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An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 1: general results

Maxence Cassier, Patrick Joly

TL;DR

This work develops a rigorous operator-theoretic framework for EM wave propagation in dispersive isotropic passive media with frequency-dependent $\varepsilon(\omega)$ and $\mu(\omega)$. It combines Nevanlinna/Herglotz representations with an augmented Maxwell system to obtain well-posed, energy-conserving formulations for general passive media and specializes to generalized Lorentz models, both non-dissipative and dissipative. A key contribution is the dispersion analysis that yields spectral bands, forward/backward mode structure, and conditions for negative index, along with a dissipative extension that leads to precise energy decay rates. The results connect physical principles (causality, passivity, energy conservation) with spectral theory, providing a solid foundation for metamaterials and time-domain simulations in dispersive media.

Abstract

We investigate in this chapter the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity $\varepsilon$ and magnetic permeability $μ$ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which $\varepsilon$ and $μ$ are rational functions of the frequency. This leads us to analyse the important class of non dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.

An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 1: general results

TL;DR

This work develops a rigorous operator-theoretic framework for EM wave propagation in dispersive isotropic passive media with frequency-dependent and . It combines Nevanlinna/Herglotz representations with an augmented Maxwell system to obtain well-posed, energy-conserving formulations for general passive media and specializes to generalized Lorentz models, both non-dissipative and dissipative. A key contribution is the dispersion analysis that yields spectral bands, forward/backward mode structure, and conditions for negative index, along with a dissipative extension that leads to precise energy decay rates. The results connect physical principles (causality, passivity, energy conservation) with spectral theory, providing a solid foundation for metamaterials and time-domain simulations in dispersive media.

Abstract

We investigate in this chapter the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity and magnetic permeability depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which and are rational functions of the frequency. This leads us to analyse the important class of non dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.

Paper Structure

This paper contains 18 sections, 18 theorems, 168 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Recap on the Fourier-Laplace transform in time
  3. Maxwell's equations in dispersive media: general properties
  4. Passive materials
  5. Maxwell's equations in general passive media
  6. A representation of $\varepsilon(\omega)$ and $\mu(\omega)$ in passive media
  7. Augmented PDE model for Maxwell's equations in passive media
  8. Abstract reformulation. Existence and uniqueness results.
  9. Generalized Lorentz media
  10. Definition of the models
  11. The abstract model
  12. Dispersion analysis
  13. Dissipative Lorentz models
  14. Dielectric permittivity and magnetic permeabilty
  15. Dissipative augmented formulation
  16. ...and 3 more sections

Key Result

Lemma 7

[Poles and zeros] Let $f$ be a non zero Herglotz function that can be extended to the whole complex half-space into a meromorphic function, still denoted $f$, the poles and zeros of $f$ are located in the half-space $\overline{{\mathbb{C}}^-} := \{ \operatorname{Im} \, z \leq 0 \}$

Figures (3)

  • Figure 1: graph of $\omega \mapsto \varepsilon(\omega)$ (left) and positivity property of $\omega \mapsto \omega \, \varepsilon(\omega)$ (right) for $N_e = 2$ and $\omega_{e,1}, \,\omega_{e,2}> 0$.
  • Figure 2: Plot of dispersion curves for $N=3$. The spectral bands ${\cal S}_n$ are the projections of the dispersion curves $\Gamma_n$ on the $\omega$-axis. There is one negative band: ${\cal S}_2$.
  • Figure 3: Plot of the spectrum of $\mathbb{A}_{{\boldsymbol{\alpha}}}^\perp$. The arcs ${\cal S}_n^\alpha$ play the same role as the (real) segments ${\cal S}_n$ in figure \ref{['Fig_Disp']}.

Theorems & Definitions (55)

  • Example 1: $L^1_{loc}({\mathbb{ R}}^+)$-convolutional media
  • Example 2: Local media
  • Remark 3
  • Definition 4
  • Definition 5
  • Remark 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 45 more