Statistical theory of rough boundaries for electromagnetic waves I Deterministic Green functions and reflection–transmission matrix

Abstract

Basic equations of a rough boundary are shown based on an effective boundary condition transferred from the real boundary onto two reference boundary planes chosen such that the entire boundary is enclosed by the two planes. The boundary condition is written in terms of a surface impedance, which is an operator and is obtained exactly for given boundary change. Maxwell’s equations are rewritten in a simple form by a 6 × 6 matrix equation (such as Dirac’s equation) with several algebraic formulas, which help us greatly to derive various equations in a compact form and thereby to get a general version of complicated equations. A surface Green function is introduced associated with the boundary condition, and an exact reflection–transmission matrix is obtained from it in an operator form for a given boundary change. The latter is also obtainable as a solution of an ordinary integral equation perfectly free from any operator. Power conservation is directly connected with the Hermitian condition imposed on the surface impedance and is investigated in detail so optical relations eventually can be obtained from it. Other symmetries inherent among associated quantities are also shown. A set of operator equations for the surface Green function is obtained to the first order of the boundary change, consistently with the power conservation, to be applied in Part II of this work [ J. Opt. Soc. Am. A2, 2260 ( 1985)].