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The radiation condition is the key question in the mathematical modelling for scattering problems in unbounded domains. Mathematically, it plays the role as the "boundary condition" at the infinity, which guarantees the well-posedness of the mathematical problem; physically, it describes the far-field asymptotic behaviour of the physical waves. In this paper, we focus on the radiation conditions for scattering problems above (locally perturbed) periodic surfaces. According to Hu et al. (2021), the radiating solution satisfies the Sommerfeld radiation condition: $$\frac{\partial u}{\partial r}-i k u=o(r^{-1/2}).$$ Although there are literature which have studied this problem, there is no specific method for dealing with periodic structures. Due to this reason, the important properties for the periodic structures may be ignored. Moreover, the existing method is not extendable to bi-periodic structures in three dimensional spaces. In this paper, we study the radiation condition for the time-harmonic scattering problem with periodic surfaces, which is modelled by the Helmholtz equation. We introduce a novel method based on the Floquet-Bloch transform, which, to the best of the author's knowledge, is the first method that works particularly for periodic media. With this method, we improve the Sommerfeld radiation condition for the scattered field from periodic media to: $$\frac{\partial u}{\partial r}-i k u=O(r^{-3/2}).$$ More importantly, the prospect of extending this method to 3D cases is optimistic.

The perfectly matched layers method is a well known truncation technique for its efficiency and convenience in numerical implementations of wave scattering problems in unbounded domains. In this paper, we study the convergence of the perfectly matched layers (PML) for wave scattering from a local perturbation of an open waveguide in the half space above the real line, where the refractive index is a function which is periodic along the axis of the waveguide and equals to one above a finite height. The problem is challenging due to the existence of guided waves, and a typical way to deal with the difficulty is to apply the limiting absorption principle. Based on the Floquet-Bloch transform and a curve deformation theory, the solution from the limiting absorption principle is rewritten as the integral of a coupled family of quasi-periodic problems with respect to the quasi-periodicity parameter on a particularly designed curve. By comparing the Dirichlet-to-Neumann maps on a straight line above the locally perturbed periodic layer, we finally show that the PML method converges exponentially with respect to the PML parameter. Finally, the numerical examples are shown to illustrate the theoretical results.

The PML method is well-known for its exponential convergence rate and easy implementation for scattering problems with unbounded domains. For rough-surface scattering problems, authors in [5] proved that the PML method converges at most algebraically in the physical domain. However, the authors also asked a question whether exponential convergence still holds for compact subsets. In [25], one of our authors proved the exponential convergence for periodic surfaces via the Floquet-Bloch transform when the wavenumber is positive and not a half integer; when the wavenumber is a positive half integer, a nearly fourth-order convergence rate was shown in [26]. The extension of this method to locally perturbed cases is not straightforward, since the domain is no longer periodic thus the Floquet-Bloch transform doesn't work, especially when the domain topology is changed. Moreover, the exact decay rate when the wavenumber is a half integer remains unclear. The purpose of this paper is to address these two significant issues. For the first topic, the main idea is to reduce the problem by the DtN map on an artificial curve, then the convergence rate of the PML is obtained from the investigation of the DtN map. It shows exactly the same convergence rate as in the unperturbed case. Second, to illustrate the convergence rate when the wavenumber is a half integer, we design a specific periodic structure for which the PML converges at the fourth-order, showing that the algebraic convergence rate is sharp. We adopt a previously developed high-accuracy PML-BIE solver to exhibit this unexpected phenomenon.

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.

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.

In this paper, we propose a new spectral decomposition method to simulate waves propagating in complicated waveguides. For the numerical solutions of waveguide scattering problems, an important task is to approximate the Dirichlet-to-Neumann map efficiently. From previous results, the physical solution can be decomposed into a family of generalized eigenfunctions, thus we can write the Dirichlet-to-Neumann map explicitly by these functions. From the exponential decay of the generalized eigenfunctions, we approximate the Dirichlet-to-Neumann (DtN) map by a finite truncation and the approximation is proved to converge exponentially. With the help of the truncated DtN map, the unbounded domain is truncated into a bounded one, and a variational formulation for the problem is set up in this bounded domain. The truncated problem is then solved by a finite element method. The error estimation is also provided for the numerical algorithm and numerical examples are shown to illustrate the efficiency of the algorithm.

