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On Uniqueness in Electromagnetic Scattering from Biperiodic Structures

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional non-trapping assumptions on the material parameter, this identity allows us to establish uniqueness of solution for all wave numbers.


Introduction
Scattering of electromagnetic waves from periodic structures is not only an interesting mathematical topic in its own right but also of great interest in applications, e.g., for the construction and optimization of optical filters, lenses, and beam-splitters in optics. An overview about this and further topics in applied mathematics related to wave propagation in periodic structures can be found in, e.g., [5]. In this paper we consider scattering of time-harmonic electromagnetic waves from a dielectric biperiodic structure mounted on a perfectly conducting plate in three dimensions. By biperiodic, we mean that the structure is periodic in the, say, x 1 -and x 2 -direction, while it is bounded in the x 3 direction. In contrast to scattering from bounded structures, uniqueness of solution for this scattering problem does in general not hold for all positive wave numbers. Instead, non-trivial solutions to the homogeneous problem might exist for a discrete set of exceptional wave numbers, and these solutions turn out to be exponentially localized surface waves. Our study in the present paper focuses on conditions guaranteeing the well-posedness of the full three-dimensional electromagnetic scattering problem mentioned above. We establish non-trapping and smoothness conditions on the (non-absorbing) dielectric such that uniqueness of solution holds for all positive wave numbers. This means that materials satisfying the latter conditions cannot guide surface waves. Mathematical formulation for the well-posedness of electromagnetic scattering problem for periodic structures has been an active area of research in the last years. For the scalar case, the authors in [2,17] studied uniqueness of solution for all wave numbers (or, equivalently, all frequencies), under geometrical conditions on the scatterer, for impenetrable structures with Dirichlet and Neumann conditions. Similar results are obtained in the paper [7] for more complicated periodic structures which are constituted of conducting and dielectric materials. The latter paper further gave examples of structures for which non-uniqueness of solution occurs at the so-called singular wave numbers. These wave numbers were shown to be related to guided waves (surface waves) that are exponentially localized along the structure.
For the case of Maxwell's equations, the authors of [10] studied well-posedness of the scattering problem from a medium consisting of two homogeneous materials separated by a smooth biperiodic surface using an integral equation approach. In [4,6,11] the authors studied existence and uniqueness of solution for the scattering problem from penetrable biperiodic structures using a variational approach for the magnetic field. Nevertheless, unlike the scalar cases, the uniqueness results in cited cases of Maxwell's equations were proven for all but possibly a discrete set of wave numbers. Furthermore, all the cited papers above considered the non-magnetic case, i.e, the coefficient magnetic permeability is assumed to be the same constant outside and inside the structure. The case of variable magnetic permeability were investigated in the paper [1] for Maxwell's equations where the biperiodic structure consists of conducting and dielectric materials. That paper studied a variational approach, formulated in terms of the electric field, and showed that the obtained saddle point problem satisfies the Fredholm alternative, and again uniqueness of solution was proven for all but possibly a discrete set of wave numbers. More recently, the paper [16] analyzed the well-posedness of the scattering problem for penetrable anisotropic biperiodic structures with a restriction on the non-magnetic case again. The latter paper also proved that the scattering problem is uniquely solvable for all wave numbers if the structure contains absorbing materials, and if the dielectric tensor is piecewise analytic. Hence, to the best of our knowledge, uniqueness results for all wave numbers for the vectorial scattering problem still remains open if the biperiodic materials is non-absorbing.
The aim of the present work is to prove that the electromagnetic scattering problem for nonabsorbing biperiodic dielectric structures mounted on a perfectly conducting plate is uniquely solvable for all positive wave numbers if the material parameter satisfies non-trapping and smoothness conditions. We formulate the Maxwell's equations variationally in terms of the magnetic field in a suitable Sobolev space. We further restrict ourselves to the case of nonmagnetic and isotropic materials. The variational problem is well-known to fit into a Fredholm framework, see, e.g., [4,11,16]. (These papers deal with periodic scattering in the full space, but can be adapted to the half-space setting that we consider here.) As mentioned in the paper [7] on the corresponding scalar scattering problems, non-uniqueness phenomena indeed arise at certain singular wave numbers if the non-absorbing material parameter satisfies suitable trapping conditions. In this paper we show a converse result for the full three-dimensional periodic Maxwell equations: uniqueness of solution holds for all positive wave numbers if the material parameter is non-absorbing and satisfies suitable non-trapping and smoothness conditions. To prove the uniqueness result we derive a so-called Rellich identity for a solution to the homogeneous variational problem. The solution estimates resulting from this integral identity allow us to show that the homogeneous variational problem has only the trivial solution for all positive wave numbers.
