# All-optical Quantum Vacuum Signals in Two-Beam Collision

Holger Gies<EMAIL_ADDRESS>Helmholtz-Institut Jena, Fröbelstieg 3,
07743 Jena, Germany  GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, 64291 Darmstadt, Germany  Theoretisch-Physikalisches Institut,
Abbe Center of Photonics,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Felix Karbstein<EMAIL_ADDRESS>Helmholtz-Institut Jena,
Fröbelstieg 3, 07743 Jena, Germany  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany  Theoretisch-
Physikalisches Institut, Abbe Center of Photonics,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Leonhard Klar<EMAIL_ADDRESS>Helmholtz-Institut Jena, Fröbelstieg
3, 07743 Jena, Germany  GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, 64291 Darmstadt, Germany  Theoretisch-Physikalisches Institut,
Abbe Center of Photonics,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany

###### Abstract

The fundamental theory of quantum electrodynamics predicts the vacuum to
resemble a polarizable medium. This gives rise to effective nonlinear
interactions between electromagnetic fields and light-by-light scattering
phenomena. We study the collision of two optical laser pulses in a pump-probe
setup using beams with circular and elliptic cross section and estimate the
number of discernible signal photons induced by quantum vacuum nonlinearities.
In this analysis we study strategies to optimize the quantum vacuum signal
discernible from the background of the driving lasers. One of the main results
is that the collision of two maximally focused lasers does not lead to the
best discernible signal. Instead, widening the focus typically improves the
signal to background separation in the far field.

##  I Introduction

As the quantitatively best verified quantum field theory within the Standard
Model of particle physics, quantum electrodynamics (QED) still offers
parameter regimes untested in experiment. The present study is devoted to the
regime pioneered by Heisenberg and Euler introducing an effective Lagrangian
Heisenberg:1935qt , which encodes the effects of QED vacuum fluctuations in
effective nonlinear interactions between macroscopic electromagnetic fields.
An in-depth study of this regime does not only expand our understanding of
this fundamental theory but could also shed light on physics beyond the
Standard Model (BSM), e.g., Born34 ; Gies:2006ca ; Ahlers:2007qf ; Gies:2008wv
; Jaeckel:2010ni ; Baker:2013zta ; Davila:2013wba ; Villalba-Chavez:2016hxw ;
Rebhan:2017zdx ; Agrawal:2021dbo .

The quantum vacuum can be interpreted as a quantum state characterized by
fluctuations of particle and antiparticle pairs on short space and time
scales. In the case of QED, these are electrons and positrons interacting with
photons. As fields couple to charges these vacuum fluctuations induce the
effective nonlinear interactions between external electrodynamic fields in the
Heisenberg-Euler Lagrangian. The nonlinear terms in the Lagrangian give rise
to effects like photon-photon scattering or birefringence, which become
sizable only at sufficiently high field strengths Euler:1935zz ; Heinzl:2006xc
. Thus, the quantum vacuum closely resembles a nonlinear medium in solid state
physics.

In previous theoretical studies, many different signatures of the QED vacuum
have already been analyzed; see the reviews Dittrich:2000zu ; Dunne:2004nc ;
Marklund:2006my ; DiPiazza:2011tq ; Battesti:2012hf ; King:2015tba ; Hill:2017
; Inada:2017lop ; Karbstein:2019oej ; Schoeffel:2020svx ; Fedotov:2022ely  and
references therein. The main problem, however, is to distinguish between the
small amount of signal photons and the comparably huge number of background
photons of the driving electromagnetic fields.

In this article, we study the prospects of inducing a discernible quantum
vacuum signal in a two-beam pump-probe setup with optical ultrashort $\rm PW$
laser pulses, such as previously studied, e.g., by Tommasini:2010fb ;
King:2012aw ; Gies:2017ygp ; Robertson:2020nnc . A key parameter to enhance
the signal is the choice of optimal beam waists. Typically, tightly focused
beams down to the diffraction limit are considered in order to maximize the
involved field strength. While this does, in fact, generically maximize the
scattering amplitudes, it does not necessarily optimize the signal-to-
background ratio which is the relevant quantity to identify the quantum vacuum
signature; cf. also Mosman:2021vua ; Roso:2021hfo . A crucial idea in this
context is that the specific use of larger beam waists can scatter the quantum
signal into lower-noise regions. This is, because the signal amplitude
decreases only with a powerlaw for increasing waists, whereas the noise
decreases exponentially in the relevant spacetime regions.

In addition to studying beams with circular cross section, we also allow the
probe pulse to have an elliptic focus cross section such as pioneered in the
context of vacuum birefringence Karbstein:2015xra ; Karbstein:2016lby . We
discuss the advantages of the circularly or elliptically focused probe beam
settings for variable probe beam waists and determine the respective number of
discernible signal photons mediated by quantum vacuum nonlinearites.

Our article is organized as follow. In section II, we briefly introduce the
theoretical framework to reliably derive the number of signal photons arising
from QED nonlinearites. Section III specifies the experimental scenario, based
on parameters available at state-of-the-art strong laser facilities. In
section V, we identify the phase space regions where the number of signal
photons dominates over the amount of background photons. We then analyze the
effects of different focusing on the yield for different collision angles
between the pump and probe beams, both for circularly focused and elliptically
focused probe beams. Finally, we conclude with a summary of our results and a
brief outlook in Sec. VI.

##  II Theoretical background: key to the quantum vacuum

A convenient way to study the signal photons induced by quantum vacuum
nonlinearities is the vacuum emission picture Galtsov:1971xm ;
Karbstein:2014fva . Here we assume the vacuum nonlinearites to be governed by
the one-loop Heisenberg-Euler effective Lagrangian ${\cal L}_{\rm HE}$
Euler:1935zz ; Heisenberg:1935qt ; Weisskopf:1936kd ; Karplus:1950zz ;
Schwinger:1951nm ; Ritus:1975cf ; Dittrich:1985yb ; Dittrich:2000zu ;
Dunne:2004nc  and focus on the leading nonlinear correction to classical
Maxwell theory. In this limit, the relevant process can be illustrated by a
closed fermion loop with four photon lines: three photon lines represent the
driving laser pulses and the remaining line characterizes the photonic output
of the nonlinear interaction.

In this section, we briefly summarize this approach and derive an expression
for photonic signatures of quantum vacuum nonlinearities. We use the metric
convention $g_{\mu\nu}=\mathrm{diag}(-,+,+,+)$ and Heaviside-Lorentz units
with $\hbar=1=c$.

In order to induce a sizable signal, the driving electromagnetic fields should
reach extremely high peak field strength values $E$ and $B$ at the feasible
end of current technology. On the other hand these fields are still weak
compared to the field-strength scale constructed from the fundamental
parameters of QED, i.e. $\\{E,B\\}\ll\frac{m_{e}^{2}}{e}$ with the elementary
charge $e$ and the electron mass $m_{e}$. In addition, we limit ourselves to
optical and near-infrared frequencies with $\omega\ll m_{e}$.

It is convenient to introduce the field-strength invariants of
electrodynamics,

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{1}{4}F^{\mu\nu}F_{\mu\nu}=\frac{1}{2}\left(\mathbf{B}^{2}-{\bf
E}^{2}\right)\,,$ $\displaystyle\mathcal{G}$
$\displaystyle=\frac{1}{4}\tilde{F}^{\mu\nu}F_{\mu\nu}=-\mathbf{B}\cdot{\bf
E}\,,$ (1)

where
$\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is
the dual field strength tensor. Then, the Heisenberg-Euler action ${\cal
L}_{\rm HE}={\cal L}_{\rm M}+{\cal L}_{\rm int}$ consists of the classical
Maxwell term $\mathcal{L}_{\text{M}}=-\mathcal{F}$ and the nonlinear
interaction term induced by quantum fluctuations,

$\displaystyle{\cal L}_{\rm int}$
$\displaystyle=\frac{m_{e}^{4}}{360\pi^{2}}\left(\frac{e}{m_{e}^{2}}\right)^{4}\left(4\mathcal{F}^{2}+7\mathcal{G}^{2}\right)+\dots$
$\displaystyle=\\!\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}\\!+\\!\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}\\!+\\!\quad\dots$
(2)

Eqrefeq:Lint is additionally depicted in terms of Feynman diagrams. In the
following, we limit ourselves to the lowest order photon-photon interaction,
i.e. the contribution proportional to the square of the scalar invariants.
Generalizations to higher orders are straightforward.

The number of quantum vacuum signal photons induced by the interactions of the
driving fields can straightforwardly be derived within the vacuum emission
picture. For this, we introduce the signal photon field $\hat{a}^{\mu}$ and
treat it as an operator on the Fock space. Replacing the electromagnetic
fields according to $F^{\mu\nu}\to F^{\mu\nu}+\hat{f}^{\mu\nu}$ with the
classical background field of the driving lasers $F^{\mu\nu}$ and the signal
photon field
$\hat{f}^{\mu\nu}=\partial^{\mu}\hat{a}^{\nu}-\partial^{\nu}\hat{a}^{\mu}$, we
obtain the effective action $\Gamma_{\rm int}\left[\hat{a}\right]$ for the
signal photons. This action mediates transitions between the vacuum state
$\left|0\right>$ and the signal photon state
$\left|\gamma_{(p)}(\mathbf{k})\right>$. The induced signal photon of unit
wave vector $\hat{\bf k}$ is characterized by a polarization vector
$\epsilon^{\mu}_{\left(p\right)}\bigl{(}\hat{\bf k}\bigr{)}=(0,{\bf e}_{(p)})$
spanned by two transverse vectors ${\bf e}_{(p)}$ with $p\in\\{1,2\\}$,
fulfilling $\hat{\bf k}\times{\bf e}_{(p)}={\bf e}_{(p+1)}$ and ${\bf
e}_{(3)}=-{\bf e}_{(1)}$.

To evaluate the effective action $\Gamma_{\rm int}[\hat{a}(x)]=\int{\rm
d}^{4}x\,{\cal L}_{\rm int}|_{F\to F+\hat{f}}$ we use the locally constant
field approximation. This is quantitatively controlled by the spatiotemporal
variation scales of the optical driving laser pulses compared to the time and
space scale of QED; the Compton wavelength of an electron is
$\lambdabar_{C}=3.86\times 10^{-13}\,{\rm m}$ and the Compton time is
$\tau_{\rm C}=1.29\times 10^{-21}\,{\rm s}$. Finally, the information about
the signal photon distribution far outside the interaction region is encoded
in the signal photon amplitude

$\displaystyle S_{(p)}\left(\mathbf{k}\right)$
$\displaystyle=\bigl{\langle}\gamma_{\left(p\right)}\left(\mathbf{k}\right)\bigr{|}\Gamma_{\text{int}}[\hat{a}(x)]\bigl{|}0\bigr{\rangle}$
$\displaystyle=\mathrm{i}\frac{\epsilon^{*\mu}_{\left(p\right)}\bigl{(}\hat{\bf
k}\bigr{)}}{\sqrt{2k^{0}}}\int\mathrm{d}^{4}x\;\mathrm{e}^{\mathrm{i}k_{\alpha}x^{\alpha}}$
$\displaystyle\phantom{=}\times\Bigl{(}k^{\nu}F_{\nu\mu}\frac{\partial\mathcal{L}_{\text{int}}}{\partial\mathcal{F}}+k^{\nu}\tilde{F}_{\nu\mu}\frac{\partial\mathcal{L}_{\text{int}}}{\partial\mathcal{G}}\Bigr{)}\biggl{|}_{k^{0}={\rm
k}}$
$\displaystyle=\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}\,.$
(3)

Here, we use $*$ to denote the complex conjugation. Upon insertion of the
corresponding derivatives of $\mathcal{L}_{\text{int}}$ into Eq. (3), we
obtain

$\displaystyle S_{(p)}\left(\mathbf{k}\right)$
$\displaystyle=\frac{1}{\mathrm{i}}\frac{e}{4\pi^{2}}\frac{m_{e}^{2}}{45}\sqrt{\frac{\rm
k}{2}}\left(\frac{e}{m_{e}^{2}}\right)^{3}\int\mathrm{d}^{4}x\;\mathrm{e}^{\mathrm{ik}(\hat{\bf
k}-t)}$
$\displaystyle\phantom{=}\times\Bigl{(}4\left[\mathbf{e}_{\left(p\right)}\cdot{\bf
E}-\mathbf{e}_{\left(p+1\right)}\cdot\mathbf{B}\right]\mathcal{F}$
$\displaystyle\phantom{=\times\Bigl{(}}+7\left[\mathbf{e}_{\left(p\right)}\cdot\mathbf{B}+\mathbf{e}_{\left(p+1\right)}\cdot{\bf
E}\right]\mathcal{G}\Bigr{)}\,.$ (4)

