Birefringence and dichroism of the vacuum in the field of a standing electromagnetic wave

Vacuum birefringence and dichroism are investigated in the setup involving a probe photon traversing a strong standing electromagnetic wave formed by two counterpopagating plane-wave laser beams. The analysis is based on the evaluation of the polarization tensor. We consider both the regime of relatively low laser frequency and photon energy and the domain where the energies are of the order of the electron rest energy. In the former case, if the external field is sufficiently weak, one can obtain very accurate predictions by means of the local values of the leading-order contribution to the Heisenberg–Euler effective Lagrangian. However, to address the high-energy and strong-field domains, one has to employ different methods. Here we utilize the locally-constant field approximation (LCFA) and compute the real and imaginary parts of the polarization tensor varying the propagation direction of the probe photon. It is demonstrated that if the propagation axis of the photon is parallel to that of the laser beams, then the effects are governed entirely by the counterpropagating beam, while the copropagating one is irrelevant. If the photon travels perpendicularly to the laser beam axis, the two plane waves are equally significant. In this case, within the Heisenberg–Euler approximation, it is sufficient to multiply the corresponding single-wave result by a factor of two, whereas the LCFA predictions are less trivial as they incorporate the higher-order nonlinear contributions.