Variational principles for guided electromagnetic waves in anisotropic materials

It is the purpose of this paper to develop a general method for obtaining approximate solutions to the problems of the propagation of waves in a guide which may only be partially filled with material having tensor electromagnetic properties. With the introduction of an appropriate dyadic Green’s function we are able to obtain a formal solution to the problem in terms of an integral involving the field vectors in the perturbing rod. The reformulation of the original problem in terms of an integral equation enables us to construct a variational principle whose extremal value is insensitive to a great range of trial functions. The extremal value of the variational expression from which the integral equations for the field are derivable is shown to be proportional to $(\gamma _0^2 - {\gamma ^2})$, the difference between the square of the propagation constant of the wave in the empty and loaded guide. Normal modes in terms of which an expansion of the dyadic Green’s function is obtained are defined. In the last section we demonstrate the ability of the variational method to improve the results obtained by perturbation methods obtaining a first order approximation for the propagation constant of a wave in a rectangular guide containing an infinite ferrite slab.