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# A novel method in determining a layered periodic structure

*Boundary Value Problems*
**volume 2020**, Article number: 177 (2020)

## Abstract

This paper is concerned with the inverse scattering of time-harmonic waves by a penetrable structure. By applying the integral equation method, we establish the uniform \(L^{p}_{\alpha }\ (1< p\leq 2)\) estimates for the scattered and transmitted wave fields corresponding to a series of incident point sources. Based on these a priori estimates and a mixed reciprocity relation, we prove that the penetrable structure can be uniquely identified by means of the scattered field measured only above the structure induced by a countably infinite number of quasi-periodic incident plane waves.

## Introduction

In this paper, we consider the inverse problem of determining a penetrable periodic structure in \({\mathbb{R}}^{3}\) from the scattered data measured only above the structure. This kind of problem occurs in various applications such as in radar imaging, modern diffractive optics, and non-destructive testing. For convenience, we write a point *x* in \({\mathbb{R}}^{3}\) for \((\widetilde{x},x_{3})\) with \(\widetilde{x}:=(x_{1},x_{2})\in {\mathbb{R}}^{2}\). Assume that the penetrable profile is described by

where *f* is a periodic function with respect to the variable *x̃*, that is, \(f(\widetilde{x}) = f( \widetilde{x}+2n\pi )\) for \(n:=(n_{1},n_{2})\in {\mathbb{Z}}^{2}\). Assume further that the homogeneous media above and below Γ are described by

with the wave numbers \(k_{1}\) and \(k_{2}\), respectively.

Consider the incident plane waves in the form of

which propagate downward from \(\Omega _{+}\) with \(\alpha =(\alpha _{1},\alpha _{2}): = k_{1}(\sin \theta _{1}\cos \theta _{2}, \sin \theta _{1}\sin \theta _{2})\) with the incident angle \(\theta _{1}\in [0, \pi /2), \theta _{2}\in [0, 2\pi )\), and \(\beta _{j}^{+}\in {\mathbb{C}}\) is given by

Then the scattering of the incident \(u^{i}\) by the periodic structure can be formulated in determining the total field \(u_{1}:=u^{i}+ u^{s} \) with the scattered field \(u^{s} \) and the transmitted field \(u_{2}\) to the following problem:

Here, \(u_{n}^{\pm }\in {\mathbb{C}}\) are the solution sequences, *λ* is the transmission coefficient and the unit normal vector *ν* on Γ is directed into the interior of \(\Omega _{-}\). Notice that the incident wave \(u^{i}(\cdot )\) satisfies such an *α*-quasi-periodic condition \(u^{i}(\widetilde{x}+2n\pi , x_{3})=e^{i 2\alpha \cdot n\pi }u^{i}( \widetilde{x}, x_{3})\) for all \(n\in {\mathbb{Z}}^{2}\). Then the solution \(u_{l}, l=1,2\), is also required to satisfy the same *α*-quasi-periodic condition, i.e., \(u_{l}(\widetilde{x}+2n\pi , x_{3})=e^{i 2\alpha \cdot n\pi }u_{l}( \widetilde{x}, x_{3})\) in \({\mathbb{R}}^{3}\). Conditions (1.5) and (1.6) are known as the Rayleigh expansion conditions of the scattered field \(u^{s}\) in \(\Omega _{+}\) and the transmitted field \(u_{2}\) in \(\Omega _{-}\), respectively, with \(\beta _{n}^{-}\) defined similarly as \(\beta _{n}^{+}\) by the wave number \(k_{2}\).

The well-posedness of problem (1.2)–(1.6) can be established by the variational method (cf. [31]) or the integral equation method (cf. [32, 33]). In the current paper we first establish the \(L_{\alpha }^{p}\ (1< p \leq 2 )\) estimates for the scattered field \(u^{s}\) and the transmitted field \(u_{2}\). Based on these a priori estimates, we focus on the unique identification of the penetrable periodic structure from the scattered field \(u^{s}\) measured only on a straight line above the periodic structure induced by a countably infinite number of quasi-periodic incident plane waves.

