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The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

Drossos Gintides, Leonidas Mindrinos

TL;DR

The paper tackles the inverse electromagnetic scattering problem for a penetrable cylinder under oblique incidence by formulating a nonlinear boundary-integral equation system that, through an indirect representation, yields four equations on the unknown boundary $\Gamma$ and one on the far-field circle. A two-step, regularized Newton-type iteration is employed: first solve a well-posed boundary-density subsystem, then update the boundary via a Fréchet-derivative–based linearization of the far-field equation using Tikhonov regularization. The authors derive explicit integral-operator formulations, establish the role of the Fréchet derivatives, and implement a Nyström discretization with trig-polynomial parametrization of the boundary. Numerical experiments with peanut- and apple-shaped geometries demonstrate feasibility and robustness, especially when leveraging multiple incident directions, even in the presence of measurement noise.

Abstract

In this work we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

TL;DR

The paper tackles the inverse electromagnetic scattering problem for a penetrable cylinder under oblique incidence by formulating a nonlinear boundary-integral equation system that, through an indirect representation, yields four equations on the unknown boundary and one on the far-field circle. A two-step, regularized Newton-type iteration is employed: first solve a well-posed boundary-density subsystem, then update the boundary via a Fréchet-derivative–based linearization of the far-field equation using Tikhonov regularization. The authors derive explicit integral-operator formulations, establish the role of the Fréchet derivatives, and implement a Nyström discretization with trig-polynomial parametrization of the boundary. Numerical experiments with peanut- and apple-shaped geometries demonstrate feasibility and robustness, especially when leveraging multiple incident directions, even in the presence of measurement noise.

Abstract

In this work we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.

Paper Structure

This paper contains 6 sections, 3 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The direct problem
  3. The inverse problem
  4. The integral equation method
  5. Numerical implementation
  6. Numerical results

Key Result

Theorem 1

If $\kappa_1^2$ is not an interior Dirichlet eigenvalue and $\kappa_0^2$ is not an interior Dirichlet and Neumann eigenvalue, then the direct transmission problem maxwell_third -- radiation admits a unique solution.

Figures (4)

  • Figure 1: The geometry of the scattering problem.
  • Figure 2: Reconstruction of a peanut-shaped boundary for exact (left) and noisy (right) data.
  • Figure 3: Reconstruction of a peanut-shaped boundary for noisy data and different initial guesses.
  • Figure 5: Reconstruction of an apple-shaped boundary for exact data and different initial guesses.

Theorems & Definitions (5)

  • Theorem 1
  • Proof 1
  • Theorem 2
  • Theorem 3
  • Remark 1