The electromagnetic scattering problem in the time domain for a dissipative slab and a point source using invariant imbedding

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Abstract
A vertical magnetic dipole source above an inhomogeneous slab is considered where the permittivity ε=ε(z) and conductivity σ=σ(z) vary with the depth z. Using the axial symmetry of the problem, a Hankel transform is introduced. Besides physical scattering kernels, ‘‘nonphysical’’ scattering kernels are introduced and the direct and inverse transformed problem is solved by means of wave splitting in an invariant imbedding approach. The imbedding equations for the nonphysical scattering kernels are simpler than the corresponding equations for the physical kernels and they are solved using existing techniques for the imbedding equations for normally incident plane waves. The solution for the physical kernels is obtained by establishing relations between the two sets of kernels. These relations are given by Volterra equations of the second kind and thus are very easy to solve numerically. Some applications and extensions of the results derived in the paper are discussed.