There are lots of results concerning the uniqueness issue for the inverse periodic transmission problems (cf. [5, 7, 12, 13, 18, 19, 23, 24, 33, 34]) and for the inverse scattering by the polygonal periodic structure (cf. [6, 11, 14]). For the special case when the medium has the energy absorption property, a uniqueness theorem was obtained in [5] from the measured scattered field for one incident plane wave in a two-dimensional space. The result of [5] was then extended to the three-dimensional case in [2]. It should be remarked that the uniqueness with one incident wave does not hold true for the inverse periodic problem for a real wave number case, that is, the medium does not has a property of energy absorption. See also [7] for a uniqueness theorem on the recovery of a smooth periodic structure with one incident plane wave under some a priori assumptions on the structure. For the case when a priori restrictions on the height of the grating surface are known in advance, a uniqueness result can be found in [18] on the identification of a smooth perfectly reflecting periodic structure from many measurements corresponding to a finite number of incident plane waves. The method of [18] was extended to the periodic transmission problem [13]. There also exist some numerical methods in reconstructing periodic structures. For example, a linear sampling method was developed in [20, 22] for determining the shape of partially coated bi-periodic structures, and in [35] a novel linear sampling method was introduced for simultaneously reconstructing dielectric grating structures in an inhomogeneous periodic medium. See also [10] for a finite element method or [3, 4, 17] for the factorization method in determining the periodic structures, or [30] for the uniquely reconstruction of a locally perturbed infinite plane. Recently, by making use of the differential sampling method, the anisotropic periodic layer can be uniquely determined in [25] under the assumption that the complement of the periodic layer in one period is connected. The analysis of sampling methods for the recovery of a local perturbation in a periodic layer can be found in [16].

For the scattering by general periodic structures case, there are several uniqueness results. We refer to [23] for a uniqueness theorem for the inverse Dirichlet problem, and to [21, 24, 32] for uniqueness results for the inverse transmission problem by means of all quasi-periodic incident plane waves. The reader is referred to [19] for a partially coated perfectly grating case with respect to infinitely many point sources, and to [34] for uniqueness results for both the partially coated perfectly reflecting grating and the periodic transmission case in a two-dimensional space, corresponding to a countably infinite number of quasi-periodic incident plane waves. In this paper we intend to develop a novel method, which differs from the approach used in [34], to prove the uniqueness on the identification of the penetrable periodic structure in the three-dimensional space from the measured data only above the structure with respect to a countably infinite number of quasi-periodic incident plane waves. The technique developed in this paper can date back to the work [27, 36] on the inverse scattering problems of determining the support of penetrable electromagnetic obstacles or to [28] for the fluid-solid interaction problem of identifying the bounded solid obstacle, [29] for the cavity scattering case.

The paper is organized as follows. In Sect. 2, the a priori estimates in the sense of \(L_{\alpha }^{p}\ (1< p \leq 2)\) norm for the solution of the direct scattering problem in \({\mathbb{R}}^{3}\) are established by applying the integral equation method. Section 3 is devoted to the inverse problem of uniquely determining the periodic structure from the measured data only above the structure produced by a countably infinite number of quasi-periodic incident plane waves.

## A priori estimates

In this section we establish some a priori estimates for the solution of the direct scattering problem by employing the integral equation method. Eliminating the incident field \(u^{i}\), it is easily found that the scattered field \(w_{1}:=u_{1}-u^{i}\) in \(\Omega _{+}\) and the transmitted field \(w_{2}:=u_{2}\) in \(\Omega _{-}\) satisfy the following boundary value problem:

in the general case \(f_{1}, f_{2}\in L^{p}_{\alpha }(\Gamma )\) with \(1< p\leq 2\). \(Here, L^{p}_{\alpha }(\Gamma ) (p\ge 1)\) denotes the Sobolev space of scalar functions on Γ which is assumed to be *α*-quasi-periodic with respect to the variable *x̃*, equipped with the norm in the usual Sobolev space \(L^{p}(\Gamma )\).

Before going further we first introduce the basic notations that are used in the rest of this paper. For simplicity, we use \(\Omega _{\pm }\) and Γ again to denote the same sets restricted to one period \(0< x_{1}, x_{2}<2\pi \). For each \(h>0\), denote by \(\Omega _{+}(h):=\{x\in \Omega _{+}: x_{3}< A_{1}+h\}\), \(\Omega _{-}(h):=\{x\in \Omega _{-}: x_{3}> A_{2}-h\}\), \(\Gamma _{+}(h):=\{x\in \Omega _{+}: x_{3}= A_{1}+h\}\), and \(\Gamma _{-}(h):=\{x\in \Omega _{-}: x_{3}=A_{2}-h\}\), respectively. Then, let \(H^{1}_{\alpha }(\Omega _{\pm }(h))\) and \(L^{p}_{\alpha }(\Omega _{\pm }(h)) (p\ge 1)\) denote the Sobolev spaces of scalar functions on \(\Omega _{\pm }(h)\) which are assumed to be *α*-quasi-periodic with respect to the variable *x̃*, equipped with the norms in the usual Sobolev spaces \(H^{1}(\Omega _{\pm }(h))\) and \(L^{p}(\Omega _{\pm }(h))\), respectively. Let \(H^{1/2}_{\alpha }(\Gamma _{\pm }(h))\) denote the trace space of \(H^{1}_{\alpha }(\Omega _{\pm }(h))\), and \(H^{-1/2}_{\alpha }(\Gamma _{\pm }(h))\) is the dual space of \(H^{1/2}_{\alpha }(\Gamma _{\pm }(h))\).

