Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 2: transmission problems

Maxence Cassier, Patrick Joly

TL;DR

This work develops a rigorous operator-theoretic framework for Maxwell’s equations in dispersive metamaterials and analyzes two key transmission scenarios: a vacuum–Drude half-space interface and a Drude slab embedded in vacuum. By reformulating the dynamics as a self-adjoint evolution problem, constructing a spectral density via a generalized Fourier transform, and establishing limiting absorption and limiting amplitude principles, the authors reveal how dispersion and negative-index behavior induce novel interfaces resonances and slow-light phenomena. The slab analysis further shows a rich set of guided modes arising from Sturm–Liouville reductions and dispersion relations, including plasmonic and propagative branches with critical points that slow wave packet decay. Collectively, the results provide a mathematically rigorous foundation for plasmonics and metamaterial-based waveguiding, linking time-domain dynamics to spectral data and uncovering conditions under which unusual resonances and slow-light effects occur. These insights have potential implications for designing devices leveraging surface plasmons and slow-light propagation in dispersive, negative-index media.

Abstract

In this second chapter, we analyse transmission problems between a dielectric and a dispersive negative material. In the first part, we consider a transmission problem between two half-spaces, filled respectively by the vacuum and a Drude material, and separated by a planar interface. In this setting, we answer to the following question: does this medium satisfy a limiting amplitude principle? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. In the second part, we consider the transmission problem of an infinite strip of Drude material embedded in the vacuum and analyse the existence and dispersive properties of guided waves. In both problems, our spectral analysis enlighten new and unusual physical phenomena for the considered transmission problems due to the presence of the dispersive negative material. In particular, we prove the existence of an interface resonance in the first part and the existence of slow light phenomena for guiding waves in the second part.

An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 2: transmission problems

TL;DR

This work develops a rigorous operator-theoretic framework for Maxwell’s equations in dispersive metamaterials and analyzes two key transmission scenarios: a vacuum–Drude half-space interface and a Drude slab embedded in vacuum. By reformulating the dynamics as a self-adjoint evolution problem, constructing a spectral density via a generalized Fourier transform, and establishing limiting absorption and limiting amplitude principles, the authors reveal how dispersion and negative-index behavior induce novel interfaces resonances and slow-light phenomena. The slab analysis further shows a rich set of guided modes arising from Sturm–Liouville reductions and dispersion relations, including plasmonic and propagative branches with critical points that slow wave packet decay. Collectively, the results provide a mathematically rigorous foundation for plasmonics and metamaterial-based waveguiding, linking time-domain dynamics to spectral data and uncovering conditions under which unusual resonances and slow-light effects occur. These insights have potential implications for designing devices leveraging surface plasmons and slow-light propagation in dispersive, negative-index media.

Abstract

In this second chapter, we analyse transmission problems between a dielectric and a dispersive negative material. In the first part, we consider a transmission problem between two half-spaces, filled respectively by the vacuum and a Drude material, and separated by a planar interface. In this setting, we answer to the following question: does this medium satisfy a limiting amplitude principle? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. In the second part, we consider the transmission problem of an infinite strip of Drude material embedded in the vacuum and analyse the existence and dispersive properties of guided waves. In both problems, our spectral analysis enlighten new and unusual physical phenomena for the considered transmission problems due to the presence of the dispersive negative material. In particular, we prove the existence of an interface resonance in the first part and the existence of slow light phenomena for guiding waves in the second part.

Paper Structure

This paper contains 25 sections, 12 theorems, 185 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Transmission problem involving a dispersive media
  3. Description of the mathematical model
  4. Large time behavior and limiting absorption principle
  5. The limiting absorption principle
  6. Limiting amplitude principle and interface resonance phenomenon
  7. Spectral density of the propagative operator
  8. The notion of spectral density
  9. Main results on the spectral density
  10. Construction of the spectral density
  11. Reduced operator ${\mathbb{A}}_{\mathbf{k}}$
  12. The generalized eigenfunctions
  13. The diagonalization Theorem
  14. Construction of $\mathbb{M}_{\omega}$
  15. Proof of the limiting absorption and limiting amplitude principles
  16. ...and 10 more sections

Key Result

Proposition 4

$\sigma({\mathbb{A}})={\mathbb{R}}$ and $\sigma_{p}({\mathbb{A}})$ is composed of eigenvalues of infinite multiplicity: The eigenspaces $\ker({\mathbb{A}})$ and $\ker({\mathbb{A}}\pm \, \Omega_{\rm m})$ are respectively given by: where $\widetilde{\boldsymbol{\Pi}}$ is the extension operator by $0$ of a 2D vector fields defined on ${\mathbb{R}}^2_-$ to ${\mathbb{R}}^2$ and $W_0^1({\mathbb{R}}^2_

Figures (7)

  • Figure 1: Left figure: Transmission problem between the vacuum and a non-dissipative Drude filling respectively $\mathbb{R}^3_-$ and $\mathbb{R}^3_+$. Right figure: Relative permeability function $\omega \mapsto \mu^{+}(\omega)/\mu_0$.
  • Figure 2: Spectral zones represented on ${\mathbb{R}}^+ \times {\mathbb{R}}^+$ for $\Omega_{\rm e}<\Omega_{\rm m}$ (left) and $\Omega_{\rm e}=\Omega_{\rm m}$ (right).
  • Figure 3: Slab of Drude non-dissipative material $\mathcal{L}$ of width $2L$ embedded in the vacuum.
  • Figure 4: Dispersion curves $\mathcal{C}_n$ for $n\geq 2$ for a slab of width $2L$ of dispersive non-dissipative Drude medium embedded in the vacuum, corresponding to the figure \ref{['fig.medslab']}.
  • Figure 5: Dispersion curve $\mathcal{C}_0$ when $\rho\geq 1$ (left) and $\rho<1$ (left) $S_{0,\rho,\Omega_m}(\tau_c)>0$ (right).
  • ...and 2 more figures

Theorems & Definitions (25)

  • Remark 1: Critical case
  • Remark 2
  • Remark 3: Physical justification
  • Proposition 4
  • Theorem 5: Limiting absorption principle
  • Corollary 6
  • proof
  • Theorem 7
  • Theorem 8
  • Remark 9
  • ...and 15 more