Appendix D - Far-field Green functions

Summary

In this appendix we state the asymptotic far-field Green functions for a planarly layered medium. It is assumed that the source point r₀ = (x₀, y₀, z₀) 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 ε₁, μ₁ and ε_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, z₀ 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 r₀ and G₀ 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.