Reconstruction of Local Perturbations in Periodic Surfaces

# Reconstruction of Local Perturbations in Periodic Surfaces

Armin Lechleiter Center for Industrial Mathematics, University of Bremen; lechleiter@math.uni-bremen.de    Ruming Zhang  Center for Industrial Mathematics, University of Bremen; rzhang@uni-bremen.decorresponding author
###### 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.

## 1 Introduction

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 [Coa12]. The direct scattering problems with non-periodic incident fields (Herglotz wave functions) have been studied in [LN15] and [Lec17] theoretically, for numerical method see [LZ17a, LZ16]; problems with locally perturbed periodic surfaces have also been studied, for theoretical part see [Lec17] and for numerical method see [LZ17b]. The Bloch transform could also be applied to waveguide problems, see [HN16]. 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 [IJZ12a], 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 [IJZ12b, IJZ13, LZ13]. 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 [EHN96]) 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.

## 2 Direct Scattering Problem

### 2.1 Formulation

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

 Ω={(x1,x2)∈R2:x2>ζ(x1)};Ωp={(x1,x2)∈R2:x2>ζp(x1)}; ΩH={(x1,x2)∈Ω:x2

Define , and the define periodic cell s

 WΛ=(−Λ2,Λ2] and WΛ∗=(−Λ∗2,Λ∗2]=(−πΛ,πΛ].

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

###### Assumption 1.

For simplicity, we assume that the functions and are smooth enough, and the support of lies in one periodic cell, i.e., there is some such that .

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

 ui(x)=∫π/2−π/2eik(x1cosθ−x2sinθ)g(θ)dθ,

where , is the closure of the space in the norm

 ∥φ∥L2cos(−π/2,π/2):=[∫π/2−π/2|φ(θ)|2/cosθdθ]1/2.

Then the total field satisfies the equations

 Δup+k2up=0 in Ωp, (1) up=0 on Γp, (2)

with the scattered field is the upward propogating field, satisfies

 ∂usp∂x2=Tusp on ΓH, (3)

where is defined by

 (Tv)(x1)=i√Λ∫R√k2−|ξ|2eix1Λ∗ξ^v(ξ)dξ,^v(ξ)=1√Λ∫Re−iΛ∗ξx1v(x1,H)dx1.

It is a bounded operator from to for all , see [CE10]. Thus the total field satisfies the following boundar condition on :

 ∂up∂x2=T(up|ΓH)+[∂ui∂x2−T(ui|ΓH)] on ΓH. (4)

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

 ∥φ∥Hsr(R)=∥∥[(1+|x|2)r/2φ(x)]∥∥Hs(R).

for any . Define the space and as

the variational formulation of (1)-(4) is to find such that

 ∫ΩpH[∇up⋅∇¯¯¯v−k2up¯¯¯v]dx−∫ΓHT+(up|ΓH)¯¯¯vds=∫ΓHf¯¯¯vds, (5)

for all , where . From [CE10], the problem is uniquley soluable in for any and .

###### Theorem 2.

For and any incident field , the variational problem (5) is uniquely solvable in .

###### Remark 3.

We can also define the total and scattered fields with the non-perturbed interface , and the unique solvability in Theorem 2 also holds for this situation. The total field is denoted by and the scattered field is .

### 2.2 Bloch Transform

Define the Bloch transform in the periodic domain for

 JΩφ(α,x)=[Λ2π]1/2∑j∈Zφ(x+(Λj0))eiΛjα.

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

 ∥ψ∥Hr0(WΛ∗;Hsα(ΩΛH))=[r∑γ=0∫WΛ∗∥∥∂γαψ(α,⋅)∥∥2Hsα(ΩΛH)dα]1/2.

The definition could be extended to all by interpolation and duality arguments. With this function space, the operator has the following properties, see [Lec17].

###### Theorem 4.

The Bloch transform extends to an isomorphism between and . Further, is an isometry for with inverse

 (J−1Ωw)(x+(Λj0))=[Λ2π]1/2∫WΛ∗w(α,x)eiαΛjdα,x∈ΩΛH

and the inverse transform equals to the adjoint operator of .

Bloch transform only defined on periodic domains, so following [LZ17b], define the diffeomorphism mapping onto as

 Φp:x↦(x1,x2+(x2−H)3(ζ(x1)−H)3(ζp(x1)−ζ(x1))),

then . Define the transformed total field , then from (5) it satisfies the following variational problem

 (6)

for all , where

 Ap(x):=∣∣det∇Φp(x)∣∣[(∇Φp(x))−1((∇Φp(x))−1)T]∈L∞(ΩH,R2×2) cp(x):=∣∣det∇Φp(x)∣∣∈L∞(ΩH).

