3.1 Scattering matrix

In what follows, we will consider a modification of the 4-photon scattering amplitude at low energy by assuming that two of the photons involved are stemming from a high-intensity laser which is probed by a dynamical photon ‘passing through’. This is visualized in Figure 4.

We assume that an incoming probe photon with four-momentum $k$ and four-polarization ${\it\varepsilon}$ scatters off a laser background described by a field strength tensor $F_{{\it\mu}{\it\nu}}$ resulting in an outgoing photon with quantum numbers $k^{\prime }$ and ${\it\varepsilon}^{\prime }$. The resulting scattering amplitude is given by the $S$-matrix element

(10)$$\begin{eqnarray}\displaystyle \langle {\it\varepsilon}^{\prime },k^{\prime };\text{out}|{\it\varepsilon},k;\text{in}\rangle =\langle {\it\varepsilon}^{\prime },k^{\prime }|{\hat{S}}|{\it\varepsilon},k\rangle \equiv S_{\text{fi}}({\it\varepsilon}^{\prime },k^{\prime },{\it\varepsilon},k). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Using the leading-order Lagrangian (Equation (8)), using $S_{\text{fi}}(q)$ to denote $S_{fi}({\it\varepsilon}^{\prime },k^{\prime },{\it\varepsilon},k)$, the $S$-matrix element takes on the simple form of a Fourier integral

(11)$$\begin{eqnarray}\displaystyle S_{\text{fi}}(q)=-\text{i}\int d^{4}x\,\text{e}^{\text{i}q\cdot x}\,S_{\text{fi}}(x), & & \displaystyle\end{eqnarray}$$

where $q=k^{\prime }-k$ is the momentum transfer and

(12)$$\begin{eqnarray}\displaystyle S_{\text{fi}}(x)=c_{1}(k^{\prime },F{\it\varepsilon}^{\prime })(k,F{\it\varepsilon})+c_{2}(k^{\prime },\tilde{F}{\it\varepsilon}^{\prime })(k,\tilde{F}{\it\varepsilon}),\qquad & & \displaystyle\end{eqnarray}$$

employing the abbreviated scalar products $(k,F{\it\varepsilon})\equiv k_{{\it\mu}}F^{{\it\mu}{\it\nu}}{\it\varepsilon}_{{\it\nu}}$, etc. Hence, one may introduce an intensity form factor,

(13)$$\begin{eqnarray}\displaystyle W^{{\it\mu}{\it\alpha},{\it\nu}{\it\beta}}(q)\equiv -\text{i}\int d^{4}x\,\text{e}^{\text{i}q\cdot x}(c_{1}F^{{\it\alpha}{\it\mu}}F^{{\it\beta},{\it\nu}}+c_{2}\tilde{F}^{{\it\alpha}{\it\mu}}\tilde{F}^{{\it\beta},{\it\nu}}), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

which is the Fourier transformation of the background intensity distribution. In terms of the latter, the scattering amplitude may be written as

(14)$$\begin{eqnarray}\displaystyle S_{\text{fi}}(q)={\it\varepsilon}_{{\it\alpha}}^{\prime }k_{{\it\mu}}^{\prime }W^{{\it\mu}{\it\alpha},{\it\nu}{\it\beta}}(q)k_{{\it\nu}}{\it\varepsilon}_{{\it\beta}}. & & \displaystyle\end{eqnarray}$$

The results above are reminiscent of elastic electron nucleus scattering, where the scattering amplitude is proportional to the nuclear charge form factor which is nothing but the Fourier transform of the nuclear charge distribution. In photon–photon scattering, one is naturally probing an intensity, rather than a charge, distribution. To proceed, one has to choose a suitable laser background field, $F_{{\it\mu}{\it\nu}}(x)$, and calculate its intensity form factor (Equation (13)).

3.2 Polarization operator

An equivalent representation is obtained in terms of a quantity aptly called the polarization operator, denoted ${\rm\Pi}^{{\it\mu}{\it\nu}}$. In its simplest incarnation it is just the mathematical expression for the Feynman diagram of Figure 1, namely

where the trace $\text{tr}_{{\it\gamma}}$ extends over the Dirac matrices ${\it\gamma}^{{\it\mu}}$. One may generalize this to the polarization tensor in an external field $A_{\text{ext}}$, where one trades the free fermion propagators for interacting ones through the standard minimal substitution $p\rightarrow p-eA_{\text{ext}}$. Indeed, this method has a long history^{[Reference Toll20–Reference Meuren, Keitel and Di Piazza23]} as reviewed by Ref. [Reference Dittrich and Gies24]. For our purposes it is sufficient to just employ the first-order weak-field Heisenberg–Euler Lagrangian (Equation (8)) once again and rewrite it as

(16)$$\begin{eqnarray}\displaystyle {\mathcal{L}}_{\text{HE}}^{(2)}={\textstyle \frac{1}{2}}A_{{\it\mu}}{\rm\Pi}^{{\it\mu}{\it\nu}}[A_{\text{ext}}]A_{{\it\nu}}, & & \displaystyle\end{eqnarray}$$

with the polarization tensor thus defining the second-order term. From Equation (8) one can straightforwardly read off that

(17)$$\begin{eqnarray}\displaystyle {\rm\Pi}^{{\it\mu}{\it\nu}}[A_{\text{ext}}]=\frac{c_{1}}{2}k_{{\it\alpha}}F^{{\it\alpha}{\it\mu}}F^{{\it\beta}{\it\nu}}k_{{\it\beta}}+\frac{c_{2}}{2}k_{{\it\alpha}}\tilde{F}^{{\it\alpha}{\it\mu}}\tilde{F}^{{\it\beta}{\it\nu}}k_{{\it\beta}},\qquad & & \displaystyle\end{eqnarray}$$

where the background field strength $F^{{\it\mu}{\it\nu}}=\partial ^{{\it\mu}}A_{\text{ext}}^{{\it\nu}}-\partial ^{{\it\nu}}A_{\text{ext}}^{{\it\mu}}$.

