Vacuum birefringence in strong inhomogeneous electromagnetic
fields
Karbstein, F., Gies, H., Reuter, M., & Zepf, M. (2015). Vacuum birefringence in strong inhomogeneous
electromagnetic fields.
Physical Review D - Particles, Fields, Gravitation and Cosmology
,
92
(7), [071301].
https://doi.org/10.1103/PhysRevD.92.071301
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Download date:09. Aug. 2022
Vacuum birefringence in strong inhomogeneous electromagnetic fields
Felix Karbstein,
1,2
Holger Gies,
1,2
Maria Reuter,
1,3
and Matt Zepf
1,3,4
1
Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany
2
Theoretisch-Physikalisches Institut, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena,
Max-Wien-Platz 1, 07743 Jena, Germany
3
Institut für Optik und Quantenelektronik, Max-Wien-Platz 1, 07743 Jena, Germany
4
Centre for Plasma Physics, School of Mathematics and Physics, Queen’s University Belfast,
Belfast BT7 1NN, United Kingdom
(Received 8 July 2015; revised manusc ript received 24 August 2015; published 26 October 2015)
Birefringence is one of the fascinating properties of the vacuum of quantum electrodynamics (QED)
in strong electromagnetic fields. The scattering of linearly polarized incident probe photons into a
perpendicularly polarized mode provides a distinct signature of the optical activity of the quantum vacuum
and thus offers an excellent opportunity for a precision test of nonlinear QED. Precision tests require
accurate predictions and thus a theoretical framework that is capable of taking the detailed experimental
geometry into account. We derive analytical solutions for vacuum birefringence which include the spatio-
temporal field structure of a strong optical pump laser field and an x-ray probe. We show that the angular
distribution of the scattered photons depends strongly on the interaction geometr y and find that scattering
of the perpendicularly polarized scattered photons out of the cone of the incident probe x-ray beam
is the key to making the phenomenon experimentally accessible with the current generation of
FEL/high-field laser facilities.
DOI: 10.1103/PhysRevD.92.071301 PACS numbers: 12.20.Ds, 42.50.Xa, 12.20.−m
In strong electromagnetic fields the vacuum of quantum
electrodynamics (QED) has peculiar properties. Fluctuations
of virtual charged particles give rise to nonlinear, effective
couplings between electromagnetic fields [1–3],which,e.g.,
can impact and modify the propagation of light, and e ven
trigger the spontaneous decay of the vacuum via Schwinger
pair production in electric fields [2,4,5]. Even though subject
to high-energy experiments [6], so far the pure electromag-
netic nonlinearity of the quantum vacuum has not been
verified directly on macroscopic scales. In particular, the
advent of petawatt class laser facilities has stimulated various
proposals to probe quantum vacuum nonlinearities in high-
intensity laser experiments; see the pertinent reviews [7–11]
and references therein. One of the most famous optical
signatures of vacuum nonlinearity in strong electromagnetic
fields is vacuum birefringence [12–16], which is so far
searched for in experiments using macroscopic magnetic
fields [17,18]. A proposal to verify vacuum birefringence
with the aid of high-intensity lasers has been put forward
by [19], envisioning the combination of an optical high-
intensity laser as pump and a linearly polarized x-ray pulse
as probe; cf. also [20] who study x-ray diffraction by a strong
standing electromagnetic wave. While [19] proposed the
experimental scenario depicted in Fig. 1 for the first time, its
theoretical description does not account for the possibility of
momentum transfer from the pump field inhomogeneity—
incorporating this in the present work entails an enhance-
ment of the birefringence signal-to-noise ratio of several
orders of magnitude.