This signal amplitude can be related to the differential number of signal
photons ${\rm d}^{3}N_{\left(p\right)}$ of polarization $p$ which have an
energy ${\rm k}=|\bf{k}|$ in the differential energy interval ${\rm dk}$ and
are emitted into the solid angle ${\rm d}\Omega$ around $\hat{{\bf k}}$ as

${\rm d}^{3}N_{(p)}=\frac{{\rm k^{2}}{\rm dk}\,{\rm
d}\Omega}{(2\pi)^{3}}\,|{\cal S}_{(p)}({\bf k})|^{2}\,.$ (5)

This differential form leads to a polarization sensitive angular resolved
signal photon density

$\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})=\frac{1}{(2\pi)^{3}}\\!\int_{\mathrm{k}_{\rm min}}^{\mathrm{k}_{\rm
max}}\\!\\!\\!\mathrm{dk}\,{\rm k}^{2}\bigl{|}S_{(p)}(\mathbf{k})\bigr{|}^{2}$
(6)

depending on the emission angles $\varphi$ and $\vartheta$ and integrated over
a given frequency range $\rm k_{\rm min}$ to $\rm k_{\rm max}$. The total
number of signal photons $N_{(p)}$ with polarization $(p)$ in a given solid
angle regime $\cal A$ reads

$N_{(p)}({\cal A}|{\rm k}_{\rm min},{\rm k}_{\rm max})=\int_{\cal A}{\rm
d}\Omega\,\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})\,.$ (7)

If the measurement is polarization insensitive we have to sum both
polarizations, i.e. $N({\cal A}|{\rm k}_{\rm min},{\rm k}_{\rm
max})=\sum_{p=1}^{2}N_{(p)}({\cal A}|{\rm k}_{\rm min},{\rm k}_{\rm max})$.
Further details can be found in the Refs. Karbstein:2014fva ; Gies:2021ymf .

##  III Experimental scenario

We study the interaction of two strong laser pulses colliding under an angle
$\vartheta_{\rm col}>90^{\circ}$. In this section, we introduce the
experimental setup based on state-of-the-art techniques. Further we choose an
example set of parameters for experimentally accessible quantum vacuum
signals.

Here, we focus on laser pulses in the optical regime with the same photon
energy of $\omega_{0}=1.55\,{\rm eV}$, as is available at the Advanced
Titanium-Sapphire Laser (ATLAS) in the Centre for Advanced Laser Applications
(CALA) facility (Garching, Germany) CALA ; Hartmann:2021iyf  and the Jenaer
Titan:Saphir 200 Terawatt Laser System (JETI-200) laser at Helmholtz Institut
Jena (Jena, Germany) JETI-200 ; Doyle:2021mdt . These facilities operate
state-of-the-art high-intensity lasers and illustrate which field strengths
are already available in strong-field laser experiments.

Accordingly, we use optical laser pulses with a pulse energy of $W=25\,{\rm
J}$ here. The wavelength is $\lambda=800\,{\rm nm}$ and the pulse duration is
$\tau_{\rm FWHM}=25\,{\rm fs}$ measured at full width half maximum (FWHM) of
the intensity. Assuming that the temporal shape of the pulse is Gaussian, we
can convert the FHWM pulse duration $\tau_{\rm FWHM}$ into an $1/e^{2}$ pulse
duration $\tau=2\sqrt{\ln 2}\,\tau_{\rm FWHM}$. The calculations are performed
with the $1/e^{2}$ pulse duration; if in the following the FWHM duration is
mentioned, then this is indicated with the addition (FWHM) in brackets.

Figure 1: Illustration of the experimental scenario. The red shades illustrate
the forward/backward cones of two laser pulses colliding under the angle
$\vartheta_{\rm col}$ with a circular focus cross section. Additionally, the
green (blue) lines represent a typical snapshot of the electric (magnetic)
field amplitude.

Moreover, we model the spatial structure of the laser pulses by Gaussian beams
with a focus waist $w_{0}$, which is determined by the radial aperture angle
of the beams in the far field $\Theta$ via the relation
$\Theta=\frac{w_{0}}{z_{R}}$ with the Rayleigh range
$z_{R}=\pi\frac{w_{0}^{2}}{\lambda}$. According to the laws of optics, the
minimum focus width is $w_{0}\simeq\lambda$, which corresponds to an radial
aperture of $\Theta=\frac{1}{\pi}\approx 18.24^{\circ}$. Because of the
scaling $\Theta\sim\frac{1}{w_{0}}$, setups with a focal width larger than the
minimum yield smaller aperture angles in the far field. Consequently, the
photon distribution of the driving lasers constituting a large background is
less wide.

In this work, we consider the collision of two Gaussian laser pulses under the
angle $\vartheta_{\rm col}$. For our calculations, we vary this angle in the
range from $100^{\circ}$ to $160^{\circ}$ in $10^{\circ}$ steps. For
convenience, we refer to one laser as the probe field, labeled by subscript
$1$, and the other as the pump field, labeled by the subscript $2$. We
deliberately avoid head-on collisions which would maximize the signal photon
yield for both practical and theoretical reasons; see below. Head-on
collisions are very challenging and may not be practical due to experimental
constraints. We will focus mainly on the signal emitted in the vicinity of the
forward direction of the probe beam. The purpose of the pump beam is to
provide a localized strong field region inducing the signal. The propagation
directions of the laser pulses are $\hat{\bf k}_{1}=\hat{\bf e}_{z}$ and
$\hat{\bf k}_{2}=\sin\vartheta_{\rm col}\,\hat{\bf e}_{x}+\cos\vartheta_{\rm
col}\,\hat{\bf e}_{z}$, respectively. Their wave vectors are ${\bf
k}_{1}=\omega_{1}\hat{\bf k}_{1}$ and ${\bf k}_{2}=\omega_{2}\hat{\bf k}_{2}$
with $\omega_{1}=\omega_{2}=\omega_{0}=2\frac{2\pi}{\lambda}$. These laser
pulses collide at their focal spots at ${\bf x}=0$ and reach their peak fields
at $t=0$, which defines the origin of our coordinate system.

The laser pulses are all linearly polarized. Here, we choose the polarization
of the probe laser such that the electric field $\mathbf{E}_{1}({\bf x},t)$
points into $x$ direction; correspondingly the magnetic field
$\mathbf{B}_{1}({\bf x},t)$ is orientated along the $y$ axis. To allow for a
generic linearly polarized pump laser, we introduce the angle $\beta_{2}$
between $\mathbf{E}_{1}({\bf 0},0)$ and $\mathbf{E}_{2}({\bf 0},0)$ for the
collinear collision with $\vartheta_{\rm col}=0^{\circ}$. Using
$\vartheta_{\rm col}$ as the general collision angle, we obtain

$\displaystyle\mathbf{E}_{2}({\bf x},t)$ $\displaystyle=\mathcal{E}_{2}({\bf
x},t)\left(\begin{array}[]{c}\cos\beta_{2}\,\cos\vartheta_{\rm col}\\\
\sin\beta_{2}\\\ -\cos\beta_{2}\,\sin\vartheta_{\rm col}\end{array}\right)\,,$
(11)

$\mathbf{B}_{2}({\bf x},t)\sim\hat{\bf k}_{2}\times\mathbf{E}_{2}({\bf x},t)$
for the pump fields, where $\mathcal{E}_{i}({\bf x},t)$ describes the general
amplitude function. The maximum of the total polarization insensitive signal
occurs for an angle $\beta_{2}=90^{\circ}$ which is our angle of choice unless
otherwise stated. In a special scenario, we consider the signal of
birefringence maximized at an angle of $\beta_{2}=45^{\circ}$, see
Karbstein:2021hwc . Further references discuss vacuum birefringence from a
general point of view, cf. DiPiazza:2006pr ; Heinzl:2006xc ; Karbstein:2015xra
; Mosman:2021vua ; Ataman:2018ucl .

Figure 2: Illustration of the beam radii in the focus of the experimental
scenario. Besides the wave vectors of probe $\hat{\rm k}_{1}$ and pump
$\hat{\rm k}_{2}$, the red dashed curves illustrate the radii of the waist in
the focus. The rotationally symmetric pump has a waist size of $w_{0}$ while
the probe with elliptical cross section has the waists $w_{x}$ in the
collision plane marked by the yellow plain and $w_{y}$ perpendicular to it.

Throughout this work we consider a radially symmetric pump, but use both
elliptical and circular focus cross sections for the probe laser. To this end,
we introduce two independent beam waists for the probe laser $w_{x}$ in the
collision plane and $w_{y}$ normal to this plane. For convenience, we
parameterize $w_{x}=w_{y}=\mu\,w_{0}$ in the case of a rotationally symmetric
probe beam, using a dimensionless parameter $\mu\geq 1$. In the case of
elliptical cross sections, we analogously introduce $w_{x}=\mu_{x}\,w_{0}$ and
$w_{y}=\mu_{y}\,w_{0}$ with $\\{\mu_{x},\mu_{y}\\}\geq 1$. Fig. 1 illustrates
this scenario with rotationally symmetric pump and probe beams.

To determine the rotationally symmetric quantum vacuum signal, it is
sufficient to consider the Gaussian beams in the limit of infinite Rayleigh
ranges. This approximation is justified if the dimensions of the interaction
volume are small compared to the Rayleigh lengths $z_{R}$. This is true in the
case of very small pulse durations $\tau_{i}\ll z_{R,i}$ compared to the
Rayleigh ranges for all involved beams. Another way to justify the
approximation is a small ratio of focal width $w_{0,i}$ and Rayleigh length
$z_{R,j}$ of two colliding beams $i$ and $j$ with respect to the sine of their
collision angle, i.e. $w_{0,i}/z_{R,j}\ll\left|\sin\vartheta_{\rm col}\right|$
for $i\neq j$. If any of the above conditions is satisfied, then the
effectively interacting regions of the laser beams can be approximated as
tubes in the interaction volume Gies:2017ygp ; King:2018wtn ; Yariv75book ;
Siegman86 . Figure 2 shows the beam radii in the focus by using a probe with
elliptical cross section. For numerical approaches beyond infinite Rayleigh
range approximation see Grismayer:2016cqp ; Blinne:2018uyn ; Blinne:2018nbd ;
Lindner:2021krv ; Sainte-Marie:2022efx .

Correspondingly, the field amplitudes can be approximated as

$\mathcal{E}_{1}\left(x\right)=\frac{\mathcal{E}_{0}}{\sqrt{\mu_{x}\mu_{y}}}\,\mathrm{e}^{-(\frac{{\bf
x}\cdot\hat{\bf
k}_{1}-t}{\tau_{1}/2})^{2}-\frac{x^{2}}{w_{x}^{2}}-\frac{{y}^{2}}{w_{y}^{2}}}\cos\bigl{(}\omega_{0}({\bf
x}\cdot\hat{\bf k}_{1}-t)\bigr{)}$ (12)

and

$\mathcal{E}_{2}\left(x\right)={\mathcal{E}_{0}}\,\mathrm{e}^{-(\frac{{\bf
x}\cdot\hat{\bf k}_{2}-t}{\tau_{2}/2})^{2}-\frac{{\bf x}^{2}-\left(\bf
x\cdot\hat{k}_{2}\right)^{2}}{w_{0}^{2}}}\cos\bigl{(}\omega_{0}({\bf
x}\cdot\hat{\bf k}_{2}-t)\bigr{)}\,,$ (13)

where we have used the different propagation directions $\hat{\bf{k}}_{1}$ and
$\hat{\bf{k}}_{2}$. Here, $\mathcal{E}_{0}$ is the field amplitude maximum
depending on the laser pulse properties. It fulfills Karbstein:2017jgh

$\mathcal{E}_{0}=\sqrt{8\sqrt{\frac{2}{\pi}}\frac{W}{\pi
w_{0}^{2}\tau}}\approx 8.5\times 10^{15}\,{\rm V/m}\,,$ (14)

using the parameters given above.

##  IV Calculation of the signature of quantum vacuum nonlinearites

###  IV.1 The background of the driving laser pulses

Before we calculate the signal, let us determine the distribution of the
driving laser photons far outside the interaction volume.

