}\frac{\omega_{\rm X}}{m}\right)^{2}\left(\frac{{\mathchoice{\hbox{\raise
5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\displaystyle\lambda$}}}{\hbox{\raise
5.06941pt\hbox to0.0pt{\kern 1.16664pt\vrule
width=4.08333pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\textstyle\lambda$}}}{\hbox{\raise
3.54857pt\hbox to0.0pt{\kern 0.81665pt\vrule
width=2.85832pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptstyle\lambda$}}}{\hbox{\raise
2.5347pt\hbox to0.0pt{\kern 0.58331pt\vrule
width=2.04166pt,height=0.1pt,depth=0.1pt\hss}\hbox{$\scriptscriptstyle\lambda$}}}}_{e}}{w_{0,{\rm
L}}}\right)^{4}\sqrt{g(0)g(\tfrac{w_{0,{\rm X}}}{w_{0,{\rm
L}}})}\,\,\exp\left\\{-2\left(\frac{r_{0}}{w_{0,{\rm
X}}}\right)^{2}\left(1-\sqrt{\frac{g(\tfrac{w_{0,{\rm X}}}{w_{0,{\rm
L}}})}{g(0)}}\right)\right\\}\,,$ (24)

where $c_{1}$ and $c_{2}$ are the low-energy constants introduced in Eq. (9),
$W_{\rm L}$ is the laser pulse energy of the pump, $w_{0}$ denotes the beam
waists, $r_{0}$ is the transverse impact parameter, and we used the shorthand
notation

$\displaystyle g(b):=\frac{1}{(1+2b^{2})^{2}}\,F\left(\tfrac{4z_{{\rm R},{\rm
L}}\sqrt{1+2b^{2}}}{\sqrt{\tau_{\rm X}^{2}+\tau_{\rm
L}^{2}/2}},\tfrac{2(z_{0}-t_{0})}{\sqrt{\tau_{\rm X}^{2}+\tau_{\rm
L}^{2}/2}},\tfrac{\tau_{\rm X}}{\tau_{\rm L}}\right),\quad\text{with}$
$\displaystyle
F(\chi,\chi_{0},\rho):=\sqrt{\frac{1+2\rho^{2}}{3}}\,\chi^{2}\,{\rm
e}^{2(\chi^{2}-\chi_{0}^{2})}\int_{-\infty}^{\infty}{\rm d}K\,{\rm
e}^{-K^{2}}\,\biggl{|}\sum_{s=\pm 1}{\rm e}^{2s(\rho K-i\chi_{0})\chi}\,{\rm
erfc}\Bigl{(}s(\rho K-i\chi_{0})+\chi\Bigr{)}\biggr{|}^{2}\,.$ (25)

The entire dependence of Eq. (24) on the longitudinal beam parameters, namely
the Rayleigh length of the pump $z_{{\rm R,L}}$, the $1/{\rm e}^{2}$ pulse
durations $\tau_{\rm X}$, $\tau_{\rm L}$ 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 Eq. (25). Note that in the limit of
$w_{0,{\rm X}}\ll w_{0,{\rm L}}$ and $\\{\tau_{\rm X},\tau_{\rm L}\\}\gg
z_{\rm R,L}$ we have $g(b)\sim(z_{\rm R,L}/\tau_{\rm L})^{2}$ [100], and the
form of Eq. (19) with characteristic distance $z\sim z_{\rm R,L}$ 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 characterised by spatial fluctuations of the order of
the beam waist, i.e., $r_{0}\sim w_{0,{\rm L}}$. Another important signal
parameter is the $1/{\rm e}^{2}$ radial divergence,

$\theta_{{\rm signal}}=\theta_{\rm X}\,\biggl{(}\frac{g(\tfrac{w_{0,{\rm
X}}}{w_{0,{\rm L}}})}{g(0)}\biggr{)}^{-\frac{1}{4}}\,,$ (26)

where $\theta_{\rm X}=2/(\omega_{\rm X}w_{0,{\rm X}})$ is the radial
divergence of the probe beam, and thus also the divergence of the background
$N_{\perp,{\rm bgr}}\sim{\cal P}N_{\rm X}$ 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 fixed other parameters to be reached for small
probe waists, $w_{0,{\rm X}}\ll w_{0,{\rm L}}$. In this case, one obtains the
polarisation-flip counts

$\begin{tabular}[]{c||c|c}$f$/\\#&$f/1$&$f/2$\\\
\hline\cr$N_{\perp}$&$0.044$&$0.0045$\end{tabular}\;\;,$ (27)

for an XFEL energy of $\omega_{\rm X}=9835\,{\rm eV}$ and the standard self-
seeding parameters in Tab. 2 with $f/1$ ($f/2$) focusing. Self-seeding is the
option of choice for experiments aiming at the detection of polarisation-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
SASE bandwidth is typically much larger, cf. Tab. 2. In the determination of
$N_{\perp}$ we also took into account that the quasi-channel-cut crystal
acting as polariser increases the probe pulse duration from its original value
given in Tab. 2 to $\tau_{\rm FWHM}=50\,{\rm fs}$ [94]. Using ${\cal
P}=10^{-11}$ [71] and neglecting real-word imperfections (such as inevitable
losses coming with the lenses used for focusing), the values in Eq. (27) imply
the associated signal-to-background ratios to be given by

$\begin{tabular}[]{c||c|c}$f$/\\#&$f/1$&$f/2$\\\ \hline\cr$N_{\perp}/({\cal
P}N_{\rm X})$&$0.022$&$0.0023$\end{tabular}\;\;.$ (28)

These values are less than unity so the signal is background-dominated.
Moreover, in the limit considered, we have $\theta_{\rm
signal}\approx\theta_{\rm X}$, and the dependence of the signal on the impact
parameter $r_{0}$ simplifies to $N_{\perp}\sim\exp\\{-2(r_{0}/w_{0,{\rm
L}})^{2}\\}$.

