Solutions to Maxwell's Equations using Spheroidal Coordinates

TL;DR

This paper derives simplified asymptotic solutions to Maxwell's equations in spheroidal coordinates, creating a basis for improved optical wave modeling beyond the paraxial approximation.

Contribution

It introduces a new set of vector solutions to Maxwell's equations based on scalar spheroidal wave functions, extending optical wave analysis.

Findings

Simplified asymptotic solutions for spheroidal wave functions.

Expressions for Cartesian derivatives of spheroidal functions.

Foundation for calculating corrections to the paraxial approximation.

Abstract

Analytical solutions to the wave equation in spheroidal coordinates in the short wavelength limit are considered. The asymptotic solutions for the radial function are significantly simplified, allowing scalar spheroidal wave functions to be defined in a form which is directly reminiscent of the Laguerre-Gaussian solutions to the paraxial wave equation in optics. Expressions for the Cartesian derivatives of the scalar spheroidal wave functions are derived, leading to a new set of vector solutions to Maxwell's equations. The results are an ideal starting point for calculations of corrections to the paraxial approximation.