Since we need extremely strong laser fields to stimulate QED vacuum
nonlinearites, the challenge in experiment is to discern the small quantum
vacuum signal from the enormous number of photons of the driver lasers. In the
current research, there are many ideas to suppress this background such as an
elastic scattering signal outside the forward cones of the driving lasers,
Klar:2020ych . Another idea is to use inelastic scattering to obtain signal at
a frequency different from the background and thus achieve a spectral signal-
to-background separation Gies:2021ymf . Furthermore, in experiments on
birefringence, the different polarization of signal and background can be the
key to measurability Heinzl:2006xc .

Let us describe the background quantitatively, in order to assess where the
signal dominates. In contrast to the signal, the background in the far field
cannot be reliably determined in the limit of infinite Rayleigh length.

We can estimate the total number of photons per laser pulse as $N^{\rm
tot}=W/\omega_{0}\approx 10^{20}$. This results in a differential number of
background photons of the $i$th laser ${\rm d}N^{\rm Bg}_{i}$ per solid angle
element ${\rm d}\Omega$ which satisfies the relation $N^{\rm
tot}=\int_{\Omega}\\!{\rm d}N^{\rm Bg}_{i}$, since we use $W_{1}=W_{2}=W$ and
frequency $\omega_{1}=\omega_{2}=\omega_{0}$ for both lasers. For the far-
field distribution of a Gaussian beam with normalization $N^{\rm
Bg}_{0}=\frac{w_{0}^{2}\omega_{0}}{2\pi}W$, we obtain

$\frac{{\rm d}N^{\rm Bg}_{i}}{{\rm d}\Omega}=\mu_{1i}\mu_{2i}\,N^{\rm
Bg}_{0}\,{\rm
e}^{-\frac{1}{2}\omega_{0}^{2}w_{0}^{2}\left(\mu_{1i}^{2}\cos^{2}\\!\phi_{i}\;+\mu_{2i}^{2}\sin^{2}\\!\phi_{i}\right)\theta_{i}^{2}}$
(15)

for a laser pulse with elliptical cross section. Here,

$\phi_{1}=\varphi$ (16)

and

$\phi_{2}=\arctan\left(\cos\varphi\,\sin\vartheta\,\cos\vartheta_{\rm
col}-\sin\varphi\,\sin\vartheta\,\sin\vartheta_{\rm col}\right)$ (17)

parameterize the rotations around the beam axis, and

$\displaystyle\theta_{1}$ $\displaystyle=$
$\displaystyle\vartheta-\arccos\left(\hat{\rm k}_{1}\cdot\hat{\rm
e}_{z}\right),$ (18) $\displaystyle\theta_{2}$ $\displaystyle=$
$\displaystyle\arccos\left(\cos\varphi\,\sin\vartheta\,\sin\vartheta_{\rm
col}+\cos\vartheta\,\cos\vartheta_{\rm col}\right)$ (19)

are the polar angles measured from the forward beam axis; $\varphi$ and
$\vartheta$ are azimuthal and polar angle used to parameterize the emission
direction in a spherical coordinate system with the north pole along the
outgoing direction of the probe laser $\mathbf{k}_{1}$.

For the pump laser with a circular cross section in the far field, we have

$\frac{{\rm d}N^{\rm Bg}_{2}}{{\rm d}\Omega}=N^{\rm Bg}_{0}\,{\rm
e}^{-\frac{1}{2}\omega_{0}^{2}w_{0}^{2}\theta_{2}^{2}}\,.$ (20)

Equation (15) implies that the density of background photons decreases more
rapidly with $\theta_{i}$ the larger $\mu_{1i}$ and $\mu_{2i}$. In the present
scenario, the full density of the background photons is given by

$\rho_{\bar{\mu}}^{\rm Bg}(\varphi,\vartheta)=\sum_{i=1}^{2}\frac{{\rm
d}N^{\rm Bg}_{i}}{{\rm d}\Omega}\,.$ (21)

Since we keep the total pulse energy $W$, the frequency $\omega_{0}$, and the
pulse duration $\tau$ constant, we write $\rho_{\bar{\mu}}^{\rm
Bg}(\varphi,\vartheta)$ with the tuple $\bar{\mu}=\\{\mu_{x},\mu_{y}\\}$
indicating the choice of foci of the probe beam. In the case of circularly
focused pulses, we use $\bar{\mu}\rightarrow\mu$, i.e. $\mu_{x}=\mu_{y}=\mu$.

The spectral distribution of the driving laser pulses is centered around
$\omega_{0}$, therefore the background is also dominated by the frequency
$\omega_{0}$.

Finally, the number of background photons in a given solid angle regime
$\mathcal{A}$, analogous to Eq. (7), is given by

$N_{\rm Bg}(\mathcal{A},\bar{\mu})=\int_{\cal A}{\rm
d}\Omega\,\rho_{\bar{\mu}}^{\rm Bg}(\varphi,\vartheta)\,.$ (22)

###  IV.2 Quantum vacuum signal

For the signal amplitude $S_{(p)}\left({\bf k}\right)$, let us express the
wave vector in spherical coordinates as ${\bf k}={\rm k}\,\hat{\bf k}$ with
$\hat{\bf k}=\cos\varphi\,\sin\vartheta\,\hat{\bf
e}_{x}+\sin\varphi\,\sin\vartheta\,\hat{\bf e}_{y}+\cos\vartheta\,\hat{\bf
e}_{z}$.

The vectors orthogonal to $\hat{\bf k}$ can then be parameterized by a single
angle $\beta$ as

${\bf e}_{\beta}=\sin\beta\left.\hat{\bf
k}\right|_{\varphi\rightarrow\varphi+\frac{\pi}{2},\vartheta=\frac{\pi}{2}}+\cos\beta\left.\hat{\bf
k}\right|_{\vartheta\rightarrow\vartheta+\frac{\pi}{2}}\,.$ (23)

Without loss of generality we associate the polarization $p=1$ with $\beta$
and $p=2$ with $\beta+\frac{\pi}{2}$, respectively.

The leading quantum vacuum signal arises from the effective coupling of three
electromagnetic fields. For convenience, we express the electric and the
magnetic field of the $i$th laser pulse field strength and direction according
to ${\bf E}_{i}({\bf x},t)=\mathcal{E}_{i}({\bf x},t)\hat{\bf E}_{i}$ and
${\bf B}_{i}({\bf x},t)=\mathcal{E}_{i}({\bf x},t)\hat{\bf B}_{i}$,
respectively. In turn, Eq. (4) can be written as Gies:2021ymf

$\displaystyle S_{(p)}\left(\mathbf{k}\right)$
$\displaystyle=\frac{1}{\mathrm{i}}\frac{e}{4\pi^{2}}\frac{m_{e}^{2}}{45}\sqrt{\frac{\rm
k}{2}}\left(\frac{e}{m_{e}^{2}}\right)^{3}\left(\mathcal{I}_{121}g_{121}+\mathcal{I}_{212}g_{212}\right)$
(24)

with

$\displaystyle g_{iji}(\varphi,\vartheta)$ $\displaystyle=2\left({\bf
e}_{\beta}\cdot\hat{\bf E}_{i}-{\bf e}_{\beta+\frac{\pi}{2}}\cdot\hat{\bf
B}_{i}\right)$ $\displaystyle\phantom{=-}\times\left(\hat{\bf
B}_{i}\cdot\hat{\bf B}_{j}-\hat{\bf E}_{i}\cdot\hat{\bf E}_{j}\right)$
$\displaystyle\phantom{=}-\frac{7}{2}\left({\bf e}_{\beta}\cdot\hat{\bf
B}_{i}+{\bf e}_{\beta+\frac{\pi}{2}}\cdot\hat{\bf E}_{i}\right)$
$\displaystyle\phantom{=-}\times\left(\hat{\bf B}_{i}\cdot\hat{\bf
E}_{j}+\hat{\bf B}_{j}\cdot\hat{\bf E}_{i}\right)\,,$ (25)

and

$\displaystyle\mathcal{I}_{iji}\left({\bf
k}\right)=\int\mathrm{d}^{4}x\;\mathrm{e}^{\mathrm{ik}(\hat{\bf
k}-t)}\mathcal{E}_{i}({\bf x},t)\mathcal{E}_{j}({\bf x},t)\mathcal{E}_{i}({\bf
x},t)\,.$ (26)

The function $g_{ijl}(\varphi,\vartheta)$ depends on the orientation of the
electromagnetic fields of the driving laser fields and the signal;
$I_{ijl}\left({\bf k}\right)$ is the Fourier transform of the product of three
field amplitudes evaluated on shell.

In the conventions used here, the relevant contributions read

$\displaystyle g_{121}(\varphi,\vartheta)$
$\displaystyle=\frac{1}{4}\left(1-\cos\vartheta\right)\left(1-\cos\vartheta_{\rm
col}\right)$
$\displaystyle\phantom{=}\times\left[3\cos\left(\beta+\beta_{2}-\varphi\right)-11\cos\left(\beta-\beta_{2}-\varphi\right)\right]$
(27)

and

$\displaystyle g$ ${}_{212}(\varphi,\vartheta)$
$\displaystyle=\frac{1}{4}\left(1-\cos\vartheta_{\rm
col}\right)\Big{(}[11\cos\beta-3\cos\left(\beta+2\beta_{2}\right)]$
$\displaystyle\phantom{=}\times\left[\left(\cos\vartheta\,\cos\vartheta_{\rm
col}-1\right)\cos\varphi+\sin\vartheta\,\sin\vartheta_{\rm col}\right]$
$\displaystyle\phantom{=}+\left[11\sin\beta-3\sin\left(\beta+2\beta_{2}\right)\right]\left(\cos\vartheta-\cos\vartheta_{\rm
col}\right)\sin\varphi\Big{)}\,.$ (28)

We aim at the density of signal photons $\rho_{\bar{\mu}}^{\rm
Sig}(\varphi,\vartheta)$ regardless of their frequency. The latter lies within
a small frequency range around $\omega_{0}$, see also the detailed analysis in
Gies:2021ymf . Apart from Sect. V.3, we consider the inclusive average over
polarizations. We are mainly interested in the parametric dependence of the
signal density on the focus width (along both orthogonal axes) explicitly
indicated by the parameter tuple $\bar{\mu}$.

We begin by substituting the signal amplitude from Eq. (24) into Eq. (6),

$\displaystyle\rho_{\bar{\mu}}^{\rm Sig}(\varphi,\vartheta)$
$\displaystyle=\sum_{p=1}^{2}\frac{1}{(2\pi)^{3}}\\!\int_{0}^{\infty}\\!\\!\\!\mathrm{dk}\,{\rm
k}^{2}\bigl{|}S_{(p)}(\mathbf{k})\bigr{|}^{2}$
$\displaystyle=\frac{1}{2\left(2\pi\right)^{7}}\frac{1}{45^{2}}\left(\frac{e}{m_{e}}\right)^{8}$
$\displaystyle\phantom{=}\times\\!\int_{0}^{\infty}\\!\\!\\!\mathrm{dk}\,{\rm
k}^{3}\sum_{p=1}^{2}\left(\mathcal{I}_{121}g_{121}+\mathcal{I}_{212}g_{212}\right)^{2},$
(29)

taking advantage of the fact that the Fourier transform of a real symmetric
function is also a real symmetric function, i.e.,
$\mathcal{I}^{*}_{ijl}=\mathcal{I}_{ijl}$. All contributions
$\mathcal{I}_{121}g_{121}$ and $\mathcal{I}_{212}g_{212}$ give rise to non-
vanishing contributions for complementary values of
$\left(\varphi,\vartheta\right)$. The interference term
$2\mathcal{I}_{121}\mathcal{I}_{212}g_{121}g_{212}$ is exponentially
suppressed in comparison to the “direct” channels and can be safely neglected.
Therefore, we neglect the interference term in the remainder. This also
increases the performance of the numerical integration over the solid angle.

