Far-field Green functions

Abstract

In this appendix we state the asymptotic far-field Green functions for a planarly layered medium. It is assumed that the source point r 0 = ( x 0 , y 0 , z 0 ) is in the upper half-space ( z > 0). The field is evaluated at a point r = ( x , y , z ) in the far-zone, i.e. r ≫ λ . The optical properties of the upper half-space and the lower half-space are characterized by e 1 , μ 1 and e n , μ n , respectively. The planarly layered medium in between the two halfspaces is characterized by the generalized Fresnel reflection and transmission coefficients. We choose a coordinate system with origin on the topmost surface of the layered medium with the z -axis perpendicular to the interfaces. In this case, z 0 denotes the height of the point source relative to the topmost layer. In the upper half-space, the asymptotic dyadic Green function is defined as where p is the dipole moment of a dipole located at r 0 and G 0 and G ref are the primary and reflected parts of the Green function. In the lower half-space we define with G tr being the transmitted part of the Green function. The asymptotic Green functions can be derived by using the far-field forms of the angular spectrum representation. The primary Green function in the far-zone is found to be The reflected part of the Green function in the far-zone is where the potentials are determined in terms of the generalized reflection coefficients of the layered structure as The transmitted part of the Green function in the far-zone is where δ denotes the overall thickness of the layered structure.