In this paper, we propose a new nonuniform mesh method to simulate acoustic scattering problems in two dimensional periodic structures with non-periodic incident fields numerically. As existing methods are difficult to extend to higher dimensions, we have designed the new method with such extensions in mind. With the help of the Floquet-Bloch transform, the solution to the original scattering problem is written as an integral of a family of quasi-periodic problems. These are defined in bounded domains for each value of the Floquet parameter which varies in a bounded interval. The key step in our method is the numerical approximation of the integral by a quadrature rule adapted to the regularity of the family of quasi-periodic solutions. We design a nonuniform mesh with a Gaussian quadrature rule applied on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.

The main task in this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter, for scattering problems with periodic surfaces. In [5], a linear convergence is proved for the PML method for scattering problems with rough surfaces. At the end of that paper, three important questions are asked, and the third question is if exponential convergence holds locally. In our paper, we answer this question for a special case, which is scattering problems with periodic surfaces. The result can also be easily extended to locally perturbed periodic surfaces or periodic layers. Due to technical reasons, we have to exclude all the half integer valued wavenumbers. The main idea of the proof is to apply the Floquet-Bloch transform to write the problem into an equivalent family of quasi-periodic problems, and then study the analytic extension of the quasi-periodic problems with respect to the Floquet-Bloch parameters. Then the Cauchy integral formula is applied for piecewise analytic functions to avoid linear convergent points. Finally the exponential convergence is proved from the inverse Floquet-Bloch transform. Numerical results are also presented at the end of this paper.

In this paper, two high order complex contour discretization methods are proposed to simulate wave propagation in locally perturbed periodic closed waveguides. As is well known the problem is not always uniquely solvable due to the existence of guided modes. The limiting absorption principle is a standard way to get the unique physical solution. Both methods are based on the Floquet-Bloch transform which transforms the original problem to an equivalent family of cell problems. The first method, which is designed based on a complex contour integral of the inverse Floquet-Bloch transform, is called the CCI method. The second method, which comes from an explicit definition of the radiation condition, is called the decomposition method. Due to the local perturbation, the family of cell problems are coupled with respect to the Floquet parameter and the computational complexity becomes much larger. To this end, high order methods to discretize the complex contours are developed to have better performances. Finally we give the convergence results which we confirm with numerical examples.

In this paper, we propose a new numerical method for scattering problems in periodic waveguide, based on the newly established contour integral representation of solutions in a previous paper by the author (see [Zhadf]). For this kind of problems, solutions are obtained via the Limiting Absorption Principle and we all them LAP solutions. Based on the Floquet-Bloch transform and analytic Fredholm theory, an LAP solution could be written explicitly as an integral of quasi-periodic solutions on a contour, which depends on the periodic structure. Compared to previous numerical methods for this kind of problems, we do not need to adopt the LAP for approximation, thus a standard error estimation is easily carried out. Based on this method, we also develop a numerical solver for the halfguide problems. This method is also based on the result from [Zhadf], which shows that any LAP solution of a halfguide problem could be extended into a fullguide problem with a non-vanishing source term(not uniquely!). So first we approximate the source term from the boundary data by a regularization method, and then the LAP solution could be obtained from the corresbonding fullguide problem. At the end of this paper, we also show some numerical results to present the efficiency of our numerical methods.

Scattering problems in periodic waveguides are interesting but also challenging topics in mathematics, both theoretically and numerically. Due to the existence of eigenvalues, the unique solvability of these problems is not always guaranteed. To obtain a unique solution that is "physically meaningful", the limiting absorption principle (LAP) is a commonly used method. LAP assumes that the limit of a family of solutions with absorbing media converges, as the absorption parameter tends to 0, and the limit is the "physically meaningful solution". It is also called the LAP solution in this paper. It has been proved that the LAP holds for periodic waveguides in [Hoa11]. In this paper, we consider the spectrum decomposition of periodic translation operators. With the curve integral formulation and a generalized Residue theorem, the operator is explicitly described by its eigenvalues and generalized eigenfunctions, which are closely related to Bloch wave solutions. Then the LAP solution is decomposed into generalized eigenfunctions. This gives a better understanding of structure of the scattered fields.

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.