With these approximations, we can calculate the Fourier integrals
$\mathcal{I}_{iji}$ analytically, since their evaluation amounts to performing
Gaussian integrals. Then, Eq. (29) can be expressed as

$\displaystyle\rho_{\bar{\mu}}^{\rm Sig}(\varphi,\vartheta)$
$\displaystyle=\frac{1}{\left(2\pi\right)^{7}}\frac{2}{45^{2}}\left(\frac{e}{m_{e}}\right)^{8}$
$\displaystyle\phantom{=}\times\Big{[}\left(\sum_{p=1}^{2}g_{121}^{2}(\varphi,\vartheta)\right)\\!\int_{0}^{\infty}\\!\\!\\!\mathrm{dk}\,{\rm
k}^{3}\,\mathcal{I}_{121}^{2}({\bf k})$
$\displaystyle\phantom{=\times}+\left(\sum_{p=1}^{2}g_{212}^{2}(\varphi,\vartheta)\right)\\!\int_{0}^{\infty}\\!\\!\\!\mathrm{dk}\,{\rm
k}^{3}\,\mathcal{I}_{212}^{2}({\bf k})\Big{]}\,.$ (30)

Subsequently we are mainly interested in the case of
$\beta_{2}=\frac{\pi}{2}$, where we obtain

$\displaystyle\sum_{p=1}^{2}\left.g_{121}^{2}(\varphi,\vartheta)\right|_{\beta_{2}=\frac{\pi}{2}}$
$\displaystyle=196\,\sin^{4}\frac{\vartheta}{2}\,\sin^{4}\frac{\vartheta_{\rm
col}}{2},$ (31)
$\displaystyle\sum_{p=1}^{2}\left.g_{212}^{2}(\varphi,\vartheta)\right|_{\beta_{2}=\frac{\pi}{2}}$
$\displaystyle=49\,\sin^{4}\frac{\vartheta_{\rm
col}}{2}\Big{[}\cos\vartheta\cos\vartheta_{\rm col}-1$
$\displaystyle\qquad\qquad\qquad+\cos\varphi\sin\vartheta\sin\vartheta_{\rm
col}\Big{]}^{2}\,.$ (32)

So far, all calculations could be performed analytically. The number of signal
photons $N_{\rm Sig}(\mathcal{A},\bar{\mu})$ emitted into a given solid angle
area $\mathcal{A}$ is finally evaluated by a numerical integration over
$\mathcal{A}$ as

$N_{\rm Sig}(\mathcal{A},\bar{\mu})=\int_{\cal A}{\rm
d}\Omega\,\rho_{\bar{\mu}}^{\rm Sig}(\varphi,\vartheta)\,.$ (33)

##  V Discernibly analysis of quantum vacuum signal

Our goal is to distinguish the weak quantum vacuum signal from the strong
laser background. This requires a comparison of angular resolved densities or
particle numbers of the signal and background. In this section, we discuss
ways to achieve optimal signal-to-background separation.

Let is first introduce a discernibility criterion.

We call a signal discernible if the density of signal photons $\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)$ in a given solid-angle region
$\mathcal{A}_{\bar{\mu}}$ is larger than the density $\rho^{\rm
Bg}_{\bar{\mu}}(\varphi,\vartheta)$ of the background provided by the driver
lasers Karbstein:2019dxo . The union of all such regions defines the solid-
angle region

$\mathcal{A}_{\rm
d,\bar{\mu}}=\left\\{(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\pi\right)\left.\right|\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)\geq\rho^{\rm
Bg}_{\bar{\mu}}(\varphi,\vartheta)\right\\}\,.$ (34)

For simply connected $\mathcal{A}_{\rm d,\bar{\mu}}$, we describe the boundary
$\partial\mathcal{A}_{\rm d,\bar{\mu}}$ using the upper boundary
$\vartheta_{\rm u}(\varphi)$ and lower boundary $\vartheta_{\rm l}(\varphi)$,
such that

$\partial\mathcal{A}_{\rm d,\bar{\mu}}=\left\\{\vartheta_{\rm
l}(\varphi)\left.\right|\varphi\in\left[\varphi_{\rm i},\varphi_{\rm
f}\right]\right\\}\cup\left\\{\vartheta_{\rm
u}(\varphi)\left.\right|\varphi\in\left[\varphi_{\rm i},\varphi_{\rm
f}\right]\right\\}\,,$ (35)

where $\vartheta_{\rm u}(\varphi_{\rm i})=\vartheta_{\rm l}(\varphi_{\rm i})$
and $\vartheta_{\rm u}(\varphi_{\rm f})=\vartheta_{\rm l}(\varphi_{\rm f})$,
respectively. In appendix A, a method for numerically determining region
$\mathcal{A}_{\rm d,\bar{\mu}}$ including its boundary is detailed. In
addition, we also show there how to compute the boundary functions
$\vartheta_{\rm l/u}(\varphi)$ for non-simply-connected regions.

###  V.1 Probe with circular cross section

Figure 3: Area of the solid angle region $\mathcal{A}_{\rm d,\mu}$ where
$\rho^{\rm Sig}_{\mu}>\rho^{\rm Bg}_{\mu}$ as a function of $\mu$ for a
collision angle of $\vartheta_{\rm col}=160^{\circ}$. Here we consider the
collision of two frequency $\omega_{0}$ beams of equal pulse duration
$\tau=25\,{\rm fs}$ (FWHM) and pulse energy $W=25\,{\rm J}$. The pump (probe)
beam has a circular cross section with waist radius
$w_{0}=\frac{2\pi}{\omega_{0}}$ ($w_{x}=w_{y}=\mu w_{0}$). Figure 4: Mollweide
plot (longitude $\varphi$, latitude $\vartheta$) of the signal photon density
$\rho_{5}^{\rm Sig}(\varphi,\vartheta)$ and the background of the driving
laser fields $\rho_{5}^{\rm Bg}(\varphi,\vartheta)$. The region where
$\rho^{\rm Sig}>\rho^{\rm Bg}$ is highlighted by the green frame with the
green color function and for the complementary region $\rho^{\rm
Sig}<\rho^{\rm Bg}$ the purple color function is chosen. We consider the
collision of two frequency $\omega_{0}$ beams of equal pulse duration
$\tau=25\,{\rm fs}$ (FWHM) and pulse energy $W=25\,{\rm J}$ for
$\vartheta_{\rm col}=160^{\circ}$. The pump (probe) beam has a circular cross
section with waist radius $w_{0}=\frac{2\pi}{\omega_{0}}$
($w_{x}=w_{y}=5w_{0}$).

Let us begin with the collision of pump and probe laser pulses with circular
cross section. First we turn our attention to the collision under the angle
$\vartheta_{\rm col}=160^{\circ}$. The pump laser is focused to its
diffraction limit with $w_{0}=w_{0}=w_{0}=2\pi/\omega_{0}$, whereas the probe
beam has a variable waist size $w=w_{x}=w_{y}=\mu\,w_{0}$ with $\mu\in[1,10]$.
The lower bound corresponds to the diffraction limit. Larger values of $\mu$
are not considered, because a saturation is observed already before $\mu=10$.
In the remainder of this work, we stick to the definitions and laser
parameters as introduced in Sect. III, i.e., both pulses have a pulse duration
of $\tau_{1}=\tau_{2}=25\,{\rm fs}$ (FWHM), a pulse energy of
$W_{1}=W_{2}=25\,{\rm J}$ and $\omega_{1}=\omega_{2}=\omega_{0}=1.55\,{\rm
eV}$.

At first we determine the solid angle regions where the signal is discernible;
see App. A. For convenience, we define the area of a given solid angle region
$\cal A$ as

$A(\mathcal{A})=\int_{\cal A}{\rm d}\Omega\,.$ (36)

The dependence of this quantity on the parameter $\mu=w/w_{0}$ is shown in
Fig. 3. From Fig. 3, we infer that there is no discernible signal for $\mu\leq
1.6$. For $\leq\mu\lesssim 4$ the area of ${\cal A}_{\rm d}$ grows very
strongly, but the increase with $\mu$ becomes much slower beyond $\mu\approx
4$.

In Fig. 4, we highlight the photon density for the full solid angle region of
$4\pi$ for the example $\mu=5$. Here, the region where the signal dominates
the background is framed in green. Inside this region, the density of the
signal photons is shown in the corresponding color scale (from black via green
to yellow). Outside this area, the background of the driving photons is shown;
for this a different color scale (from black via purple to ocher) is used.

Figure 5: Number of discernible signal photons $N_{\rm Sig}(\mathcal{A}_{\rm
d,\mu})$ as a function of $\mu$ for a collision angle of $\vartheta_{\rm
col}=160^{\circ}$. The maximum number of background photons $N_{\rm
Bg}(\mathcal{A}_{\rm d,3.8})\approx 0.16$ is reached at $\mu=3.8$. See Figure
3 for the laser parameters employed here.

$\vartheta_{\rm col}=100^{\circ}$ $\qquad\vartheta_{\rm col}=110^{\circ}$
$\vartheta_{\rm col}=120^{\circ}$ $\qquad\vartheta_{\rm col}=130^{\circ}$
$\vartheta_{\rm col}=140^{\circ}$ $\qquad\vartheta_{\rm col}=150^{\circ}$
$\vartheta_{\rm col}=160^{\circ}$

Figure 6: Mollweide plot (longitude $\varphi$, latitude $\vartheta$) showing
the different direction of the pump laser pulses for different collision
angles $\vartheta_{\rm col}$. The forward beam axis of the pump laser is
marked by different colors and the forward beam axis of the probe is marked by
the black color. The bounds $\vartheta_{\rm l}(\varphi)$ and $\vartheta_{\rm
u}(\varphi)$ highlighting $\rho^{\rm Sig}=\rho^{\rm Bg}$ are marked by colored
lines for different values of $\mu$: $2$, $3$, $4$, $5$, $6$, $7$. The value
of $\mu$ is encoded by the lines style. Lager $\mu$ means smaller spacing
between dashes. The pump (probe) is focused to a waist of
$w_{0}=\frac{2\pi}{\omega_{0}}$ ($w_{x}=w_{y}=\mu\omega_{0}$). Both
frequencies $\omega_{0}=1.55\,{\rm eV}$ lasers have the same pulse energy
$W=25\,{\rm J}$ and duration $\tau=25\,{\rm fs}$ (FWHM). Figure 7: Number of
signal (solid) and background photons (dashed) in solid angle region
$\mathcal{A}_{\rm d,\mu}$ as a function of $\mu=w_{x}/w_{0}=w_{y}/w_{0}$ for
different collision angles $\vartheta_{\rm col}$. See Fig. 6 for the laser
parameters employed here.

In order to map the sphere onto a flat diagram, we use a Mollweide projection,
which conserves the relevant areas of surfaces but is not angle conserving.
The lower boundary $\vartheta_{\rm l}(\varphi)$ is almost constant in
$\varphi$ because this region is close to the forward beam axis of the probe
beam $\vartheta=0$ which has a circular cross section.

By contrast, the upper bound $\vartheta_{\rm u}(\varphi)$ shows a more
pronounced $\varphi$ dependence. It becomes minimal in the collision plane of
the driving lasers at $\varphi=0$ and maximal perpendicular to the collision
plane at $\varphi=180^{\circ}$. In between, $\vartheta_{\rm u}(\varphi)$
varies monotonically.

The color scale encodes the distribution of the signal photons. The largest
discernible signals are reached close to $\vartheta_{\rm l}(\varphi)$.
Furthermore, we observe pronounced maxima for $\vartheta\approx 17,4^{\circ}$
and $\varphi\in[100^{\circ},140^{\circ}]$ and
$\varphi\in[220^{\circ},260^{\circ}]$, respectively.

Next, we analyze the total number of discernible signal photons $N_{\rm
Sig}(\mathcal{A}_{\rm d,\mu})$ as a function of $\mu$, as depicted in Fig. 5.
Only for $\mu\gtrsim 1.6$, a discernible signal can be identified. The
increase of $N_{\rm Sig}$ differs visibly from the growth of $\mathcal{A_{\rm
d,\mu}}$ with $\mu$. At $\mu\approx 5.5$ we encounter a pronounced maximum
with $N_{\rm Sig}(\mathcal{A}_{\rm d,5.5})\approx 5.84$ photons per shot. For
comparison, the associated number of background photons yields $N_{\rm
Bg}(\mathcal{A}_{\rm d,5})\approx 0.12$ photons per shot. For $\mu\gtrsim 5.5$
$N_{\rm Sig}$ decreases approximately linearly with $\mu$. A similar behavior
was identified in the head-on collision of optical and x-ray laser pulses in
Mosman:2021vua .