To connect this approach with the $S$ matrix formalism we specialize to forward scattering by setting $k=k^{\prime }$ in Equation (12) which yields the relation

(18)$$\begin{eqnarray}\displaystyle S_{\text{fi, fwd}}(k)={\it\varepsilon}_{{\it\mu}}^{\prime }(k){\rm\Pi}^{{\it\mu}{\it\nu}}(k){\it\varepsilon}_{{\it\nu}}(k). & & \displaystyle\end{eqnarray}$$

This makes the link between the polarization operator and scattering matrix approaches manifest.

3.3 Modified Maxwell equations

In standard quantum field theory notion^{[Reference Peskin and Schroeder9]}, the total Heisenberg–Euler action, $S_{\text{eff}}=\int d^{4}x\,{\mathcal{L}}_{\text{eff}}$, is nothing but the one-loop effective (or quantum) action of QED evaluated at low energies where there are no external electron lines. The associated effective Lagrangian is the sum of the classical Maxwell term ${\mathcal{L}}_{\text{M}}=(E^{2}-B^{2})/2$ and the first quantum correction:

(19)$$\begin{eqnarray}\displaystyle {\mathcal{L}}_{\text{eff}}={\mathcal{L}}_{\text{M}}+{\mathcal{L}}_{\text{HE}}. & & \displaystyle\end{eqnarray}$$

By variation of the quantum action, one can derive the corresponding modified Maxwell equations^{[Reference Akhiezer and Berestetskii25]}:

(20)$$\begin{eqnarray}\displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\mathbf{E}={\it\rho}_{\text{vac}};\quad \boldsymbol{{\rm\nabla}}\wedge \mathbf{B}=\mathbf{J}_{\text{vac}}+\partial _{t}\mathbf{E}, & & \displaystyle\end{eqnarray}$$

in which:

(21)$$\begin{eqnarray}\displaystyle {\it\rho}_{\text{vac}}=\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\mathbf{P}_{\text{vac}};\quad \mathbf{J}_{\text{vac}}=\boldsymbol{{\rm\nabla}}\wedge \mathbf{M}_{\text{vac}}+\partial _{t}\mathbf{P}_{\text{vac}} & & \displaystyle\end{eqnarray}$$

and the vacuum polarization and magnetization are:

(22)$$\begin{eqnarray}\displaystyle \mathbf{P}_{\text{vac}}=\frac{\partial {\mathcal{L}}_{\text{HE}}}{\partial \mathbf{E}};\quad \mathbf{M}_{\text{vac}}=\frac{\partial {\mathcal{L}}_{\text{HE}}}{\partial \mathbf{B}}. & & \displaystyle\end{eqnarray}$$

The wave equations:

(23)$$\begin{eqnarray}\displaystyle \partial _{t}^{2}\mathbf{E}-\boldsymbol{{\rm\nabla}}^{2}\mathbf{E} & = & \displaystyle -\boldsymbol{{\rm\nabla}}{\it\rho}_{\text{vac}}[\mathbf{E},\mathbf{B}]-\partial _{t}\mathbf{J}_{\text{vac}}[\mathbf{E},\mathbf{B}],\end{eqnarray}$$

(24)$$\begin{eqnarray}\displaystyle \partial _{t}^{2}\mathbf{B}-\boldsymbol{{\rm\nabla}}^{2}\mathbf{B} & = & \displaystyle \boldsymbol{{\rm\nabla}}\wedge \mathbf{J}_{\text{vac}}[\mathbf{E},\mathbf{B}],\end{eqnarray}$$

can be solved using, for example, the method of Green’s functions.

4.1 Effects on probe photon polarization

Vacuum birefringence refers to the prediction that the refractive index experienced by a probe propagating through regions of intense, but weakly varying strong fields of amplitude $E_{s}$ is of the form^{[Reference Toll20, Reference Baier and Breitenlohner29]}:

(26)$$\begin{eqnarray}\displaystyle \mathsf{n}_{\text{vac}}^{\Vert ,\bot }=1+\frac{(11\mp 3){\it\alpha}}{45{\it\pi}}\frac{E_{s}^{2}}{E_{\text{cr}}^{2}}, & & \displaystyle\end{eqnarray}$$

where the $\Vert$ ($\bot$) indices apply to a probe polarized parallel (perpendicular) to the strong background. This result may be derived from the Heisenberg–Euler quantum equation of motion,

(27)$$\begin{eqnarray}\displaystyle (\partial _{{\it\lambda}}\partial ^{{\it\lambda}}g^{{\it\mu}{\it\nu}}-\partial ^{{\it\mu}}\partial ^{{\it\nu}}+{\rm\Pi}^{{\it\mu}{\it\nu}})A_{{\it\nu}}=0. & & \displaystyle\end{eqnarray}$$

A plane-wave ansatz for $A_{{\it\nu}}$ implies two secular equations or dispersion relations,

(28)$$\begin{eqnarray}\displaystyle k^{2}-{\rm\Pi}_{1,2}(k)=(g^{{\it\mu}{\it\nu}}-c_{1,2}T^{{\it\mu}{\it\nu}})k_{{\it\mu}}k_{{\it\nu}}=0, & & \displaystyle\end{eqnarray}$$

where ${\rm\Pi}_{1,2}=c_{1,2}(k,Tk)$ are the two nontrivial eigenvalues of the polarization tensor (Equation (17)), expressed in terms of the background energy–momentum tensor $T^{{\it\mu}{\it\nu}}=F_{{\it\alpha}}^{{\it\mu}}F^{{\it\alpha}{\it\nu}}$. The dispersion relations (Equation (28)) describe the change in light propagation caused by the energy–momentum density stored in the background field and have been referred to as modified light-cone conditions^{[Reference Dittrich and Gies30, Reference Shore31]}. They imply group velocities different from the vacuum speed of light, $c$, and hence the refractive indices (Equation (26)) different from unity, which can be rewritten as $\mathsf{n}_{\text{vac}}^{\Vert ,\bot }=1+{\rm\Pi}_{1,2}/2{\it\omega}_{p}^{2}$, ${\it\omega}_{p}=k^{0}c$ being the probe frequency.

