Symmetry, Unitarity, and Geometry in Electromagnetic Scattering

TLDR

The authors define vector spherical partial waves as a basis, derive a matrix equation for scattering of arbitrarily shaped objects, obtain a symmetric, unitary scattering matrix through Schmidt orthogonalization of Q, and present a secular equation for constructing eigenfunctions. For quadric surfaces, the matrix Q becomes symmetric, simplifying the analysis, and the authors present numerical results that agree with experimental measurements.

Abstract

Upon defining vector spherical partial waves {${\mathbf{\ensuremath{\Psi}}}_{n}$} 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 $S=\ensuremath{-}{\stackrel{^}{Q}}^{\ensuremath{'}*}{\stackrel{^}{Q}}^{*}$, where (perfect conductor) $\stackrel{^}{Q}$ is the Schmidt orthogonalization of ${Q}_{n{n}^{\ensuremath{'}}}=(\frac{k}{\ensuremath{\pi}})\ensuremath{\int}d\mathbf{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}[(\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathrm{Re}{\mathbf{\ensuremath{\Psi}}}_{n})\ifmmode\times\else\texttimes\fi{}{\mathbf{\ensuremath{\Psi}}}_{{n}^{\ensuremath{'}}}]$, 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.