Near-field imaging of locally perturbed periodic surfaces

This paper concerns the inverse scattering problem to reconstruct a locally perturbed periodic surface. Different from scattering problems with quasi-periodic incident fields and periodic surfaces, the scattered fields are no longer quasi-periodic. Thus the classical method for quasi-periodic scattering problems no longer works. In this paper, we apply a Floquet-Bloch transform based numerical method to reconstruct both the unknown periodic part and the unknown local perturbation from the near-field data. By transforming the original scattering problem into one defined in an infinite rectangle, the information of the surface is included in the coefficients. The numerical scheme contains two steps. The first step is to obtain an initial guess, i.e., the locations of both the periodic surfaces and the local perturbations, from a sampling method. The second step is to reconstruct the surface. As is proved in this paper, for some incident fields, the corresponding scattered fields carry little information of the perturbation. In this case, we use this scattered field to reconstruct the periodic surface. Then we could apply the data that carries more information of the perturbation to reconstruct the local perturbation. The Newton-CG method is applied to solve the associated optimization problems. Numerical examples are given at the end of this paper to show the efficiency of the numerical method

Higher order convergence of perfectly matched layers in 3D bi-periodic surface scattering problems

The perfectly matched layer (PML) is a very popular tool in the truncation of wave scattering in unbounded domains. In Chandler-Wilde & Monk et al. 2009, the author proposed a conjecture that for scattering problems with rough surfaces, the PML converges exponentially with respect to the PML parameter in any compact subset. In the author's previous paper (Zhang et al. 2022), this result has been proved for periodic surfaces in two dimensional spaces, when the wave number is not a half integer. In this paper, we prove that the method has a high order convergence rate in the 3D bi-periodic surface scattering problems. We extend the 2D results and prove that the exponential convergence still holds when the wavenumber is smaller than $0.5$. For lareger wavenumbers, although exponential convergence is no longer proved, we are able to prove that a higher order convergence for the PML method

Fast convergence for of perfectly matched layers for scattering with periodic surfaces: the exceptional case

In the author's previous paper (Zhang et al. 2022), exponential convergence was proved for the perfectly matched layers (PML) approximation of scattering problems with periodic surfaces in 2D. However, due to the overlapping of singularities, an exceptional case, i.e., when the wave number is a half integer, has to be excluded in the proof. However, numerical results for these cases still have fast convergence rate and this motivates us to go deeper into these cases. In this paper, we focus on these cases and prove that the fast convergence result for the discretized form. Numerical examples are also presented to support our theoretical results

Numerical methods for scattering problems from multi‐layers with different periodicities

In this paper, we consider a numerical method to solve scattering problems with multi-periodic layers with different periodicities. The main tool applied in this paper is the Bloch transform. With this method, the problem is written into an equivalent coupled family of quasi-periodic problems. As the Bloch transform is only defined for one fixed period, the inhomogeneous layer with another period is simply treated as a non-periodic one. First, we approximate the refractive index by a periodic one where its period is an integer multiple of the fixed period, and it is decomposed by finite number of quasi-periodic functions. Then the coupled system is reduced into a simplified formulation. A convergent finite element method is proposed for the numerical solution, and the numerical method has been applied to several numerical experiments. At the end of this paper, relative errors of the numerical solutions will be shown to illustrate the convergence of the numerical algorithm