The result for the refractive indices has been shown to hold to all perturbative orders using the polarization operator^{[Reference Toll20, Reference Shore31, Reference Meuren, Hatsagortsyan, Keitel and Di Piazza32]} and Heisenberg–Euler Lagrangian numerically^{[Reference Ferrando, Michinel, Seco and Tommasini33, Reference Böhl, King and Ruhl34]} and analytically^{[Reference Böhl, King and Ruhl34]}. When the pump field is space–time-dependent as is the case for laser pulses, the effect on the probe is calculated by integrating over the inhomogeneous refractive index of the pump background^{[Reference King, Böhl and Ruhl35]}. There has also been recent work indicating finite-time effects in an inhomogeneous background may leave a detectable signal^{[Reference Hu and Huang36]}.

Polarization flip is the underlying physical mechanism of vacuum birefringence. The term is used when an incoming photon’s polarization vector ${\it\varepsilon}^{{\it\mu}}$ is ‘flipped’ to an orthogonal one ${\it\varepsilon}^{\prime \,{\it\mu}}$ due to real photon–photon scattering. Linearly polarized probe photons can flip if the background contains some ellipticity and circularly polarized probe photons if the background contains some linear polarization. The flip amplitude (for a head-on collision of probe and background) after a propagation distance $z$ can be found from the Heisenberg–Euler forward scattering amplitude (Equation (18)) and coincides with the birefringence-induced ellipticity^{[Reference Dinu, Heinzl, Ilderton, Marklund and Torgrimsson37]},

(29)$$\begin{eqnarray}\displaystyle \mathsf{e}\equiv \langle {\it\varepsilon}^{\prime },k|S|{\it\varepsilon},k\rangle =2E_{s}^{2}{\it\omega}_{p}z(c_{2}-c_{1}), & & \displaystyle\end{eqnarray}$$

where ${\it\varepsilon}\cdot {\it\varepsilon}^{\prime }=0$. Note the dependence on the difference of the low-energy constants. This implies that a confirmation of vacuum birefringence would rule out other versions of electrodynamics popular in beyond-the-standard-model physics such as Born–Infeld theory, which has $c_{1}=c_{2}$^{[Reference Boillat38–Reference Białynicki-Birula, Jancewicz and Lukierski40]}. From Equation (26), the flip amplitude or ellipticity (Equation (29)) has the equivalent representation

(30)$$\begin{eqnarray}\displaystyle \mathsf{e}={\it\omega}_{p}z\frac{\mathsf{n}_{\text{vac}}^{\bot }-\mathsf{n}_{\text{vac}}^{\Vert }}{2}, & & \displaystyle\end{eqnarray}$$

which is proportional to the difference in refractive indices, hence the phase shift between different polarizations.

Detailed calculations have been performed for photons propagating in an arbitrary plane-wave background^{[Reference Baĭer, Mil’shteĭn and Strakhovenko22, Reference Dinu, Heinzl, Ilderton, Marklund and Torgrimsson37]}, and the kinematic low-energy limit relevant for laser-based experiments was found to be consistent with use of the Heisenberg–Euler approach for calculating birefringence and ellipticity^{[Reference Torgrimsson41]}. A study of the dependency of the flip and nonflip amplitude on spatial and timing jitter and angle of incidence^{[Reference Dinu, Heinzl, Ilderton, Marklund and Torgrimsson42]} was performed, with the results also being consistent with a previous similar study in the low-energy limit^{[Reference King and Keitel43]}. Both studies^{[Reference Dinu, Heinzl, Ilderton, Marklund and Torgrimsson42, Reference King and Keitel43]} found that modelling the background as a focused paraxial Gaussian beam without taking into account the finite pulse duration led to an order of magnitude discrepancy in the number of scattered photons.

Induced ellipticity is a consequence of birefringence as pointed out in the previous subsection, see Equations (29) and (30). The polarization of a linearly polarized probe plane wave can be described with the vector:

(31)$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}{\it\varepsilon}^{\Vert }\\ {\it\varepsilon}^{\bot }\end{array}\right)=\cos {\it\varphi}\left(\begin{array}{@{}c@{}}\cos {\it\theta}\\ \sin {\it\theta}\end{array}\right), & & \displaystyle\end{eqnarray}$$

where ${\it\varphi}$ is the probe phase. If, over some probe phase ${\it\omega}_{p}z$ the $\Vert$ and $\bot$ components experience a different refractive index, then when the phase shift ${\it\delta}{\it\varphi}^{\Vert ,\bot }=n^{\Vert ,\bot }{\it\omega}_{p}z\ll 1$, the polarization changes to:

(32)$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}{\it\varepsilon}^{\Vert }\\ {\it\varepsilon}^{\bot }\end{array}\right)=\left[\begin{array}{@{}cc@{}}\cos {\it\theta} & -\!\cos {\it\theta}~{\it\delta}{\it\varphi}^{\Vert }\\ \sin {\it\theta} & -\!\sin {\it\theta}~{\it\delta}{\it\varphi}^{\bot }\end{array}\right]\left(\begin{array}{@{}c@{}}\cos {\it\varphi}\\ \sin {\it\varphi}\end{array}\right), & & \displaystyle\end{eqnarray}$$

and the originally linearly polarized probe is now elliptically polarized. If the background is constant, the ellipticity can be written^{[Reference Di Piazza, Hatsagortsyan and Keitel44]}:

(33)$$\begin{eqnarray}\displaystyle \mathsf{e}={\it\omega}_{p}z\frac{\mathsf{n}_{\text{vac}}^{\bot }-\mathsf{n}_{\text{vac}}^{\Vert }}{2}\sin 2{\it\theta}, & & \displaystyle\end{eqnarray}$$

which generalizes Equation (30). The induced ellipticity in the interaction of an x-ray probe plane wave of wavelength ${\it\lambda}_{p}=0.4~\text{nm}$ counterpropagating with a Gaussian pump beam of intensity $10^{23}~\text{W}~\text{cm}^{-2}$ and wavelength ${\it\lambda}_{s}=745~\text{nm}$ focused to $8\,{\rm\mu}\text{m}$ was calculated^{[Reference Di Piazza, Hatsagortsyan and Keitel44]} to experience an ellipticity of $\mathsf{e}\approx 5\times 10^{-9}~\text{rad}$ when measured at a distance of $0.25~\text{m}$ from the pump–probe collision. By considering the same pump energy distributed over two pump Gaussian laser beams counterpropagating with a Gaussian probe beam, a modest improvement of around $\sqrt{2}$ was found, and the near-field-induced ellipticity^{[Reference King, Di Piazza and Keitel45]}

(34)$$\begin{eqnarray}\displaystyle \mathsf{e}=\frac{2{\it\pi}{\it\alpha}}{15}\frac{I_{s}}{I_{\text{cr}}}\frac{z_{\text{eff.}}}{{\it\lambda}_{p}}\,\sin 2{\it\theta};\quad z_{\text{eff.}}=\frac{z_{r,p}z_{r,s}}{z_{r,p}+z_{r,s}}, & & \displaystyle\end{eqnarray}$$

with the effective interaction length between the two Gaussians $z_{\text{eff.}}$ depending on the probe $z_{r,p}$ and pump $z_{r,s}$ Rayleigh lengths. This agrees with the expressions calculated for a monochromatic probe plane wave counterpropagating with a Gaussian pump^{[Reference Heinzl, Liesfeld, Amthor, Schwoerer, Sauerbrey and Wipf46]} in the limit $z_{r,p}\rightarrow \infty$.

Polarization rotation is the macroscopic consequence of coherent polarization flipping at the photon level. The effect on the transverse photon polarization states in Equations (31) and (32) has the consequence that the polarization angle ${\it\theta}$ will rotate as the initially linearly polarized probe acquires an ellipticity. The ellipse traced out by the probe field vector can be seen to be^{[Reference King47]}:

(35)$$\begin{eqnarray}\displaystyle x^{2}-2xy\cos ({\it\delta}{\it\varphi}^{\bot }-{\it\delta}{\it\varphi}^{\Vert })+y^{2}=\sin ^{2}({\it\delta}{\it\varphi}^{\bot }-{\it\delta}{\it\varphi}^{\Vert }),\qquad & & \displaystyle\end{eqnarray}$$

where $x\cos {\it\theta}={\it\varepsilon}^{\Vert }$ and $y\sin {\it\theta}={\it\varepsilon}^{\bot }$. For an x-ray probe counterpropagating with an optical Gaussian pump beam, the rotation angle was found to be the same order of magnitude as the induced ellipticity^{[Reference Di Piazza, Hatsagortsyan and Keitel44, Reference King, Di Piazza and Keitel45]}.

4.2 Effects on probe photon wavevector

On the photon level, four-wave mixing as depicted in Figure 5 can be understood as two incoming photons, one from the probe and one from the pump, being scattered to two outgoing photons, one being back into the pump field and the other being the signal of the vacuum interaction. Conservation of momentum permits the scattered photons having a wider transverse distribution than the probe and strong background, hence allowing one to spatially separate the photon–photon scattering signal from the large background of pump and probe laser photons.

On the classical level, a refractive index $\mathsf{n}_{\text{vac}}$ different from unity, implies altered transmitted wavevectors via Snell’s law, and altered transmission $T$ and reflection coefficient $R$ via Fresnel’s law at perpendicular incidence^{[Reference Jackson48]}:

(36)$$\begin{eqnarray}\displaystyle T=\frac{4\,\mathsf{n}_{\text{vac}}}{(1+\mathsf{n}_{\text{vac}})^{2}};\quad R=\left(\frac{1-\mathsf{n}_{\text{vac}}}{1+\mathsf{n}_{\text{vac}}}\right)^{2}. & & \displaystyle\end{eqnarray}$$

If the vacuum refractive index is written as $\mathsf{n}_{\text{vac}}=1+{\it\delta}\mathsf{n}_{\text{vac}}$, the effect on probe transmission ${\sim}O({\it\delta}\mathsf{n}_{\text{vac}})$ whereas the effect on reflection ${\sim}O({\it\delta}\mathsf{n}_{\text{vac}}^{2})$.

Figure 6.

Predicted diffracted electric field in a collision of two counterpropagating Gaussian beams. Adapted from ^{[Reference King47]}.

If the probe beam is considered to be much wider than the pump background, the region of polarized vacuum can be considered to ‘diffract’ the probe. An example of such a ‘single-slit’ diffraction pattern is given in Figure 6. It is well known that the far-field diffracted field is related to the Fourier transform of the aperture function^{[Reference Levi49]}, and via Babinet’s principle, this can be related to an integral over the region of refractive index different from unity. We underline the connection of this classical analogue to the intensity form factor of the scattering matrix approach Equation (13).