On the other hand, for matching waists, $w_{0,{\rm X}}=w_{0,{\rm L}}$, the
signal is reduced by roughly a factor of $3$ to $N_{\perp}\simeq 0.017$
($0.0015$) for $f/1$ ($f/2$) focusing. At the same time the divergence of the
signal, $\theta_{\rm signal}\approx 1.6\,\theta_{\rm X}$, 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 ${\cal E}_{\rm X}(x)\,{\cal E}^{2}_{\rm L}(x)$ of the focus field
profiles of probe and pump, the Fourier transform of which governs the far-
field distribution of the signal. When $w_{0,{\rm X}}\gtrsim w_{0,{\rm
L}}/\sqrt{2}$, the transverse extent of the signal emission region is smaller
than the focal spot of the probe, and in turn $\theta_{\rm signal}>\theta_{\rm
X}$; the factor of $\sqrt{2}$ is due to the linear (quadratic) dependency on
the probe (pump) profile. Furthermore, as
$N_{\perp}\sim\exp\\{-1.2(r_{0}/w_{0,{\rm L}})^{2}\\}$ for $f/1$ focusing, the
dependence on $r_{0}$ – and thus the sensitivity to beam jitter in the
experiment – becomes milder. Clearly, when $\theta_{\rm sig}>\theta_{\rm X}$,
a measurement of the full signal requires collecting photons scattered outside
the forward cone of the probe on the detector. However, as $\theta_{\rm X}\sim
1/w_{0,{\rm X}}$, the most tightly focussed 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 emphasise
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 $\theta_{\rm sig}>\theta_{\rm X}$ suggests that
angular cuts can be used to improve the signal-to-background ratio in
experiment: Collecting only those photons scattered outside a minimum
diffraction angle $\vartheta_{\rm min}$, the signal-to-background ratio can be
enhanced to $N_{\perp}/({\cal P}N_{\rm X})|_{\vartheta>\vartheta_{\rm min}}>1$
at the expense of reduced absolute photon numbers. Following this route, it is
even possible to achieve $N_{\parallel}/N_{\rm X}|_{\vartheta>\vartheta_{\rm
min}}>1$, such that one can hope to also measure the $\parallel$-polarised
signal component [43, 102]. However, an experimental implementation of such a
diffraction-assisted measurement concept [16, 86] requires that any unwanted
background, e.g., stray light from the lenses used for focusing, can be
controlled and inhibited from reaching the detector, cf. [103] for a related
background study at optical frequencies. This would necessitate setting up a
dedicated filtering and imaging system along the lines outlined in Sec. 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 experiment each reflection at a diamond surface of the quasi-channel-cut
polariser and analyzer comes with a reduction of the number of photons by
about $2\,\%$ [94] (the setup in Fig. 8 employs $8$ reflections in total).
(ii) The transmission of lenses preserving the polarisation purity can be
estimated as $25\,\%$. (iii) The acceptance bandwidth of the polariser,
$\Delta\omega_{\rm diamond}^{400}=80\,{\rm meV}$, entails a reduction of its
throughput by $\Delta\omega_{\rm diamond}^{400}/\Delta\omega=80/300$ for self-
seeding. In consequence, both the signal and background photon numbers,
$N_{\perp}$ and ${\cal P}N_{\rm X}$, 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 Eq. (27) above.

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

$n>\\#^{2}\,\frac{1}{2}\Bigl{[}(N_{\rm sig}+N_{\rm
bgr})\ln\Bigl{(}1+\frac{N_{\rm sig}}{N_{\rm bgr}}\Bigl{)}-N_{\rm
sig}\Bigr{]}^{-1}=\\#^{2}\,\frac{N_{\rm bgr}}{N_{\rm sig}^{2}}\Bigl{[}1+{\cal
O}\Bigl{(}\frac{N_{\rm sig}}{N_{\rm bgr}}\Bigr{)}\Bigr{]}$ (29)

shots are required for a measurement with a significance of $\\#\,\sigma$
[104], where $N_{\rm sig}$ and $N_{\rm bgr}$ 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 $1\,{\rm Hz}$, this translates into the requirement of
more than $\\#^{2}\times 21\,{\rm hours}$ ($\\#^{2}\times 80\,{\rm days}$)
optimal shots with zero spatiotemporal offset.

###  3.2 Dark-field scenario

A prospective variation of the conventional two-beam scenario detailed in Sec.
3.1 envisions modifying the probe beam such 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 [105, 22]. In the experiment the shadow in the
probe beam is generated by a well-defined beamstop inserted into the incident
beam; see Fig. 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 [106, 107] 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
polarisation components of the nonlinear vacuum response. To this end, the
shadow in the probe beam is to be imaged onto a polarisation-sensitive
detector employing an advanced filtering and imaging system. This will consist
of appropriately designed and placed beamstops and apertures such 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

${\cal S}=\frac{1}{N_{\rm X}}\int_{A_{\rm det}}{\rm d}^{2}x\,\frac{{\rm
d}^{2}N_{\rm X}(x,y)}{{\rm d}x\,{\rm d}y}$ (30)

that measures the fraction of the total number of input probe photons $N_{\rm
X}$ registered by a detector of acceptance area $A_{\rm det}$ in the shadow in
the absence of a quantum vacuum signal, i.e., for a vanishing pump field. In a
theoretically idealised calculation using ideal fat-top far-field profiles and
neglecting diffraction at the obstacles and apertures put into the beam path,
we have ${\cal S}=0$. However, in any real experiment, we inevitably have
${\cal S}>0$, i.e., a finite background in the shadow, due to diffraction and
scattering off the optical elements (cf. Sec. 4.2 below). The discussion of
this set-up follows [22] and more detail can be found in this work.

Figure 9: Schematic layout of the dark-field scenario. The XFEL is focussed
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 ${\cal E}_{\rm X}(x){\cal E}^{2}_{\rm
L}(x)$ is sufficiently similar to that in the conventional scenario in Fig. 8,
a scattering signal is induced in the shadow. A crystal polariser directs its
different polarisation components to separate detectors.

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 ${\cal E}_{\rm X}(x)\,{\cal
E}^{2}_{\rm L}(x)$ effectively matches the focus profile of a beam without
shadow [108], one can induce a signal component scattered into the shadow
located in the expanding probe beam (cf. Sec. 3.1 above).

Note that this dependency immediately implies a signal in the dark field for
$w_{0,{\rm X}}\gtrsim w_{0,{\rm L}}/\sqrt{2}$. 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
${\cal E}_{\rm X}(x)\,{\cal E}^{2}_{\rm L}(x)$ is characterised 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 $w_{0,{\rm L}}$, because the far-
field divergence scales inversely with the source size. On the other hand, it
is clear that for $w_{0,{\rm X}}\ll w_{0,{\rm L}}$ 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 ${\cal E}_{\rm X}(x)\,{\cal
E}^{2}_{\rm L}(x)$ on the transverse coordinate $r$ is governed by the $r$
dependence of ${\cal E}_{\rm X}(x)$. 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 maximise
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 [105, 108].) To be specific, we
limit ourselves to $w_{0,{\rm X}}\gtrsim w_{0,{\rm L}}/\sqrt{2}$ 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 [108]. In
addition, we assume a relative polarisation of $\pi/4$ between the colliding
beams and a detector of radial opening angle $\theta_{\rm det}$ placed
centrally in the dark zone of the outgoing far field. The number
$N_{\perp}^{\bullet}$ of $\perp$ polarised signal photons registered (per
shot) by this detector is then well-approximated by [108]