We introduce the free space *α*-quasi-periodic Green function

and the *α*-quasi-periodic layer-potential operators \(S_{1}\), \(K_{1}\), \(K'_{1}\), and \(T_{1}\) defined by

Noting that \(G_{1}(x,y;k_{1})-\Phi (x,y;k_{1})\) is smooth, it follows from [8] that the operators \(S_{1}:H^{-\frac{1}{2}}_{\alpha }(\Gamma )\rightarrow H^{\frac{1}{2}}_{\alpha }(\Gamma )\), \(K_{1}:H^{\frac{1}{2}}_{\alpha }(\Gamma )\rightarrow H^{\frac{1}{2}}_{\alpha }(\Gamma )\), \(K'_{j}:H^{-\frac{1}{2}}_{\alpha }(\Gamma )\rightarrow H^{-\frac{1}{2}}_{\alpha }(\Gamma )\), and \(T_{1}:H^{\frac{1}{2}}_{\alpha }(\Gamma )\rightarrow H^{-\frac{1}{2}}_{\alpha }(\Gamma )\) are all bounded, where \(\Phi (x,y;k_{1})=\frac{1}{4\pi }\frac{e^{ik_{1}|x-y|}}{|x-y|}\) is the fundamental solution of the Helmholtz equation \(\triangle \Phi +k_{1}^{2}\Phi =-\delta _{y}\) in the free space \({\mathbb{R}}^{3}\).

### Theorem 2.1

*For* \(f_{1},f_{2}\in L^{p}_{\alpha }(\Gamma )\) *with* \(1< p\leq 2\), *there exists a unique solution* \((w_{1},w_{2})\in L^{p}_{\alpha }(\Omega _{+}(h))\times L^{p}_{\alpha }( \Omega _{-}(h))\) *to the transmission problem* (2.1)*–*(2.5) *satisfying the estimate*

*where* \(C>0\) *is a constant independent of* \(f_{1}, f_{2}\), *and depending on* \(G_{j}(\cdot ,y;k_{j}), \Omega _{+}(h)\) *with* \(j=1,2\) *and the boundedness of the operators* \(S_{j},K_{j}, K'_{j}, j=1,2\), *and* \(T_{2}-T_{1}\) *in* \(L^{p}_{\alpha }(\Gamma )\).

*Moreover*, *if* \(f_{1},f_{2}\in L^{p}_{\alpha }(\Gamma )\) *with* \(\frac{4}{3}< p\leq 2\), *we have*

*with a positive constant* \(C>0\), *which is independent of* \(f_{1}, f_{2}\), *and depending on* \(G_{j}(\cdot ,y;k_{j}), \Omega _{+}(h)\) *with* \(j=1,2\) *and the boundedness of the operators* \(S_{j},K_{j}, K'_{j}, j=1,2\) *and* \(T_{2}-T_{1}\) *in* \(L^{p}_{\alpha }(\Gamma )\).

### Proof

We seek a solution of problem (2.1)–(2.5) in the form of combined single- and double-layer potential

where \(G_{2}(x,y;k_{2})\) is defined as (2.6) with the wave number \(k_{1}\) replaced by \(k_{2}\).

With the aid of the jump relations of the layer potentials (see [26] for the case in the \(L^{p}\) norm), we obtain that the transmission problem (2.1)–(2.5) can be reduced to the system of integral equations

where the operator *L* is given by

It is easily shown that (2.15) is of Fredholm type due to the compactness of the operators \(S_{j},K_{j}, K'_{j}, j=1,2\), and \(T_{2}-T_{1}\) in \(L^{p}_{\alpha }(\Gamma )\). This, together with the uniqueness of the scattering problem (1.2)–(1.6), implies that (2.15) has a unique solution \((\varphi _{2},\varphi _{1})^{T}\in L^{p}_{\alpha }(\Gamma )\times L^{p}_{\alpha }(\Gamma )\) with the estimate

