Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semi classical measures and Maxwell's system

Hassan Taha

TL;DR

This work develops a semiclassical (Wigner) measure framework to rigorously describe high-frequency energy propagation for Maxwell’s equations under diverse boundary conditions, including perfect conductor, Calderón transmission, and curved interfaces. By formulating Maxwell as a symmetric hyperbolic system and employing microlocal test functions, the authors derive dispersion relations, decompose the limiting measures into polarization modes, and obtain transport equations with explicit boundary/interfacial source terms. The results show energy propagation along the characteristic sets defined by $\omega_{\pm}=\omega$ with two transverse polarizations, while boundary conditions dictate how energy transfers across interfaces via boundary measures. The curved-interface case is reduced to the planar setting through a coordinate transform, demonstrating the robustness of the approach for heterogeneous electromagnetic media in complex geometries.

Abstract

We are interested in the homogenization of energy like quantities for electromagnetic waves in the high frequency limit for Maxwell's equations with various boundary conditions. We use a scaled variant of H-measures known as semi classical measures or Wigner measures. Firstly, we consider this system in the half space of $\R^3$ in the time harmonic and with conductor boundary condition at the flat boundary $x_3=0$. Secondly we consider the same system but with Calderon boundary condition. Thirdly, we consider this system in the curved interface case.

Semi classical measures and Maxwell's system

TL;DR

This work develops a semiclassical (Wigner) measure framework to rigorously describe high-frequency energy propagation for Maxwell’s equations under diverse boundary conditions, including perfect conductor, Calderón transmission, and curved interfaces. By formulating Maxwell as a symmetric hyperbolic system and employing microlocal test functions, the authors derive dispersion relations, decompose the limiting measures into polarization modes, and obtain transport equations with explicit boundary/interfacial source terms. The results show energy propagation along the characteristic sets defined by with two transverse polarizations, while boundary conditions dictate how energy transfers across interfaces via boundary measures. The curved-interface case is reduced to the planar setting through a coordinate transform, demonstrating the robustness of the approach for heterogeneous electromagnetic media in complex geometries.

Abstract

We are interested in the homogenization of energy like quantities for electromagnetic waves in the high frequency limit for Maxwell's equations with various boundary conditions. We use a scaled variant of H-measures known as semi classical measures or Wigner measures. Firstly, we consider this system in the half space of in the time harmonic and with conductor boundary condition at the flat boundary . Secondly we consider the same system but with Calderon boundary condition. Thirdly, we consider this system in the curved interface case.

Paper Structure

This paper contains 6 sections, 6 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Prerequesites on semi classical measures
  3. Proofs of the Theorems
  4. Proof of Theorem (\ref{['hsdaa']}), Perfect boundary condition case
  5. Proof of Theorem \ref{['helen2']}, Calderon type boundary condition
  6. Remarks on the curved interface case

Key Result

Theorem 1.1

Perfect conductor case Consider time harmonic Maxwell's system in the half space x_3\geq 0 with a perfect boundary condition, written in the form (3.4), with solution vector u^{\varepsilon}. Let \theta(x) be a test function with compact support that is equal to one on a compact set K\subset {\mathbb where v(x)=\frac{1}{\sqrt{\epsilon(x)\eta(x)}}\; is the propagation speed. Above \omega_{0}= \omega

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 1
  • Lemma 2
  • Lemma 3