The appearance of a local maximum in $N_{\rm Sig}$ shows that the probe waist
provides a handle to amplify or decrease the signal for given other
parameters. Since the pulse energy and duration are kept constant, the probe
intensity decreases for a wider focus. At the same time, a wider focus tends
to increase $\mathcal{A}_{\rm d,\mu}$ because the probe divergence diminishes.
Even though the area $A(\mathcal{A}_{\rm d,\mu})$ does not change too much,
the value of the lower limit $\vartheta_{\rm l}$ changes significantly; this
can be seen in Fig. 6. At some point, however, the intensity of the probe
eventually becomes too weak for large values of $\mu$ such that the number of
signal photons decreases.

In Fig. 5, apart from $N_{\rm Sig}(\mathcal{A}_{\rm d,\mu})$, the maximum
number of background photons ${\rm max}\\{N_{\rm Bg}(\mathcal{A}_{\rm
d,\mu})|{\mu\in[1.6,10]}\\}\approx 0.16$ is marked with a dashed line. This
maximum is reached for $\mu=3.3$.

In Fig. 6, we highlight our results for $\mathcal{A}_{\rm d,\mu}$ for other
collision angles. Here we analyze the collision angles from $100^{\circ}$ to
$160^{\circ}$ in $10^{\circ}$ steps.

This figure shows the boundaries $\vartheta_{\rm l}(\varphi)$ and
$\vartheta_{\rm d}(\varphi)$ of the regions $\mathcal{A}_{\rm d,\mu}$.
Besides, the directions of the driving lasers are indicated: the colored
shading indicates the direction of the pump laser, the black one the direction
of the probe beam.

For each value of $\vartheta_{\rm col}$, we plot $\partial\mathcal{A}_{\rm
d,\mu}$ for six different values of $\mu\in\\{2,3,4,5,6,7\\}$. These are
indicated by the spacing of the dashing; the larger $\mu$, the smaller the
spacing of the dashes.

We observe that the upper bound $\vartheta_{\rm u}$ is almost independent of
$\mu$ for $\vartheta_{\rm col}>100^{\circ}$. The reason for this is that the
choice of $\mu$ affects only the focusing of the probe laser propagating along
$\vartheta=0$. At the same time, we find that the lower bound $\vartheta_{\rm
l}$ becomes smaller, i.e., $A(\mathcal{A}_{\rm d,mu})$ grows with increasing
$\mu$.

Analogous to Fig. 5 for $\vartheta_{\rm col}=160^{\circ}$, we plot the number
of signal and background photons per shot in the regions $\mathcal{A}_{\rm
d,\mu}$ as a function of $\mu$ in Fig. 7. Here the different photon numbers
are compared adapting a logarithmic scale. In each case, the number of signal
photons (solid) surpasses the number of background photons (dashed) in line
with the discernibility criterion adopted to determine $\mathcal{A}_{\rm
d,\mu}$. For each value of $\vartheta_{\rm col}$ considered here, a local
maximum in the numbers of signal or background photons is observed. The
positions $\mu_{\rm max}$ of these maxima shift to smaller values of $\mu$ as
the collision angle $\vartheta_{\rm col}$ increases. Moreover, we find that
the position of the maximum of the background varies slower with $\mu$ than
that for the signal when changing $\vartheta_{\rm col}$. In addition,
$\mu^{\rm Bg}_{\rm max}<\mu^{\rm Sig}_{\rm max}$ always holds in the
considered cases.

$\vartheta_{\rm col}$ | $\mu=\mu_{\rm max}^{\rm Sig}$ | $N_{\rm Sig}({\cal A}_{{\rm d},\mu})$ | $N_{\rm Bg}({\cal A}_{{\rm d},\mu})$
---|---|---|---
$100^{\circ}$ | $7.1$ | $0.237$ | $0.004$
$110^{\circ}$ | $6.9$ | $0.414$ | $0.006$
$120^{\circ}$ | $6.5$ | $0.694$ | $0.012$
$130^{\circ}$ | $6.4$ | $1.139$ | $0.017$
$140^{\circ}$ | $6.1$ | $1.898$ | $0.032$
$150^{\circ}$ | $5.8$ | $3.376$ | $0.062$
$160^{\circ}$ | $5.5$ | $5.839$ | $0.118$
Table 1: Maximum numbers of discernible signal photons $N_{\rm Sig}({\cal A}_{{\rm d},\mu})$ for different collision angles $\vartheta_{\rm col}$. The maximum is reached at $\mu=\mu_{\rm max}^{\rm Sig}$. For comparison, the corresponding number of background photons $N_{\rm Bg}({\cal A}_{{\rm d},\mu})$ is also given here. See Fig. 6 for the laser parameters employed here. $\vartheta_{\rm col}$ | $\mu=\mu_{\rm max}^{\rm Bg}$ | $N_{\rm Sig}({\cal A}_{{\rm d},\mu})$ | $N_{\rm Bg}({\cal A}_{{\rm d},\mu})$
---|---|---|---
$100^{\circ}$ | $4.5$ | $0.169$ | $0.005$
$110^{\circ}$ | $4.5$ | $0.309$ | $0.009$
$120^{\circ}$ | $4.4$ | $0.524$ | $0.016$
$130^{\circ}$ | $4.2$ | $0.838$ | $0.022$
$140^{\circ}$ | $4.0$ | $1.367$ | $0.041$
$150^{\circ}$ | $4.0$ | $2.669$ | $0.082$
$160^{\circ}$ | $3.8$ | $4.340$ | $0.161$
Table 2: Maximum numbers of background photons $N_{\rm Bg}({\cal A}_{{\rm
d},\mu})$ for different collision angles $\vartheta_{\rm col}$. The maximum is
reached at $\mu=\mu_{\rm max}^{\rm Bg}$. For comparison, the corresponding
number of signal photons $N_{\rm Sig}({\cal A}_{{\rm d},\mu})$ is also given
here. See Fig. 6 for the laser parameters employed here.

Tables 1 and 2 list the numbers of signal and background photons per shot at
the corresponding values $\mu_{\rm max}^{\rm Sig}$ and $\mu_{\rm max}^{\rm
Sig}$, respectively.

A comparison of these values confirms that the number of discernible signal
photons increases with $\vartheta_{\rm col}$. Note, however, that the infinite
Rayleigh range approximation allows for reliable insights only as long as
$2/\left(\mu w_{0}\omega_{0}\right)<\left|\sin\vartheta_{\rm col}\right|$; cf.
also Sect. III.

Interestingly, the area of the regions $\mathcal{A}_{\rm d,\mu}$ decreases
with increasing angle $\vartheta_{\rm col}$ – except for the results with
collision angle $\vartheta_{\rm col}=100^{\circ}$. For example, with
$\vartheta_{\rm col}=130^{\circ}$ at $\mu_{\rm max}^{\rm Sig}=6.4$ we find
$A(\mathcal{A}_{\rm d,6.4})\approx 1.10\pi$, whereas for $\vartheta_{\rm
col}=160^{\circ}$ we have $A(\mathcal{A}_{\rm d,6.4})\approx 0.85\pi$. This
indicates that the strength of the signal is more important than the size of
the region where the signal surpasses the background.

###  V.2 Probe with elliptical cross section

In a next step we turn to a probe beam with an elliptical cross section. To
this end, we introduce two independent waist sizes $w_{x}=\mu_{x}w_{0}$ and
$w_{y}=\mu_{y}w_{0}$ for the probe beam. These values determine the two semi-
axes of the elliptic cross section.

From the analysis of circular cross sections, see Section V.1, we know that
the signal becomes maximal for $w_{x}=w_{y}=\mu_{\rm max}^{\rm Sig}w_{0}$. The
tighter the focusing, the larger the intensity, which results in a stronger
signal. However, at the same time the divergence of the beam is increased, so
that the discernible signal generically decreases. If this beam is now focused
harder along only the one of the semi-axes, the signal is still expected to
increase, whereas it remains discernible in the perpendicular direction
characterized by keeping the value $\mu_{\rm max}^{\rm Sig}$ fixed. Keeping
other parameters fixed, such an elliptical cross section is expected to result
in a larger yield of discernible signal photons $N_{\rm Sig}$
Karbstein:2015xra ; Karbstein:2016lby .

For generic choices of $\mu_{x}$ and $\mu_{y}$, the parameter space is
significantly increased in comparison to the rotationally symmetric case
characterized by a single parameter $\mu=\mu_{x}=\mu_{y}$. For convenience and
for simplifying the following discussion, we fix the parameter $\mu_{x}$ to
the value maximizing the signal in section V.1 for the corresponding collision
angle $\vartheta_{\rm col}$, i.e. set $\mu_{x}=\mu_{\rm max}^{\rm
Sig}(\vartheta_{\rm col})$, cf. table 1. We choose $\mu_{y}(\vartheta_{\rm
col})\in[1,\mu_{\rm max}^{\rm Sig}(\vartheta_{\rm col})]$ as a free parameter,
since this leads to the maximum possible value of discernible signal photons
at $\vartheta_{\rm col}=160^{\circ}$. For comparison, we discuss afterwards
the case with $\mu_{y}=\mu_{\rm max}^{\rm Sig}$ and variable
$\mu_{x}\in[1,\mu_{\rm max}^{\rm Sig}(\vartheta_{\rm col})]$.

In a first step, we focus on a collision of angle $\vartheta_{\rm
col}=160^{\circ}$. We set $\mu_{x}=\mu_{\rm max}^{\rm Sig}(160^{\circ})=5.5$,
maximizing the discernible signal for the circular case, and keep $\mu_{y}$ as
a free parameter. All other parameters remain unchanged.

Figure 8: Area of solid angle regime $\mathcal{A}_{\rm d,\mu_{y}}$ as function
of $\mu_{y}$ for a collision angle of $\vartheta_{\rm col}=160^{\circ}$. For
low values of $\mu_{y}$ the region $\mathcal{A}_{\rm d\mu_{y}}$ is separated
into two parts, as illustrated by the red and blue colored curves. Their sum
yields the dashed green curve. The inlays show Mollweide plots of the areas
for the example values of $\mu_{y}=2$ and $\mu_{y}=3$. For $\mu_{y}=3$ the
region is simply connected; see the green solid line in the corresponding
plot. Both pulses have the same frequency $\omega_{0}=1.55\,{\rm eV}$, energy
$W=25\,{\rm J}$ and duration $\tau=25\,{\rm fs}$ (FWHM). Here, we depict
results for a probe with an elliptical cross section ($w_{x}=5.5w_{0}$,
$w_{y}=\mu_{y}w_{0}$) and a pump focused to $w_{0}=2\pi/\omega_{0}$.

First we consider the angular regions $\mathcal{A}_{\rm d,\mu_{y}}$ where the
signal dominates the background. Here the index $\mu_{y}$ indicates the
parameter to be varied.

In Fig. 8 we study the discernible solid angle area $A(\mathcal{A}_{\rm
d,\mu_{y}})$ as function of $\mu_{y}$. Unlike before, there exist two distinct
angular regions for small values of $\mu_{y}$, where the signal becomes
discernible. This is illustrated in Fig. 8 by the red and blue curves
representing the areas of these regions; the red (blue) curve measures the
area around $\varphi=0$ ($\varphi=180^{\circ}$). We denote the former (latter)
region by $\mathcal{A}_{\rm d,\mu_{y}}^{(1)}$ ($\mathcal{A}_{\rm
d,\mu_{y}}^{(2)}$). In Fig. 8 the third, dashed green curve indicates the sum
of both areas, $A(\mathcal{A}_{\rm d,\mu_{y}}^{(1)}\cup\mathcal{A}_{\rm
d,\mu_{y}}^{(2)})=A(\mathcal{A}_{\rm d,\mu_{y}})$. For $\mu_{y}\approx 2.3$
the two regions unite and beyond this value only a single angular region where
the signal is discernible persists; cf. the green solid line. Interestingly,
the value of $\mu_{y}$ where the two regions merge amounts to an inflection
point in the angular area. Figure 8 also shows two representative Mollweide
projections of the boundaries of these regions for $\mu_{y}=2$ and
$\mu_{y}=3$.

Tracing these areas in Mollweide projections as a function of $\mu_{y}$, it
can be observed how the two regions $\mathcal{A}_{\rm d,\mu_{y}}^{(1)}$ and
$\mathcal{A}_{\rm d,\mu_{y}}^{(2)}$ slowly converge and finally merge. In this
process the upper bounds $\vartheta_{\rm u}^{(m)}(\varphi)$ do not change
significantly; only the lower bounds $\vartheta_{\rm l}^{(m)}(\varphi)$
visibly change with $\mu_{y}$ predominantly in the vicinity of
$\varphi=90^{\circ}$ and $\varphi=270^{\circ}$.

