  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, the 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.
