Uniqueness and numerical method for phaseless inverse diffraction grating problem with known superposition of incident point sources

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems Pub Date : 2024-07-03 DOI:10.1088/1361-6420/ad5b81

Tian Niu, Junliang Lv and Jiahui Gao

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引用次数: 0

Abstract

In this paper, we establish the uniqueness of identifying a smooth grating profile with a mixed boundary condition (MBC) or transmission boundary conditions (TBCs) from phaseless data. The existing uniqueness result requires the measured data to be in a bounded domain. To break this restriction, we design an incident system consisting of the superposition of point sources to reduce the measurement data from a bounded domain to a line above the grating profile. We derive reciprocity relations for point sources, diffracted fields, and total fields, respectively. Based on Rayleigh’s expansion and reciprocity relation of the total field, a grating profile with a MBC or TBCs can be uniquely determined from the phaseless total field data. An iterative algorithm is proposed to recover the Fourier modes of grating profiles at a fixed wavenumber. To implement this algorithm, we derive the Fréchet derivative of the total field operator and its adjoint operator. Some numerical examples are presented to verify the correctness of theoretical results and to show the effectiveness of our numerical algorithm.

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已知入射点源叠加的无相位反衍射光栅问题的唯一性和数值方法

在本文中，我们通过无相数据确定了具有混合边界条件（MBC）或传输边界条件（TBC）的光栅轮廓的唯一性。现有的唯一性结果要求测量数据位于有界域中。为了打破这一限制，我们设计了一个由点源叠加组成的入射系统，将测量数据从有界域减少到光栅轮廓上方的一条线上。我们分别推导出了点源、衍射场和总场的互易关系。根据雷利展开和总场的互易关系，可以从无相总场数据唯一确定具有 MBC 或 TBC 的光栅轮廓。我们提出了一种迭代算法来恢复光栅轮廓在固定波长下的傅里叶模式。为了实现这一算法，我们推导出了总场算子的弗雷谢特导数及其邻接算子。为了验证理论结果的正确性，并显示我们的数值算法的有效性，我们给出了一些数值示例。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Inverse Problems 数学-物理：数学物理

CiteScore

4.40

自引率

14.30%

发文量

115

审稿时长

2.3 months

期刊介绍： An interdisciplinary journal combining mathematical and experimental papers on inverse problems with theoretical, numerical and practical approaches to their solution. As well as applied mathematicians, physical scientists and engineers, the readership includes those working in geophysics, radar, optics, biology, acoustics, communication theory, signal processing and imaging, among others. The emphasis is on publishing original contributions to methods of solving mathematical, physical and applied problems. To be publishable in this journal, papers must meet the highest standards of scientific quality, contain significant and original new science and should present substantial advancement in the field. Due to the broad scope of the journal, we require that authors provide sufficient introductory material to appeal to the wide readership and that articles which are not explicitly applied include a discussion of possible applications.