$N_{\perp}^{\bullet}=N_{\perp}\,(1-\nu)^{2}\,\frac{1-{\rm
e}^{-1}}{1+\nu}\left(1-{\rm e}^{-2\bigl{(}\theta_{\rm det}/\theta_{\rm
signal}\bigr{)}^{2}}\right)\,,$ (31)

with $N_{\perp}$ and $\theta_{\rm signal}\leq\theta_{\rm in}$ as given in Eqs.
(24) and (26). For the same scenario, the complementary signal is somewhat
larger, namely

$N_{\parallel}^{\bullet}=\biggl{(}\frac{c_{1}+c_{2}}{c_{1}-c_{2}}\biggr{)}^{2}\,N_{\perp}^{\bullet}\simeq
13.4\,N_{\perp}^{\bullet}\,.$ (32)

Here, $\nu=(\theta_{\rm in}/\theta_{\rm out})^{2}$ measures the fraction of
the transverse area of the incident XFEL beam that is blocked by the beamstop.
This blocking fraction can be parameterised in terms of the inner and outer
radial divergences, $\theta_{\rm in}$ and $\theta_{\rm out}$, of the ‘hollow’
probe beam featuring the central shadow [22],

$\theta_{\rm out}=\frac{2}{\omega_{\rm X}w_{0,{\rm X}}}\sqrt{\frac{2(1-{\rm
e}^{-1})}{1+\nu}}\quad\text{and}\quad\theta_{\rm in}=\sqrt{\nu}\,\theta_{\rm
out}\,.$ (33)

Equation (31) follows upon integrating the differential number of signal
photons derived in Refs. [100, 101] over solid angle. In addition, one has to
account for the reduction of the peak field of the probe [105] to match the
central focus peak of a flat-top beam with a perfect central shadow in its far
field.

From Eq. (31) we infer that the choice $\nu=\sqrt{5}-2\simeq 0.24$ maximises
the signal in the shadow for optimal collisions. We also choose $w_{0,{\rm
X}}=w_{0,{\rm L}}/\sqrt{2}\simeq 0.7\,w_{0,{\rm L}}$ (which should be close to
the optimal waist spot ratio [22]) and an XFEL energy of $\omega_{\rm
X}=8766\,{\rm eV}$ compatible with the use of a Germanium crystal to separate
the polarisation components of the signal, cf. also Sec. 4.2.4 below for the
details. Adopting $\theta_{\rm det}=\theta_{\rm in}$ to collect the maximum
number of signal photons scattered into the dark-field on the detector (for
the parameters in Tab. 2 attainable with the self-seeding option), the second
identity in Eq. (33) then results in $\theta_{\rm in}\simeq 28.3\,\upmu{\rm
rad}/\\#$, for $f/\\#$ focusing. For these parameters, Eq. (31) predicts a
signal yield which can be tabulated as

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

We also note that for a detector with half opening angle, i.e., for
$\theta_{\rm det}=\theta_{\rm in}/2$, these photon numbers are reduced by a
factor of $\approx 0.3$. As in Sec. 3.1 we can also assess the dependence of
the signal photon number on the impact parameter using Eq. (31). For the
present parameters and $f/1$ focusing we infer
$N_{\perp}^{\bullet}\sim\exp\\{-1.8(r_{0}/w_{0,{\rm L}})^{2}\\}$. 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
$r_{0}\gtrsim w_{0,{\rm X}}$ and, in turn, is expected to somewhat
overestimate the drop of the signal with $r_{0}$.

In the dark-field scenario, the polarisation 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 $10^{-6}$ [109]. Therefore, the analyser, which separates the
differently polarised 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
polarisation purity of the order of $10^{-3}$ can be readily achieved.
Employing _several_ such reflections, the polariser can be further enhanced,
thus achieving a purity on par with the polarisation purity of the XFEL. In
consequence, the losses associated with the setup in Fig. 9 are dominated by
the lenses and not the analyser. Due to the relaxed polarisation 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 polariser in Fig. 9 to
achieve a polarisation purity of $10^{-6}$, the associated background in the
dark field then consists of $N_{\rm bgr}={\cal S}N_{\rm X}\times 0.25\times
10^{-6}$ (${\cal S}N_{\rm X}\times 0.25$) XFEL photons for the $\perp$
($\parallel$) polarised mode. Identifying $N_{\rm sig}=N_{\perp}^{\bullet}$,
and given that $N_{\rm sig}\ll N_{\rm bgr}$, Eq. (29) then implies the need of
$n>\\#^{2}\times{\cal S}\times 3.2\times 10^{11}$ ($\\#^{2}\times{\cal
S}\times 5.5\times 10^{16}$) optimal shots to achieve a $\\#\,\sigma$
measurement of the $\perp$ polarised component of the nonlinear vacuum
response with $f/1$ ($f/2$) focusing. For $N_{\rm
sig}=N^{\bullet}_{\parallel}$, the analogous values for the $\parallel$
polarised component are $n>\\#^{2}\times{\cal S}\times 1.8\times 10^{15}$
($\\#^{2}\times{\cal S}\times 3.0\times 10^{17}$).

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

Finally, we note that the measurement of both polarisation components of the
nonlinear vacuum response allows to determine the low-energy constants $c_{1}$
and $c_{2}$ in Eq. (6) individually, and not just their difference as in the
conventional scenario. Moreover, we emphasise that a simultaneous measurement
of both components gives direct access to the ratio

$\frac{N_{\perp}}{N_{\parallel}}=\biggl{(}\frac{c_{1}-c_{2}}{c_{1}+c_{2}}\biggr{)}^{2}=\frac{9}{121}\biggl{(}1+\frac{260}{99}\frac{\alpha}{\pi}+{\cal
O}\bigl{(}\alpha^{2}\bigr{)}\biggr{)}\,,$ (35)

where we have also included higher-order (two-loop) contributions [110, 111].
Importantly, the ratio (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 [22].