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

 (7)

where

 aα(u,v)=∫ΩΛH[∇u⋅∇¯¯¯v−k2u¯¯¯v]dx−∫ΓΛH(Tαu)¯¯¯vds, b(u,v)=∫ΩΛH+J(Λ,0)T[(Ap−I)∇u⋅∇¯¯¯v−k2(cp−1)u¯¯¯v]dx, f(α,⋅)=∂(JΩui)(α,⋅)∂x2−Tα(JΩui)(α,⋅), Tα(φ)=i∑j∈Z√k2−|Λ∗j−α|2^φ(j)ei(Λ∗j−α)x1,φ=∑j∈Z^φ(j)ei(Λ∗j−α)x1.

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

 wT(x,α)=∑j∈Z^wT(j,α)ei(Λ∗j−α)x1+iβjx2,βj=√k2−|Λ∗j−α|2. (8)

From [LZ17b], the variational problem is uniquely solvable in certain conditions.

###### Theorem 5.

Suppose is the graph of a Liptshitz continuous function.

1. The variational problem (7) is uniquely solvable in .

2. If for and , then .

###### Remark 6.

For the fields with non-perturbed surface , there is no need to do the transformation . We denote the Bloch transform of by , and Theorem 5 also holds for this problem.

### 2.3 Green’s Functions in the Periodic Domain

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

 ΔG(x,y)+k2G(x,y) = 0,x∈ΩH∖{y}, (9) G(x,y) = 0,x∈Γ, (10) ∂G(x,y)∂x2 = TG(x,y),x∈ΓH, (11)

For we note that is the incident half-space Green’s function, where , and . From Theorem 5, with the fact that , the Green’s function is well-defined in . Moreover, .

###### Theorem 7 (Theorem III.1, [Lec08]).

The Green’s function is symmetric, i.e., with .

Suppose and , then from the boundary condition (11), if has the following form of Fourier transform, i.e.,

 G(x,xs)=∫Reix1Λ∗ξφs(ξ,x2)dξ, (12)

then its normal derivative on

 ∂G(x,xs)∂x2=i∫R√k2−|ξ|2eix1Λ∗ξφs(ξ,x2)dξ. (13)
###### Theorem 8.

Suppose are two different points in . Then the Green’s function satisfies the following property

 Im[G(xp,xq)]=−Λ∫k−k√k2−|ξ|2φp(ξ,h)¯¯¯¯φq(ξ,h)dξ. (14)
###### Proof.

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

 Δ˜Gs(x,xp)+k2˜Gs(x,xp)=0,ΔG(x,xq)+k2G(x,xq)=0 in Ωεp.

With these properties,

 ∫∂Ωεp⎛⎝G(x,xp)∂¯¯¯¯G(x,xq)∂ν(x)−∂G(x,xp)∂ν(x)¯¯¯¯G(x,xq)⎞⎠dx = ∫∂Ωεp⎛⎝Φ(x,xp)∂¯¯¯¯G(x,xq)∂ν(x)−∂Φ(x,xp)∂ν(x)¯¯¯¯G(x,xq)⎞⎠dx = i4∫2π0⎡⎣H(1)0(kε)∂¯¯¯¯G(x,xq)∂ν(x)+kH(1)1(kε)¯¯¯¯G(x,xq)⎤⎦εdθ.

From the asympototic behaviers of and ,

 ∫∂Ωεp(G(x,xp)∂¯G(x,xq)∂ν(x)−∂G(x,xp)∂ν(x)¯¯¯¯G(x,xq))dx→¯¯¯¯G(xp,xq), as ε→0.

Similarly,

 ∫∂Ωεp(G(x,xp)∂¯G(x,xq)∂ν(x)−∂G(x,xp)∂ν(x)¯¯¯¯G(x,xq))dx→−G(xp,xq), % as ε→0.

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

 ∫ΓH⎛⎝G(x,xp)∂¯¯¯¯G(x,xq)∂x2−∂G(x,xp)∂x2¯¯¯¯G(x,xq)⎞⎠dx = (∫∂Ωεp+∫∂Ωεq)⎛⎝G(x,xp)∂¯¯¯¯G(x,xq)∂ν−∂G(x,xp)∂ν¯¯¯¯G(x,xq)⎞⎠dx,

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

 2iIm[G(xp,xq)]=∫ΓH⎛⎝G(x,xp)∂¯¯¯¯G(x,xq)∂x2−∂G(x,xp)∂x2¯¯¯¯G(x,xq)⎞⎠dx. (15)