Our analysis extends the approach in [12] that was motivated by an existence and uniqueness proof for solutions to rough surface scattering problems via Rellich identities in [8]. For scalar periodic problems, a related technique has been used in [7]. The paper [12] studied electromagnetic scattering from rough, unbounded penetrable layers. Such scattering problems are considered to be more complicated than those for periodic structures since the problem to find the scattered field cannot reduced, e.g., to a bounded domain. The applications of rough scattering problems include for instance outdoor noise propagation, oceanography or even optical technologies when the dielectric lacks periodicity. The authors in [12] formulated the latter scattering problem in terms of the electric field. We will instead choose a formulation in terms of the magnetic field, which somewhat changes the role of the dielectric material parameter in the integral identities since the material is non-magnetic. The paper [12] establishes existence and uniqueness of solution under non-trapping and smoothness conditions on the material parameter. While a priori estimates resulting from the Rellich identity allowed the authors in [12] to deduce uniqueness of solution, existence of solution has been obtained using a limiting absorption argument. The approach studied in the present paper is, from the technical point of view, somewhat similar to the one introduced in [12]. However, the analysis of the biperiodic case is definitely simpler since uniqueness of solution directly implies existence. Therefore, one only needs to investigate the Rellich identity and estimates for solutions to the homogeneous problem. It turns out also that this procedure produces weaker assumptions on the material parameter than those found in [12]. More precisely, uniqueness and existence of solution for all wave numbers are obtained under the following (non-trapping and smoothness) assumptions on the biperiodic relative material parameter ε r : + ) equals one in {x 3 > h} for some h > 0 and possesses essentially bounded and measurable first weak derivatives. Second, we require that where ∇ T ε −1 r := (∂ε −1 r /∂x 1 , ∂ε −1 r /∂x 2 , 0) ⊤ . Under these conditions, the existence of surface waves is automatically ruled out. While conditions (a) and (c) are similar to conditions (a) and (d) in [12,Eq. (7.2)], condition (b) is weaker and clearly simpler than the corresponding conditions (b) and (c) in [12,Eq. (7.2)].
The half-space setting that we consider in this paper is somewhat special, and it seems worth to mention that the Rellich identity itself generalizes to a corresponding periodic scattering problem in full space. The resulting estimate for a solution H to the scattering problem has a similar structure to the estimate in Lemma 5.4. However, in the half-space setting, the term 2Re Ω (∂ε −1 r /∂x 3 ) (∂H 3 /∂x 3 )H 3 dx can be treated without integration by parts using a Poincaré lemma. In contrast, in the full-space setting the only obvious way of treating this term is to integrate by parts. Since we seek for solution estimates, this introduces the condition that x 3 → ε −1 r (x 1 , x 2 , x 3 ) needs to be concave to conclude. Since this is a somewhat unnatural condition, we do not present this result in more detail.
One can further generalize the results presented here to certain anisotropic structures. However, already for the simpler case of isotropic coefficients the derivation of the Rellich identity is a technical matter. Again, we have opted to try to keep the presentation simple instead of treating the most general setting that could be considered.
The paper is organized as follows: In Section 2 we present setting of the problem. Section 3 is dedicated to a variational formulation and to the Fredholm property of the latter. Section 4 contains a couple of technical lemmas. We derive the integral inequalities resulting from the Rellich identity in Section 5. Finally, the uniqueness of the variational problem for all wave numbers is proven in Section 6.