Vacuum diffraction was considered in the collision of a plane probe and a focused Gaussian pump beam^{[Reference Di Piazza, Hatsagortsyan and Keitel44]}, and extended to the collision of focused Gaussian probe and pump beams^{[Reference King, Piazza and Keitel50]}. The advantage of this signal is that for increasing scattering angle, while the focused laser background is exponentially suppressed, the scattered photon vacuum signal is power-law suppressed. In the detector plane then, the number of scattered photons can be calculated in ‘measurable’ regions, where the signal-to-noise ratio is much larger than unity. One interesting scenario was calculated of colliding two parallel, highly focused Gaussian pump beams with a wide weakly focused Gaussian probe beam, such that the photons scattered in the two slit-like polarized regions around the pump beams would interfere and hence together form an all-optical double-slit experiment^{[Reference King, Piazza and Keitel50]}. For the case of two colliding Gaussian pulses, the dependency of the diffracted photon signal on experimental parameters such as the total beam power, spatial and timing jitter, angle of collision, pulse duration, probe wavelength and focal width has been carried out^{[Reference King and Keitel43]}. With 10 PW total laser power split into pump and probe focused optical pulses, of the order of a few photons were predicted to be diffracted into measurable regions on a detector place 1 m from the interaction centre. These results were verified in a study by different authors^{[Reference Tommasini and Michinel51]}, who used a different beam model. The diffraction paradigm was extended from single and double slits to a ‘diffraction grating’ of having a probe beam diffract off a regular series of pump beams^{[Reference Kryuchkyan and Hatsagortsyan52]}. Only on positions of the detector where the Bragg condition:

$$\begin{eqnarray}nq=2k_{p}\sin \frac{{\it\theta}}{2},\end{eqnarray}$$

for integer $n$, probe wavenumber $k_{p}$, wavenumber of the pump beam structure $q$ and angle between incoming and diffracted probe ${\it\theta}$, is there constructive interference of the signal of scattered photons. Since the addition of diffracted waves occurs at the level of the field, and since the number of photons scattered depends upon the total diffracted field squared, there is an enhancement in such a setup proportional to the square of the number of modulation periods. Alternatively, rather than using many beams, a single, wide-angle beam diffracting with itself at the focus has also been studied^{[Reference Monden and Kodama53]}, with the conclusion that the number of diffracted photons increases exponentially with the angular aperture. Since only the near-field signal was presented, more work is required to determine measurability in this scheme.

The idea of using the diffracted photons’ flipped polarization as well as their altered wavevector in an experimental measurement was explored for the wide-angled single-beam setup^{[Reference Monden and Kodama53]}, a single propagating Gaussian beam taking into account higher orders in a Hermite–Gauss expansion^{[Reference Paredes, Novoa and Tommasini54]} and has been most recently applied to the upcoming HIBEF experiment^{[Reference Karbstein, Gies, Reuter and Zepf55]}.

Vacuum reflection refers to the back-scattering of photons in real photon–photon scattering. Static magnetic inhomogeneities of the form of a Lorentzian, Gaussian and oscillating Gaussian have been studied^{[Reference Gies, Karbstein and Seegert56]} and more recently static electromagnetic inhomogeneities but most significantly scattering in a Gaussian beam^{[Reference Gies, Karbstein and Seegert57]}, although calculations for pulses of a finite duration are still to be performed.

Figure 7.

Parametric frequency upshifting (left) and downshifting (right) can occur between pump and probe through the vacuum interaction.

4.3 Effects on probe photon frequency

The frequency of probe photons can change via interaction with the polarized vacuum. However, this effect is much more difficult to measure experimentally because of the limited range of energy and momenta for which it is permitted. Suppose via the four-photon interaction, two photons from the strong pump background merge with a probe photon as depicted in Figure 7. Then via energy–momentum conservation:

(37)$$\begin{eqnarray}\displaystyle {\it\omega}_{p}+{\it\omega}_{s,1}+{\it\omega}_{s,2}={\it\omega}^{\prime };\quad \mathbf{k}_{p}+\mathbf{k}_{s,1}+\mathbf{k}_{s,2}=\mathbf{k}^{\prime },\qquad & & \displaystyle\end{eqnarray}$$

but at the same time, the photon must be real to propagate to the detector so ${\it\omega}^{\prime 2}=\mathbf{k}^{\prime }\boldsymbol{\cdot }\mathbf{k}^{\prime }$. This constrains the allowed frequencies, momenta and angles that can be combined. Similar relations occur for Raman and Brillouin scattering^{[Reference Moloney and Newell58]}, except all the waves here are electromagnetic.

Vacuum parametric frequency shifting has been calculated for special beam configurations. Combining three monochromatic plane waves at right angles, whose wavelengths are 800, 800 and 400 nm, was predicted to produce a signal that is spatially and frequentially (at 267 nm) separated from the background^{[Reference Lundström, Brodin, Lundin, Marklund, Bingham, Collier, Mendonça and Norreys59]}. For respective beam powers 0.1, 0.1 and 0.5 PW, taking the interaction region to be cuboidal, on average 0.07 photons would be frequency-upshifted per collision of the beams, which is predicted to be larger than the Compton-scattering background. A signature of the frequency-shifting four-wave mixing process on the number of total measurable diffracted photons for a collision of two ultra-short focused Gaussian pulses was also calculated^{[Reference King and Keitel43]}. For 10 PW total beam power split into a probe with wavelength 228 nm and duration 2 fs, as the duration of the 910 nm pump is reduced to 1 fs, the total number of diffracted photons is predicted to change by around 20%, equal to one photon per shot. Calculations beyond the paraxial approximation recently performed^{[Reference Fillion-Gourdeau, Lefebvre and MacLean60]} for two co-propagating beams of different frequencies incident on a parabolic mirror suggest 1–10 PW laser beams are required to observe vacuum frequency mixing, although the method of detecting the signal needs to be given more attention.