In the present letter, we reanalyze vacuum birefringence
in manifestly inhomogeneous fields, rephrasing the
phenomenon in terms of a vacuum emission process
[21]. We use new theoretical insights into photon propa-
gation in slowly varying inhomogeneous electromagnetic
fields [22], allowing us to overcome previous limitations
and to calculate the angular divergence of the cross-
polarized photons for the first time. Our study provides
a new twist and perspective on the feasibility of future
vacuum birefringence experiments. The key idea—so far
completely unappreciated in this context—is to exploit the
diffraction spreading of the outgoing signal photons. We
detail on a realistic experimental setup combining a high-
intensity laser system and an XFEL source as envisioned at
the Helmholtz International Beamline for Extreme Fields
(HIBEF) [23] at the European XFEL [24] at DESY. The
FIG. 1 (color online). Sketch of the pump-probe type scenario
intended to verify vacuum birefringence. A linearly polarized
optical high-intensity laser pulse—wavevector κ, electric (mag-
netic) field E (B)—propagates along the positive z axis. Its strong
electromagnetic field couples to the charged particle-antiparticle
fluctuations in the qua ntum vacuum, and thereby effectively
modifies its properties to be probed by a counter-propagating
x-ray beam (wavevector k, polarization ϵ). Vacuum birefringence
manifests itself in an ellipticity of the outgoing x-ray photons
(wavevector k
0
, polarization components along ϵ
ð1Þ
and ϵ
ð2Þ
).
PHYSICAL REVIEW D 92, 071301(R) (2015)
1550-7998=2015=92(7)=071301(6) 071301-1 © 2015 American Physical Society
RAPID COMMUNICATIONS
potential of diffraction effects for novel experimental
signatures of quantum vacuum nonlinearity in other
all-optical scenarios has been appreciated before; cf.,
e.g., [25–28].
From a theoretical perspective vacuum birefringence is
most conveniently analyzed within the effective theory
describing the propagation of macroscopic photon fields A
μ
in the quantum vacuum. The corresponding effective action
is S
eff
½A; A¼S
MW
½AþS
int
½A; A, where S
MW
½A¼
−
1
4
R
x
F
μν
ðxÞF
μν
ðxÞ is the Maxwell action of classical
electrodynamics, with F
μν
denoting the field strength
tensor of both the pump A
μ
and the probe A
μ
field, i.e.,
A
μ
¼ A
μ
þ A
μ
. The additional term S
int
½A; A¼
−
1
2
R
x
R
x
0
A
μ
ðxÞΠ
μν
ðx; x
0
jAÞA
ν
ðx
0
Þ encodes quantum cor-
rections to probe photon propagation and vanishes for
ℏ → 0. All the effective interactions with the pump laser
field are encoded in the photon polarization tensor
Π
μν
ðx; x
0
jAÞ, evaluated in the A
μ
background. In momen-
tum space, Π
μν
in inhomogeneous fields A
μ
ðxÞ generically
mediates between two independent momenta k
μ
and k
0μ
,
i.e., Π
μν
≡ Π
μν
ðk
0
;kjAÞ, This prevents a straightforward
diagonalization of the probe photons’ equation of motion,
as possible in homogeneous fields [12,14], where Π
μν
only
depends on the momentum transfer ðk
0
− kÞ
μ
due to trans-
lational invariance.
A particular convenient way to analyze vacuum birefrin-
gence in inhomogeneous fields is to phrase it as a v acuum
emission process [21]. Viewing the pump and probe beams
as macroscopic electromagnetic fields, and not resolving
the indi vidual photons constituting the beams, the vacuum
subjected to these electromagnetic fields can be interpreted
as a source term for outgoing photons. From this perspective,
the induced photons correspond to the signal photons
imprinted by the effective interaction of the pump and probe
beams. For a linearly polarized x-ray beam brought into
head-on collision with a linearly polarized optical high-
intensity laser pulse, the induced x-ray signal generically
encompasses photons whose polarization characteristics
differ from the original probe beam. This results in induced
photons polarized perpendicularly to the incident probe
beam, which constitutes a signal of vacuum birefringence;
cf. [29]. The outgoing x-ray beam, made up of the induced
x-ray photons as well as the original probe beam which has
traversed the pump laser pulse, then effectively picks up a
tiny ellipticity.
