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Inverse scattering for waveguides in topological insulators

Guillaume Bal, Xixian Wang, Zhongjian Wang

TL;DR

The paper addresses the inverse scattering problem for a topological Dirac waveguide created by a domain-wall interface between two 2D insulators, modeled by H = H_0 + V with V compactly supported. It develops both linearized and finite-dimensional nonlinear theories, proving injectivity and stability results for reconstructing V from scattering data, and presents an adjoint-based algorithm to perform the reconstruction. Numerical experiments validate convergence, robustness to noise, and behavior with incomplete data, and illustrate non-reconstructible scenarios when data are insufficient. The results demonstrate that edge topology does not prevent recovery of compactly supported perturbations and provide a practical strategy for identifying V in topological waveguide systems, with implications for imaging and characterization in related physical settings.

Abstract

This paper concerns the inverse scattering problem of a topologically non-trivial waveguide separating two-dimensional topological insulators. We consider the specific model of a Dirac system. We show that a short-range perturbation can be fully reconstructed from scattering data in a linearized setting and in a finite-dimensional setting under a smallness constraint. We also provide a stability result in appropriate topologies. We then solve the problem numerically by means of a standard adjoint method and illustrate our theoretical findings with several numerical simulations.

Inverse scattering for waveguides in topological insulators

TL;DR

The paper addresses the inverse scattering problem for a topological Dirac waveguide created by a domain-wall interface between two 2D insulators, modeled by H = H_0 + V with V compactly supported. It develops both linearized and finite-dimensional nonlinear theories, proving injectivity and stability results for reconstructing V from scattering data, and presents an adjoint-based algorithm to perform the reconstruction. Numerical experiments validate convergence, robustness to noise, and behavior with incomplete data, and illustrate non-reconstructible scenarios when data are insufficient. The results demonstrate that edge topology does not prevent recovery of compactly supported perturbations and provide a practical strategy for identifying V in topological waveguide systems, with implications for imaging and characterization in related physical settings.

Abstract

This paper concerns the inverse scattering problem of a topologically non-trivial waveguide separating two-dimensional topological insulators. We consider the specific model of a Dirac system. We show that a short-range perturbation can be fully reconstructed from scattering data in a linearized setting and in a finite-dimensional setting under a smallness constraint. We also provide a stability result in appropriate topologies. We then solve the problem numerically by means of a standard adjoint method and illustrate our theoretical findings with several numerical simulations.
Paper Structure (25 sections, 9 theorems, 140 equations, 6 figures, 3 algorithms)

This paper contains 25 sections, 9 theorems, 140 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Forward and inverse scattering problem
  3. Spectral decomposition.
  4. Scattering matrix.
  5. Inverse Scattering Problem.
  6. Small potential linearization.
  7. Small potential nonlinear inverse problem.
  8. Explicit linearized scattering data.
  9. Main results
  10. Reconstruction of a scalar potential.
  11. Nonlinear reconstruction of finite-dimensional potential.
  12. Adjoint method and TR matrices
  13. Adjoint-based optimization
  14. Integral formulation for adjoint problem
  15. Computation of $g^p_{in}$.
  16. ...and 10 more sections

Key Result

Lemma 3.1

Fixing $(n,q)\in \mathbb{N}\times \mathbb{N}\setminus\{(0,0)\}$, for any $\xi\in \mathbb{R}\setminus\{\pm\sqrt{2|n-q|},0\}$, there exists $m=(n,\epsilon_m),p=(q,\epsilon_p)$ and $E_{n,q}(\xi)$ such that $\xi_{m,p}(E_{n,q}(\xi)) = \xi.$ More precisely, and

Figures (6)

  • Figure 1: Relative reconstruction errors $\mathcal{E}$ against iterations for different noise levels $\sigma$.
  • Figure 2: Reconstruction of the potential from TR matrices with no noise at iteration $600$ .
  • Figure 3: Reconstructions (top) and absolute errors (bottom) under different noise levels, shown at iteration $\mathrm{i}_{\max}=600$.
  • Figure 4: Relative reconstruction errors $\mathcal{E}$ against iterations
  • Figure 5: Reconstruction error measured by $\mathcal{E}$ and $\mathcal{E}_{\mathrm{avg}}$ with respect to iteration steps.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 9 more