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Trapped modes in electromagnetic waveguides

Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Sonia Fliss

TL;DR

This work analyzes trapped electromagnetic modes in 3D unbounded waveguides formed by a resonator attached to semi-infinite guides. It develops two main strategies: exact construction of trapped modes via complete separation of variables in geometries with separable cross-sections, and variational min–max methods with carefully crafted divergence-free test fields for more general, non-separable geometries. The authors derive explicit criteria for the existence of discrete eigenvalues, including embedded ones, across multiple geometries (e.g., 6-leg, tripode, L-shaped cross-sections) and extend results to locally perturbed and tensor-valued material coefficients, including approaches based on magnetic formulations and symmetry arguments. The work highlights Maxwell-specific trapping mechanisms (not present in scalar Dirichlet/Neumann settings) and discusses implications for scattering ill-posedness, bound states in the continuum, and resonance phenomena, while outlining open questions and future directions.

Abstract

We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. $L^2$ solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.

Trapped modes in electromagnetic waveguides

TL;DR

This work analyzes trapped electromagnetic modes in 3D unbounded waveguides formed by a resonator attached to semi-infinite guides. It develops two main strategies: exact construction of trapped modes via complete separation of variables in geometries with separable cross-sections, and variational min–max methods with carefully crafted divergence-free test fields for more general, non-separable geometries. The authors derive explicit criteria for the existence of discrete eigenvalues, including embedded ones, across multiple geometries (e.g., 6-leg, tripode, L-shaped cross-sections) and extend results to locally perturbed and tensor-valued material coefficients, including approaches based on magnetic formulations and symmetry arguments. The work highlights Maxwell-specific trapping mechanisms (not present in scalar Dirichlet/Neumann settings) and discusses implications for scattering ill-posedness, bound states in the continuum, and resonance phenomena, while outlining open questions and future directions.

Abstract

We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.

Paper Structure

This paper contains 28 sections, 20 theorems, 226 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The electric operator
  3. Waveguides with complete separation of variables
  4. Construction from eigenvalues of the 2D Dirichlet Laplacian
  5. Construction from eigenvalues of the 2D Neumann Laplacian
  6. Waveguides with separation of variables in the resonator
  7. Strategy
  8. A first simple case
  9. Generalization: working with TE modes in the resonator
  10. Absence of monotonicity of the spectrum with respect to the geometry
  11. Working with TEM modes in the resonator
  12. Working with TM modes in the resonator
  13. Waveguides without separation of variables
  14. Existence of discrete spectrum for resonators large enough
  15. The 6 legs geometry
  16. ...and 13 more sections

Key Result

Proposition 2.1

If $S$ is simply connected, then $\sigma_{\mathrm{ess}}(A)=[\lambda_N;+\infty)$. If $S$ is not simply connected, then $\sigma_{\mathrm{ess}}(A)=[0;+\infty)$.

Figures (14)

  • Figure 1: Example of waveguide $\Omega$ (the guide $\Pi$ extends to infinity).
  • Figure 2: Example of $\Omega_{\mathrm{2D}}$ (left), trapped mode for the 2D L-shaped domain (center), 3D L-shaped domain (right).
  • Figure 3: Examples of trapped modes for the 2D Neumann Laplacian. The last two come respectively from ChPa18, ChPa19.
  • Figure 4: Waveguides where the resonator is a product domain.
  • Figure 5: Classical examples of bounded domains $\omega_1\subset\omega_2$ such that $\inf\,\sigma(A_1)<\inf\,\sigma(A_2)$.
  • ...and 9 more figures

Theorems & Definitions (44)

  • Proposition 2.1
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 34 more