In this paper, we develop a high order numerical method for the numerical solutions of scattering problems with slightly perturbed periodic surfaces in two dimensional spaces. Based on the regularity property introduced in Part I, the decaying rate of the incident field could be transferred directly to the total field for small perturbations. Thus the finite section method could reach a high accuracy rate. With the help of a modification of the truncated problem, the problem is solved by a finite element method. The convergence of the finite element method is proved and numerical examples have been carried out to show the efficiency of the numerical scheme.

Rough surface scattering problems are always very challenging both theoretically and numerically. In this paper, we adopt the Bloch transform and the perturbation theory to investigate a special case, i.e., when the rough surface is a slight perturbation of a periodic one. Based on known results from the Bloch transform, the problem could be written into an equivalent bounded variational problem in higher dimensional spaces. The first step is to consider the non-perturbed problem. From the perturbation theory, the solution could be written as a Neumann series. The second step is to consider the slightly perturbed problems. The key point in the process is the relationship between the Dirichlet-to-Neumann map. With the study between these two operators, it is proved that when the right hand side belongs to a certain space, the problem is equivalent to a modified one. Thus a high regularity result is obtained., Comment: Fundamental mistake in the paper. Equation (65) is wrong, which makes the result in Lemma 22 impossible to be proved. Then the main theorem in Section 5, Theorem 23 and 24 could not be proved

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., Comment: arXiv admin note: text overlap with arXiv:1805.11484

In this paper, we will study the Bloch transformed rough surface scattering problems, and propose a numerical method based on the Bloch transformed problems. Based on the mathematical theory of the scattering problems from locally perturbed periodic surfaces, the same techniques will be applied to the rough surface scattering problems, and an equivalent coupled family of quasi-periodic scattering problems in one periodic cell will be established. The most important result obtained in this paper is on the finite Fourier series approximation of the Bloch transformed field with respect to the quasi-periodicity parameter. It will be proved that the finite series is exactly the Bloch transformed solution corresponds to truncated rough surfaces. Thus the truncation provides a reasonable approximation, and could be applied to the numerical solutions. Based on the approximation, a numerical method is proposed for the rough surface scattering problems. The convergence of the numerical method is proved and illustrated by the numerical experiments. The method provides a completely new perspective for the rough surface scattering problems. There is possibility that some high order method will be developed based on this new method.

The integral equation method is widely used in numerical simulations of 2D/3D acoustic and electromagnetic scattering problems, which needs a large number of values of the Green's functions. A significant topic is the scattering problems in periodic domains, where the corresponding Green's functions are quasi-periodic. The quasi-periodic Green's functions are defined by series that converge too slowly to be used for calculations. Many mathematicians have developed several efficient numerical methods to calculate quasi-periodic Green's functions. In this paper, we will propose a new FFT-based fast algorithm to compute the 2D/3D quasi-periodic Green's functions for both the Helmholtz equations and Maxwell's equations. The convergence results and error estimates are also investigated in this paper. Further, the numerical examples are given to show that, when a large number of values are needed, the new algorithm is very competitive.

Waves scattering from unbounded structures are always complicated problems for numerical simulations. For the case that the non-periodic incident field scattered by (locally perturbed) periodic surfaces, with the help of the Bloch transform, the problem could be solved by some finite element methods, if the incident fields decay at certain rate at the infinity. For faster decaying incident fields, a high order numerical method is also available. However, in these cases, the plain waves, which belong to a very important family of incident fields but do not decay at the infinity, are not included. In this paper, we aim to develop the Bloch transform based standard finite method for this certain case, and then establish the high order method afterwards. Numerical experiments have been carried out for both the standard and high order numerical methods. Based on the algorithms for incident plain waves, we could also extend the numerical methods to more generalized cases when only the not so efficient standard method is available.

This paper concerns the inverse scattering problem to reconstruct a local perturbation in a periodic structure. Unlike the periodic problems, the periodicity for the scattered field no longer holds, thus classical methods, which reduce quasi-periodic fields in one periodic cell, are no longer available. Based on the Floquet-Bloch transform, a numerical method has been developed to solve the direct problem, that leads to a possibility to design an algorithm for the inverse problem. The numerical method introduced in this paper contains two steps. The first step is initialization, that is to locate the support of the perturbation by a simple method. This step reduces the inverse problem in an infinite domain into one periodic cell. The second step is to apply Newton-CG method to solve the associated optimization problem. The perturbation is then approximated by a finite spline basis. Numerical examples are given at the end of this paper, shows the efficiency of the numerical method.

