Symmetry, Unitarity, and Geometry in Electromagnetic Scattering

Upon defining vector spherical partial waves {${\mathbf{\Psi }}_n$} as a basis, a matrix equation is derived describing scattering for general incidence on objects of arbitrary shape. With no losses present, the scattering matrix is then obtained in the symmetric, unitary form $S=-{\hat{Q}}^{\prime *}{\hat{Q}}^{*}$, where (perfect conductor) $\hat{Q}$ is the Schmidt orthogonalization of $Q_{n{n}^{\prime }}=\left(\frac{k}{\pi }\right)\int d\mathbf{\sigma }·\left[\right(\nabla \times \mathrm{Re}{\mathbf{\Psi }}_n)\times {\mathbf{\Psi }}_{n^{\prime }}]$, integration extending over the object surface. For quadric (separable) surfaces, $Q$ itself becomes symmetric, effecting considerable simplification. A secular equation is given for constructing eigenfunctions of general objects. Finally, numerical results are presented and compared with experimental measurements.