[Paper Review] Decoupled field integral equations for electromagnetic scattering from homogeneous penetrable obstacles

Felipe Vico, Leslie Greengard+1 more|arXiv (Cornell University)|Apr 22, 2017

Electromagnetic Scattering and Analysis28 references4 citations

TL;DR

This paper introduces decoupled field integral equations for electromagnetic scattering from homogeneous penetrable obstacles by reformulating Maxwell's equations into two independent vector Helmholtz systems for electric and magnetic fields. The method yields resonance-free, second-kind Fredholm equations that are stable at all frequencies, insensitive to scatterer topology, and valid for all passive materials—including those with negative permittivity or permeability—enabling robust and invertible solutions across challenging material regimes.

ABSTRACT

We present a new method for the analysis of electromagnetic scattering from homogeneous penetrable bodies. Our approach is based on a reformulation of the governing Maxwell equations in terms of two uncoupled vector Helmholtz systems: one for the electric feld and one for the magnetic field. This permits the derivation of resonance-free Fredholm equations of the second kind that are stable at all frequencies, insensitive to the genus of the scatterers, and invertible for all passive materials including those with negative permittivities or permeabilities. We refer to these as decoupled field integral equations.

Motivation & Objective

• To address the instability and resonance issues in conventional integral equation methods for electromagnetic scattering from penetrable obstacles.
• To develop a formulation that remains well-conditioned and invertible for all passive materials, including those with negative permittivity or permeability.
• To decouple the electric and magnetic field equations to enable independent solution of each system.
• To ensure numerical stability and convergence across all frequencies, including the static limit.
• To provide a robust framework valid for scatterers of arbitrary genus and complex material properties.

Proposed method

• Reformulate Maxwell's equations into two uncoupled vector Helmholtz equations—one for the electric field and one for the magnetic field—by introducing auxiliary potentials and currents.
• Define surface electric and magnetic currents on the boundary ∂D, and use single-layer potential operators with appropriate Green's functions to represent the fields.
• Derive boundary integral equations by enforcing continuity of tangential electric and magnetic fields across the interface, leading to a system of Fredholm equations of the second kind.
• Introduce auxiliary fields using exterior material parameters in the interior and interior parameters in the exterior to analyze jump conditions and prove uniqueness.
• Apply the Fredholm alternative and prove that the only solution to the homogeneous problem is the trivial one, ensuring invertibility of the system.
• Use Hölder continuity and compactness arguments in C^{0,α} spaces to establish uniform boundedness and continuity of the inverse operator across all frequencies.

Experimental results

Research questions

• RQ1 Can a stable, resonance-free integral equation formulation be developed for electromagnetic scattering from penetrable obstacles with arbitrary material properties?
• RQ2 Does decoupling the electric and magnetic field equations eliminate ill-conditioning and numerical instability at characteristic frequencies?
• RQ3 Can the method remain invertible and well-posed for materials with negative permittivity or permeability, including lossy and metamaterial regimes?
• RQ4 Is the formulation independent of the topological genus of the scatterer, ensuring robustness for complex geometries?
• RQ5 Can the system be proven to have a unique solution using functional analysis tools such as the Fredholm alternative and compactness arguments?

Key findings

• The proposed decoupled field integral equations are Fredholm equations of the second kind, which are inherently more stable than first-kind formulations.
• The system is resonance-free and remains well-conditioned at all frequencies, including the static limit and high-frequency regimes.
• The method is insensitive to the genus of the scatterer, enabling application to multiply connected or complex-shaped obstacles.
• The formulation is valid for all passive materials, including those with negative permittivity or permeability, provided the imaginary part is positive or the material is real and positive.
• The inverse operator is bounded and continuous uniformly in frequency over [0, ω_max], ensuring robust numerical solution with iterative solvers.
• Uniqueness of the solution is proven by showing that the only solution to the homogeneous problem is the trivial one, using auxiliary fields and jump conditions.