Full Vectorial Maxwell Equations with Continuous Angular Indices

TL;DR

This paper develops a mathematical framework for solving Maxwell's equations with continuous angular spectral indices in cylindrical and spherical geometries, capturing singular behaviors and ensuring finite-energy solutions.

Contribution

It introduces a continuous spectral representation for Maxwell's equations in symmetric geometries, extending beyond discrete harmonic decomposition and including explicit spectral kernels.

Findings

Finite energy solutions for bcl > -1/2

Explicit spectral kernels constructed via biorthogonal systems

Validated through Galerkin methods and numerical spectral integration

Abstract

This article presents a mathematical framework for solving Maxwell's equations in cylindrical and spherical geometries with continuous angular indices. We extend beyond standard discrete harmonic decomposition to a continuous spectral representation using generalized spectral integrals, capturing electromagnetic solutions that exhibit singular behavoiur yet yield finite-energy fields at the geometric center. For continuous angular indices $\ell, m \in \mathbb{R}$, we study existence and uniqueness of solutions in weighted Sobolev spaces $H^s_{\alpha(\ell,m)}(\Omega)$ following the framework established in ~\cite{adams2003, reed1975}, prove finite energy for $\ell > -\frac{1}{2}$, and construct explicit spectral kernels via biorthogonal function systems. The framework encompasses both separable cylindrical modes with continuous azimuthal index $\nu \in (0,1)$ and non-separable spherical modes where field components couple through vectorial curl operations. We present asymptotic analysis of singular field behavior, investigate convergence rates for spectral approximations, and validate the theoretical framework through Galerkin projection methods and numerical spectral integration.