GENERALIZATION OF THE PARABOLIC EQUATION FOR EM WAVES IN A DIELECTRIC LAYER OF NONUNIFORM THICKNESS

The parabolic equation method by Leontovich and Fock is usually applied to scalar wave equations or to such vector problems that admit introduction of scalar potentials. Here, we seek an approximate description of three-dimensional wave propagation in a nonuniform dielectric layer. It is supposed that the characteristic thickness B is large compared with the wavelength lambda=2 pi/k, and small compared with the characteristic scale of horizontal nonuniformity L. In many cases of practical interest, the Fresnel parameter sigma=kB(2)/L is of the order of unity. Under this assumption, we construct an asymptotic solution of Maxwell's equations as a series in powers of the smoothness parameter nu=B/L much less than 1. Its leading term can be expressed via solutions of two parabolic equations for slowly changing scalar amplitudes.