We assume a linearly polarized x-ray probe beam,
A
ν
ðxÞ¼
1
2
E
ω
ϵ
ν
ð
ˆ
kÞe
iωð
ˆ
kxþt
0
Þ−ð
ˆ
kxþt
0
T=2
Þ
2
, of frequency ω ¼
2π
λ
probe
,
peak field amplitude E and pulse duration T; ϵ
ν
ð
ˆ
kÞ is the
polarization vector of the beam and
ˆ
k
μ
¼ð1;
ˆ
kÞ, with unit
wavevector
ˆ
k. In momentum space, the x-ray photon
current generated by the pump and probe laser fields
can be expressed as j
μ
ðk
0
Þ¼
ffiffi
π
p
2
E
ω
T
2
R
d ~ω
2π
e
−
1
4
ð
T
2
Þ
2
ð
~
ω−ωÞ
2
þit
0
~
ω
Π
μν
ð−k
0
;
~
ω
ˆ
k jAÞϵ
ν
ð
ˆ
kÞ, and the associated single photon
emission amplitude as S
ðpÞ
ðk
0
Þ¼
ϵ
ðpÞ
μ
ð
ˆ
k
0
Þ
ffiffiffiffiffi
2ω
0
p
j
μ
ðk
0
Þ [21], where
the polarization vectors ϵ
ðpÞ
μ
ð
ˆ
k
0
Þ, with p ∈ f1; 2g, span the
transverse polarizations of the induced photons of four-
momentum k
0μ
¼ ω
0
ð1;
ˆ
k
0
Þ. Employing E ¼
ffiffiffiffiffiffiffiffiffi
2hIi
p
, with
the probe mean intensity given by hIi¼Jω, where J ≡
N
σT
is the probe photon current density, i.e., the number N of
incident frequency-ω photons per area σ and time interval
T, the differential number of induced photons with polari-
zation p determined with Fermi’s golden rule, d
3
N
ðpÞ
¼
d
3
k
0
ð2π Þ
3
jS
ðpÞ
ðk
0
Þj
2
, becomes
d
3
N
ðpÞ
¼
d
3
k
0
ð2πÞ
3
ϵ
ðpÞ
μ
ð
ˆ
k
0
Þ
ffiffiffiffiffiffiffi
2ω
0
p
M
μν
ð−k
0
;kjAÞ
ϵ
ν
ð
ˆ
kÞ
ffiffiffiffiffiffi
2ω
p
2
J; ð1Þ
where we defined
M
μν
ð−k
0
;kjAÞ¼
ffiffiffi
π
p
T
2
Z
d
~
ω
2π
e
−
1
4
ð
T
2
Þ
2
ð~ω−ωÞ
2
þit
0
~ω
× Π
μν
ð−k
0
;
~
ω
ˆ
k jAÞ: ð2Þ
For a plane-wave probe beam, recovered in the limit
T → ∞,wehaveM
μν
ð−k
0
;kjAÞj
T→∞
¼ Π
μν
ð−k
0
;kjAÞ.
Choosing ϵ
ðpÞ
ð
ˆ
k
0
Þ perpendicular to ϵð
ˆ
kÞ, the modulus
squared term in Eq. (1) can be interpreted as polarization
flip probability [29].
We assume the high-intensity laser with normalized
four wave vector
ˆ
κ
μ
¼ð1; e
z
Þ to be linearly polarized
along e
E
¼ðcos ϕ; sin ϕ; 0Þ (cf. Fig. 1, where ϕ ¼ 0).
The choice of the angle parameter ϕ fixes the directions
of the electric and magnetic fields (e
B
¼ e
E
j
ϕ→ϕþ
π
2
).
Moreover, k
μ
⊥
¼ð0; k − ð
ˆ
κ · kÞ
ˆ
κÞ and k
0μ
⊥
¼ k
μ
⊥
j
k→k
0
.
