Uniqueness for phaseless inverse elastic scattering problem for periodic structures

Abstract

This paper establishes uniqueness results of inverse elastic scattering problem with phaseless near-field data in periodic structures in $\mathbb{R}^2$ and periodic/biperiodic structures in $\mathbb{R}^3$. We use a superposition of two point sources in each periodic unit with different positions as the incident field, and measures the phaseless near-field data on a line parallel to $x_1$-axis in 2D, or on a plane parallel to $(x_1,x_2)$-plane in 3D. We first calculate the explicit formula of quasi-periodic/biperiodic Green's functions of Lam\'{e} system in $\mathbb{R}^3$. Then, to establish the uniqueness results, the reciprocity relations for point sources, scattered fields, and total fields are derived. Finally, with the help of Rayleigh's expansion, the uniqueness results are proved. The quasi-periodic/biperiodic Green's functions of Lam\'{e} system in $\mathbb{R}^3$, the reciprocity relations, and Rayleigh's expansion in $\mathbb{R}^3$ are novel results as important by-products in the proof process.