Green's function of periodic scatterers applied to scattering by finite periodic arrays of scatterers

Abstract

The Green's function is an important physical concept and is key to the formulation of integral equations. In the past, the free space Green's function and the Green's function of an empty lattice were used to study wave behavior in periodic structures such as metamaterials and photonic crystals. In this paper, the Green's function of periodic scatterers are calculated and next used to formulate integral equations for scattering by finite periodic arrays. The Green's function of periodic scatterers is expressed in terms of band solutions. A low wavenumber component is subtracted out from the Green's function. The low wavenumber component represents the slowly converging reactive near field. The remaining part is shown to have no singularity and to converge rapidly with respect to the number of modes included. It is efficient to be evaluated over a wide range of wavenumbers. Multiple band solutions are derived from a linear eigenvalue problem, converted from a surface integral equation, where the technique of broadband Green's function with low wavenumber extraction (BBGFL) is used to represent the propagator. The Green's function of periodic scatterers satisfies boundary conditions on the scatterers. When it is used to formulate a surface integral equation of scattering from a finite periodic array, the unknowns are limited to the boundaries of the finite structure. We demonstrate such applications by computing the reflections from a half-space of periodic scatterers. The results are compared to that of half space Green's function of an effective medium for the half space medium.