Uniqueness in inverse diffraction grating problems with infinitely many plane waves at a fixed frequency

TL;DR

This paper proves the unique determination of a periodic curve in inverse diffraction problems using near-field data from infinitely many incident plane waves at a fixed frequency, including phaseless data, based on Schiffer's idea.

Contribution

It introduces a new uniqueness result for inverse diffraction problems with infinitely many incident directions at a fixed frequency, including phaseless data, using phase retrieval techniques.

Findings

Unique determination of periodic curves from near-field data.

Phase retrieval from phaseless near-field data.

Extension of inverse diffraction results to fixed incident directions.

Abstract

This paper is concerned with the inverse diffraction problems by a periodic curve with Dirichlet boundary condition in two dimensions. It is proved that the periodic curve can be uniquely determined by the near-field measurement data corresponding to infinitely many incident plane waves with distinct directions at a fixed frequency. Our proof is based on Schiffer's idea which consists of two ingredients: i) the total fields for incident plane waves with distinct directions are linearly independent, and ii) there exist only finitely many linearly independent Dirichlet eigenfunctions in a bounded domain or in a closed waveguide under additional assumptions on the waveguide boundary. Based on the Rayleigh expansion, we prove that the phased near-field data can be uniquely determined by the phaseless near-field data in a bounded domain, with the exception of a finite set of incident angles. Such a phase retrieval result leads to a new uniqueness result for the inverse grating diffraction problem with phaseless near-field data at a fixed frequency. Since the incident direction determines the quasi-periodicity of the boundary value problem, our inverse issues are different from the existing results of [Htttlich & Kirsch, Inverse Problems 13 (1997): 351-361] where fixed-direction plane waves at multiple frequencies were considered.