Modal Expansions and Orthogonal Complements in the Theory of Complex Media Waveguide Excitation by External Sources for Isotropic, Anisotropic, and Bianisotropic Media

A unified electrodynamic approach to the guided wave excitation by external sources in the waveguiding structures with bianisotropic media is developed. Effect of electric, magnetic, and magneto-electric losses in such media manifests itself in the special form of eigenmode orthogonality referred to as the quasi-orthogonality relation. It reflects the existence of the cross-power flow Pkl and loss Qkl for any pair (k, l) of modes which are rigidly linked to each other by this relation. The quasi-orthogonality relation remains true in the limiting case of lossless waveguides yielding the customary relations of orthogonality and normalization for propagating (active) modes and also their generalization for nonpropagating (reactive) modes. It is shown that the eigenmode set for a waveguiding structure is complete only outside the region of exciting sources. Inside this region the modal expansions of fields are incomplete and must be supplemented with the orthogonal complementary fields which extend the proper Hilbert space spanned by waveguide eigenfunctions. Among exciting sources there are the external bulk sources (currents, fields, and medium perturbations) and the external surface currents. Besides, the orthogonal complementary fields generate the effective surface currents on boundaries of the bulk exciting sources. The problem of waveguide excitation by external sources is solved by means of determining both the mode amplitudes for the modal field expansions and the orthogonal complementary fields inside the source region. The equations of mode excitation are derived on the basis of three approaches applying the direct use of Maxwell’s equations, the electrodynamic analogy with the mathematical method of variation of constants, and the conjugate reciprocity theorem.