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A few techniques to achieve invisibility in waveguides

Lucas Chesnel

TL;DR

The paper develops a multifaceted framework to realize invisibility in waveguides by manipulating geometry, frequency, and material properties. It combines classical waveguide theory, shape-derivative perturbations, resonance-based techniques (notably Fano resonances), and non-selfadjoint spectral analysis (complex and conjugated complex scaling) to engineer zero reflection or perfect transmission. It also provides rigorous arguments (limiting absorption, Fredholm theory) and concrete numerical schemes (DTN maps, finite elements) to construct or detect invisible defects and to characterize reflectionless modes. The work highlights the interplay between geometry, spectral properties, and resonances and demonstrates both theoretical constructs and practical computational procedures for cloaking-like effects in unbounded waveguides.

Abstract

The aim of this lecture is to consider a concrete problem, namely the identification of situations of invisibility in waveguides, to present techniques and tools that may be useful in various fields of applied mathematics. To be more specific, we will be interested in the propagation of acoustic waves in guides which are unbounded in one direction. In general, the diffraction of an incident field in such a structure in presence of an obstacle generates a reflection and a transmission characterized by some scattering coefficients. Our goal will be to play with the geometry, the frequency and/or the index material to control these scattering coefficients. We will explain how to: - develop a continuation method based on the use of shape derivatives to construct invisible defects; - exploit complex resonances located closed to the real axis to hid obstacles; - construct a non self-adjoint operator whose eigenvalues coincide with frequencies such that there are incident fields whose energy is completely transmitted. Our approaches will mainly rely on techniques of asymptotic analysis as well as spectral theory for self-adjoint and non self-adjoint operators. Most of the results will be illustrated by numerical experiments.

A few techniques to achieve invisibility in waveguides

TL;DR

The paper develops a multifaceted framework to realize invisibility in waveguides by manipulating geometry, frequency, and material properties. It combines classical waveguide theory, shape-derivative perturbations, resonance-based techniques (notably Fano resonances), and non-selfadjoint spectral analysis (complex and conjugated complex scaling) to engineer zero reflection or perfect transmission. It also provides rigorous arguments (limiting absorption, Fredholm theory) and concrete numerical schemes (DTN maps, finite elements) to construct or detect invisible defects and to characterize reflectionless modes. The work highlights the interplay between geometry, spectral properties, and resonances and demonstrates both theoretical constructs and practical computational procedures for cloaking-like effects in unbounded waveguides.

Abstract

The aim of this lecture is to consider a concrete problem, namely the identification of situations of invisibility in waveguides, to present techniques and tools that may be useful in various fields of applied mathematics. To be more specific, we will be interested in the propagation of acoustic waves in guides which are unbounded in one direction. In general, the diffraction of an incident field in such a structure in presence of an obstacle generates a reflection and a transmission characterized by some scattering coefficients. Our goal will be to play with the geometry, the frequency and/or the index material to control these scattering coefficients. We will explain how to: - develop a continuation method based on the use of shape derivatives to construct invisible defects; - exploit complex resonances located closed to the real axis to hid obstacles; - construct a non self-adjoint operator whose eigenvalues coincide with frequencies such that there are incident fields whose energy is completely transmitted. Our approaches will mainly rely on techniques of asymptotic analysis as well as spectral theory for self-adjoint and non self-adjoint operators. Most of the results will be illustrated by numerical experiments.

Paper Structure

This paper contains 38 sections, 23 theorems, 296 equations, 47 figures.

Table of Contents

  1. Waveguide problems
  2. Setting
  3. Dirichlet problem for $0<k<\pi$
  4. Dirichlet problem for $k>\pi$
  5. Computation of modes
  6. Ill-posedness in $\mathrm{H}^1_0(\Omega)$
  7. Problem with dissipation
  8. Problem without dissipation
  9. Limiting absorption principle
  10. Scattering problem
  11. Numerical approximation
  12. Neumann problem
  13. Invisibility questions
  14. Invisible perturbations of the reference geometry
  15. General scheme
  16. ...and 23 more sections

Key Result

Theorem I.1

Pick $k\in(0;\pi)$. The operator $A(k)$ decomposes as where $B:\mathrm{H}^1_0(\Omega)\to\mathrm{H}^1_0(\Omega)$ is an isomorphism and $K:\mathrm{H}^1_0(\Omega)\to\mathrm{H}^1_0(\Omega)$ is compact ($B$ and $K$ are allowed to depend on $k$).

Figures (47)

  • Figure 1: Examples of waveguides (source Wikipedia).
  • Figure 2: Left: reference strip $\mathcal{S}$. Right: perturbed waveguide $\Omega$.
  • Figure 3: Left: example of domain where $A(k)$ is an isomorphism for all $k\in(0;\pi)$. Right: example of domain where $A(k)$ is not an isomorphism for all $k\in(0;\pi)$.
  • Figure 4: Graphs of the cut-off function $\psi_m$.
  • Figure 5: Domain $\Omega_L$.
  • ...and 42 more figures

Theorems & Definitions (47)

  • Theorem I.1
  • proof
  • Lemma I.2
  • proof
  • Proposition I.3
  • Remark I.4
  • proof
  • Definition I.5
  • Lemma I.6
  • Proposition I.7
  • ...and 37 more