Full vectorial maxwell equations with continuous angular indices—part II: numerical solvers and convergence analysis

Source abstract

Abstract In part I of this work, we established an analytic framework for solving Maxwell's equations in spherical geometry with continuous angular indices (ℓ,m)∈R2, proving finite-energy conditions for singular solutions and formulating continuous spectral integrals in weighted Sobolev spaces. In this companion article (part II), we develop and validate numerical methods that operationalize this framework. We introduce a Galerkin discretization of the coupled angular operator on divergence-free continuous-index vector bases. To reconstruct physical fields, we complement this with an Eisenstein-type spectral integral, evaluated by mapped Gauss–Legendre quadrature. These methods explicitly account for non-orthogonality of the basis and the singular behaviour near the origin. The block formulation acts on the full matrix-valued angular Maxwell operator and is not restricted to the wedge and cone geometries used below as analytically tractable benchmarks. We analyse and numerically demonstrate convergence. The hybrid pipeline is validated by reproducing eigenmode frequencies of spherical cavities with wedge and cone modifications to within 1−2% of full-wave high-frequency structure simulator (HFSS) simulations. Together with part I, this paper completes a unified analytic–computational framework for singular electromagnetic fields with continuous angular indices.