## ABSTRACT

In this chapter, the authors examine the Maxwell’s equation. They provide the orthogonality relation for lateral eigenmodes using the Lorentz reciprocity theorem. Whereas the power orthogonality relation only holds in geometries with real-valued permittivity profiles, the orthogonality relation is valid in arbitrary geometries, including those with complex permittivity profiles, and is thus more general. The eigenmode problems are equivalent in that they describe electric and magnetic field components of the same optical transverse electric field. The eigenmode profile describes the mode in the (x, y) plane; however, its z dependence is governed by its propagation constant β alone. It is thus useful to classify different types of eigenmodes according to their eigenvalues.