For the following discussion it is convenient to turn to
spherical coordinates and express the unit momentum
vectors as
ˆ
k ¼ðcos φ sin ϑ; sin φ sin ϑ; − cos ϑÞ, such that
ˆ
kj
ϑ¼0
¼−e
z
, and likewise
ˆ
k
0
¼
ˆ
kj
φ→φ
0
;ϑ→ϑ
0
. Correspond
ingly, the polarization vectors can be expressed as
ϵ
μ
ð
ˆ
kÞ¼ð0; e
φ;ϑ;β
Þ, with e
φ;ϑ;β
≡ sin β
ˆ
kj
ϑ¼
π
2
;φ→φþ
π
2
−
cos β
ˆ
kj
ϑ→ϑþ
π
2
, and ϵ
ðpÞ
μ
ð
ˆ
k
0
Þ¼ð0;e
φ
0
;ϑ
0
;β
0
Þ. Without loss of
generality ϵ
ð1Þ
μ
ð
ˆ
k
0
Þ is fixed by a particular choice of β
0
, and
the perpendicular vector by ϵ
ð2Þ
μ
ð
ˆ
k
0
Þ¼ϵ
ð1Þ
μ
ð
ˆ
k
0
Þj
β
0
→β
0
þ
π
2
.
On shell, i.e., for
ˆ
k
2
¼
ˆ
k
02
¼ 0, Π
μν
in a linearly polarized,
pulsed Gaussian laser beam reads (cf. Eq. (16) of [22])
Π
μν
ð−k
0
;
~
ω
ˆ
kÞ¼−ω
0
~
ω
α
π
1
45
I
0
I
cr
gðk
0
−
~
ω
ˆ
kÞ
×½4ð
ˆ
k
0
ˆ
FÞ
μ
ð
ˆ
k
ˆ
FÞ
ν
þ7ð
ˆ
k
0
ˆ
FÞ
μ
ð
ˆ
k
ˆ
FÞ
ν
; ð3Þ
where gðk
0
−
~
ω
ˆ
kÞ¼
R
x
e
−iðk
0
−
~
ω
ˆ
kÞx
gðxÞ is the Fourier trans-
form of the normalized intensity profile of the pump laser
in position space, gðxÞ¼IðxÞ=I
0
, with peak intensity I
0
;
KARBSTEIN et al. PHYSICAL REVIEW D 92, 071301( R) (2015)
071301-2
RAPID COMMUNICATIONS
α ¼
e
2
4π
≃
1
137
,andI
cr
¼ð
m
2
e
Þ
2
≈ 4.4 × 10
29
W
cm
2
. Here, the tensor
structure is expressed in terms of ð
ˆ
k
ˆ
FÞ
μ
¼ ε
μ
1
ð
ˆ
kÞcos ϕ þ
ε
μ
2
ð
ˆ
kÞsin ϕ and ð
ˆ
k
ˆ
FÞ
μ
¼ð
ˆ
k
ˆ
F Þ
μ
j
ϕ→ϕþ
π
2
,withε
μ
1
ð
ˆ
kÞ¼
ð−
ˆ
k
x
;
ˆ
κ
ˆ
k;0; −
ˆ
k
x
Þ and ε
μ
2
ð
ˆ
kÞ¼ð−
ˆ
k
y
;0;
ˆ
κ
ˆ
k;−
ˆ
k
y
Þ. The polari-
zation dependence of the induced photon signal (1) is
encoded in the tensor structure in Eq. (3) contracted with
the polarization vector of the probe beam and ϵ
ðpÞ
μ
ð
ˆ
k
0
Þ,
ϵ
ðpÞ
μ
ð
ˆ
k
0
Þ½4ð
ˆ
k
0
ˆ
F Þ
μ
ð
ˆ
k
ˆ
F Þ
ν
þ 7ð
ˆ
k
0
ˆ
FÞ
μ
ð
ˆ
k
ˆ
FÞ
ν
ϵ
ν
ð
ˆ
kÞ
¼ð1 þ cos ϑ
0
Þð1 þ cos ϑÞ½4 cos γ
0
cos γ þ 7 sin γ
0
sin γ;
ð4Þ
where γ ¼ φ − β − ϕ and γ
0
¼ φ
0
− β
0
− ϕ. Equation (4)
depends on the direction (φ, ϑ) and polarization (β)ofthe
probe, the polarization of the pump (ϕ), as well as
the emission direction (φ
0
, ϑ
0
) and polarization (β
0
)ofthe
induced photons.