We next prove the \(L^{p}_{\alpha }, 1< p\leq 2\) estimates for the solution of the transmission problem (2.1)–(2.5). In fact, it can be checked that

and

with \(\frac{1}{p}+\frac{1}{q}=1\). Here, we have used the fact that the volume potential operator is bounded from \(L^{q}_{\alpha }(\Omega _{+}(h))\) into \(W^{2,q}_{\alpha }(\Omega _{+}(h))\) with \(2\leq q\leq 4\) (see [15, Theorem 9.9]), and the boundary trace operator is bounded from \(W^{1,q}_{\alpha }(\Omega _{+}(h))\) into \(L^{q}_{\alpha }(\Gamma )\) with \(2\leq q\leq 4\) (see [1, Theorem 5.36]). It is noted that (2.17)–(2.18) still holds true, with \(G_{1}(x,\cdot;k_{1})\) replaced by \(G_{2}(x,\cdot;k_{2})\) and \(\Omega _{+}(h)\) replaced by \(\Omega _{-}(h)\), respectively. Now the desired estimate (2.11) follows from (2.13)–(2.14) and (2.16)–(2.18). Furthermore, if \(f_{1},f_{2}\in L^{p}_{\alpha }(\Gamma )\) with \(\frac{4}{3}< p\leq 2\), by the similar arguments as those in (2.17)–(2.18), one can derive the required result (2.13). This completes the proof of the theorem. □

### Corollary 2.2

*For* \(y_{0}\in \Gamma \), *define the sequence* \(y_{j}:=y_{0}- \frac{1}{j}\nu (y_{0})\in \Omega _{+}\), \(j\in {\mathbb{N}}\). *Let* \((u_{1j},u_{2j})\) *be the solution of the scattering problem* (1.2)*–*(1.6) *with the incident point source* \(u^{i}=G_{1}(x,y_{j};k_{1})\). *Then*, *for any* \(h\in {\mathbb{R}}\), *we have*

*uniformly for* \(j\in {\mathbb{N}}_{+}\), *where* \(C>0\) *is a constant depending on* \(G_{j}(\cdot ,y;k_{j}), \Omega _{+}(h)\) *with* \(j=1,2\).

### Proof

It is obvious that \((u_{1j}^{s},u_{2j})\) satisfies problem (2.1)–(2.5) with the boundary data

It is easy to see that \(f_{1}(j),f_{2}(j)\in L^{p}_{\alpha }(\Gamma )\) are uniformly bounded for \(j\in {\mathbb{N}}\) with \(\frac{4}{3}< p<\frac{3}{2}\). Then the required result (2.19) follows from Theorem 2.1. This proves the corollary. □

### Theorem 2.3

*Let* \((u_{1j},u_{2j})\) *be the solution of the scattering problem* (1.2)*–*(1.6) *corresponding to the incident point source* \(u^{i}=G_{1}(x,y_{j};k_{1})\) *with* \(y_{j}\) *defined in Corollary *2.2. *Then*, *for any* \(h\in {\mathbb{R}}\), *it holds that*

*uniformly for* \(j\in {\mathbb{N}}_{+}\). *Here*, \(C>0\) *is a constant depending on* \(G_{j}(\cdot ,y;k_{j}), \Omega _{+}(h)\) *with* \(j=1,2\) *and the uniform boundedness of* \(S_{\Gamma \setminus {B}}(j)\) *and* \(K_{\Gamma \setminus {B}}(j)\) *in the corresponding Hilbert spaces*, *B* *is a ball satisfying that* \(B\supset B_{\delta }\), *and* \(B_{\delta }\) *is a small ball centered at* \(y_{0}\) *with the radius* \(\delta >0\).

### Proof

Define \(\tilde{y}_{j}:=y_{0}+\frac{1}{j}\nu (y_{0})\in \Omega _{-}\), let \(w_{1}(j):=u_{1j}^{s}-G_{1}(x,\tilde{y}_{j};k_{1})\) and \(w_{2}(j):= u_{2j}\), it follows that \((w_{1}(j),w_{2}(j))\) satisfies problem (2.1)–(2.5) with the boundary data

Obviously, \(f_{1}(j)\in L^{p}_{\alpha }(\Gamma )\) is uniformly bounded for \(j\in {\mathbb{N}}\), where \(1< p<2\). Furthermore, it is seen from [9, Lemma 4.2] that \(f_{2}(j)\in C(\Gamma )\) is uniformly bounded for \(j\in {\mathbb{N}}\). So \(f_{2}(j)\in L^{p}_{\alpha }(\Gamma )\) is uniformly bounded for \(j\in {\mathbb{N}}\), where \(1< p<2\). Then, by (2.16) in Theorem 2.1, one obtains that the solution \((\varphi _{1},\varphi _{2})^{T}\) of (2.15) satisfies

We next prove that the operator \(S_{1j}:L^{p}_{\alpha }(\Gamma )\rightarrow L^{2}_{\alpha }(\Gamma \setminus {B})\) is uniformly bounded for \(j\in {\mathbb{N}}\), where \(1< p<2\). Indeed, by direct calculations, we can deduce that

