论文标题
反向衍射光栅问题的独特性无限多个平面波以固定频率
Uniqueness in inverse diffraction grating problems with infinitely many plane waves at a fixed frequency
论文作者
论文摘要
本文通过具有二维的Dirichlet边界条件的周期性曲线与衍射问题有关。事实证明，周期曲线可以由近场测量数据唯一地确定，这些数据对应于无限的许多入射平面波，这些飞机波在固定频率下具有不同的方向。我们的证明是基于Schiffer的想法，该想法由两种成分组成：i）具有不同方向的事件平面波的总字段是线性独立的，ii）仅在有限的域中或在封闭的封闭的波导中，在波导边界上的其他假设下，仅存在有限的许多线性独立的dirichlet eigenfunctions。基于瑞利的扩展，我们证明，除了有限的入射角集外，可以通过界面域中的无斑点近场数据来唯一确定分阶段的近场数据。这种相位检索结果会导致在固定频率下与无相位近场数据的反光栅衍射问题产生新的唯一性结果。由于事件方向确定了边界价值问题的准周期性，因此我们的反问题与[Htttlich＆Kirsch的现有结果不同，而逆问题13（1997）：351-361]，其中考虑了在多个频率下的固定方向波。
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.
