[Paper Review] Vector Potential Electromagnetic Theory with Generalized Gauge for Inhomogeneous Anisotropic Media

Weng Cho Chew|arXiv (Cornell University)|Jun 9, 2014

Electromagnetic Scattering and Analysis24 references3 citations

TL;DR

This paper proposes a generalized gauge formulation of vector potential electromagnetic theory for inhomogeneous anisotropic media, eliminating the low-frequency catastrophe inherent in traditional E–H formulations. By introducing a flexible gauge condition that decouples scalar and vector potential equations, the method enables stable, multi-scale solutions across broad bandwidths using standard differential and integral solvers, with consistent interface conditions and surface integral equations.

ABSTRACT

Vector and scalar potential formulation is valid from quantum theory to classical electromagnetics. The rapid development in quantum optics calls for electromagnetic solutions that straddle quantum physics as well as classical physics. The vector potential formulation is a good candidate to bridge these two regimes. Hence, there is a need to generalize this formulation to inhomogeneous media. A generalized gauge is suggested for solving electromagnetic problems in inhomogenous media which can be extended to the anistropic case. The advantages of the resulting equations are their absence of low-frequency catastrophe. Hence, usual differential equation solvers can be used to solve them over multi-scale and broad bandwidth. It is shown that the interface boundary conditions from the resulting equations reduce to those of classical Maxwell's equations. Also, classical Green's theorem can be extended to such a formulation, resulting in similar extinction theorem, and surface integral equation formulation for surface scatterers. The integral equations also do not exhibit low-frequency catastrophe as well as frequency imbalance as observed in the classical formulation using E-H fields. The matrix representation of the integral equation for a PEC scatterer is given.

Motivation & Objective

• Address the low-frequency catastrophe in classical E–H formulations that hinders multi-scale and broadband electromagnetic simulations.
• Extend vector potential formulation—already essential in quantum electrodynamics—to inhomogeneous and anisotropic media.
• Develop a generalized gauge condition that decouples scalar and vector potential equations, ensuring numerical stability across all frequencies.
• Ensure consistency with classical Maxwell’s equations at interfaces and enable extension of Green’s theorem and surface integral equations.
• Provide a robust framework for multi-scale electromagnetic problems in quantum optics and classical electromagnetics using a unified A–Φ formulation.

Proposed method

• Formulate Maxwell’s equations using vector potential A and scalar potential Φ, satisfying ∇×E = −∂B/∂t and ∇·B = 0 by construction.
• Introduce a generalized gauge condition: ∇·(εA) = −χ∂tΦ, where χ = αε²μ, allowing decoupling of scalar and vector potential equations.
• Derive modified wave equations for A and Φ that avoid low-frequency breakdown, with the scalar potential equation becoming ∇·(ε∇Φ) − χ∂t²Φ = −ρ.
• Derive the vector potential equation: −∇×(μ⁻¹∇×A) − ε∂t²A + ε∇(χ⁻¹∇·(εA)) = −J.
• Apply the generalized gauge to derive surface integral equations for PEC scatterers, with matrix representations that remain well-conditioned at low frequencies.
• Extend classical Green’s theorem and extinction theorems to the A–Φ formulation, preserving physical consistency at boundaries.

Experimental results

Research questions

• RQ1 Can a generalized gauge condition be formulated for inhomogeneous anisotropic media that eliminates the low-frequency catastrophe in electromagnetic solvers?
• RQ2 How can the vector potential formulation be extended to maintain consistency with classical Maxwell’s equations at material interfaces?
• RQ3 Does the generalized gauge formulation allow for stable, multi-scale solutions across broad bandwidths without frequency imbalance?
• RQ4 Can classical integral equation frameworks—such as surface integral equations and extinction theorems—be consistently extended to the A–Φ formulation?
• RQ5 What is the matrix representation of the surface integral equation for a PEC scatterer in this generalized A–Φ framework, and does it remain numerically stable at low frequencies?

Key findings

• The generalized gauge condition ∇·(εA) = −χ∂tΦ successfully decouples the scalar and vector potential equations, enabling stable solution of the system across all frequencies.
• The resulting equations do not exhibit the low-frequency catastrophe or frequency imbalance seen in classical E–H formulations, allowing standard differential solvers to be used over multi-scale and broad bandwidth problems.
• Interface boundary conditions derived from the generalized formulation reduce to those of classical Maxwell’s equations, ensuring physical consistency at material boundaries.
• Classical Green’s theorem and the extinction theorem can be extended to the A–Φ formulation, enabling the derivation of surface integral equations for scatterers.
• The matrix representation of the surface integral equation for a PEC scatterer is well-conditioned and free from low-frequency breakdown, enabling robust numerical implementation.
• The formulation supports the unification of quantum and classical electromagnetic regimes, particularly beneficial for quantum optics and multi-scale electromagnetic modeling.