Here, we have used the fact that \(G_{1}(\cdot ,y;k_{1})\) is smooth on the boundary \(\Gamma \setminus {B}\) in the first inequality. Then we have that \(S_{1j}:L^{p}_{\alpha }(\Gamma )\rightarrow L^{2}_{\alpha }(\Gamma \setminus {B})\) is uniformly bounded for \(j\in {\mathbb{N}}_{+}\). Moreover, by using similar arguments as those in the proof of (2.22), it is seen that the operators \(S_{ij}\), \(K_{ij}\), \(K'_{ij}\), and \(T_{ij}\) are all uniformly bounded from \(L^{p}_{\alpha }(\Gamma )\) into \(L^{2}_{\alpha }(\Gamma \setminus {B})\) for \(j\in {\mathbb{N}}_{+}\), \(i=1,2\). Also notice that \(f_{1}(j), f_{2}(j)\in L^{2}_{\alpha }(\Gamma \setminus {B})\) are uniformly bounded for \(j\in {\mathbb{N}}_{+}\). This, combined with equation (2.15), gives that the unique solution \((\varphi _{1},\varphi _{2})^{T}\) of (2.15) satisfies that \((\varphi _{1},\varphi _{2})^{T}\in L^{2}_{\alpha }(\Gamma \setminus {B}) \times L^{2}_{\alpha }(\Gamma \setminus {B})\). It is noted from (2.14) that the solution \(u_{2j}\) of the transmission problem (2.1)–(2.5) can be rewritten in the form of

Define

It is easily seen that \(S_{\Gamma \setminus {B}}(j):H^{-\frac{1}{2}}_{\alpha }(\Gamma \setminus {B})\rightarrow H^{\frac{1}{2}}_{\alpha }(\Gamma \setminus {B})\) is uniformly bounded for \(j\in {\mathbb{N}}\). This in combination with the fact that \(\varphi _{1}\in L^{2}_{\alpha }(\Gamma \setminus {B})\) implies that \(q_{1j}(x):=S_{\Gamma \setminus {B}}(j)\varphi _{1}\) satisfies the following Dirichlet problem:

where \(\tilde{\Gamma }=(\Gamma \setminus {B})\cup (\partial B\cap \Omega _{-})\). Then the well-posedness of the Dirichlet problem (2.24) yields that, for any \(h\in {\mathbb{R}}\), \(q_{1j}\in H^{1}(\Omega _{-}(h)\setminus {\overline{B}})\) uniformly for \(j\in {\mathbb{N}}_{+}\).

We now define

Since the region \(\Omega _{-}\setminus {B}\) has a positive distance from \(y_{0}\), it is found that \(q_{2j}(x)\in H^{1}(\Omega _{-}(h)\setminus {\overline{B}})\) uniformly for \(j\in {\mathbb{N}}_{+}\). We further define

Obviously, \(K_{\Gamma \setminus {B}}(j):H^{-\frac{1}{2}}_{\alpha }(\Gamma \setminus {B})\rightarrow H^{\frac{1}{2}}_{\alpha }(\Gamma \setminus {B})\) is uniformly bounded for \(j\in {\mathbb{N}}_{+}\). Then, by the fact that \(\varphi _{2}\in L^{2}_{\alpha }(\Gamma \setminus {B})\), we obtain that \(q_{3j}(x):=K_{\Gamma \setminus {B}}(j)\varphi _{2}\) satisfies the Dirichlet problem (2.24), with the boundary data \(w=q_{1j}\) replaced by \(w=q_{3j}\) on Γ̃. Then using similar arguments as those in the proof of \(q_{1j}\in H^{1}_{\alpha }(\Omega _{-}(h)\setminus {\overline{B}})\) yields that \(q_{3j}\in H^{1}_{\alpha }(\Omega _{-}(h)\setminus {\overline{B}})\) uniformly for \(j\in {\mathbb{N}}_{+}\). We also define

The uniform boundedness of \(q_{4j}\in H^{1}_{\alpha }(\Omega _{-}(h)\setminus {\overline{B}})\) for \(j\in {\mathbb{N}}_{+}\) can be concluded from the positive distance between the region \((\Omega _{-}(h)\setminus {\overline{B}})\) and \(y_{0}\). Finally, the desired result (2.20) follows from the discussions below (2.24). The proof of the theorem is thus completed. □

