Electromagnetic Scattering by Unbounded Rough Surfaces

This paper is concerned with the analysis of the electromagnetic wave scattering in inhomogeneous medium with infinite rough surfaces. Consider a time-harmonic electromagnetic field generated by either a magnetic dipole or an electric dipole incident on an infinite rough surface. Throughout the dielectric permittivity is assumed to have a positive imaginary part which accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Maxwell equations with transparent boundary conditions proposed on plane surfaces with the inhomogeneity in between. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. The perfectly matched layer (PML) method is investigated to truncated the unbounded rough surface electromagnetic scattering problem in the direction away from the rough surfaces. It is shown that the truncated PML problem attains a unique solution. An explicit error estimate is given between the solution of the scattering problem and that of the truncated PML problem. The error estimate implies that the PML solution converges exponentially to the scattering solution by increasing either the PML medium parameter or the PML layer thickness. The convergence result is expected to be useful for determining the PML medium parameter in the computational electromagnetic scattering problem.