Now, we compare the properties of the two disjoint regions for the example
value of $\mu_{y}=2$. It is obvious that the area $A(\mathcal{A}_{\rm
d,2}^{(1)})$ is much larger than $A(\mathcal{A}_{\rm d,2}^{(2)})$;
$A(\mathcal{A}_{\rm d,2}^{(2)})/A(\mathcal{A}_{\rm d,2})\approx 22.3\%$.
Interestingly, only $3.0\%$ of the background photons and $0.5\%$ of the
signal photons are located in this $22.3\%$ of the total area. In absolute
numbers, we count around 0.02 signal photons per shot in $\mathcal{A}_{\rm
d,2}^{(2)}$ and 7.22 in $\mathcal{A}_{\rm d,2}^{(1)}$. With rotationally
symmetric focusing, cf. section V.1 and in particular Fig. 4, the signal at
$\varphi=0^{\circ}$ is significantly weaker compared to $\varphi=180^{\circ}$.
An analogous behavior is observed for beams with elliptical average, the
effect is even amplified by the different beam waists.

Figure 9: Mollweide plot (longitude $\varphi$, latitude $\vartheta$) of the
signal $\rho_{1}^{\rm Sig}(\varphi,\vartheta)$ and background $\rho_{1}^{\rm
Bg}(\varphi,\vartheta)$ photon densities for $\vartheta_{\rm
col}=160^{\circ}$. The angular region where the signal is discernible is
highlighted by solid green lines (left color scales). Outside these regions,
the background dominates (right color scale). Both lasers deliver pulses of
energy $W=25\,{\rm J}$ and duration $\tau=25\,{\rm fs}$ (FWHM) at a frequency
of $\omega_{0}=1.55\,{\rm eV}$. The pump (probe) has a circular (elliptical)
cross section. The pump waist is $w_{0}=2\pi/\omega_{0}$ and the probe waists
are $w_{x}=5.5w_{0}$ and $w_{y}=\mu_{y}w_{0}$, respectively. Figure 10:
Mollweide plot (longitude $\varphi$, latitude $\vartheta$) showing angular
regions where the signal is discernible for different collision angles
$\vartheta_{\rm col}$: $100^{\circ}$ to $160^{\circ}$ in $10^{\circ}$ steps
labeled by the color order: blue, red, yellow, magenta, light green, cyan, and
brown. Additionally, the probe laser is marked by the black shade. This waist
sizes of the elliptically focused probe $w_{x}=\mu_{\rm max}^{\rm
Sig}(\vartheta_{\rm col})\,2\pi/\omega_{0}$ and
$w_{y}=\frac{1}{5}(5+l(\mu_{\rm max}^{\rm Sig}(\vartheta_{\rm
col})-1))\,\pi/\omega_{0}$. The contours for which $\rho_{\rm Sig}=\rho_{\rm
Bg}$ holds for a given collision angle are marked by the differently colored
lines associated with different values of $\mu_{y}=\frac{1}{5}(5+l(\mu_{\rm
max}^{\rm Sig}(\vartheta_{\rm col})-1))$ with $l\in\mathbb{N}^{[0,5]}$:
encoded by the type of lines; with higher wideness parameter, the spacing of
the dashes is smaller. The driving laser pulses have the same energy
$W=25\,\rm{J}$, duration $\tau=25\,{\rm fs}$, and frequency
$\omega_{0}=1.55\,{\rm eV}$. The pump waist is $w_{0}=2\pi/\omega_{0}$.

Figure 9 illustates the densities $\rho_{1}^{\rm Sig}(\varphi,\vartheta)$ and
$\rho_{1}^{\rm Bg}(\varphi,\vartheta)$ for the example $\mu_{x}=5.5$,
$\mu_{y}=1$ and $\vartheta_{\rm col}=160^{\circ}$. In contrast to Fig. 4, we
use here a logarithmic scale for the background. Figure 9 clearly shows how
the elliptical probe cross section affects the discernible signal. The ocher
colored area in the lower half sphere reflects the elliptical probe cross
section. Its shape is reminiscent to a strongly curved banana, which overlays
parts of the region of the signal. In the region around
$\vartheta=180^{\circ}$ the signal is much stronger than in the region around
$\varphi=0$ separated from it. This is consistent with previous observations
at $\mu_{y}=2$: the former region accounts for $21.2\%$ of the total area and
contains only $3\%$ of the background and $0.5\%$ of the discernible signal
photons. Considering the union of both regions, where the signal dominates the
background, 11.63 signal photons per shot can be counted, and 0.25 background
photons per shot.

Figure 10 shows the regions of the dominant signals for different collision
angles $\vartheta_{\rm col}$. The parameter $\mu_{x}=\mu_{\rm max}^{\rm
Sig}(\vartheta_{\rm col})$ is fixed for each collision angle $\vartheta_{\rm
col}$ to a corresponding value listed in table 1. At the same time, the
parameter $\mu_{y}$ is varied in the interval $[1,\mu_{\rm max}^{\rm
Sig}(\vartheta_{\rm col})]$. Figure 10 clearly shows that the upper bound
$\vartheta_{\rm u}(\varphi)$ typically does not change appreciably with
$\mu_{y}$ for constant $\vartheta_{\rm col}$. The only exception is the angle
$\vartheta_{\rm col}=100^{\circ}$. Here the collision angle is close to
$\vartheta_{\rm col}\approx 90^{\circ}$; thus the influence of the width of
the probe beam on the upper bond $\vartheta_{\rm u}(\varphi)$ varies more, cf.
also Fig. 6.

At the same time the lower boundary $\vartheta_{\rm l}(\varphi)$ changes
strongly with $\vartheta_{\rm col}$. Here we observe that pronounced local
maxima around $\varphi=90^{\circ}$ and $\varphi=270^{\circ}$ appear for
decreasing $\mu_{y}$. If the value of $\vartheta_{\rm l}(\varphi)$ at the
maxima reaches the corresponding upper bound $\vartheta_{\rm u}(\varphi)$,
then two separate angular regions are formed. However, this effect appears
only at larger collision angles – starting from $\vartheta_{\rm col}\simeq
120^{\circ}$ – and becomes more pronounced with larger angles $\vartheta_{\rm
col}$.

The discernible-signal area increases with $\mu_{y}$ for each $\vartheta_{\rm
col}$. At larger collision angles $\vartheta_{\rm col}$ the total area
$A(\mathcal{A}_{\rm d,\mu_{y}})$ is smaller than for smaller $\vartheta_{\rm
col}$.

Figure 11: Discernible number of signal photons in the solid angle regime
$\mathcal{A}_{\rm d,\bar{\mu}}$ as function of $\mu_{y}$ ($\mu_{x}$) with
constant $\mu_{x}=\mu_{\rm max}^{\rm Sig}$ ($\mu_{y}=\mu_{\rm max}^{\rm Sig}$)
for different collision angles $\vartheta_{\rm col}$ illustated by solid
(dashed) lines. Both lasers deliver frequency $\omega_{0}=1.55\,{\rm eV}$
photons at a pulse energy $W=25\,{\rm J}$ and duration $\tau=25\,{\rm fs}$
(FWHM). The pump is focused to $w_{0}=2\pi/\omega_{0}$. The probe waists are
$w_{x}=\mu_{\rm max}^{\rm Sig}\,w_{0}$ and $w_{y}=\mu_{y}w_{0}$
($w_{x}=\mu_{x}w_{0}$ and $w_{y}=\mu_{\rm max}^{\rm Sig}\,w_{0}$).

In Fig. 11, we highlight the dependence of the number of discernible signal
photons $N_{\rm Sig}(\mathcal{A}_{\rm d,\mu_{y}})$on $\mu_{y}$ for different
collision angles by solid lines. Additionally, we mark the number of
discernible signal photons as function of $\mu_{x}$ by dashed lines with a
constant choice of $\mu_{y}=\mu_{\rm max}^{\rm Sig}$. Obviously, the
discernible signal $N_{\rm Sig}(\mathcal{A}_{\rm d,\bar{\mu}})$ increases with
increasing collision angle $\vartheta_{\rm col}$ for each value of $\mu_{y}$
and $\mu_{x}$, respectively.

First we consider the results with fixed $\mu_{x}$ and varying $\mu_{y}$. It
is interesting to note that two (local) maxima can occur in the $\mu_{y}$
intervals studied. In any case the value $N_{\rm Sig}(\mathcal{A}_{\rm
d,\mu_{\rm max}^{\rm Sig}})$ for $\mu_{y}=\mu_{\rm max}^{\rm Sig}$ is a
maximum; cf. also discussion in section V.1. For all graphs with
$\vartheta_{\rm col}>100^{\circ}$ and variable $\mu_{y}$, there exists a local
minimum in the interval $\mu_{y}\in[1,\mu_{\rm max}^{\rm Sig}]$. Therefore,
the function increases with $\mu_{y}$ smaller than $\mu_{y}$ at this local
minimum. This leads to a maximum value (not necessarily a local maximum) at
the position $\mu_{y}=1$. However, this value is only for the collision angles
$\vartheta_{\rm col}=150^{\circ}$ and $\vartheta_{\rm col}=160^{\circ}$ larger
than the signal photon number at $\mu_{y}=\mu_{\rm max}^{\rm Sig}$. This
implies that for smaller collision angles $\vartheta_{\rm col}<150^{\circ}$
elliptical cross sections do not result in an increased discernible signal as
long as the semi-axes $\mu_{y}<\mu_{\rm max}^{\rm Sig}$ and $\mu_{x}=\mu_{\rm
max}^{\rm Sig}$ are fixed. On the other hand, for large collision angles
$\vartheta_{\rm col}$ elliptical probe cross sections can increase the
discernible signal. For $\vartheta_{\rm col}=150^{\circ}$ we determine $N_{\rm
Sig}(\mathcal{A}_{\rm d,1})=4.90$ signal photons per shot, for $\vartheta_{\rm
col}=160^{\circ}$ it is even $N_{\rm Sig}(\mathcal{A}_{\rm d,1})=11.63$.

If, on the other hand, we consider $\mu_{x}$ as a free parameter and fix
$\mu_{y}=\mu_{\rm max}^{\rm Sig}$ for the corresponding angle $\vartheta_{\rm
col}$, we find in all cases that the number of discernible signal photons
increases with decreasing $\mu_{x}$. Except for $\vartheta_{\rm
col}=140^{\circ}$, $\vartheta_{\rm col}=150^{\circ}$ and $\vartheta_{\rm
col}=160^{\circ}$, the maximum value is always found at $\mu_{x}=1$ and
$\mu_{y}=\mu_{\rm max}^{\rm Sig}$, see Tab. 3. Moreover, except for the angle
$\vartheta_{\rm col}=160^{\circ}$, the maxima are always larger than in the
case of variable $\mu_{y}$ and fixed $\mu_{x}$. With decreasing collision
angle $\vartheta_{\rm col}$, this is because the pronounced maxima of
$\rho(\varphi,\vartheta_{\rm l}(\varphi))$ of the signal are closer to
$\varphi=0$ or $\varphi=180^{\circ}$, i.e. the collision plane. Only at the
angle $\vartheta_{\rm col}=160^{\circ}$ with $\varphi=114.5^{\circ}$ and
$\varphi=245.5^{\circ}$ the pronounced maxima are closer to the plane of
$w_{y}$ than $w_{x}$. At the same time, the maximum of $N(\mathcal{A}_{{\rm
d},\bar{\mu}})$ for $\vartheta_{\rm col}=160^{\circ}$ is significantly larger
for $\mu_{y}=1$, $\mu_{x}=\mu_{\rm max}^{\rm Sig}$, which is why we focus here
on the case of variable $\mu_{y}$.

The kinks visible in some graphs in Fig. 11 are numerical artifacts; they
become more pronounced as the collision angle $\vartheta_{\rm col}$ increases.
They occur when two regions merge, as shown in Fig. 8, affecting the precision
of the numerical integration. Also, the artifacts appear at the value for
$\mu_{y}$ where $\phi_{i}=0$ and $\varphi_{f}=2\pi$ for the boundary functions
$\vartheta_{\rm l/u}(\varphi)$ of $\mathcal{A}_{\rm d,\mu_{y}}$ appears for
the first time.