Combining Eqs. (1)–(4), we obtain
d
3
N
ðpÞ
¼
d
3
k
0
ð2πÞ
3
1
π
ω
0
ω
α
90
I
0
I
cr
T
2
2
ð1 þ cos ϑ
0
Þ
2
ð1 þ cos ϑÞ
2
× ½4 cos γ
0
cos γ þ 7 sin γ
0
sin γ
2
×
Z
d
~
ω
2π
e
−
1
4
ð
T
2
Þ
2
ð
~
ω−ωÞ
2
þit
0
~
ω
~
ω gðk
0
−
~
ω
ˆ
kÞ
2
J: ð5Þ
The total number of induced photons is obtained upon
summation over the two photon polarizations p, i.e.,
P
p
d
3
N
ðpÞ
. It can be inferred from Eq. (5) by substituting
½4 cos γ
0
cos γ þ 7 sin γ
0
sin γ
2
→ ½16 þ 33sin
2
γ.
So far our considerations were valid for arbitrary
collision geometries of pump and probe. Subsequently,
we stick to counter-propagating pump and probe beams,
i.e.,
ˆ
k
μ
¼ð1; −e
z
Þ, as typically adopted in proposals for
all-optical vacuum birefringence experiments [19] to maxi-
mize the overall factor of ð1 þ cos ϑÞ
2
j
ϑ¼0
¼ 4 in Eq. (5).
In this limit it is convenient to set φ ¼ 0, such that
ϵ
μ
ð
ˆ
kÞ¼ð0; − cos β; sin β; 0Þ. Besides, we assume t
0
¼ 0,
such that the two laser pulses have an optimal temporal
overlap and the effect is maximized. We adopt the standard
choice β ¼
π
4
− ϕ (cf. [19]) for scenarios aiming at the
detection of vacuum birefringence in a high-intensity laser
experiment, implying that the polarization vector of the
incident probe photons forms an angle of
π
4
with respect to
both the electric and magnetic field vectors of the pump;
see Fig. 1. This ensures an equal overlap with the two
photon polarization eigenmodes featuring different phase
velocities in constant crossed and plane wave backgrounds.
Note that the intensity profile of a linearly polarized,
focused Gaussian laser pulse (frequency Ω ¼
2π
λ
, beam
waist w
0
, pulse duration τ and phase φ
0
), depending on
the longitudinal (z) as well as the transverse (x, y)
coordinates, is given by [30]
gðxÞ¼
e
−
ðz−tÞ
2
ðτ=2Þ
2
w
0
wðzÞ
e
−
x
2
þy
2
w
2
ðzÞ
× cos
Ωðz − tÞþ
x
2
þy
2
w
2
ðzÞ
z
z
R
− arctan
z
z
R
þφ
0
2
;
ð6Þ
with wðzÞ¼w
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þð
z
z
R
Þ
2
q
and Rayleigh range z
R
¼
πw
2
0
λ
.
In this paper, we consider three generic cases (cf. Fig. 2),
namely (a) the radius of the x-ray probe beam is signifi-
cantly smaller than the beam waist of the pump, (b) the
probe beam radius is substantially larger than the beam
waist of the pump [31], and (c) an asymmetric x-ray beam
profile which is substantially smaller than the beam waist of
the pump laser in one, and larger in the other transverse
direction.
To tackle case (a) theoretically, we assume that the radius
of the probe beam is so tiny that basically all probe photons
propagate on the beam axis of the pump laser beam where
the laser field becomes maximum. To this end, for gðxÞ we
adopt the on-axis profile of a focused Gaussian laser pulse,
gðxÞ¼gðxÞj
x¼y¼0
, i.e., do not account for the transverse
structure of the beam. Correspondingly, the transverse
momentum components remain unaffected and we hav e
ˆ
k
0
¼
ˆ
k ¼ −e
z
, implying that the photons carrying the
birefringence signal will be propagating in the direction
of the original probe beam, and we can set φ
0
¼ 0.This
kinematic restriction requires ω
0
¼jk
z
0
j, but allows for k
0
z
≠
k
z
and ω
0
≠ ω. In the evaluation of Eq. (5) we can then make
use of the homogeneity in the transverse directions, implying
gðk
0
−
~
ω
ˆ
kÞ ∼ ð2πÞ
2
δ
ð2Þ
ðk
⊥
Þ,and
R
k
⊥
d
3
N
ðpÞ
∼ Jσ ¼
N
T
.
