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Uniqueness for phaseless inverse elastic scattering problem for periodic structures

Youzi He, Wei Wu, Hongyi Dang

TL;DR

The paper addresses the problem of uniqueness in phaseless inverse elastic scattering from periodic structures in two and three dimensions. It advances the theory by deriving explicit quasi-periodic/biperiodic Green's functions for the Lamé system, establishing reciprocity relations for point sources, scattered and total fields, and obtaining Rayleigh-type expansions in 3D. Using these tools, the authors prove uniqueness results for reconstructing periodic boundary surfaces from intensity-only near-field data, applicable to both 2D and 3D periodic and biperiodic geometries. The work provides novel mathematical constructs (Green's functions and Rayleigh expansions) and foundational uniqueness results that support phaseless reconstruction in periodic elastic media and can inform future algorithm development.

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 Lamé 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 Lamé 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.

Uniqueness for phaseless inverse elastic scattering problem for periodic structures

TL;DR

The paper addresses the problem of uniqueness in phaseless inverse elastic scattering from periodic structures in two and three dimensions. It advances the theory by deriving explicit quasi-periodic/biperiodic Green's functions for the Lamé system, establishing reciprocity relations for point sources, scattered and total fields, and obtaining Rayleigh-type expansions in 3D. Using these tools, the authors prove uniqueness results for reconstructing periodic boundary surfaces from intensity-only near-field data, applicable to both 2D and 3D periodic and biperiodic geometries. The work provides novel mathematical constructs (Green's functions and Rayleigh expansions) and foundational uniqueness results that support phaseless reconstruction in periodic elastic media and can inform future algorithm development.

Abstract

This paper establishes uniqueness results of inverse elastic scattering problem with phaseless near-field data in periodic structures in and periodic/biperiodic structures in . 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 -axis in 2D, or on a plane parallel to -plane in 3D. We first calculate the explicit formula of quasi-periodic/biperiodic Green's functions of Lamé system in . 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 Lamé system in , the reciprocity relations, and Rayleigh's expansion in are novel results as important by-products in the proof process.

Paper Structure

This paper contains 15 sections, 17 theorems, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formulation of Problems
  3. Formulation of problem in $\mathbb{R}^2$
  4. Formulation of problem in $\mathbb{R}^3$
  5. Rayleigh expansion of quasi-periodic/biperiodic system in $\mathbb{R}^3$
  6. Quasi-periodic Green's functions in 2D and 3D
  7. 2D quasi-periodic Green's function
  8. 3D quasi-periodic Green's function
  9. 3D quasi-biperiodic Green's function
  10. General properties of quasi-periodic/biperiodic Green's functions
  11. Uniqueness theorem of quasi-periodic/biperiodic system
  12. Uniqueness of quasi-periodic elastic system in $\mathbb{R}^2$
  13. Uniqueness of quasi-periodic/biperiodic elastic system in $\mathbb{R}^3$
  14. Acknowledgements
  15. Uniqueness theorem in three-dimensional space

Key Result

Lemma 3.1

$\bm{c}_l$ satisfies for $j=1,2,3$.

Figures (3)

  • Figure 1: Quasi-periodic system in $\mathbb{R}^2$
  • Figure 2: Quasi-periodic system in $\mathbb{R}^3$
  • Figure 3: Quasi-biperiodic system in $\mathbb{R}^3$

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 1
  • Remark 2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 24 more