$\vartheta_{\rm col}$ | $\mu_{x}$ | $\mu_{y}$ | $N_{\rm Sig}({\cal A}_{{\rm d},\bar{\mu}})$ | $N_{\rm Bg}({\cal A}_{{\rm d},\bar{\mu}})$
---|---|---|---|---
$100^{\circ}$ | $1$ | $7.1$ | $0.294$ | $0.009$
$110^{\circ}$ | $1$ | $6.9$ | $0.586$ | $0.009$
$120^{\circ}$ | $1$ | $6.5$ | $1.044$ | $0.016$
$130^{\circ}$ | $1$ | $6.4$ | $1.864$ | $0.032$
$140^{\circ}$ | $1.113$ | $6.1$ | $3.205$ | $0.061$
$150^{\circ}$ | $1.32$ | $5.8$ | $5.243$ | $0.116$
$160^{\circ}$ | $5.5$ | $1.$ | $11.633$ | $0.248$
Table 3: Maximum numbers of discernible signal photons $N({\cal A}_{{\rm
d},\bar{\mu}})$ and its corresponding relative probe waist sizes $\mu_{y}$ and
$\mu_{y}$ for two-beam collision with elliptic probe for different collision
angles $\vartheta_{\rm col}$. These parameters are selected according to the
maximum signal, whereby at least one parameter fulfills $\mu_{x}=\mu_{\rm
max}^{\rm Sig}(\vartheta_{\rm col})$ or $\mu_{y}=\mu_{\rm max}^{\rm
Sig}(\vartheta_{\rm col})$. For comparison, the corresponding number of
background photons $N_{\rm Bg}({\cal A}_{{\rm d},\bar{\mu}})$ is also given
here. See Fig. 11 for the laser parameters employed here.

In table 3 we list the maximal discernible signal photon number for the
parameters considered here. Only for $\vartheta_{\rm col}=160^{\circ}$, the
maxmimum is detected for the fixed choice $\mu_{x}=\mu_{\rm max}^{\rm
Sig}(\vartheta_{\rm col})$. Since this signal is much larger than the other
maximum, we limit our discussion to this parameter choice. In addition, table
3 lists the corresponding number of background photons. Here, we want to
emphasize again that in the determined regions the signal significantly
dominates the background. d thus this focus can contribute to a larger maximum
background.

###  V.3 Quantum vacuum birefringence

Let us finally consider the phenomenon of birefringence. For this, we
concentrate exclusively on the collision angle $\vartheta_{\rm
col}=160^{\circ}$ and the two parameter sets of circular or elliptical cross
sections where polarization insensitive measurements maximize the discernible
signal, i.e. we use the width parameters $\mu=5.5$ or $\mu_{x}=5.5$,
$\mu_{y}=1$, respectively.

For birefringence, we now set the polarization angle $\beta_{2}$ of the pump
beam to $\beta_{2}=45^{\circ}$, as explained earlier in the section III. We
consider only signal photons whose polarization is perpendicular to the
original linear polarization of the probe beam with flipped polarization angle
$\beta_{\perp}$ Dinu:2013gaa ; Karbstein:2015xra . This implies ${\bf
e}_{\beta_{\perp}}\cdot{\bf E}_{1}({\bf x},t)=0$ such that
$\beta_{\perp}=\arctan\left(\cos\vartheta\cot\varphi\right)$ holds.

We determine the number of signal photons as before with the restriction that
no sum over the polarizations is performed. Instead, we use
$(p)\rightarrow\beta_{\perp}$ and derive the corresponding signal photon
density

$\displaystyle\rho_{\bar{\mu}}^{\rm Sig,\perp}(\varphi,\vartheta)$
$\displaystyle=\frac{1}{(2\pi)^{7}}\frac{2}{45^{2}}\left(\frac{e}{m_{e}}\right)^{8}$
$\displaystyle\phantom{=}\times\Big{[}g_{\beta_{\perp}\frac{\pi}{4},121}^{2}(\varphi,\vartheta)\\!\int_{-\infty}^{\infty}\\!\\!\\!\mathrm{dk}\,{\rm
k}^{3}\,\mathcal{I}_{121}^{2}({\bf k})$
$\displaystyle\phantom{=\times}+g_{\beta_{\perp}\frac{\pi}{4},212}^{2}(\varphi,\vartheta)\\!\int_{-\infty}^{\infty}\\!\\!\\!\mathrm{dk}\,{\rm
k}^{3}\,\mathcal{I}_{212}^{2}({\bf k})\Big{]}\,.$ (37)

Compared with Eq. 30, only the value of the function
$g_{\beta_{\perp}\frac{\pi}{4},iji}$ changes. The number of polarization
flipped signal photons $N_{\rm Sig,\perp}(\mathcal{A},\bar{\mu})$ follows
analogously to the Eq. 33.

Figure 12: Discernible number of polarization flipped signal photons in the
solid angle regime $\mathcal{A}_{\rm d,\bar{\mu}}$ as function pof the purity
$\mathcal{P}$ of the polarization of the probe beam in logarithmic scale. Here
only the collision scenarios for a collision angles $\vartheta_{\rm
col}=160^{\circ}$ are analyzed. Both linear polarized driving lasers deliver
frequency $\omega_{0}=1.55\,{\rm eV}$ photons at a pulse energy $W=25\,{\rm
J}$ and duration $\tau=25\,{\rm fs}$ (FWHM). The pump is focused to
$w_{0}=2\pi/\omega_{0}$. The probe waists are $w_{x}=\mu_{x}\,w_{0}$ and
$w_{y}=\mu_{y}w_{0}$.

In order to estimate the background, we introduce the polarization purity
$\mathcal{P}$ characterizing the quality of the probe polarimetry polarizer-
analyzer system Bernhardt:2020vxa . Here we consider realistic purities on the
range from $\mathcal{P}=10^{-1}$ to $\mathcal{P}=10^{-10}$ in logarithmic
steps. For a conservative estimate of the solid angle region of the
discernible signal, we use

$\displaystyle\mathcal{A}_{\rm d,\bar{\mu}}$
$\displaystyle=\Big{\\{}(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\pi\right)\left.\right|$
$\displaystyle\phantom{=\Big{\\{}}\rho^{\rm
Sig,\perp}_{\bar{\mu}}(\varphi,\vartheta)\geq\mathcal{P}\frac{{\rm d}N^{\rm
Bg}_{1}}{{\rm d}\Omega}+\frac{{\rm d}N^{\rm Bg}_{2}}{{\rm
d}\Omega}\Big{\\}}\,,$ (38)

where we have not applied any polarization constraints to the background of
the pump beam. The latter has no influence in the relevant region of the
flipped signal and additionally gives us an upper bound $\vartheta_{\rm
u}(\varphi)$ for easier evaluation of the integrations.

In Fig. 12, the number of signal photons in the dominant regions $N_{\rm
Sig,\perp}(\mathcal{A}_{\rm d,\bar{\mu}})$ is plotted as a function of the
purity $\mathcal{P}$. We also distinguish between an elliptical and a circular
probe beam. In both cases there is always a sufficient number of discernible
signal photons; as expected the number decreases with the purity, however,
only rather mildy in accordance with the findings of Karbstein:2021ldz .

Let us discuss the results for $\mathcal{P}=10^{-10}$. Considering a probe
beam with circular cross section we obtain $N_{\rm Sig,\perp}(\mathcal{A}_{\rm
d,5.5})=0.44$ signal photons per shot against a background of $N_{\rm
Bg,\perp}(\mathcal{A}_{\rm d,5.5})=0.01$ on an area of $A(\mathcal{A}_{\rm
d,5.5})=0.85\pi$. In the case of the elliptically focused probe beam we get
$N_{\rm Sig,\perp}(\mathcal{A}_{\rm d,1})=1.00$ discernible signal photons per
shot and a background of $N_{\rm Bg,\perp}(\mathcal{A}_{\rm d,1})=0.03$. The
area of the region of the discernible signal is $A(\mathcal{A}_{\rm
d,1})=0.66\pi$. In order to put these values into context, let us compare them
with the number of signal photons in a forward cone
$\mathcal{A}_{\frac{\pi}{4}}=\left\\{(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\frac{\pi}{4}\right]\right\\}$.
Here we count $N_{\rm Sig,\perp}(\mathcal{A}_{\frac{\pi}{4}})=0.86$ by using a
probe beam with circular cross section or $N_{\rm
Sig,\perp}(\mathcal{A}_{\frac{\pi}{4}})=3.67$ with an elliptical probe.
Accordingly, it is possible to measure a significant fraction $0.51\%$
($0.27\%$) of the whole signal for a circular (elliptical) probe.

However, the number of driving laser photons beyond the output polarity
occurring in $\mathcal{A}_{\frac{\pi}{4}}$ exceeds these numbers by far
($N_{\rm Bg,\perp}(\mathcal{A}_{\frac{\pi}{4}})\approx 10^{20}$), which shows
that our method for optimizing the measurement region of the signature of the
quantum vacuum is also useful for the effect of birefringence.

The rather weak dependence of the discernible-photon number on the
polarization purity might seem surprising but is completely consistent with
the observations of vacuum birefringence studies with x-ray probes such as,
e.g., Schlenvoigt:2016 ; Karbstein:2021ldz ; Mosman:2021vua : while a
substantial degree of polarization purity is absolutely essential for a
birefringence measurement on axis, a less pure setup can be compensated for by
a scenario that includes scattering into low-background regions.

##  VI Conclusions and Outlook

In an effort to increase the signal of nonlinear vacuum-induced interactions,
we have studied the relevance of the choice of beam waists in the collision of
two optical laser pulses. Focusing on collision angles in the range
$100^{\circ}\leq\vartheta_{\rm col}\leq 160^{\circ}$, we see a decisive
dependence of the detectable signal strength in terms of discernible photons
on the beam waist of the probe beam both for circular as well as elliptical
beam cross section.

In order to simplify the experimental scenario, we consider the two pulses as
originating from the same ultra-intense laser source, having similar
properties with respect to pulse duration, frequency, and total energy; the
common source is taken to resemble the parameters available in Doyle:2021mdt ;
Hartmann:2021iyf ; CALA ; JETI-200 . For a mostly analytically accessible and
efficient theoretical modeling, the laser pulses are considered in the
infinite Rayleigh range approximation. This allows for a controlled
determination and analysis of discernible signals and their angular emission
regimes in the considered parameter ranges.

The essential mechanisms become already visible for the simpler case of a
probe beam with circular cross section. One of our key findings is that the
maximization of the discernible signal requires a comprise between increasing
the intensity in the collision region and decreasing the background in the
detection region. While the signal increases quadratically with the intensity,
the background decreases exponentially beyond the outgoing laser cone which in
turn decreases for larger waist sizes. In the range of collision angles
$100^{\circ}<\theta_{\rm col}<160^{\circ}$, we observe that an optimal beam
waist is about $\sim 5\dots 7$ times bigger than the diffraction limit. Our
results are in line with those found in a counterpropagating setup
Mosman:2021vua .

For the largest scattering angle studied in this work, elliptical cross
sections come with further mechanisms to increase the signal: suitable choices
of the ellipticity can increase the intensity in the collision region while
maintaining the background suppression in the detection region.

Additionally, we investigated the discernible photon signal including a
polarization flip in promising laser configurations. This signal of
birefringence was determined as a function of the polarization purity; the
results are promising for probe beams with circular cross section as well as
with elliptical cross section.

For completeness, let us point out that further sources of background photons
can become relevant in experiments. Apart from the photons comprising the
driving laser beams included in our present study, there is generically a
scattering background. Such scattering originates in diffration of laser
photons off the walls of the vacuum chamber and optical apertures both
upstream from the beam guidance as well as downstream in the collection
optics. Furthermore (relativistic) Thomson and (relativistic) Rayleigh
scattering by residual gas ions and their electrons can occur and need to be
controlled and reduced in experiment. First ideas of reducing these background
sources through space and time resolved detection have already been
successfully implemented Doyle:2021mdt .

We expect that the mechanisms studied in this work that maximize the signal
can also be operative in collisions of a larger number of pulses Klar:2020ych
; Moulin:1999hwj ; Lundstrom:2005za ; Lundin:2006wu ; Gies:2017ezf ;
Aboushelbaya:2019ncg ; in addition, also scattering into different frequency
channels can lead to further enhancements in such scenarios Fillion-
Gourdeau:2014uua ; Huang:2019ojh ; Gies:2021ymf ; Sasorov:2021anc ;
Sangal:2021qeg .

In summary, the present work demonstrates that a detailed modeling of laser
pulse collisions based on theory from first principles can identify unexpected
sweet spots for future discovery experiments of QED vacuum nonlinearities.