Vacuum high-harmonic generation can take place if the colliding laser pulses have the same frequency. Then via the four-wave mixing process in Equation (37), if ${\it\omega}_{p}={\it\omega}_{s,1}={\it\omega}_{s,2}={\it\omega}$, the signal of the vacuum process has a frequency ${\it\omega}^{\prime }=3{\it\omega}$ and so is at the third harmonic of the probe. By considering six-, eight- and in general $2n$-wave mixing as depicted in Figure 8, it can be seen that a harmonic spectrum for the vacuum interaction can be produced. As each extra interaction between the virtual pair and a laser photon is weighted at the amplitude level with a factor $E/E_{\text{cr}}\ll 1$, higher harmonics are in general exponentially suppressed. Nevertheless, the harmonic spectrum produced by a standing wave formed of two monochromatic pump laser beams was calculated for subcritical ($E<E_{\text{cr}}$) strengths where higher harmonic orders $j$ were found^{[Reference Di Piazza, Hatsagortsyan and Keitel61]} to follow the hierarchy $(E/E_{\text{cr}})^{4j}$. In a setup involving three beams, the minimum power of each laser required to scatter one photon was found to be:

(38)$$\begin{eqnarray}\displaystyle P_{\min }\approx 33.5\,\frac{{\it\lambda}}{1~\text{nm}}\frac{w_{0}}{1~\text{nm}}\left(\frac{1~\text{fs}}{{\it\tau}}\right)^{1/3}\left(\frac{1~\text{fs}}{{\it\tau}_{c}}\right)^{2/3}~\text{GW},\qquad & & \displaystyle\end{eqnarray}$$

for typical beam cross-sectional dimension $w_{0}$, interaction duration ${\it\tau}$ and coherence time ${\it\tau}_{c}$. The most likely frequency of the scattered photon is, however, the fundamental harmonic. The intensity at which a single focused laser pulse will begin to produce harmonics via self-interaction has been studied^{[Reference Fedotov and Narozhny62]}, with the conclusion that a pulse of 1000 nm photons focused within a cone of angle 0.1 rad will produce one photon per period at $5\times 10^{27}~\text{W}~\text{cm}^{-2}$. A recent calculation of an alternative route to high-harmonic generation through having many scattering events involving low numbers of photons^{[Reference Lutzky and Toll63–Reference Heyl and Hernquist68]} (as in Figure 9) has recently been suggested to be more efficient. For the collision of a Gaussian probe at much higher frequency than the background, if the parameter $(64{\it\alpha}/105{\it\pi})(E_{s}^{3}E_{p}/E_{\text{cr}}^{4}){\it\omega}_{p}{\it\tau}_{s}$, where ${\it\tau}_{s}$ is the duration of the pump, can be made close to unity, harmonic generation will dominate, with the spectrum displaying a power-law behaviour and the appearance of a corresponding electromagnetic shock^{[Reference Böhl, King and Ruhl34]}.

Figure 8.

Vacuum high-harmonic generation of the $n$th harmonic of the probe via $2n$-photon scattering.

Figure 9.

Vacuum high-harmonic generation of the $n$th harmonic of the probe via a chain of six-photon scattering.

Figure 10.

An incoming probe photon can split into $k$ outgoing ones, due to interaction with the background.

Photon splitting as depicted in Figure 10, is sometimes thought of as the opposite of high-harmonic generation, but unlike harmonic generation, the emitted photons can have a continuum of energies. If one considers splitting to two photons via four-wave mixing then via energy and momentum conservation, one possibility is:

(39)$$\begin{eqnarray}\displaystyle {\it\omega}_{p}+{\it\omega}_{s}={\it\omega}_{1}^{\prime }+{\it\omega}_{2}^{\prime };\quad \mathbf{k}_{p}+\mathbf{k}_{s}=\mathbf{k}_{1}^{\prime }+\mathbf{k}_{2}^{\prime }, & & \displaystyle\end{eqnarray}$$

where now two constraints on these equations are $({\it\omega}_{1,2}^{\prime })^{2}=\mathbf{k}_{1,2}^{\prime }\boldsymbol{\cdot }\mathbf{k}_{1,2}^{\prime }$. The continuum of allowed energies and the possibility for a wide angular distribution of emitted photons makes this process worthy of study. The process has been comprehensively studied for a probe photon propagating through a plane-wave background of arbitrary form and polarization^{[Reference Di Piazza, Milstein and Keitel69]}, which was found to depend on the two parameters ${\it\eta}={\it\omega}_{p}{\it\omega}_{s}/m^{2}$ and ${\it\chi}=({\it\omega}_{p}/m)(E/E_{\text{cr}})$. Two events per hour were predicted using $10^{8}$ 250 MeV tagged photons per second almost counterpropagating with 100 fs $10^{15}~\text{W}~\text{cm}^{-2}$ 1 keV XFEL beams separated by 93 ns. Alternatively, two events per hour were also predicted using $10^{8}$ 100 MeV tagged photons counterpropagating with a 1 Hz 1 eV optical pump of intensity $10^{25}~\text{W}~\text{cm}^{-2}$. The conclusion was that a different experimental setup must be considered if this effect is to be observed in the near future^{[Reference Di Piazza, Milstein and Keitel69]}.

Figure 11.

Cerenkov-like radiation (right) generated by pulse collapse into photon bullets (left) against longitudinal $z$ and transverse $r$ co-ordinates of an initially Gaussian pulse of central wavenumber $k_{0}$. Reproduced with permission^{[Reference Marklund, Brodin and Stenflo75]}.