## Uniqueness of the inverse problem

In this section we mainly focus on the inverse problem of determining the periodic interface by means of the near-field data measured from one side of the periodic interface. To address this issue, we first introduce a mixed-reciprocity relation between the incident plane wave (1.1) and the incident point source (2.6). To accomplish this, we let \(\hat{\alpha }: = -\alpha \) and consider an incident point source located at \(z\in \Omega _{+}\) taking the form

with the coefficients \(\hat{\alpha }_{n}, \hat{\beta }_{n}^{+}\) defined by \(\alpha _{n}, \beta _{n}^{+}\) with *α* replaced by *α̂*, respectively. Then the inverse scattering of the incident point source \(G_{1}(\cdot ,z;k_{1})\) by the two-layered periodic interface can be formulated as the following *α̂*-quasi-periodic problem:

Here, both \(\hat{v}_{1}\) in \(\Omega _{+}\) and \(\hat{v}_{2}\) in \(\Omega _{-}\) satisfy the *α̂*-quasi-periodic condition

Moreover, we write the scattered field \(\hat{v}^{s}( \cdot , z):=\hat{v}_{1} (\cdot , z)-G_{1}(\cdot ,z;k_{1})\) indicates the dependance of the wave field on the location of the point source, and let \(v(\cdot;m)\) and \(u^{s}(\cdot;m)\) be the scattered solution to problem (1.2)–(1.6) with respect to the incident wave \(u^{i}(x;m)=\exp (i\alpha _{m}\cdot \widetilde{x}-i\beta _{m}^{+}x_{3}), m\in {\mathbb{Z}}^{2}\). Therefore, we have the following mixed-reciprocity relation (for a proof, we refer to [34, Lemma 4.1]).

### Lemma 3.1

*For* \(z_{0}\in \Omega _{+}\), *let* \(\hat{v}_{n}^{+}(z_{0})\) *be the Rayleigh coefficients of* \(\hat{v}_{1}^{s}(\cdot;z_{0})\). *Then it holds that*

Now we are in a position to present a uniqueness theorem for our inverse problem. The proof mainly depends on the a priori estimates established in Sect. 2 and a construction of a well-posed transmission problem in a sufficiently small domain.

### Theorem 3.2

*Let* \(u_{1}^{s}(\cdot;m)\) *and* \(\widetilde{u}_{1}^{s}(\cdot;m)\) *be the scattered fields corresponding to problem* (1.2)*–*(1.6) *with respect to the different bi*-*periodic interfaces* Γ *and* Γ̃, *respectively*, *induced by the same incident field* \(u^{i}(x;m)=\exp (i\alpha _{m}\cdot \widetilde{x}-i\beta _{m}^{+}x_{3}), m\in {\mathbb{Z}}^{2}\). *If* \(u_{1}^{s}(\cdot;m)|_{\Gamma _{+}(h)}=\widetilde{u}_{1}^{s}(\cdot;m)|_{ \Gamma _{+}(h)}\) *for all incident fields* \(u^{i}(x;m) m\in {\mathbb{Z}}^{2}\), *then we have* \(\Gamma =\widetilde{\Gamma }\).

### Proof

We shall prove the assertion by contradiction. Assume contrarily that \(\Gamma \neq \widetilde{\Gamma }\). Without loss of generality, we can choose a point \(z^{*}\in \Gamma \setminus \widetilde{\Gamma }\) satisfying that \(f(\widetilde{z}^{*}) >\widetilde{f}(\widetilde{z}^{*})\) with \(z^{*}=(\widetilde{z}^{*},z_{3})\). Then we define the sequence

with sufficiently small \(\delta >0\) such that \(z_{j}\in B_{\varepsilon _{0}}(z^{*})\subseteq (\Omega _{+}\cap \widetilde{\Omega }_{+})\) for all \(j\in {\mathbb{N}}_{+}\), where \(B_{\varepsilon _{0}}(z^{*})\) is a small ball centered at \(z^{*}\) with the radius \(\varepsilon _{0}>0\).

Let \((\hat{v}_{1}(\cdot;z_{j}),\hat{v}_{2}(\cdot;z_{j}))\) and \((\hat{\widetilde{v}}_{1}(\cdot;z_{j}),\hat{\widetilde{v}}_{2}( \cdot;z_{j}))\) be the solutions to problem (3.2)–(3.6) corresponding to the same *α̂*-quasi-periodic incident point source \(\hat{v}^{i} = \hat{G}(\cdot ,z_{j})\) with \(z_{j}\) defined by (3.8). Then one obtains from Lemma 3.1 that

for all \(m\in {\mathbb{Z}}^{2}\), where \(\hat{v}_{-m}^{+}(z_{j})\) and \(\hat{\widetilde{v}}_{-m}^{+}(z_{j})\) denote the Rayleigh coefficients of the scattered fields \(\hat{v}^{s}(\cdot;z_{j})\) and \(\hat{\widetilde{v}}^{s}(\cdot;z_{j})\), respectively. By the assumption that \(u_{1}^{s}(\cdot;m)|_{\Gamma _{+}(h)}=\widetilde{u}_{1}^{s}(\cdot;m)|_{ \Gamma _{+}(h)}\) for all incident fields \(u^{i}(x;m) m\in {\mathbb{Z}}^{2}\), we arrive at that \(\hat{v}_{-m}^{+}(z_{j}) = \hat{\widetilde{v}}_{-m}^{+}(z_{j})\), \(m\in {\mathbb{Z}}^{2}\). This in combination with the Rayleigh expansions and the unique continuation principle implies that

for all \(j\in {\mathbb{N}}_{+}\).

