Transfer of polarized light in an isotropic medium: Biorthogonality and the solution of transfer problems in semi infinite media

Abstract

Sets of adjoint function vectors, which can be interpreted as describing the emergent radiation for generalized Milne's problems, are constructed by means of Case's eigenvectors and the albedo matrix. It is shown that the adjoints allow us to transform the full range orthogonality relation for eigenvectors of the positive half spectrum into a half range biorthogonality relation. A half range closure relation is derived. Half range integral equations are obtained for the eigenvectors, as well as for its adjoints. By separating the variables, the adjoints and the albedo-matrix are reduced to fundamental H-matrices and sets of generalized Busbridge polynomials, where the H-matrices obey linear singular integral equations. The half range biorthogonality is applied to the solution of half space standard problems. The complete radiation fields for Milne's problem and the albedo problem are obtained. A spectral representation of the Green's function matrix for a semi infinite medium is derived in terms of eigenmode and generalized Milne's problems.