###  3.3 Planar three-beam set-up

From a scattering point of view, vacuum birefringence is explained by
polarisation flips without momentum transfer and hence corresponds to a 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 [89, 82]. 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, $F_{\textsf{x}}$, two optical
beams, $F_{1}$ and $F_{2}$, and the scattered photon
$F_{\textsf{x}^{\prime}}$, such that
$F=F_{\textsf{x}}+F_{1}+F_{2}+F_{\textsf{x}^{\prime}}$. Then, for the low-
energy HE Lagrangian given in Eq. (6), the scattering matrix element,
$\textsf{S}=-i\int d^{4}x\,\mathcal{L}_{\textrm{int}}$ contains many channels:

$\displaystyle\textsf{S}=\underbrace{\textsf{S}_{\textsf{x}11\textsf{x}^{\prime}}+\textsf{S}_{\textsf{x}22\textsf{x}^{\prime}}}_{\textrm{x-ray}+\textrm{one
optical
beam}}+\underbrace{\textsf{S}_{\textsf{x}12\textsf{x}^{\prime}}}_{\textrm{x-ray}+\textrm{two
optical
beams}}+\cdots+\underbrace{\textsf{S}_{\textsf{x}\textsf{x}1\textsf{x}^{\prime}}}_{\textrm{x-ray
merging}}+\cdots+\underbrace{\textsf{S}_{\textsf{x}1\textsf{x}^{\prime}\textsf{x}^{\prime}}}_{\textrm{x-ray
splitting}}+\cdots\;,$ (36)

where the subscripts on each of the amplitudes count the number of fields
included (e.g. $\textsf{S}_{\textsf{x}11\textsf{x}^{\prime}}$ 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’ spacetime dependency from their tensor structure, and writing
kinematic factors $\Gamma(q)$ defined in terms of the momentum transfer $q$.
For example,
$\textsf{S}_{\textsf{x}11\textsf{x}^{\prime}}=\mathfrak{S}\Gamma_{\textsf{x}11\textsf{x}^{\prime}}(q)$
where $\mathfrak{S}$ is a geometrical factor depending on the polarisation of
the photons and $\Gamma_{\textsf{x}11\textsf{x}^{\prime}}(q)=\int
F_{1}^{2}(x)\exp(-iq\cdot x)d^{4}x$. We note this has a form analogous to a
far-field diffraction integral, where the x-ray photon ‘diffracts’ on regions
of the vacuum polarised by the intense optical beams [15, 85]. These kinematic
factors can be used to demonstrate the channels in the three-beam scenario;
here it will suffice to consider monochromatic beams, e.g.
$F_{j}(x)=F_{j}\cos(k_{j}\cdot x)$ for $j\in\\{1,2\\}$. Then we see that

$\displaystyle\Gamma_{\textsf{x}11\textsf{x}^{\prime}}(q)\propto\underbrace{\delta\left(2k_{1}-q\right)}_{\textrm{frequency
upshift}}+\underbrace{\delta\left(2k_{1}+q\right)}_{\textrm{frequency
downshift}}+\underbrace{2\delta\left(q\right)}_{\textrm{elastic}}$ (37)

i.e., the ‘x-ray+one optical beam’ channel in principle includes frequency
shifting parts. However, since the scattered photons with momentum
$\ell^{\prime}$ must obey the vacuum dispersion relation
$\ell^{\prime}\cdot\ell^{\prime}=0$, the frequency shifting channels are
effectively inaccessible [89]. Compare this to the ‘x-ray+two optical beams’
channel, for which the corresponding kinematic factor is

$\displaystyle\Gamma_{\textsf{x}12\textsf{x}^{\prime}}(q)\propto\underbrace{\delta\left(k_{1}+k_{2}-q\right)}_{\textrm{frequency
upshift}}+\underbrace{\delta\left(k_{1}+k_{2}+q\right)}_{\textrm{frequency
downshift}}+\underbrace{\delta\left(k_{1}-k_{2}+q\right)}_{\textrm{quasi-
elastic}}+\underbrace{\delta\left(-k_{1}+k_{2}+q\right)}_{\textrm{quasi-
elastic}}.$ (38)

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
$\ell^{\prime}\cdot\ell^{\prime}=0.$ 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 [42, 13, 14, 88, 89, 90, 82, 91].
However, for the x-ray + two optical photon 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 $\Delta$, i.e., for non-monochromatic beams, leading to a momentum
transfer in the scattering process of $q=\pm(k_{1}-k_{2})+\Delta$. 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 Fig. 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 [86, 112].

Here we focus on a planar collision geometry for the three beams, depicted in
Fig. 10.

Figure 10: The planar three-beam configuration of a focussed XFEL beam
colliding in a plane with two focussed optical beams at a collision angle
$\psi$.

Using the kinematic factors in Eq. (37) and Eq. (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,

$\displaystyle\textsf{S}\approx\underbrace{\textsf{S}_{\textsf{x}11\textsf{x}^{\prime}}+\textsf{S}_{\textsf{x}22\textsf{x}^{\prime}}}_{\textrm{central
peak}}+\underbrace{\textsf{S}_{\textsf{x}12\textsf{x}^{\prime}}}_{\textrm{side
peaks}}\;.$ (39)

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 Eq.
(38), at a scattering angle approximately twice the ratio of the transverse
optical photon momentum to the x-ray momentum:
$\theta_{x}=\pm\arctan(2\omega_{\textsf{optical}}\sin\psi/\omega_{\textsf{x-ray}})$,
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
$\textsf{N}=V\int\frac{d^{3}\ell^{\prime}}{(2\pi)^{3}}|\textsf{S}|^{2}$, where
$\ell^{\prime}$ is the scattered photon momentum, and only the terms in Eq.
(39) are included in S. The beams are modelled using the leading order
paraxial Gaussian solution from Eq. (23), and parameters for the optical and
EuXFEL beams taken from Tab. (2). An example of the distribution of scattered
photons is given in Fig. 11, which assumes perfect alignment of the three
beams, where the grid lines indicate the predicted central positions of the
side peaks.

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.

Fig. 11 demonstrates the key utility of the three-beam set-up: 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 Fig. 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 12: The number of scattered photons, parallel ($N_{\parallel}$) and
perpendicular ($N_{\perp}$) to the XFEL probe, as a function of 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
($N^{>200\upmu\textrm{rad}}_{\parallel,\perp}$), refer to the signal falling
on the detector outside a central exclusion region of radius
$200\,\upmu\textrm{rad}$. The SASE and self-seeded options are taken from Tab.
(2). Left: Example results for total scattered photons. Right: Photon
scattering and birefringence (polarisation 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 Sec.
4.2.1) we determine that a $200\,\upmu\textrm{m}$ radius for the central
exclusion region on the detector would be sufficient. The entailed reduction
in signal photon count is demonstrated in Fig. 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, e.g. the $6\,\textrm{keV}$ mode. On the other hand, the cross-
section, $\sigma$, has a strong scaling with centre-of-mass energy,
$\omega_{\ast}$ as $\sigma\propto\omega^{6}_{\ast}$, and so to maximise the
total number of photons scattered, we would use the higher energy XFEL photons
e.g. the $12.9\,\textrm{keV}$ mode. From our analyses using the HED-HIBEF
parameters, we find that the $9.8\,\textrm{keV}$ 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 bandwidth narrow enough such that scattered photons are
accepted by the analyser.