4.4 Effects on probe pulse form

In addition to the effects on single photons, one can consider the consequence of real photon–photon scattering on the propagation of an ensemble of photons. A probe laser pulse can be understood as a superposition of photons with a range of frequencies and phases. From the study of nonlinear dispersive media, it is well known that a refractive index that depends on a probe’s intensity directly or indirectly can lead to pulse shape effects^{[Reference Moloney and Newell58]}. In particular, for the interaction with vacuum, probe pulse effects can occur if the next-to-leading-order effect of a probe-dependent refractive index is taken into account.

Nonlinear phase shift is a term used to denote the relative difference in phases between parts of a probe beam that have experienced different vacuum refractive indices. For a constant refractive index, the relative phase difference compared to a unitary refractive index is:

$$\begin{eqnarray}{\it\delta}{\it\phi}=(\mathsf{n}_{\text{vac}}-1){\it\omega}_{p}z,\end{eqnarray}$$

where ${\it\omega}_{p}z$ is the phase over which ${\it\delta}{\it\phi}$ has been accrued. For two counterpropagating initially monochromatic plane waves, with the envisaged ELI parameters of 800 nm wavelength, $10^{25}~\text{W}~\text{cm}^{-2}$ intensity, 10 fs duration and $10~{\rm\mu}\text{m}$ focal spot diameter, a phase shift of the order of ${\it\delta}{\it\phi}\approx 10^{-7}~\text{rad}$ has been calculated^{[Reference Ferrando, Michinel, Seco and Tommasini33, Reference Tommasini, Ferrando, Michinel and Seco70]}. This nonlinear phase shift can be enhanced by using multiple crossings of the interacting beams. For $N_{r}$ reflections from plasma mirrors of reflectivity $R_{\text{mir}}$ of two beams crossing each other at an angle ${\it\theta}_{c}$, the gain factor has been calculated to be^{[Reference Tommasini, Ferrando, Humberto and Seco71]}:

$$\begin{eqnarray}\sin ^{4}\left(\frac{{\it\theta}_{c}}{2}\right)\mathop{\sum }_{n=0}^{N_{r}+1}R_{\text{mir}}^{n}.\end{eqnarray}$$

The measurement of this phase shift using Fourier imaging has also been explored^{[Reference Homma, Habs and Tajima72]}.

Vacuum self-focusing is an analogue to the well-known plasma self-focusing or ‘Benjamin–Weir’ instability^{[Reference Moloney and Newell58]} in which there is positive feedback between a refractive index increasing the intensity of a pulse via focusing, and a higher intensity resulting from that focusing in turn increasing the refractive index. Mutual channelling of counterpropagating laser pulses and large-scale focusing have been considered, but either YW powers are predicted as necessary^{[Reference Rozanov66]} or intensities above critical^{[Reference Fedotov73]}, before which vacuum pair-creation would have set in. In considering the idealized geometry of a Gaussian plane-wave probe pulse counterpropagating and interacting via six-wave mixing with a much slower varying pump, the probe-dependent refractive index:

(40)$$\begin{eqnarray}\displaystyle \mathsf{n}_{\text{vac}}^{\Vert }=1+\frac{{\it\alpha}}{{\it\pi}}\frac{E_{s}^{2}}{E_{\text{cr}}^{2}}\left[\frac{8}{45}+\frac{64}{105}\frac{E_{s}}{E_{\text{cr}}}\frac{E_{p}}{E_{\text{cr}}}\right] & & \displaystyle\end{eqnarray}$$

was predicted to lead to the generation of a shock wave, a signature of self-focusing, when the phase difference due to the probe-dependent refractive index tended to a quarter wavelength^{[Reference Böhl, King and Ruhl34]}. The mutual attraction between two photons due to mutual exchange of virtual photons on two vacuum loops has also been considered^{[Reference Kharzeev and Tuchin74]}, and a self-focusing angle of ${\it\theta}=\sqrt{157/16{\it\pi}^{3}}({\it\alpha}^{2}/180m^{4}R^{4})$ for photon separation $R$.

Pulse collapse is predicted to occur for high-intensity probe pulses propagating through an even higher-intensity background. The wave equation for the probe can be recast as a nonlinear Schrödinger equation^{[Reference Moloney and Newell58]} with the consequence that the pulse envelope becomes space–time-dependent, even if assumed initially homogeneous. Unlike typical optically nonlinear dispersive media, the nonlinearity of the vacuum is ‘formed’ by the pump laser background, which is then probed by a second pulse. Even when the leading-order effect on the probe is a nonlinear refractive index that is independent of the probe pulse, because of its effect on the pump’s evolution, it can indirectly effect the probe’s propagation. This interplay between a Gaussian probe distribution propagating through a radiation gas has been demonstrated to lead to self-focusing and collapse of the probe into ‘photon bullets’, thereby driving acoustic waves^{[Reference Marklund, Brodin and Stenflo75]} as demonstrated in Figure 11. Depending on initial parameters, probe collapse can occur before or after the critical Schwinger limit is reached^{[Reference Marklund, Eliasson and Shukla76]}.

4.5 Finite-time effects

Similar to the case for regular plasmas, there are effects on the probe when propagating through regions of the polarized ‘vacuum plasma’ that do not persist long enough to be directly detected.

Photon acceleration is well known from plasma physics^{[Reference Wilks, Dawson, Mori, Katsouleas and Jones77]} and corresponds to the frequency downshift (upshift) as probe photons traverse an increasing (decreasing) plasma gradient. The possibility of measuring this effect in vacuum has been considered for a probe photon propagating almost parallel with a pump pulse^{[Reference Mendonca, Marklund, Shukla and Brodin78]}, with a frequency up (down) shift occurring at the rear (front) of the pump beam.

