Do vector plane waves form complete basis of solutions to Maxwell's equations? Introduction to Generalized Plane Wave Solutions

Abstract

As the title says, this article questions the well established belief that the vector plane waves form a complete basis of solutions to Maxwell's equations. Vector solutions to Maxwell's equations are presented here, which have planar wave-fronts and transverse electric and magnetic fields but spatially varying polarization. They form a one-parameter family specified by integer $n$, and are termed as \textit{generalized vector plane waves}. The known vector plane wave solution with spatially uniform polarization, referred to as \textit{conventional vector plane waves} in this article, is a subset of this family obtained for $n = 0$. In contradiction to the established belief, it is shown that these \textit{generalized vector plane waves} with spatially varying polarization (for $n \neq 0$) cannot be expressed as superposition of conventional vector plane waves. The family of solutions also includes the interesting cases of radially and azimuthally polarized plane waves for $n=1$.