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Modified Rayleigh Conjecture for scattering by periodic structures

A. G. Ramm, S. Gutman

TL;DR

This work develops a self-contained scattering theory for 2-D periodic structures, introducing a resolvent framework with a Limiting Absorption Principle (LAP) and a spectral decomposition for quasiperiodic boundary conditions. It then proves the Modified Rayleigh Conjecture (MRC) as a rigorous foundation for a finite-mode expansion that approximates the scattered field with controllable error, and provides a practical minimization scheme to compute the expansion coefficients. An integral-equation approach is used to construct the Dirichlet resolvent kernel $G(x,y,\xi,\eta,k)$ via a Fredholm equation, enabling stable existence/uniqueness results and LAP, even in the presence of obstacles. The paper also develops and tests a numerical MRC-based method, combining outgoing Green's functions and SVD-regularized least-squares to achieve accurate scattering computations for multiple periodic profiles with modest computational cost, highlighting potential applications to inverse problems.

Abstract

This paper contains a self-contained brief presentation of the scattering theory for periodic structures. Its main result is a theorem (the Modified Rayleigh Conjecture, or MRC), which gives a rigorous foundation for a numerical method for solving the direct scattering problem for periodic structures. A numerical example illustrating the procedure is presented.

Modified Rayleigh Conjecture for scattering by periodic structures

TL;DR

This work develops a self-contained scattering theory for 2-D periodic structures, introducing a resolvent framework with a Limiting Absorption Principle (LAP) and a spectral decomposition for quasiperiodic boundary conditions. It then proves the Modified Rayleigh Conjecture (MRC) as a rigorous foundation for a finite-mode expansion that approximates the scattered field with controllable error, and provides a practical minimization scheme to compute the expansion coefficients. An integral-equation approach is used to construct the Dirichlet resolvent kernel via a Fredholm equation, enabling stable existence/uniqueness results and LAP, even in the presence of obstacles. The paper also develops and tests a numerical MRC-based method, combining outgoing Green's functions and SVD-regularized least-squares to achieve accurate scattering computations for multiple periodic profiles with modest computational cost, highlighting potential applications to inverse problems.

Abstract

This paper contains a self-contained brief presentation of the scattering theory for periodic structures. Its main result is a theorem (the Modified Rayleigh Conjecture, or MRC), which gives a rigorous foundation for a numerical method for solving the direct scattering problem for periodic structures. A numerical example illustrating the procedure is presented.

Paper Structure

This paper contains 5 sections, 5 theorems, 77 equations, 1 table.

Table of Contents

  1. Introduction
  2. Periodic Scattering Problem
  3. Integral equations method
  4. Modified Rayleigh Conjecture (MRC)
  5. Numerical solution of the scattering problem

Key Result

Lemma 2.1

Let $\ 0<\epsilon<1$, and $a>2$. Then where $c=const>0$ does not depend on $\epsilon$, and $(x,y)$ is running on compact sets.

Theorems & Definitions (11)

  • Lemma 2.1
  • proof : Proof of Lemma \ref{['s2_l1']}
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['s2_l2']}
  • Remark 2.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Remark 4.3
  • ...and 1 more