Denote \(D_{0}: = B_{\varepsilon _{0}}(z^{*})\cap \Omega ^{-}\) with sufficiently small \(\varepsilon _{0}>0\) such that \(D_{0}\subseteq (\Omega _{-}\cap \widetilde{\Omega }_{+})\). Let \(U_{j}:=\hat{\widetilde{v}}_{1}(\cdot;z_{j})\) and \(W_{j}:=\hat{v}_{2}(\cdot;z_{j})\), it is observed that \((U_{j}, W_{j})\) satisfies the following modified interior transmission problem:

with the right terms and the boundary data

Clearly, one has that \(h_{1,j}=h_{2,j}\) on \(\partial D_{0}\cap \Gamma \). Since \(Z^{*}\) has a positive distance from Γ̃, we obtain that \(\hat{\widetilde{v}}^{s}(\cdot;z_{j})\in H^{1}(D_{0})\) uniformly for all \(j\in {\mathbb{N}}_{+}\). In view of the fact that \(\hat{G}(\cdot ,z_{j})\in L^{2}(D_{0})\) uniformly for all \(j\in {\mathbb{N}}_{+}\), it is deduced that \(g_{1,j}\in L^{2}(D_{0})\) uniformly for all \(j\in {\mathbb{N}}_{+}\). The uniform boundedness of \(g_{2,j}\) in \(L^{2}(D_{0})\) for all \(j\in {\mathbb{N}}_{+}\) is a direct consequence of Corollary 2.2 in Sect. 2. Moreover, arguing similarly as in [36, Theorem 2.9], one derives from the fact that \(h_{1,j}=h_{2,j}\) on \(\partial D_{0}\cap \Gamma \) that \(h_{1,j}\in H^{1/2}(\partial D_{0})\) and \(h_{2,j}\in H^{-1/2}(\partial D_{0})\), respectively, uniformly for all \(j\in {\mathbb{N}}_{+}\). Therefore, by the well-posedness of problem (3.11), we have

However, the above inequality is a contradiction since \(\|\hat{\widetilde{v}}^{s}(\cdot;z_{j})\|_{H^{1}(D_{0})}\) is uniformly bounded and \(\|\hat{G}(\cdot ,z_{j})\|_{H^{1}(D_{0})}\to \infty \) as \(j\to \infty \). Therefore, one concludes that \(\Gamma =\widetilde{\Gamma }\). This completes the proof of the theorem. □

## Availability of data and materials

Not applicable.

## References

- 1.
Adams, A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Elsevier, Singapore (2003)

- 2.
Ammari, H.: Uniqueness theorems for an inverse problem in a doubly periodic structure. Inverse Probl.

**11**, 823–833 (1995) - 3.
Arens, T., Grinberg, N.: A complete factorization method for scattering by periodic surfaces. Computing

**75**, 111–132 (2005) - 4.
Arens, T., Kirsch, A.: The factorization method in inverse scattering from periodic structures. Inverse Probl.

**19**, 1195–1211 (2003) - 5.
Bao, G.: A uniqueness theorem for an inverse problem in periodic diffractive optics. Inverse Probl.

**10**, 335–340 (1994) - 6.
Bao, G., Zhang, H., Zou, J.: Unique determination of periodic polyhedral structures by scattered electromagnetic fields. Trans. Am. Math. Soc.

**363**, 4527–4551 (2011) - 7.
Bao, G., Zhou, Z.: An inverse problem for scattering by a doubly periodic structure. Trans. Am. Math. Soc.

**350**, 4089–4103 (1998) - 8.
Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley, New York (1983)

- 9.
Colton, D., Kress, R., Monk, P.: Inverse scattering from an orthotropic medium. J. Comput. Appl. Math.

**81**, 269–298 (2007) - 10.
Elschner, J., Hsiao, G.C., Rathsfeld, A.: Grating profile reconstruction based on finite elements and optimization techniques. SIAM J. Appl. Math.

**64**, 525–545 (2003) - 11.
Elschner, J., Hu, G.: Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves. Inverse Probl.