As the Green’s function has the integral representation,

 ∫ΓHG(x,xp)∂¯¯¯¯G(x,xq)∂x2dx = −i∫R[∫Reix1Λ∗ξφp(ξ,h)dξ][∫R¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯√k2−|η|2e−ix1Λ∗η¯¯¯¯φq(η,h)dη]dx1 = −i∫R(∫R[∫Reix1Λ∗(ξ−η)dx1]¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯√k2−|η|2¯¯¯¯φq(η,h)dη)φp(ξ,h)dξ = −Λi∫R∫Rδ(ξ−η)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯√k2−|η|2¯¯¯¯φq(η,h)dηφp(ξ,h)dξ = −Λi∫R¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯√k2−|ξ|2¯¯¯¯φq(ξ,h)φp(ξ,h)dξ,

thus

 Im[G(xp,xq)] =−Λ∫RRe[√k2−|ξ|2]φp(ξ,h)¯¯¯¯φq(ξ,h)dξ =−Λ∫k−k√k2−|ξ|2φp(ξ,h)¯¯¯¯φq(ξ,h)dξ.

Define the function

 I(xp,xq):=∫ΓhG(x,xp)¯¯¯¯G(x,xq)dx, (16)

the following property is also easy to prove

 I(xp,xq)=Λ∫Rφp(ξ,h)¯¯¯¯φq(ξ,h)dξ. (17)

When is large enough, from the definition of UPRC,

 I(xp,xq)→Λ∫k−kφp(ξ,h)¯¯¯¯φq(ξ,h)dξ.

When and are close enough, , thus

 Im[G(xp,xq)]→−Λ∫k−k√k2−|ξ|2|φp(ξ)|2dξ I(xp,xq)→Λ∫k−k|φp(ξ)|2dξ≥k|Im[G(xp,xq)]|.

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.

## 3 The 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 5, . Then the functions , . Define the following scattering operator

 S:X →L2(Γh) p ↦usp|Γh

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

 ∥Sp−U∥2L2(Γh)=minp∗∈X∥Sp∗−U∥2L2(Γh). (18)

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

###### Theorem 9.

The operator is differentiable, and its derivative has the following representation, i.e.,

 DS:X → L2(Γh) (19) h ↦ u′|Γh

where satisfies

 Δu′+k2u′=0 in Ωp, (20) u′=−∂up∂νhν2%onΓp, (21) ∂u′∂x2=Tu′% on Γh. (22)

is the total field of the scattering problem (1)-(4).

The proof is similar to that in [Kir93] so we omit it here.

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

###### Lemma 10.

is the duality pairing defined by the inner product of . The DtN operator satisfies .

###### Proof.

Suppose , then from the definition of the DtN map,

Then we have the following calculations.

 (Tu,¯¯¯v) = ∫R(Tu)(x1)v(x1)dx1 = i√Λ∫R∫R√k2−|ξ|2eix1Λ∗ξ^u(ξ)v(x1)dξdx1 = i√Λ∫R√k2−|ξ|2^u(ξ)[∫Reix1Λ∗ξv(x1)dx1]dξ = i∫R√k2−|ξ|2^u(ξ)^v(−ξ)dξ.

From similar procedure,

 (u,¯¯¯¯¯¯¯Tv)=i∫R√k2−|ξ|2^v(ξ)^u(−ξ)dξ=(Tu,¯¯¯v),

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.

###### Theorem 11.

The adjoint operator of , denote by , is then given by

 Mφ=−Re[∂¯¯¯¯¯up∂ν∂¯¯¯z∂ν]ν2, (23)

where is the normal derivative upwards, is the total field of (1)-(4), satisfies

 Δz+k2z=0 in Ωp, (24) z=0 on Γp, (25) ∂z∂x2−Tz=¯¯¯¯φ%onΓh. (26)
###### Proof.

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

 (h,Mφ)X = Re((DS)h,φ)Γh=Re(u′,φ)Γh=Re(u′,∂¯¯¯v∂x2−¯¯¯¯¯¯¯Tv)Γh = Re[(u′,∂¯¯¯z∂x2)Γh−(Tu′,¯¯¯z)Γh]=Re[(u′,∂¯z∂x2)Γh−(∂u′∂x2,¯¯¯z)Γh] = Re∫Γh[u′∂z∂x2−∂u′∂x2z]ds=Re∫Γp[u′∂z∂ν−∂u′∂νz]ds.

Use the Helmholtz equation with the homogeneous Dirichlet boundary condition, i.e., on , and the condition (21),

 (h,Mφ)X=Re∫Γp[u′∂z∂ν]ds=−Re∫Γp[hν2∂up∂ν∂z∂ν]ds,

then we have the final results

 Mφ=−Re[∂¯¯¯¯¯up∂ν∂¯¯¯z∂ν]ν2.

###### Remark 12.

To solve the problem (24)-(26), we will study the variational forms of the Bloch transform of . Followed by the skills in Section 2, satisfies the following variaitonal form

 ∫WΛ∗aα(wB(α,⋅),vB(α,⋅))dα+[Λ2π]1/2b(J−1ΩwB,