Conversely, for case (b), where the probe inherently
senses the transverse structure of the pump, such that in
general k
0μ
≠ k
μ
, we take into account the full intensity
profile of a focused Gaussian laser pulse, i.e., adopt
gðxÞ¼gðxÞ.
For case (c) we do not account for the transverse
structure of the Gaussian beam along one direction, say
y, but fully take it into account in x direction. Hence, we
adopt gðxÞ¼gðxÞj
y¼0
, implying k
0
y
¼ k
y
, but generically
k
0μ
≠ k
μ
for μ ∈ f0;1;3g. Here, we have gðk
0
−
~
ω
ˆ
kÞ∼
ð2πÞδðk
y
Þ, such that
R
k
y
d
3
N
ðpÞ
∼ JL
y
¼
N
TL
, with L ≡ L
x
.
(a) (b) (c)
FIG. 2 (color online). Pictogram of the three different cases
(a)–(c) considered in this paper. The circle (orange) is the cross
section of the pump beam at its waist, and the filled circle/ellipse
(purple) is the cross section of the probe beam.
VACUUM BIREFRINGENCE IN STRONG INHOMOGENEOUS … PHYSICAL REVIEW D 92, 071301( R) (2015)
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RAPID COMMUNICATIONS
As demonstrated below, in all of the cases (a)–(c) the
induced x-ray photons are emitted in directions
ˆ
k
0
very
close to the propagation direction
ˆ
k of the probe beam,
and hence fulfill ϑ
0
≪ 1. Within at most a few tens of
μrad the signal falls off rapidly to zero, and we have
ϵ
ðpÞ
μ
ð
ˆ
k
0
Þ¼ð0; − cosðβ
0
− φ
0
Þ; sinðβ
0
− φ
0
Þ; 0ÞþOðϑ
0
Þ. This
implies that one can decompose the induced
photon signal into photons polarized parallel, ϵ
ð∥Þ
μ
ð
ˆ
k
0
Þ (β
0
¼
π
4
− ϕ þ φ
0
), and perpendicular, ϵ
ð⊥Þ
μ
ð
ˆ
k
0
Þ (β
0
¼
3π
4
− ϕ þ φ
0
),
to the probe. In turn, Eq. (5) becomes φ
0
independent,
and
R
dφ
0
→ 2π. The term ½4 cos γ
0
cos γ þ 7 sin γ
0
sin γ
2
encoding the polarization dependence in Eq. (5), becomes
121
4
(
9
4
) for the ∥ (⊥) polarization mode. The ⊥-polarized
photons constitute the birefringence signal [29].
To obtain realistic estimates for the numbers of induced
photons for a state-of-the-art laser system, we assume the
pump laser to be of the 1 PW class (pulse energy W ¼ 30 J,
pulse duration τ ¼ 30 fs and wavelength λ ¼ 800 nm)
focused to w
0
¼ 1 μm. The associated peak intensity is
I
0
¼ 2
0.86W
τπw
2
0
; the effective focus area contains 86% of the
beam energy (1=e
2
criterion). For the x-ray probe we
choose ω ¼ 12914 eV, for which the presently most
sensitive x-ray polarimeter [32] was benchmarked. The
polarization purity of x-rays of this frequency can be
measured to the level of 5.7 × 10
−10
. Assuming also the
probe beam to be well described as a focused Gaussian
beam of waist r, its divergence is given by θðrÞ¼
λ
probe
πr
.
Neglecting diffraction and curvature effects for the probe in
the actual calculation is nevertheless well justified as long
as z
R;probe
≫ z
R
. Measuring r in units of w
0
, i.e., r ¼ ρw
0
,
we have
z
R;probe
z
R
¼ ρ
2
ω
Ω
, implying that z
R;probe
≥ z
R
for
ρ
2
≥
Ω
ω
. Generically, we find
Ω
ω
< Oð10
−3
Þ [33].
