A novel method in determining a layered periodic structure

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 Lα (1 < p ≤ 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.

which propagate downward from + with α = (α 1 , α 2 ) := k 1 (sin θ 1 cos θ 2 , sin θ 1 sin θ 2 ) with the incident angle θ 1 ∈ [0, π/2), θ 2 ∈ [0, 2π), and β + j ∈ 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: 3) Here, u ± n ∈ C are the solution sequences, λ is the transmission coefficient and the unit normal vector ν on is directed into the interior of -. Notice that the incident wave u i (·) satisfies such an α-quasi-periodic condition u i ( x + 2nπ, x 3 ) = e i2α·nπ u i ( x, x 3 ) for all n ∈ 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 ( x + 2nπ, x 3 ) = e i2α·nπ u l ( x, x 3 ) in R 3 . Conditions (1.5) and (1.6) are known as the Rayleigh expansion conditions of the scattered field u s in + and the transmitted field u 2 in -, respectively, with βn defined similarly as β + 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 p α (1 < p ≤ 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 quasiperiodic 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 p α (1 < p ≤ 2) norm for the solution of the direct scattering problem in 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 1u i in + and the transmitted field w 2 := u 2 insatisfy the following boundary value problem: Here, L p α ( )(p ≥ 1) denotes the Sobolev space of scalar functions on which is assumed to be α-quasi-periodic with respect to the variable x, equipped with the norm in the usual Sobolev space L p ( ).
Before going further we first introduce the basic notations that are used in the rest of this paper. For simplicity, we use ± and again to denote the same sets restricted to one period 0 < x 1 , x 2 < 2π . For each h > 0, denote by + (h) := {x ∈ + : denote the Sobolev spaces of scalar functions on ± (h) which are assumed to be α-quasi-periodic with respect to the variable x, equipped with the norms in the usual Sobolev spaces H 1 ( ± (h)) and L p ( ± (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 ) -(x, y; k 1 ) is smooth, it follows from [8] that the operators where C > 0 is a constant independent of f 1 , f 2 , and depending on G j (·, y; k j ), + (h) with j = 1, 2 and the boundedness of the operators S j , K j , K j , j = 1, 2, and T 2 - with a positive constant C > 0, which is independent of f 1 , f 2 , and depending on G j (·, y; k j ), + (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 α ( ).
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 .
Proof It is obvious that (u s 1j , u 2j ) satisfies problem (2.1)-(2.5) with the boundary data It is easy to see that f 1 (j), f 2 (j) ∈ L p α ( ) are uniformly bounded for j ∈ N with 4 3 < p < 3 2 . Then the required result (2.19) follows from Theorem 2.1. This proves the corollary. (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 ∈ R, it holds that

Theorem 2.3 Let
uniformly for j ∈ N + . Here, C > 0 is a constant depending on G j (·, y; k j ), + (h) with j = 1, 2 and the uniform boundedness of S \B (j) and K \B (j) in the corresponding Hilbert spaces, B is a ball satisfying that B ⊃ B δ , and B δ is a small ball centered at y 0 with the radius δ > 0.
The uniform boundedness of q 4j ∈ H 1 α ( -(h) \ B) for j ∈ N + can be concluded from the positive distance between the region ( -(h) \ 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 letα := -α and consider an incident point source located at z ∈ + taking the form with the coefficientsα n ,β + n defined by α n , β + n with α replaced byα, respectively. Then the inverse scattering of the incident point source G 1 (·, z; k 1 ) by the two-layered periodic interface can be formulated as the followingα-quasi-periodic problem: Here, bothv 1 in + andv 2 insatisfy theα-quasi-periodic condition v j ( x + 2nπ, x 3 ) = e i2α·nπv j ( x, x 3 ), j = 1, 2.
Moreover, we write the scattered fieldv s (·, z) :=v 1 (·, z) -G 1 (·, z; k 1 ) indicates the dependance of the wave field on the location of the point source, and let v(·; m) and u s (·; m) be the scattered solution to problem (1.2)-(1.6) with respect to the incident wave u i (x; m) = exp(iα m · xiβ + m x 3 ), m ∈ Z 2 . Therefore, we have the following mixed-reciprocity relation (for a proof, we refer to [34, Lemma 4.1]). 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. Proof We shall prove the assertion by contradiction. Assume contrarily that = . Without loss of generality, we can choose a point z * ∈ \ satisfying that f ( z * ) > f ( z * ) with z * = ( z * , z 3 ). Then we define the sequence z j := z * -δ j ν z * for j = 1, 2, . . . · · · (3.8) with sufficiently small δ > 0 such that z j ∈ B ε 0 (z * ) ⊆ ( + ∩ + ) for all j ∈ N + , where B ε 0 (z * ) is a small ball centered at z * with the radius ε 0 > 0.