Crossed-beam set-up | Focussing | $\omega_{\textsf{x-ray}}$ | $N_{\parallel}$ | $N_{\perp}$ | $N_{\textsf{total}}$
---|---|---|---|---|---
‘Momentum kick’ | $f/2$ | 12.9 keV | - | - | $42\times 10^{-3}$
‘Momentum kick’ | $f/1$ | 12.9 keV | - | - | $180\times 10^{-3}$
‘Momentum kick’+200$\upmu$m hole | $f/2$ | 12.9 keV | - | - | $1.5\times 10^{-3}$
‘Momentum kick’+200$\upmu$m hole | $f/1$ | 12.9 keV | - | - | $3.7\times 10^{-3}$
Crossed beam + biref. | $f/1$ | $9.8\,\textrm{keV}$ | $49\times 10^{-3}$ | $3.5\times 10^{-3}$ | -
Crossed beam + biref.+200$\upmu$m hole | $f/1$ | $9.8\,\textrm{keV}$ | $1.3\times 10^{-3}$ | $0.1\times 10^{-3}$ | -
Table 3: Example number of signal photons scattered in collision of XFEL probe
with two optical beams.

The planar three-beam scenario is a set-up of interest for BIREF@HIBEF. As the
dark-field scenario has been selected for priority access beam time 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 set-up.

###  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 summarised 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.

Pump Field
---
Magnetic Field | Laser Focus | Nuclear Coulomb Field
$\mathcal{O}(10^{-9}B_{\mathrm{S}})$ | $\mathcal{O}(10^{-4}E_{\mathrm{S}})$ | $\mathcal{O}(E_{\mathrm{S}})$
$\delta n=\mathcal{O}(10^{-22})$ | $\delta n=\mathcal{O}(10^{-11})$ | $\delta n=\mathcal{O}(10^{-2})$
Field Strength $\rightarrow$ $\leftarrow$ Interaction Volume
Probe Field
Optical Laser | XFEL |  $\gamma$-Ray
$\mathcal{O}(\mathrm{eV})$ | $\mathcal{O}(\mathrm{keV})$ | $\mathcal{O}(\mathrm{MeV})$
$N=\mathcal{O}(10^{20})$ | $N=\mathcal{O}(10^{11})$ | $N=\mathcal{O}(1)$
Wave number $\rightarrow$ $\leftarrow$ Photon Number
PVLAS, BMV, … | HED-HIBEF | Delbrück
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.

Table 4 presents three distinct choices each for 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 Tesla. 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 $\mathcal{O}({\rm eV})$. The table in Table 4
illustrates various experiments that have utilised different combinations of
pump and probe fields. For instance, as detailed in Sec. 2, experiments like
PVLAS [77] and BMV [113] 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
focussed array of several optical lasers for the pump field and x-rays for the
probe. Additionally, for the measurement of Delbrück scattering [4, 40] have
utilised the electric field of a nucleus as pump field and high-energy gamma
photons as the probe; for more details see [114].

Figure 13: Feynman diagram for Coulomb-assisted birefringence

In this section, we utilise the combination of a nuclear Coulomb field,
denoted as ${\bf E}^{\rm ext}$, and the strong magnetic field ${\bf B}^{\rm
ext}$ provided by a focussed high-intensity laser pulse as pump field. This is
coupled with an XFEL beam, which serves as the probe field, see [115, 114]. An
illustrative Feynman diagram for this process resembles the one shown in Fig.
3 (a), with the key distinction being that one of the Coulomb fields is
replaced by an interaction with an external magnetic field ${\bf B}^{\rm
ext}$, see Fig 13. This scenario may offer several advantages: the nuclear
Coulomb field adds a significant momentum transfer $\Delta{\bf k}$ to the flip
of the XFEL polarisation [115, 114]. Compared to the past experiments
measuring Delbrück scattering at photon energies $O(100\,\textrm{MeV})$ [4,
40, 5], 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 (6) with the
low-energy constants $c_{1}$ and $c_{2}$ given in (9). In the next step, we
decompose the electric and magnetic fields in Eqs. (6) and (8) as ${\bf
E}\to{\bf E}+{\bf E}^{\rm ext}$ and ${\bf B}\to{\bf B}+{\bf B}^{\rm ext}$ and
assume that the magnetic field ${\bf B}^{\rm ext}$ is approximately constant.
In addition to the static Coulomb field ${\bf E}^{\rm ext}$ of the nucleus, we
have space-time dependent XFEL fields ${\bf E}$ and ${\bf B}$. After inserting
the field decomposition into Eq. (6) we obtain the effective Lagrangian for
the XFEL field

$\displaystyle\mathcal{L}_{\rm XFEL}=\frac{1}{2}\Big{[}{\bf
E}\cdot\epsilon^{\rm ext}\cdot{\bf E}+{\bf B}\cdot(\mu^{-1})^{\rm
ext}\cdot{\bf B}+{\bf E}\cdot\Psi^{\rm ext}\cdot{\bf B}\Big{]}\,,$ (40)

where we have introduced the notation ${\bf E}\cdot\epsilon^{\rm ext}\cdot{\bf
E}\equiv E_{i}\epsilon_{ij}^{\rm ext}E_{j}$ etc. for the quadratic forms in
the Lagrangian $\eqref{xfel-eff}$. Its Hessian defines the dynamical vacuum
response functions [58], i.e. the permittivity tensor, $\epsilon_{ij}^{\rm
ext}$, the permeability tensor, $(\mu^{-1}_{ij})^{\rm ext}$, and the magneto-
electric coupling tensor, $\Psi_{ij}^{\rm ext}$, all of which are functions of
the pump fields, ${\bf E}_{\rm ext}$ and ${\bf B}_{\rm ext}$,

$\displaystyle\epsilon_{ij}^{\rm ext}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}}{45m^{4}}\Big{[}8E^{\rm ext}_{i}E^{\rm
ext}_{j}+14B^{\rm ext}_{i}B^{\rm ext}_{j}+\delta_{ij}(({\bf E}^{\rm
ext})^{2}-({\bf B}^{\rm ext})^{2})\Big{]}\,,$
$\displaystyle(\mu_{ij}^{-1})^{\rm ext}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}}{45m^{4}}\Big{[}8B^{\rm ext}_{i}B^{\rm
ext}_{j}+14E^{\rm ext}_{i}E^{\rm ext}_{j}+\delta_{ij}(({\bf B}^{\rm
ext})^{2}-({\bf E}^{\rm ext})^{2})\Big{]}\,,$ $\displaystyle\Psi_{ij}^{\rm
ext}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}}{45m^{4}}\Big{[}-8E_{i}^{\rm ext}B_{j}^{\rm
ext}+14B^{\rm ext}_{i}E^{\rm ext}_{j}+14\delta_{ij}({\bf E}^{\rm ext}\cdot{\bf
B}^{\rm ext})\Big{]}\,.$ (41)

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

$\displaystyle\Box{\bf D}=\nabla\times[\nabla\times{\bf
D}]+\partial_{t}[\nabla\times{\bf H}]={\bf J}^{\rm eff}\,.$ (42)

