Photon merging and splitting in electromagnetic field inhomogeneities

We investigate photon merging and splitting processes in inhomogeneous, slowly varying electromagnetic fields. Our study is based on the three-photon polarization tensor following from the Heisenberg-Euler effective action. We put special emphasis on deviations from the well-known constant field results, also revisiting the selection rules for these processes. In the context of high-intensity laser facilities, we analytically determine compact expressions for the number of merged/split photons as obtained in the focal spots of intense laser beams. For the parameter range of typical petawatt class laser systems as pump and probe, we provide estimates for the numbers of signal photons attainable in an actual experiment. The combination of frequency upshifting, polarization dependence and scattering off the inhomogeneities renders photon merging an ideal signature for the experimental exploration of nonlinear quantum vacuum properties.

Figure 1

Feynman diagrams of the three-photon interactions considered in this work. (a) Photon splitting: An incoming photon with momentum $k^{\mu }$ and polarization four-vector ${\epsilon }_{\rho }^{(p)}(k)$ splits, under the assistance of the external field inhomogeneity $F^{\mu \nu }(x)$, into two photons with $\{{k^{\prime }}^{\mu },{\epsilon }_{\sigma }^{*(p^{\prime })}(k^{\prime })\}$ and $\{{k^{\prime \prime }}^{\mu },{\epsilon }_{\eta }^{*(p^{\prime \prime })}(k^{\prime \prime })\}$. (b) Photon merging: Two incoming photons with $\{{k^{\prime }}^{\mu },{\epsilon }_{\sigma }^{(p^{\prime })}(k^{\prime })\}$ and $\{{k^{\prime \prime }}^{\mu },{\epsilon }_{\eta }^{(p^{\prime \prime })}(k^{\prime \prime })\}$ merge to yield a single outgoing photon with $\{k^{\mu },{\epsilon }_{\rho }^{*(p)}(k)\}$. The coupling of the probe photons and the external field inhomogeneity $F^{\mu \nu }(x)$ to the electron-positron loop is encoded in the three-photon polarization tensor ${\mathrm{\Pi }}^{\rho \sigma \eta }$.

Figure 2

Left: geometry of photon merging in a localized background field inhomogeneity with mutually perpendicular $\mathbf{E}$, $\mathbf{B}$ and $\mathit{\kappa }\sim {\widehat{\mathbf{e}}}_{\mathrm{z}}$ for the particular choice of $\phi =0$ (cf. main text). In this field configuration, two incoming probe photons with momenta ${\mathbf{k}}^{\prime }$ and ${\mathbf{k}}^{\prime \prime }$ may merge into one photon with momentum $\mathbf{k}$. For photon splitting (not depicted) the roles are reversed: An incident photon with momentum $\mathbf{k}$ may split into two photons with momenta ${\mathbf{k}}^{\prime }$ and ${\mathbf{k}}^{\prime \prime }$. The polarization degrees of freedom of the photons are spanned by the unit vectors ${\mathit{\epsilon }}^{(1)}(q)$ and ${\mathit{\epsilon }}^{(2)}(q)$, where $q\in \{k,k^{\prime },k^{\prime \prime }\}$. This figure depicts the special case where the incident photons propagate in the x-z plane, and ${\gamma }^{\prime }={\gamma }^{\prime \prime }=0$. Right: the convention for the trihedron composed of $\widehat{\mathbf{k}}$, ${\mathit{\epsilon }}^{(1)}(k)$ and ${\mathit{\epsilon }}^{(2)}(k)$ is such that for $\gamma =0$ the polarization vector ${\mathit{\epsilon }}^{(1)}(k)$ lies in the x-y plane (shaded area).

Figure 3

Left panel: total number of merged photons ${\mathcal{N}}_{\text{Merg}}$ as a function of the angle ${\theta }^{\prime \prime }=\sphericalangle ({\mathbf{k}}^{\prime \prime },{\widehat{\mathbf{e}}}_{\mathrm{z}})$ attainable for the beam configuration sketched on right panel driven by two 1PW-class lasers as described in the main text. In the vicinity of the focal spot of a Gaussian beam, curvature effects can be neglected, justifying the simpler pump-beam profile (19) employed in this work (indicated by the dashed lines). The result includes an integration over the complete energy range and the full solid angle of the induced merged photons. The probe beams have been chosen to propagate in the x-z plane, i.e. ${\varphi }^{\prime }=0$, ${\varphi }^{\prime \prime }=0$, and ${\theta }^{\prime }=\pi$ (cf. also Fig. 2). The polarization of the pump beam is chosen as $\phi =0$. The plot depicts the number of merged photons for various polarization assignments of the probe photons, allowed according to the selection rules in Table 1 (right). For two 10PW driver lasers, the number of merged photons increases by a factor of 1000.

Figure 4

Emission characteristics of the attainable number of merged photons for our setup driven by two 1PW-class lasers. The parameters of the incoming beams are chosen as ${\theta }^{\prime }=\pi$, ${\varphi }^{\prime }=0$, ${\theta }^{\prime \prime }=1.23\mathrm{rad}$ and ${\varphi }^{\prime \prime }=0$; the polarization of the pump beam is $\phi =0$. The angle ${\theta }^{\prime \prime }$ maximizing the merged photon yield is visible in Fig. 3. Left panel: differential photon number $\mathrm{d}{\mathcal{N}}_{\text{Merg}}(\theta )$ as a function of the polar angle $\theta$, with the corresponding energy and $\varphi$ integrations performed over the full parameter regime. Right panel: $\mathrm{d}{\mathcal{N}}_{\text{Merg}}(\varphi )$ as a function of the polar angle $\varphi$, with the corresponding energy and polar angle integrals performed over their full parameter regimes. From these plots we infer that the merged signal photons are predominantly emitted in the x-z plane, at an outgoing polar angle of $\theta \approx 2.46\mathrm{rad}$. For two 10PW driver lasers, the number of merged photons increases by a factor of 1000.