As an important QED effect to detect the vacuum polarization, birefringence in the presence of a strong electric and magnetic field, E0⊥ B0, E0≤ c B0, is considered. The directional dependence of birefringence is obtained. In two special cases： E0= 0 and E0= c B0, our results are consistent with the previous ones. The refractive index of the probe wave propagating in the -E0× B0 direction decreases with E0/c B0, while that in the-E0× B0 direction increases with E0/c B0.The physics of the direction dependence of birefringence maybe the E0× B0 drift velocity of the virtual electrons and positrons.
Contrary to the superposition principle, it is well known that photorefraction exists in the vacuum with the presence of a strong static field, a laser field, or a rotational magnetic field. Different from the classical optical crystals, the refractive index also depends on the phase of the strong electromagnetic field. We obtain the phase and direction dependence of the refractive index of a probe wave incident in the strong field of a circular-polarized plane wave by solving the Maxwell equations corrected by the effective Lagrangian. It may provide a valuable theoretical basis to calculate the polarization evolution of waves in the strong electromagnetic circumstances of pulsar or neutron stars.