The resulting wave equation can be analysed using standard Green’s function
methods (see e.g. [18]). The source term ${\bf J}^{\rm eff}$, 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
${\bf D}$ into an incoming plane wave ${\bf D}^{\rm in}$ and a small
scattering component ${\bf D}^{\rm out}$. Assuming a stationary time-
dependence of ${\rm e}^{-i\omega t}$ for the XFEL, we arrive at a Helmholtz
equation [115]. This equation can be solved using Green function methods.
Focusing on the long-distance behaviour, we find the following scattering
amplitude [116],

$\mathfrak{U}=\frac{1}{4\pi\mid\bf{D}^{\rm in}_{\omega}\mid}\,{\bf e}_{\rm
out}\cdot\int d^{3}r\,\exp\\{-i\,{\bf k}_{\rm out}\cdot{\bf r}\\}\,{\bf
J}^{\rm eff}_{\omega}$ (43)

where ${\bf k}_{\rm out}$ is the wave-number and ${\bf e}_{\rm out}$ the
polarisation unit vector of the scattered XFEL radiation. The scattering
amplitude straightforwardly yields the differential cross section,
$d\sigma/d\Omega=\mid\mathfrak{U}\mid^{2}$. For Coulomb assisted Delbrück
scattering, cf. Fig. 13, the amplitude can be expressed as

$\displaystyle\mathfrak{U}$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}}{4\pi}\int d^{3}r\,{\rm e}^{i\Delta{\bf
k}\cdot{\bf r}}\big{[}{\bf e}_{\rm out}\cdot\epsilon^{\rm ext}\cdot{\bf
e}_{\rm in}+{\bf e}_{\rm out}\cdot\Psi^{\rm ext}\cdot({\bf n}_{\rm
in}\times{\bf e}_{\rm in})+{\bf e}_{\rm in}\cdot\Psi^{\rm ext}\cdot({\bf
n}_{\rm out}\times{\bf e}_{\rm out})$ (44) $\displaystyle\hskip
227.62204pt\\!+\,({\bf n}_{\rm out}\times{\bf e}_{\rm
out})\cdot(\mu^{-1})^{\rm ext}\cdot({\bf n}_{\rm in}\times{\bf e}_{\rm
in})\big{]}\,.$

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

Note that the spatial integration is effectively cut off by the momentum
transfer $\Delta{\bf k}$, implying a large interaction volume extending over
many XFEL wavelengths for small scattering angles (near-forward direction),
$\mid{\bf n}_{\rm in}-{\bf n}_{\rm out}\mid\ll 1$. 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 now longer be justified; see also [117].

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

$\mathfrak{U}_{\perp}^{\Psi^{\rm ext}}=\pm
i\,\frac{12\alpha^{2}}{45m^{4}}\,\frac{Ze}{(\Delta{\bf
k})^{2}}\frac{\omega^{2}}{4\pi}\left[({\bf e}_{\rm out}\cdot{\bf B}^{\rm
ext})({\bf e}_{\rm out}\cdot\Delta{\bf k})-({\bf e}_{\rm in}\cdot{\bf B}^{\rm
ext})({\bf e}_{\rm in}\cdot\Delta{\bf k})\right]\,.$ (45)

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

$\frac{d\sigma_{\perp}^{\Psi^{\rm
ext}}}{d\Omega}=\mid\mathfrak{U}_{\perp}^{\Psi^{\rm ext}}\mid^{2}\approx
10^{-25}\frac{Z^{2}}{(\Delta{\bf k})^{2}}\,.$ (46)

The magnitude of the signal Eq. (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: first, the large number,
$\mathcal{O}(10^{12})$, of polarised XFEL photons; second, the large number
$N$ of nuclei. In one possible scenario, a cubic carbon cluster with an edge
length of $100\,{\rm nm}$ is ionised completely using the pre-pulse of a high-
intensity laser of order $\mathcal{O}(10^{22}\,{\rm W/cm^{2}})$. For small
momentum transfer (of order $\rm eV$), 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 $Z_{\rm eff}=NZ$. The
birefringent signal would be of the order of $\mathcal{O}(10^{-5})$ photons
per shot and thus could be detected in complete analogy with the scenarios
discussed above, see also [78, 66, 43, 44, 83]. For a more detailed
discussion, we refer to the recent publications [115, 118], 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 polarisation 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 ionisation by the
polarised optical laser pulse. The x-ray dichroism of ionised krypton gas was
experimentally demonstrated in [119], [120], [121] and theoretically studied
in [122], [123] for x-rays resonant to transition between the 1s core atomic
orbital and 4p3/2 valence atomic orbital where an aligned electron hole was
produced. From these studies, the probability of polarisation flipping can be
estimated as
$\sim[\sin(2\theta)n\lambda^{2}L\Delta\rho_{\text{aniso}}]^{2}A^{2}/(\Delta\omega^{2}+\gamma^{2}/4)$,
here $\theta$ is an angle between the polarisations of the optical and x-ray
pulses; $n$ and $L$ are the concentration and the length of the gas sample;
$\lambda$ is the x-ray wavelength, $\Delta\rho_{\text{aniso}}$ 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 for the corresponding valence-
to-core transition. For the case of strong-field ionised ${\rm Kr}^{+}{\rm
4p}^{-1}$, the alignment of the valence hole $\Delta\rho_{\text{aniso}}$ can
reach the value of $64\%$, and for an atomic concentration 10${}^{18}\,{\rm
cm}^{-3}$, an interaction length of a few mm and an x-ray photon energy
resonant to valence-to-core transition, the polarisation 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 needed to suppress the atomic birefringence
background. From the 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 Plans for experimental implementation

Based on the expected signal and noise estimates presented in this paper, as
well as on the experimental feasibility, the dark-field approach was selected
to be pursued via the HIBEF user consortium priority access at the European
XFEL. The first x-ray-only beamtime was allocated for March 2024. It was
devoted to carrying out a proof-of-principle experiment of the dark-field
concept at XFEL. The outcomes of this campaign are currently being analysed
and will determine the setup to be implemented for the actual discovery
experiment.

###  4.1 Conventional scenario

The decisive bottle necks 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 on 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 polarisation purity. More detailed analyses, together
with the fact that only $300\,{\rm TW}$ are available at the European XFEL,
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 polarisation purity.

At the University of Jena and the Helmholtz Institute Jena, a research and
development program to eliminate these two bottle necks has been 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 (CC). The wavelength is chosen such that the x-rays are Bragg-reflected
at the (inner) walls of the channel at a Bragg angle of $45^{\circ}$, the
Brewster angle for x-rays. At the Brewster angle, the polarisation 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 quasi channel-cut (QCC).

