Variational Principles for Electromagnetic Potential Scattering

Abstract

Variational methods are considered for the solution of the vector wave equation describing the field due to an arbitrary source placed in the neighborhood of an inhomogeneous absorbing medium. Variational principles for the tensor Green's function satisfying the point source equation, $\nabla \times \nabla \times \Gamma (\mathrm{r}, {\mathrm{r}}^{\prime })-k^2\Gamma (\mathrm{r}, {\mathrm{r}}^{\prime })+U\left(\mathrm{r}\right)\Gamma (\mathrm{r}, {\mathrm{r}}^{\prime })=-\mathrm{I}\delta (\mathrm{r}-{\mathrm{r}}^{\prime })$, have been obtained in linear and exponential forms, analogous to the Altshuler principles for the scalar wave function. Stationary forms for the wave function and the scattering amplitude in the standard scattering problem (incident plane wave, outgoing solutions) are recovered when the point source recedes to infinity. For the special case of a spherically symmetric scatterer, the analysis leads to variational principles for the two independent $l\mathrm{th}$-order phase shifts. The method is illustrated by a calculation of the fields internal to an axially symmetric potential.