Generalized Helmholtz Decomposition for Modal Analysis of Electromagnetic Problems in Inhomogeneous Media

TL;DR

This paper generalizes the Helmholtz decomposition to address mode redundancy in potential-based electromagnetic eigenproblem formulations in inhomogeneous media, providing a mathematical foundation for numerical quantization.

Contribution

It introduces a generalized Helmholtz decomposition for electromagnetic modes in inhomogeneous media, including mathematical analysis, orthogonality proofs, and numerical validation.

Findings

Orthogonality relations between mode classes are established.

Completeness of mode sets is numerically validated.

The framework supports quantization of electromagnetic fields in complex media.

Abstract

Potential-based formulation with generalized Lorenz gauge can be used in the quantization of electromagnetic fields in inhomogeneous media. However, one often faces the redundancy of modes when finding eigenmodes from potential-based formulation. In free space, this can be explained by the connection to the well-known Helmholtz decomposition. In this work, we generalize the Helmholtz decomposition to its generalized form, echoing the use of generalized Lorenz gauge in inhomogeneous media. We formulate electromagnetics eigenvalue problems using vector potential formulation which is often used in numerical quantization. The properties of the differential operators are mathematically analyzed. Orthogonality relations between the two classes of modes are proved in both continuous and discrete space. Completeness of two sets of modes and the orthogonality relations are numerically validated in inhomogeneous anisotropic media. This work serves as a foundation for numerical quantization of electromagnetic fields in inhomogeneous media with potential-based formulation.