As early as 2010, an extinction ratio of $1.5\times 10^{-9}$ at $6.457\,{\rm
keV}$ photon energy and of $9.0\times 10^{-9}$ at $12.914\,{\rm keV}$ was
demonstrated. The (400) and (800) Bragg reflexes of Si were used [124].
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 polarisation purity to $2.4\times 10^{-10}$ and $5.7\times
10^{-10}$ for the same x-ray wavelengths as above [84]. 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 polarisation purities several (!) orders of magnitude
better than the previous state-of-the-art were quickly exploited (e.g. [84,
125, 126, 20]). Prominent applications were the detection of quantum optics
and QED effects in the x-ray range [127, 128, 129].

In order to achieve further improvements in polarisation purity, the limiting
factors had to be identified [130, 131] 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 beam
time.

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 [132]. Nevertheless, a polarisation purity of $8.9\times 10^{-10}$
at $9.837\,{\rm keV}$ ((400) Bragg reflection) was obtained, limited by the
divergence of the synchrotron radiation. With four crystals for each, the
polarizer and the analyser, and some control on the beam divergence, another
record polarisation purity of $1.4\times 10^{-10}$ was achieved [133].

The first experiment at a free-electron x-ray laser was performed at the High
Energy Density (HED) instrument of the European XFEL. The silicon 400
reflection and 6 reflections in the CC were used. A polarisation purity of
$8\times 10^{-11}$ was achieved; see Fig. 14 for an illustration. It should be
emphasised that this number is not determined by the quality of the polarizers
but 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 polarisation [71]. It should also be emphasised that
the limited photon flux was due to limited beam time, the XFEL running in SASE
mode, and other factors, i.e. they do not constitute principal road blocks.
This is highlighted by the fact that the present record polarization purity of
$(1.4\pm 0.9)\times 10^{-11}$ was achieved at a 3-rd generation synchrotron
facility [20]. The theoretical limit for the polarisation purity as determined
by the finite laser beam divergence is on the order of $10^{-14}$! It has yet
to be shown, whether this limit can be reached or whether other effects kick
in.

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
polarising the vacuum. Right: Extinction curve around $0.01^{\circ}$ of the
crossed-polarizer position. At a few data points, the corresponding detector
signal is displayed: In crossed-polarizer position not a single photon reaches
the detector. To the level tested in this experiment, the polarizers are
perfect.

Another bottle neck are 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 (CRL). 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 $500\,\upmu{\rm m}$ of
Beryllium degrade the polarisation purity to the level of $10^{-6}$ [134]!

One alternative is the use of amorphous lens materials. Two options were
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 $90^{\circ}$ 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 [134]. For glassy carbon, the
situation was exactly the opposite. Meanwhile, better components have become
available and tests will be carried out soon.

Still another alternative, on first glance maybe paradoxically, are 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 Sec. 3.2 and schematically shown in Fig. 9 was adapted
to fit the experimental conditions of the HED instrument of the European XFEL
for the proof-of-principle experiment, as can be seen in Fig. 15. The
experiment is designed to be used with XFEL photon energy of $8766\,{\rm eV}$,
so that a 440 plane of a Germanium crystal can be used as a polarisation
analyser. The XFEL beam is coming from the left side of the picture, entering
the IC1 experimental chamber about $-1.2\,{\rm m}$ upstream from the
interaction point. The whole setup is in vacuum chambers up till the kapton
window located before detectors, at about $5\,{\rm m}$ downstream from TCC. At
the entrance point ($-1.1\,{\rm m}$), obstacle O1 is located, which is made of
a vertical wire-like structure with a diameter of $125-150\,\upmu{\rm m}$ to
block the central part of the beam. The focusing lens (L1) is located
$47\,\textrm{cm}$ upstream from focus, and consists of 12 Beryllium lenses
with a $50\,\upmu{\rm m}$ central radius of curvature. The Beryllium lenses
have a $400\,\upmu{\rm m}$ 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 $50\,\upmu{\rm m}$
diameter is placed exactly in the focus, which prevents radiation scattered on
the first lens from propagating further. The image of O1 projected by the lens
L1 is at the position $85\,{\rm cm}$ downstream from the focus, where the
aperture A1 is located. The scattering on the edges of these slits is a
critical factor for the success of the setup, therefore several possibilities
for how to limit the beam are being pursued, as described further. The second
lens stack (L2), consisting of 6 lenses of the same type as lens 1, is located
$114\,{\rm cm}$ 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, i.e. about $4\,{\rm m}$ 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 test or can be used for measurement
of the polarisation distinction. Alternatively, the analyser setup hosting 2
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 are located along the beam
propagation axis. For the ultimate discovery experiment the ReLaX optical beam
will counter-propagate with the XFEL beam, focussed down by an $f/1$ off-axis
parabolic mirror (OAP).In this experiment the ReLaX beam is expected to
deliver $300\,{\rm TW}$ in $25\,{\rm fs}$, and be focussed down to a focal
spot FWHM diameter $1.3\,\upmu{\rm m}$. This focusing scheme, beam routing,
and corresponding diagnostics are yet to be developed.

Figure 15: Experimental setup for the dark-field proof-of-principle experiment
at the HED instrument of the European XFEL. For completeness, here also the
RelaX beam path for the counter-propagating geometry is indicated.

There are two key parameters to evaluate the quality of the Dark-Field setup.
The shadow factor $\cal S$ was already defined in the theory, Sec. 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 minimise the $\cal S$ 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 the vacuum birefringence scattering can be detected. Distinguishing this
area, which might be smaller than $10\,\upmu{\rm m}$ from the rest 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 $\cal S$ represents the noise. The
standard way of optimisation would be to maximise the signal-over-noise ratio,
represented by the fraction $T/{\cal S}$. 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
$T^{6}/{\cal S}$ was identified as an optimal quantity to be maximised, or
less exactly, minimising the $\cal S$ factor while keeping the $T$ at a
reasonable $10\%$ level.

####  4.2.1 Diffractive simulations

In order to design and optimise the x-ray beam path and all its elements, the
whole setup was simulated by a complex simulation using the LightPipes
framework [135], which is designed to propagate a coherent beam where
diffraction is essential. Most of the simulations presented here have a
$700\,\upmu{\rm m}$ box size, resolution of $47\,{\rm nm}$ and therefore
$15000\times 15000$ simulated points. This was tested to be sufficient to
avoid numerical imprecision for given cases. All obstacles and apertures are
modelled as 2D maps of thickness, i.e. their exact 3D shape is neglected. The
thickness is converted into transmission and phase shifts by using the online
Henke tables [136]. The full result of a typical simulation is shown and
described in Fig. 16

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, (d) 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
$\upmu{\rm m}$ and the colour scale is logarithmic over 3 orders of magnitude.