High-harmonic generation can also occur due to the inhomogeneity of the pump pulse background, in an effect distinct from standard vacuum high-harmonic generation. For a probe pulse counterpropagating with a slowly varying background, this is predicted to occur at finite time during overlap of the probe and pump pulses at an order earlier (via four-photon scattering), than for those photons that reach a detector (via six-photon scattering)^{[Reference King, Böhl and Ruhl79]}. This finite-time signal disappears when the probe and pump pulses are well separated again, but is calculated to dominate the signal of frequency-shifted photons when the pulses overlap in this setup if $(E_{s}/E_{\text{cr}})^{2}{\it\omega}_{p}{\it\tau}_{s}\ll 1$ for strong-pulse duration ${\it\tau}_{s}$.

Gradient-dependent vacuum refractive index is a way to describe the addition to the standard predicted vacuum refractive index that occurs when the pump laser is time varying. This has been calculated for a probe propagating through the electric/magnetic antinode of a pump standing wave^{[Reference Hu and Huang36]}. The change in vacuum refractive index ${\rm\Delta}\mathsf{n}_{\text{vac}}$ can be written in the form:

(41)$$\begin{eqnarray}\displaystyle {\rm\Delta}\mathsf{n}_{\text{vac}}^{\Vert ,\bot }({\it\varphi})=\frac{E_{p}}{E_{p}^{\prime }}\mathsf{n}_{\text{vac}}^{\Vert ,\bot \,\prime }({\it\varphi}). & & \displaystyle\end{eqnarray}$$

In a setup of two colliding plane waves with no transverse structure, it was shown that this term is a surface term and is zero initially and finally, when the probe and background are well separated^{[Reference King, Böhl and Ruhl79]}. The contact term was also noted in a recent study of polarization flipping in arbitrary plane waves^{[Reference Dinu, Heinzl, Ilderton, Marklund and Torgrimsson37]}. Although it has been suggested that this part of the interaction could be a useful probe of dark matter particles^{[Reference Hu and Huang36]}, a consistent finite-time calculation has yet to be performed to establish the nature of this effect.

4.6 Nonperfect vacua

In any realistic experiment, the vacuum will be synthetic and hence imperfect. Residue particles in interaction chambers will also be affected by intense laser pulses and can produce a source of background that may obscure the measurement of real photon–photon scattering. The Cotton–Mouton effect, in which a dilute gas becomes birefringent in the presence of an electromagnetic wave is just one such example^{[Reference Zavattini, Gastaldi, Pengo, Ruoso, Della Valle and Milotti80]}. In light of this, various proposals have been considered that instead use an altered vacuum to enhance the signal of vacuum polarization.

Resonant cavities can be employed in order to increase the sensitivity of whatever eigenfrequencies are resonant for that particular cavity^{[Reference Jackson48]}. For example, a cavity can be designed such that the frequency that is generated by vacuum four-wave mixing of two modes of the cavity, is resonant. This idea has been studied for the $\text{TE}_{01}$ modes of such a cavity and the growth of the mixing signal in the form of the longitudinal standing-wave magnetic field, found to increase linearly with time^{[Reference Brodin, Marklund and Stenflo81]} as

$$\begin{eqnarray}B_{3}(t)=\frac{itV}{2{\it\omega}_{3}}\,B_{1}^{2}B_{2}^{\ast },\end{eqnarray}$$

for source magnetic standing-wave strengths $B_{1},B_{2}$ and coupling constant $V$. The vacuum signal was predicted to be detectable if an electric field $2\times 10^{-8}$ times the critical Schwinger field was employed with a superconducting cavity with a resistance of $1~\text{n}{\rm\Omega}$ and a resonant, vacuum-mixing frequency of $13.2~{\rm\mu}\text{eV}$. This idea was refined^{[Reference Eriksson, Brodin, Marklund and Stenflo82]} and the prediction made that 18 photons can be produced by a magnetic field of around 0.28 T in a cylindrical cavity of length 2.5 m, radius 25 cm and quality factor $4\times 10^{10}$.

Real plasmas already have a refractive index different from unity, and this can combine with the shift of the refractive index due to vacuum polarization and lead to an enhancement. The system of equations by Akhiezer and Polovin^{[Reference Akhiezer and Polovin83]} for the propagation of a circularly polarized plane wave through a cold collisionless plasma was updated to include the vacuum current in Maxwell’s equations and also take into account collisions^{[Reference Di Piazza, Hatsagortsyan and Keitel84]}. For the collisionless case, the modified refractive index of the combined system was found to be:

$$\begin{eqnarray}\mathsf{n}=\sqrt{\mathsf{n }_{\text{pl}}^{2}+{\textstyle \frac{1}{4}}{\it\delta}\mathsf{n}_{\text{vac}}^{\bot }(1-\mathsf{n}_{\text{pl}}^{2})^{2}}\end{eqnarray}$$

with $\mathsf{n}_{\text{pl}}$ the plasma refractive index and ${\it\delta}\mathsf{n}_{\text{vac}}^{\bot }=\mathsf{n}_{\text{vac}}^{\bot }-1$ as defined in Equation (26). Another detectable signal of photon–photon scattering has been calculated to exist when an overdense plasma channel is subjected to an intense laser beam^{[Reference Shen, Yu and Wang85]}. In addition, the altered dispersion relation for electromagnetic waves due to vacuum polarization effects in a strongly magnetized cold plasma has been calculated^{[Reference Brodin, Marklund, Stenflo and Shukla86–Reference Shukla and Stenflo88]}, which is particularly relevant for the dynamics of strongly magnetized neutron stars.