###### Acknowledgements.

We are grateful to Richard Schmieden for helpful discussions. This work has
been funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant Nos.
392856280, 416607684, and 416611371 within the Research Unit FOR2783/2.

##  Appendix A Solid angle regions of discernible signal

Figure 13: Scheme for the numerical determination of the lower bound of
$\vartheta_{\rm l}$ at constant $\varphi$ for a dominant signal. The evaluated
functions $\rho_{\bar{\mu}}^{\rm Sig}(\varphi,\vartheta)$ and
$\rho_{\bar{\mu}}^{\rm Bg}(\varphi,\vartheta)$ are compared at fixed $\varphi$
on the interval $[\vartheta_{i},\vartheta_{f}]$, starting in steps
$\Delta\vartheta$ ascending from $\vartheta_{i}$. In the first iteration
$s={\rm I}$ we define $\vartheta_{i}\equiv 0$ and $\vartheta_{f}\equiv\pi$. If
always $\rho_{\bar{\mu}}^{\rm Bg}(\varphi,\vartheta)>\rho_{\bar{\mu}}^{\rm
Sig}(\varphi,\vartheta)$ is valid, the interval is scanned again with a
quarter of the step size. If there is no finding for given $\varphi$ the
signal is classified as recessive in this region. If the signal is dominant at
$\vartheta_{{\rm l},s}$, then the run is terminated prematurely and the next
iteration step $s\rightarrow s+1$ begins. Here, the interval boundaries are
adjusted to $\vartheta_{i}\equiv\vartheta_{{\rm l},s-1}-\Delta\vartheta$ and
$\vartheta_{f}\equiv\vartheta_{{\rm l},s-1}$ according to the previous
iteration step. The step size is redetermined and the procedure starts again.
After seven iterations, the result $\vartheta_{\rm l}=\vartheta_{\rm l,VII}$
is used.

A signal is discernible if the local photon density of the signal surpasses
the density of the photons of the driving lasers. The solid angle region where
the signal dominates is defined by

$\mathcal{A}_{\rm
d,\bar{\mu}}=\left\\{(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\pi\right)\left.\right|\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)\geq\rho^{\rm
Bg}_{\bar{\mu}}(\varphi,\vartheta)\right\\}\,.$ (39)

The shape of this region depends on the parameters of the interacting lasers,
in particular the choice of the beam waists, represented by $\bar{\mu}$, and
the collision angle $\vartheta_{\rm col}$. This region can be of variable
shape and is in general not simply connected. If the signal is nowhere
discernible, then $\mathcal{A}_{\rm d,\bar{\mu}}=\\{\;\\}$.

We denote the boundary of the region $\mathcal{A}_{\rm d,\bar{\mu}}$ by

$\partial\mathcal{A}_{\rm
d,\bar{\mu}}=\left\\{(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\pi\right)\left.\right|\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)=\rho^{\rm
Bg}_{\bar{\mu}}(\varphi,\vartheta)\right\\}\,.$ (40)

For simply connected regions, the boundary can be parameterized by two
functions $\vartheta_{\rm l}(\varphi)\leq\vartheta_{\rm u}(\varphi)$ depending
on the azimuthal angle $\varphi$, where $\vartheta_{\rm l}(\varphi)$ denotes
the lower bound and $\vartheta_{\rm u}(\varphi)$ the upper bound for a
discernible signal region. These functions are defined in the interval
$[\varphi_{i},\varphi_{f}]$ with $\varphi_{i}$ and $\varphi_{f}$ implicitly
determined by $\vartheta_{\rm l}(\varphi_{i})=\vartheta_{\rm u}(\varphi_{i})$
and $\vartheta_{\rm l}(\varphi_{f})=\vartheta_{\rm u}(\varphi_{f})$ for
$\varphi_{i}>0$ and $\varphi_{f}<2\pi$. If the latter condition is violated,
$\vartheta_{\rm l/u}(0)=\vartheta_{\rm l/u}(2\pi)$ applies in both cases,
where the values of the two functions on the upper and lower boundary may now
be unequal. In this case, we have $\partial\mathcal{A}_{\rm
d,\bar{\mu}}=\left\\{\vartheta_{\rm
l}(\varphi)\left.\right|\varphi\in\left[\varphi_{i},\varphi_{f}\right]\right\\}\cup\left\\{\vartheta_{\rm
u}(\varphi)\left.\right|\varphi\in\left[\varphi_{i},\varphi_{f}\right]\right\\}$.

If the region $\mathcal{A}_{\rm d,\bar{\mu}}$ is not simply connected, but at
most two mappings to $\vartheta$ can be found for any given value of
$\varphi$, then the $\varphi$ domain can be divided into $M$ connected
intervals $[\varphi_{i}^{(m)},\varphi_{f}^{(m)}]$ labeled by $m=1,2\dots,M$,
where the lower and upper bounds $\vartheta_{l/u}^{(m)}(\varphi)$ can be
defined such that $\partial\mathcal{A}_{\rm
d,\bar{\mu}}=\bigcup_{m=1}^{M}\Big{(}\left\\{\vartheta_{\rm
l}^{(m)}(\varphi)\left.\right|\varphi\in\left[\varphi_{i}^{(m)},\varphi_{f}^{(m)}\right]\right\\}\cup\left\\{\vartheta_{\rm
u}^{(m)}(\varphi)\left.\right|\varphi\in\left[\varphi_{i}^{(m)},\varphi_{f}^{(m)}\right]\right\\}\Big{)}$.
It is possible to discuss even more complicated shapes of $\mathcal{A}_{\rm
d,\bar{\mu}}$ and $\partial\mathcal{A}_{\rm d,\bar{\mu}}$, respectively, but
this is not necessary in the present case, cf. below.

In order to determine $\mathcal{A}_{\rm d,\bar{\mu}}$ explicitly, we first
calculate $\partial\mathcal{A}_{\rm d,\bar{\mu}}$ by determining all possible
functions $\vartheta(\varphi)$, which solve the equation $\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)=\rho^{\rm
Bg}_{\bar{\mu}}(\varphi,\vartheta)$ on
$(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\pi\right)$. Since
$\rho^{\rm Sig}_{\bar{\mu}}$ in general exhibits a complicated dependency on
$\vartheta$ and $\varphi$, we construct a algorithm for its determination.

Being aware of potential shortcomings of this approach discussed below, we
determine both a lower bound $\vartheta_{\rm l}(\varphi)$ and upper bound
$\vartheta_{\rm u}(\varphi)\geq\vartheta_{\rm l}(\varphi)$ in a first step.
For this end, we sample the interval $\varphi\in[0,2\pi]$ by $n_{\varphi}$
sampling points, usually using $n_{\varphi}=300$ or bigger. For each given
value of $\varphi$, we then determine the lower/upper bounds $\vartheta_{\rm
l/d}(\varphi)$ using the iterative procedure illustrated schematically in Fig.
13. In iteration I the interval $\vartheta\in[0,\pi)$ is divided into
$n_{\vartheta}$ points with $\Delta\vartheta=\frac{\pi}{n_{\vartheta}}$, using
$n_{\vartheta}=70$ or bigger. Depending on whether we are looking for the
upper or lower bound we start with $\vartheta=\pi$ or $\vartheta=0$. In the
following we concentrate on the determination of the lower bound. To this end,
we evaluate both $\rho_{\bar{\mu}}^{\rm Sig}$ and $\rho_{\bar{\mu}}^{\rm Bg}$
at each sampling point with ascending value of
$\vartheta\rightarrow\vartheta+\Delta\vartheta$ from the initial value
$\vartheta=0$ and compare them. If $\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)<\rho^{\rm
Bg}_{\bar{\mu}}(\varphi,\vartheta)$ holds, we increase $\vartheta$ by
$\Delta\vartheta$ and compare again. If the signal density becomes larger or
equal to the background density, we terminate the procedure and register this
value as $\vartheta_{\rm l,I}$. If the first iteration did not result in a
positive outcome, we decrease the step size $\Delta\vartheta$ by one-fourth,
$\Delta\vartheta\rightarrow\frac{1}{4}\Delta\vartheta$, and repeat iteration
I. If this also does not yield a result, we assume an empty solution for this
value of $\varphi$. We continue with the next iteration step.

In iteration II, we refine the outcome of iteration I by adjusting the bounds
of the $\vartheta$ domain to be studied. As initial value, we now use
$\vartheta_{i}=\vartheta_{\rm l,I}-\Delta\vartheta$, as final value
$\vartheta_{f}=\vartheta_{\rm l,I}$. From these two values a new step size
$\Delta\vartheta=\frac{\vartheta_{f}-\vartheta_{i}}{n_{\vartheta}}$ is
calculated. Starting with $\vartheta=\vartheta_{i}$ the values of
$\rho_{\bar{\mu}}^{\rm Sig}$ and $\rho_{\bar{\mu}}^{\rm Bg}$ are compared
again until the signal dominates. The corresponding value is registered and
$\vartheta_{\rm l,II}$ is used as input for the next iteration III which
resembles iteration II with $\vartheta_{\rm l,I}\rightarrow\vartheta_{\rm
l,II}$.

After seven iterations we stop and use $\vartheta_{\rm l,VII}=\vartheta_{\rm
l}$. By successively decreasing the intervals and step sizes, this value has a
maximum error of $\pi n_{\vartheta}^{-7}$, which is negligible in the context
of the accuracy of the applied approximations.

Applying this procedure to all sampling points of $\varphi$ we obtain a table
of mappings $\\{\varphi,\vartheta_{\rm l}\\}$. We use this table to
interpolate a function $\vartheta_{\rm l}(\varphi)$ (or several functions for
regions which are not simply connected). For this interpolation we fit third-
order polynomials between successive data points. Analogously, we obtain the
interpolation $\vartheta_{\rm u}(\varphi)$ of the upper bound. These results
allow us to determine the solid angle region where the signal dominates the
background as $\mathcal{A}_{\rm d,\bar{\mu}}=\mathcal{A}_{\rm
l,{\bar{\mu}}}\cap\mathcal{A}_{\rm u,{\bar{\mu}}}$ with $\mathcal{A}_{\rm
l,{\bar{\mu}}}=\left\\{(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\pi\right)\left.\right|\vartheta(\varphi)\geq\vartheta_{\rm
l}(\varphi))\right\\}$ and $\mathcal{A}_{\rm
u,{\bar{\mu}}}=\left\\{(\varphi,\vartheta)\in\left[0,2\pi\right]\times\left[0,\pi\right)\left.\right|\vartheta(\varphi)\leq\vartheta_{\rm
u}(\varphi))\right\\}$.

Figure 14: Potential source of error of the algorithm for determining the
solid angle region of the dominant signal. The frames show the range where
$\rho^{\rm Sig}_{\bar{\mu}}(\varphi,\vartheta)=\rho^{\rm
Bg}_{\bar{\mu}}(\varphi,\vartheta)$. Left is the real requested region shown;
on the other hand, right is the result according to the used numerical method
shown. Since the numerical method always assumes exactly one upper and one
lower bound, the multiple mappings of $\varphi$ to $\vartheta$, as visible
left, cannot be detected. Such a case was observed at a collision angle
$\vartheta_{\rm col}=90^{\circ}$ and circular focusing $\mu=5$ of the probe
beam.

The iterative algorithm may provide us with an erroneous result for
$\mathcal{A}_{\rm d,\bar{\mu}}$. Figure 14 illustrates a potential source of
error: if for a fixed value of $\varphi$ more than two values of $\vartheta$
exist, then the numerical method presented above fails. However, it
nevertheless makes sense to adopt this method for the following reasons:
first, this algorithm remains fast and compact in comparison to additionally
resolving distinguished regions with multiple values of $\vartheta$ for fixed
values of $\varphi$. Second – and most importantly – we search for regions
fulfilling $\int_{\mathcal{A}}\\!{\rm d}\Omega[\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)-\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)]>0$. As long the total integral over
$\mathcal{A}$ is positive, the signal dominates the background, regardless of
whether this is also the case locally in $\mathcal{A}$. A potential failure of
the algorithm therefore does not lead to wrong estimates for the signal photon
number. Third, we can make sure that region $\mathcal{A}_{\rm d,\bar{\mu}}$
satisfies the desired condition by plotting the function $\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)-\rho^{\rm
Sig}_{\bar{\mu}}(\varphi,\vartheta)>0$ and compare it with the graph of region
$\mathcal{A}_{\rm d,\bar{\mu}}$.

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