Orthogonality Properties for Modes in Passive and Active Uniform Wave Guides

Abstract

Orthogonality relations are given for the ``four‐vector'' guided modes of anisotropic uniform wave guides and for the ``six‐vector'' guided modes of uniform wave guides containing unidirectional electron beams with axially independent dc velocities. In general, these orthogonality relations involve both the modes of the given wave guide and those of an appropriate ``adjoint'' wave guide. A description is given for the adjoint wave guides associated with wave guides which contain media whose characteristics are restricted only by the requirement that they be axially independent and which, if bounded, are subject to a variety of boundary conditions arising from idealizations of actual boundaries.