We first focus on case (a), where r ≪ w
0
. In Fig. 3 we
plot
T
N
dN
ð⊥Þ
dk
0
z
. Upon integration over k
0
z
we obtain
T½fs N
ð⊥Þ
=
N
T
½
1
fs
N
ð⊥Þ
=N
(a)
∞ 1.12 × 10
−10
–
500 1.10 × 10
−10
2.20 × 10
−13
30 4.16 × 10
−11
1.39 × 10
−12
Keeping all other parameters fixed, these results can be
rescaled as ð
W½J
30
Þ
2
to any other pump laser energy. Note that
even the maximum ratio
N
ð⊥Þ
N
≈ 1.39 × 10
−12
obtained here
for T ¼ 30 fs is too small to be confirmed experimentally
with currently available x-ray polarization purity [32]
(improvement by a factor of ≳410 required).
As to be expected, in scenario (b) the numbers of induced
photons are lower. In this case the probe senses the
transverse structure of the focused pump, which results
in outgoing x-ray photons with nonvanishing transverse
momentum components. This can provide us with an
additional handle to identify the induced photon signal
as we can search for ⊥-polarized photons emitted outside
θðrÞ (cf. Fig. 4), where the demand on the polarization
purity is significantly lower due to the low photon back-
ground. Denoting the number of ⊥-polarized photons
emitted outside θðrÞ as N
ð⊥Þ
>θðrÞ
, we consider probe beams
of width r ¼ ρw
0
and exemplarily show results for
ρ ¼ 3. Identifying the cross section of the x-ray probe
with σ ¼ πðρw
0
Þ
2
, for case (b) we infer N
ð⊥Þ
=N ∼ ρ
−2
and
T½fs N
ð⊥Þ
=J½
1
μm
2
fs
N
ð⊥Þ
=N N
ð⊥Þ
>θð3w
0
Þ
=N
ð⊥Þ
(b)
∞ 1.79 × 10
−10
– 63.1%
500 1.52 × 10
−10
9.68 × 10
−14
=ρ
2
72.1%
30 3.66 × 10
−11
3.88 × 10
−13
=ρ
2
88.1%
Analogously, setting L ¼ 2ρw
0
for case (c) we find
N
ð⊥Þ
=N ∼ ρ
−1
and
T½fs N
ð⊥Þ
=
N
TL
½
1
μmfs
N
ð⊥Þ
=N N
ð⊥Þ
>θð3w
0
Þ
=N
ð⊥Þ
(c)
∞ 6.38 × 10
−11
– 51.1%
500 6.11 × 10
−11
1.22 × 10
−13
=ρ 52.6%
30 1.95 × 10
−11
6.50 × 10
−13
=ρ 61.7%
Hence, especially for large probe beam widths (ρ > 1), case
(c) is experimentally favored as it provides for the largest
number of signal photons outside θðρw
0
Þ. Finally, we
demonstrate that the ⊥-polarized photons emitted outside
θðrÞ can be measured with state-of-the-art technology.
To do this, we stick to case (c) with T ¼ 30 fs and
ρ ¼ 3. The number of ⊥-polarized photons scattered in
directions ϑ
0
≥ ϑ
min
can be estimated from Fig. 4 as
N
ð⊥Þ
>ϑ
min
≈ N
ð⊥Þ
½1 − erfð
ffiffiffiffiffiffiffiffiffi
1.13
p
ϑ
min
θðw
0
Þ
Þ, where erfð:Þ is the
error function. Similarly, the number of probe photons
outside ϑ
min
is N
>ϑ
min
¼ N½1 − erfð
ffiffiffi
2
p
ϑ
min
θð3w
0
Þ
Þ. Herefrom,
FIG. 3 (color online). Plot of
T
N
dN
ð⊥Þ
dk
0
z
for case (a) as a function of
k
0
z
. The induced photon signal is peaked at k
0
z
¼ −ω and rapidly
falls off to zero. We depict results for different probe pulse
durations T. The inlay shows a plot of the same quantity for
T → ∞ (φ
0
¼ 0) over a wider frequency range, adopting a
logarithmic scale. Here, the strongly suppres sed con tributions
to be associated with frequencies ≈ω 2Ω are clearly visible.
KARBSTEIN et al. PHYSICAL REVIEW D 92, 071301( R) (2015)
071301-4
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