Vacuum polarization and photon propagation in an electromagnetic plane wave

The QED vacuum polarization in external monochromatic plane-wave electromagnetic fields is calculated with spatial and temporal variations of the external fields being taken into account. We develop a perturbation theory to calculate the induced electromagnetic current that appears in the Maxwell equations, based on Schwinger's proper time method, and combine it with the so-called gradient expansion to handle the variation of external fields perturbatively. The crossed field, i.e., the long wavelength limit of the electromagnetic wave, is first considered. The eigenmodes and the refractive indices, as the eigenvalues associated with the eigenmodes, are computed numerically for the probe photon propagating in some particular directions. In so doing, no limitation is imposed on the field strength and the photon energy, unlike previous studies. It is shown that the real part of the refractive index becomes less than unity for strong fields, a phenomenon that has been known to occur for high-energy probe photons. We then evaluate numerically the lowest-order corrections to the crossed field resulting from the field variations in space and time. It is demonstrated that the corrections occur mainly in the imaginary part of the refractive index.