Reconstruction of Local Perturbations in Periodic Surfaces

Reconstruction of Local Perturbations in Periodic Surfaces

Abstract

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.

1Introduction

In this paper, we will introduce the numerical method of the inverse scattering problem in a locally perturbed periodic structure. Both the direct and inverse scattering problems in periodic structures have been studied in the last few years, especially for the case that the incident fields are quasi-periodic, e.g. plain waves. A classical way is to reduce the problems defined in an infinite periodic domain into one periodic cell, then the finitely defined problems could be solved in normal methods. However, if the periodicity of the solutions is destroyed, i.e., the incident field is not (quasi-)periodic, or the periodic structure is perturbed, the classical methods are no longer available and new techniques are needed.

The Floque-Bloch transform has been applied to perturbed periodic structures in [2]. The direct scattering problems with non-periodic incident fields (Herglotz wave functions) have been studied in [11] and [10] theoretically, for numerical method see [14]; problems with locally perturbed periodic surfaces have also been studied, for theoretical part see [10] and for numerical method see [15]. The Bloch transform could also be applied to waveguide problems, see [4]. In this paper, the analysis and numerical solutions of the direct problems are based on these results.

The numerical method developed in this paper is a combination of an initialization and an iteration scheme. The initialization step is to locate the perturbation from a relatively larger area. As was introduced by Ito, Jin and Zou in [5], a sampling method that only involves one incident field and a simple evaluation of an integration, could roughly locate the inhomogeneity embedded in free space. For more results for this method, see [6]. Followed by their idea, we will design an initialization algorithm to locate the perturbation, such that we could continue the iteration step in a finite domain. In the next step, we will approximate the perturbation by finite number of spline basis, and rewritten the inverse problem as an optimization problem. In this paper, we will apply Newton-CG method (see [3]) to solve the associated optimization method.

The rest of the paper is organized as follows. In Section 2, a description of the direct problem is made, together with the Green’s function in periodic structures. The inverse problem is formulated in Section 3, and the Fréchet derivative and its adjoint operator are studied. In Section 4, we will introduce the numerical methods for the inverse problem. In Section 5, several numerical examples are shown to illustrate the efficiency of the numerical method.

2Direct Scattering Problem

2.1Formulation

Suppose is a -periodic interface in defined by the function that is -periodic. is a local perturbation of , where the function has a compact support in . is a straight line above and , where . For some , define . Define the unbounded domains , , , as

Define , and the define periodic cell s

Let , , , denote the domains , , , restricted in one period .

Let the incident field be a downward propagating Herglotz wave function, i.e.,

where , is the closure of the space in the norm

Then the total field satisfies the equations

with the scattered field is the upward propogating field, satisfies

where is defined by

It is a bounded operator from to for all , see [1]. Thus the total field satisfies the following boundar condition on :

Recall the definition of the function space . The space is defined as the closure of with the norm

for any . Define the space and as

the variational formulation of - is to find such that

for all , where . From [1], the problem is uniquley soluable in for any and .

2.2Bloch Transform

Define the Bloch transform in the periodic domain for

Define the functions space by the closure of with the following norm when

The definition could be extended to all by interpolation and duality arguments. With this function space, the operator has the following properties, see [10].

Bloch transform only defined on periodic domains, so following [15], define the diffeomorphism mapping onto as

then . Define the transformed total field , then from it satisfies the following variational problem

for all , where

It is easy to deduce that the support of and are all subsets of . From the solution , the transformed function .

Let and , then it satisfies

where

For any , for each fixed , the -quasi-periodic solution has the following representation, i.e., the Rayleigh expansion

From [15], the variational problem is uniquely solvable in certain conditions.

2.3Green’s Functions in the Periodic Domain

Suppose is the Green’s function located in , then it satisfies

For we note that is the incident half-space Green’s function, where , and . From Theorem ?, with the fact that , the Green’s function is well-defined in . Moreover, .

Suppose and , then from the boundary condition , if has the following form of Fourier transform, i.e.,

then its normal derivative on

As , for any , define (and similar for ), then . Denote , then

With these properties,

From the asympototic behaviers of and ,

Similarly,

From the Green’s identity and Dirichlet boundary conditions, for small enough

where the right hand side tends to as , i.e.,

As the Green’s function has the integral representation,

thus

Define the function

the following property is also easy to prove

When is large enough, from the definition of UPRC,

When and are close enough, , thus

As has a singularity at , the imaginary part of the Green’s function may possess a sigularity, or a local maximum at that point. From the approximate behavior of when when falls in a small enough neighbourhood of , it could be expected to obtain relatively large values.

3The Inverse Problem

Suppose and is a -periodic knowns function, and define the space

In the following, we will always assume that and , and , thus from Theorem ?, . Then the functions , . Define the following scattering operator

Suppose data is measured on , which is the scattered field with some noise, denoted by , then the inverse problem is described as follows.

Inverse Problem: Given a measured data , to find such that

In the rest part of this section, we will describe some properties of the scattering operator.

The proof is similar to that in [8] so we omit it here.

Now we will study the property of , before that, the following property of the DtN operator is needed.

Suppose , then from the definition of the DtN map,

Then we have the following calculations.

From similar procedure,

the proof is finished from the denseness of in the space .

With these results, the adjoint operator of from to is given in the following theorem.

For any and , as is the space of real-valued functions,

Use the Helmholtz equation with the homogeneous Dirichlet boundary condition, i.e., on , and the condition ,

then we have the final results

4Numerical Method for Inverse Problems

The numerical method to inverse problems is devided into two parts. The first part is to initialize the location of the perturbation, i.e., to find out the integer such that , with the idea from [5]. With the known periodic cell where the perturbation located, the second step is to utilize the Newton-CG method to reconstruct the perturbed function. The two steps are introduced in the following two subsections.

4.1Initial Guess from the Sampling Method

In this subsection, assume that . From Section 2, the transformed total field in satisfies

Denote the total field of the non-perturbed surface by . Next we denote the difference between the transformed total field and by . This difference satisfies

subject to vanishing boundary values in . Indeed, vanishes on since vanishes on and vanishes on as well. It hence follows from the representation formula for solutions to the Helmholtz equation that can be represented as

Obviously, the latter representation formula extends to all points , such that we extend by this formula to a function in . Since this extension solves the Helmholtz equation and satisfies the angular spectrum representation encoded in , it is well-known that the extension belongs to for all .

Taking the inner product on some measurement line of with , involving a parameter then shows by the results from Section 2 that

From the function and note that

Choosing any (small) neighborhood of , there exists a diffeomorphism supported in , such that the volumetric domain of integration in the last formula can be concentrated to . As attains a maximum at this shows that the scalar product

can be expected to reach local maxima in , too.

When applying this method, firstly, a relatively large domain is known to contain the perturbation. Assume that there is a positive integer such that . Then the algorithm is to find the such that . Then the algorithm of this method could be concluded as follows.

4.2Newton-CG Method

In this section, we will discuss the Newton-CG method to solve , i.e., to find some such that it is a minimizer of .

From the initialization step, the integer is known. Suppose is a basis in the space , and we are looking for the approximation of the function that belongs to the subspace of , i.e.,

where is a positive integer. Let be the approximation of , i.e.,