**26**, 115002 (2010) - 12.
Elschner, J., Schmidt, G., Yamamoto, M.: An inverse problem in periodic diffractive optics: global uniqueness with a single wave number. Inverse Probl.

**19**, 779–787 (2003) - 13.
Elschner, J., Yamamoto, M.: Uniqueness results for an inverse periodic transmission problem. Inverse Probl.

**20**, 1841–1852 (2004) - 14.
Elschner, J., Yamamoto, M.: Uniqueness in determining polygonal periodic structures. Z. Anal. Anwend.

**26**, 165–177 (2007) - 15.
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (1983)

- 16.
Haddar, H., Nguyen, T.P.: Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers. Comput. Math. Appl.

**74**, 2831–2855 (2017) - 17.
Harris, I., Nguyen, D.L., Sands, J., Truong, T.: On the inverse scattering from anisotropic periodic layers and transmission eigenvalues. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2020.1836349

- 18.
Hettlich, F., Kirsch, A.: Schiffer’s theorem in inverse scattering theory for periodic structures. Inverse Probl.

**13**, 351–361 (1997) - 19.
Hu, G., Qu, F., Zhang, B.: Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric. Math. Methods Appl. Sci.

**33**, 147–156 (2010) - 20.
Hu, G., Qu, F., Zhang, B.: A linear sampling method for inverse problems of diffraction gratings of mixed type. Math. Methods Appl. Sci.

**35**, 1047–1066 (2012) - 21.
Hu, G., Yang, J., Zhang, B.: An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate. Appl. Anal.

**90**, 317–333 (2011) - 22.
Hu, G., Zhang, B.: The linear sampling method for the inverse electromagnetic scattering by a partially coated bi-periodic structure. Math. Methods Appl. Sci.

**34**, 509–519 (2011) - 23.
Kirsch, A.: Uniqueness theorems in inverse scattering theory for periodic structures. Inverse Probl.

**10**, 145–152 (1994) - 24.
Kirsch, A.: An inverse problem for periodic structures. In: Kleinman, R.E., Kress, R., Martensen, E. (eds.) Inverse Scattering and Potential Problems Mathematical Physics pp. 75–93. Peter Lang, Frankfurt (1995)

- 25.
Nguyen, T.P.: Differential imaging of local perturbations in anisotropic periodic media. Inverse Probl.

**36**, 034004 (2020) - 26.
Potthast, R.: On the convergence of a new Newton-type method in inverse scattering. Inverse Probl.

**17**, 1419–1434 (2001) - 27.
Qu, F., Yang, J.: On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Probl. Imaging

**12**, 281–291 (2018) - 28.
Qu, F., Yang, J., Zhang, B.: Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements. Inverse Probl.

**34**, 015002 (2018) - 29.
Qu, F., Yang, J., Zhang, H.: Shape reconstruction in inverse scattering by an inhomogeneous cavity with internal measurements. SIAM J. Imaging Sci.

**12**, 788–808 (2019) - 30.
Qu, F., Zhang, B., Zhang, H.: A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: the Neumann case. SIAM J. Sci. Comput.

**41**, A3673–A3702 (2019) - 31.
Strycharz, B.: An acoustic scattering problem for periodic, inhomogeneous media. Math. Methods Appl. Sci.

**21**, 969–983 (1998) - 32.
Strycharz, B.: Uniqueness in the inverse transmission scattering problem for periodic media. Math. Methods Appl. Sci.

**22**, 753–772 (1998) - 33.
Yang, J., Zhang, B.: An inverse transmission scattering problem for periodic media. Inverse Probl.

**27**, 125010 (2011) - 34.
Yang, J., Zhang, B.: Uniqueness results in the inverse scattering problem for periodic structures. Math. Methods Appl. Sci.

**35**, 828–838 (2012) - 35.
Yang, J., Zhang, B., Zhang, H.: A sampling method for the inverse transmission problem for periodic media. Inverse Probl.

**28**, 035004 (2012) - 36.
Yang, J., Zhang, B., Zhang, H.: Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles. J. Differ. Equ.

**12**, 6352–6383 (2018)

## Acknowledgements

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## Funding

This work was supported by the NNSF of China under grants 11871416, 11971273 and by the projects ZR2019MA027, ZR2018MA004, and ZR2017MA044 supported by Shandong Provincial Natural Science Foundation.

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### Cite this article

Cui, Y., Li, X. & Qu, F. A novel method in determining a layered periodic structure.
*Bound Value Probl* **2020, **177 (2020). https://doi.org/10.1186/s13661-020-01474-6

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### Keywords

- Inverse scattering
- Uniqueness
- \(L^{p}_{\alpha }\) estimate
- Periodic structures