In this paper, we will introduce a high order numerical method to solve the scattering problems with non-periodic incident fields and (locally perturbed) periodic surfaces. For the problems we are considering, the classical methods to treat quasi-periodic scattering problems no longer work, while a Bloch transform based numerical method was proposed in [LZ17b]. This numerical method, on one hand, is able to solve this kind of problems convergently; on the other hand, it takes up a lot of time and memory during the computation. The motivation of this paper is to improve this numerical method, from the regularity results of the Bloch transform of the total field, which have been studied in [Zha17]. As the set of the singularities of the total field is discrete in $\R$, and finite in one periodic cell, we are able to improve the numerical method by designing a proper integration contour with special conditions at the singularities. With a good choice of the transformation, we can prove that the new numerical method could possess a super algebraic convergence rate. \high{This new method improves the efficient significantly. At the end of this paper, several numerical results will be provided to show the fast convergence of the new method.} The method also provides a possibility to solve more complicated problems efficiently, e.g., three dimensional problems, or electromagnetic scattering problems.

Scattering problems with locally perturbed periodic surfaces have been studied both theoretically and numerically in recent years. In this paper, we will discuss the regularity results of the Bloch transform of the total fields. The idea is inspired by Theorem a in \cite{Kirsc1993}, which considered how the total field depends on the wave numbers and the incident angles, with a family of plain incident fields and a smooth enough periodic surface. We will show that when the incident field satisfies some certain conditions, the Bloch transform of the total field depends analytically on the quasi-periodicities one the straight line $\R$ except for a countable number of points, while near such points, a square-root like singularity exists. We also give some examples to show that the conditions are satisfied by a large number of commonly used incident fields. This result also provides a probability to improve the numerical solution of this kind of problems, which is expected to be discussed in our future papers.

Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we firstly rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we secondly transform this problem into a coupled family of quasiperiodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasiperiodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretization in the spatial and the quasiperiodicity's unit cells. We also provide a simple and efficient way for implementation that does not require numerical integration in the quasiperiodicity, together with numerical examples for scattering from locally perturbed periodic surfaces computed by this scheme.

Periodic surface structures are nowadays standard building blocks of optical devices. If such structures are illuminated by aperiodic time-harmonic incident waves as, e.g., Gaussian beams, the resulting surface scattering problem must be formulated in an unbounded layer including the periodic surface structure. An obvious recipe to avoid the need to discretize this problem in an unbounded domain is to set up an equivalent system of quasiperiodic scattering problems in a single (bounded) periodicity cell via the Floquet-Bloch transform. The solution to the original surface scattering problem then equals the inverse Floquet-Bloch transform applied to the family of solutions to the quasiperiodic problems, which simply requires to integrate these solutions in the quasiperiodicity parameter. A numerical scheme derived from this representation hence completely avoids the need to tackle differential equations on unbounded domains. In this paper, we provide rigorous convergence analysis and error bounds for such a scheme when applied to a two-dimensional model problem, relying upon a quadrature-based approximation to the inverse Floquet-Bloch transform and finite element approximations to quasiperiodic scattering problems. Our analysis essentially relies upon regularity results for the family of solutions to the quasiperiodic scattering problems in suitable mixed Sobolev spaces. We illustrate our error bounds as well as efficiency of the numerical scheme via several numerical examples.

Recently, a new eigenvalue problem, called the transmission eigenvalue problem, has attracted many researchers. The problem arose in inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing. The problem is numerically challenging because it is non-selfadjoint and nonlinear. In this paper, we propose a recursive integral method for computing transmission eigenvalues from a finite element discretization of the continuous problem. The method, which overcomes some difficulties of existing methods, is based on eigenprojectors of compact operators. It is self-correcting, can separate nearby eigenvalues, and does not require an initial approximation based on some a priori spectral information. These features make the method well suited for the transmission eigenvalue problem whose spectrum is complicated. Numerical examples show that the method is effective and robust., Comment: 18 pages, 8 figures