The fundamental solution of the time-dependent system of crystal optics

We set up the electromagnetic system and its plane-wave solutions with the associated slowness and wave surfaces. We treat the Cauchy initial-value problem for the electric vector and make explicit the quantities necessary for numerical evaluation. We use the Herglotz-Petrovskii representation as an integral around loops which, for each position and time form the intersection of a plane in the space of slownesses with the slowness surface. The field and especially its singularities are strongly dependent on the varying geometry of these loops; we use a level set numerical technique to compute those real loops which essentially give us second order accuracy. We give the static term corresponding to the mode with zero wave speed. Numerical evaluation of the solution is presented graphically followed by some concluding remarks.