An example of parameter optimisation is shown in Fig. 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 $\upmu{\rm m}$. It
can be seen that optimal obstacles are rather large with a diameter of around
$140\,\upmu{\rm m}$. 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 VB signal photon. That
means that a simple optimisation of the signal-to-noise ratio would lead to
prohibitively small signals. Therefore in further considerations, an approach
to minimise the noise while keeping the signal on the $10\%$ level was used.

Figure 17: Example of optimisation of the experimental parameters. The
horizontal axis is the shadow factor, 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. As the diffraction and
scattering on the edges of the wires are critical, their surfaces shall be
polished to high surface quality, either by FIB or electrochemical etching.
Commercially available diameters of $200\,\upmu{\rm m}$ and $175\,\upmu{\rm
m}$ 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 minimise
the diffraction. This so-called _trumpet_ shape 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 in the
order of a single $\upmu{\rm m}$, 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,
therefore such photons do not propagate further downstream and do not
contribute to the noise on the detector.

A variant of such object is depicted in Fig. 18. The left 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,
i.e. 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 a microfabricated obstacles. The left pane shows their
shape (thickness as a function of position perpendicular to beam), 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 then to
scatter it.

####  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
$\cal S$ and $T$ factor in the top row, and the derived quantities ${\cal
S}/T^{2}$ and ${\cal S}/T^{6}$ in the second row, for the case that both O1
and O2 are tungsten wires with respective diameters of $200\,\upmu{\rm m}$ and
$175\,\upmu{\rm m}$. The best performance, i.e. the minimum value, of ${\cal
S}/T^{6}=2\times 10^{-4}$ is achieved for openings of $150\,\upmu{\rm m}$ and
$80\,\upmu{\rm m}$, respectively. Different size of A1 increases this factor
significantly, while A2 does not have that high influence, showing that the
majority of signal reduction is done on the O1 – A1 pair, while the 02 – A2
pair plays rather a minor role.

This effect is seen even stronger in simulations with optimised O1 – A1
components, as seen in Fig. 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 $\cal S$ by a factor of
$\approx 30$, and the value for ${\cal S}/T^{6}$ by a factor of $\approx 20$.
What is interesting is to see that this value is for A2 size of
$300\,\upmu{\rm m}$, 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 19: Diffraction simulation of various openings of apertures A1 and A2
slits while using the wires as obstacles. The first two figures show the
simulated shadow and transmission factors, while the bottom figures show
derived ${\cal S}/T^{2}$ and ${\cal S}/T^{6}$ factors, which are considered
for optimisation. Minimum values of the latter factors are desirable for our
purpose.  Figure 20: Diffraction simulation of various openings of the
apertures A1 and A2 slits while using the _trumpet_ as obstacle O1 and the
phase-corrected aperture as A1.

####  4.2.4 Ge analyser

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

  1. 1.

Bandwidth covering the majority of the XFEL spectrum.

  2. 2.

Possibility to measure both polarisation states.

As we are relying on crystal-based polarisers, where the setting of Bragg
angle to $45^{\circ}$ ensures reflection of only the $s$-polarisation, we have
to admit that to our knowledge, all considered crystal reflection in their
default configuration have insufficient bandwidth, i.e. below or at $0.2\,{\rm
eV}$, while the bandwidth of self-seeded XFEL is in the order of $1\,{\rm
eV}$.

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, therefore also the integrated reflectivity is increased. A
summary of the performance of considered or common crystals is shown in Fig.
21, where the integrated reflectivity is plotted as a function of photon
energy, where the given plane can serve as a polariser. However, it is
important to realise 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
crystal cuts ending with _A or similar 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 are the crystals of choice. From those the Ge
440 was chosen, which fixes the experimental constraint to $8766\,{\rm eV}$.
Various cuts and their integrated reflectivities are summarised in Tab. (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.

Case | Angle between surface and 440 plane [∘] | Darwin width [$\upmu{\rm rad}$] | Bandwidth [$\rm eV$] | Integrated reflectivity [$\upmu{\rm rad}$]
---|---|---|---|---
standard (110) | $0$ | $21$ | $0.18$ | $18$
111 | $35.25$ | $45$ | $0.39$ | $39$
001 +$5.0^{\circ}$ miscut | $40$ | $62$ | $0.54$ | $52$
001 +$2.5^{\circ}$ miscut | $42.5$ | $88$ | $0.77$ | $69$
Table 5: Parameters of different cuts of Ge 440 crystal reflection. Figure 21:
Integrated reflectivity of considered crystal cuts.

The second condition - possibility to measure both polarisation states - could
be easily fulfilled by employing a thin, transmissive, crystal, as is done in
other spectrometers already used e.g. at the HED instrument [137]. However,
two issues arise if that would 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 [138] is employed, which
utilises a thick crystal in reflective geometry, but in such a geometry that
both polarisation states are reflected on different planes, therefore in
different directions. This setup will be adopted at XFEL such that the native
XFEL polarisation (horizontal) will be reflected above the beam, while the
polarisation-flipped photons will be reflected horizontally to another camera.
The reflectivities for both beams shall be in the predicted value of about
$50\,\upmu{\rm rad}$.

##  5 Conclusions and Outlook

The HED-HIBEF facility at the European XFEL offers a unique opportunity for
the exploration of fundamental properties of the ground state of nature. From
a microscopic viewpoint, HED-HIBEF has access to a parameter range of
sufficiently large centre of mass energy and photon number density to measure
the light-by-light scattering cross section which is induced by quantum
fluctuations of QED degrees of freedom. From an effective macroscopic
viewpoint, HED-HIBEF can create sufficiently strong fields to discover a
nonlinear response of the quantum vacuum, violating the classical
superposition principle and providing the vacuum with medium-like properties.

The experiment planned by the present collaboration aims at the discovery of
vacuum birefringence, taking advantage of several recent developments: in
addition to the HED-HIBEF facility reflecting the progress in generation and
control of ultra-intense near-IR and x-ray pulses, our concept draws from the
evolution of x-ray optics, most prominently high-purity polarimetry, as well
as novel theoretical tools for the prediction of quantum signatures generated
by generic spatiotemporal pulse distributions.

Because of the diminutiveness of the expected signal, our collaboration
envisages several prospective scenarios designed to isolate the nonlinear
quantum signature from the expected large linear background. First beam time
will be used to pursue the dark-field scenario where the discovery potential
hinges on the shadow factor $\mathcal{S}$ in combination with x-ray
polarimetry as a measure for the quality of background suppression. Currently
available facility parameters together with theory predictions, diffractive
simulations and practical feasibility suggest this scenario for a first step.
However, unforeseen obstacles or improvements in other crucial experimental
parameters may subsequently give preference to one of the other scenarios
brought forward by our collaboration.

The first discovery of vacuum birefringence would be a landmark for several