Problem Setting
We consider scattering of time-harmonic electromagnetic waves from a biperiodic structure which models a dielectric layer mounted on a perfectly conducting plate. The electric field E and the magnetic field H are governed by the time-harmonic Maxwell equations at frequency where e 3 = (0, 0, 1) ⊤ . The electric permittivity ε is a bounded measurable function that is 2πperiodic in x 1 and x 2 . Further, we assume that ε equals ε 0 > 0 outside the biperiodic structure, that is, for The magnetic permeability µ = µ 0 is assumed to be a positive constant and the conductivity is assumed to vanish. As usual, the problem (1)-(3) has to be completed by a radiation condition that we set up using Fourier series. The biperiodic structure is illuminated by an electromagnetic plane wave with wave vector With these definitions, the incident plane waves E i and H i are given by In the following we will exploit that one can explicitly compute the corresponding reflected field at {x 3 = 0}. To this end, we introduce the notationã = (a 1 , . From now on, we denote the sum of the incident and reflected plane waves by and define E ir α and H ir α by such that E ir α and H ir α are 2π-periodic in x 1 and x 2 . If we apply the same phase shift to solutions E and H of the Maxwell equations (1)- (3), and if we denote for scalar functions f and vector fields F , then E α and H α satisfy Note that we still have div α curl α = 0 and curl α ∇ α = 0. Let us denote the relative material parameter by Obviously, ε r equals one outside the biperiodic dielectric structure. Recall that the magnetic permeability µ 0 is constant which motivates us to work with the divergence-free magnetic field, that is, div α H α = 0. Note that (4) plugged in into (6) implies that e 3 × (ε −1 r curl α H α ) = 0 on {x 3 = 0} and that the condition e 3 · H α = 0 on {x 3 = 0} can be derived by plugging (6) into (5). Hence, introducing the wave number k = ω(ǫ 0 µ 0 ) 1/2 , and eliminating the electric field E α from (4)-(6), we find that We wish to reformulate the last three equations in terms of the scattered field H s α , defined by Due to the biperiodicity of the right-hand side and of ε r , we seek for a biperiodic solution H s α , and reduce the problem to the domain (0, 2π) 2 × (0, ∞). We complement this boundary value problem by a radiation condition, see also in [6,11], that we set up using Fourier series. The scattered field H s α is 2π-periodic in x 1 and x 2 and can hence be expanded as where the Fourier coefficientsĤ n (x 3 ) are defined bŷ Define  Figure 2: Geometric setting for electromagnetic scattering problem from a biperiodic dielectric structure mounted on a perfectly conducting plate (in two dimensions, for simplicity).
Since ε −1 r equals one for x 3 > h it holds that div α H s α vanishes for x 3 > h, and equation (10) Using separation of variables, and choosing the upward propagating solution, we set up a radiation condition in form of a Rayleigh expansion condition, prescribing that H s α can be written as and that the series converges uniformly in compact subsets of {x 3 > h}.
The scattering problem to find a scattered field H s α that satisfies the boundary value problem (10) and the expansion (13) is in the following section reformulated variationally in a suitable Sobolev space.

Variational Formulation
We solve the scattering problem presented in the last section variationally, and briefly recall in this section a variational formulation of the problem in a suitable Sobolev space. Our framework is an adaption of the results from [16] to our half-space setting. In contrast to the variational formulation in H(curl) in [1], the papers [4,6,11,16] set up a variational formulation in H 1 for the magnetic field. Indeed, since the latter is divergence-free, any solution that is locally H(curl) indeed belongs locally to H 1 . For our purposes, the H 1 formulation has the additional advantage that it is well-defined at Rayleigh-Wood frequencies, as it was noted in [16]. We define a bounded domain with boundaries Γ 0 := (0, 2π) 2 × {0} and Γ h := (0, 2π) 2 × {h}, and Sobolev spaces equipped with the usual integral norm, e.g., The space H 1 p,T (Ω) 3 of periodic vector fields that are tangential on Γ 0 is well-defined due to the standard trace theorem in H 1 . We also define periodic Sobolev spaces of functions with d = 1, 2, 3 components on Γ h : for s ∈ R, We define a non-local boundary operator T α (the exterior Dirichlet-Neumann operator) by It is a classical result that T α is bounded from H , see, e.g., [3]. Using T α , we define a vector of (pseudo-)differential operators R α := (∂ α /∂x 1 , ∂ α /∂x 2 , T α ). For a vector field F ∈ H Since all components of R α are bounded operators from H [16]).
Assume that H s α is a distributional periodic solution to the boundary value problem (10) such that H s α , curl α H s α , and div α H s α are locally square-integrable, such that the radiation condition (13) is satisfied, and such that ν · (H s α + H ir α ) and ν × (ε −1 r curl(H s α + H ir α )) are continuous over interfaces with normal vector ν where ε r jumps. As noted in [16], this implies that, following the above notation, H s α ∈ H 1 p,T (Ω). Then the Stokes formula [1,16] implies that for all test functions F ∈ H 1 p,T (Ω) 3 . Since we assumed that 0 = e 3 × (ε −1 r curl α H α ) = e 3 × (ε −1 r curl α (H s α + H ir α )) on Γ 0 , the above identity simplifies to By construction, e 3 × curl α H ir α vanishes on Γ 0 , that is, we can neglect the last term in the last equation. The divergence constraint div α H s α = 0 that follows from (10) shows that where ρ is some complex constant with Re (ρ) ≥ c > 0 and Im (ρ) < 0. We next prove that the bounded sesquilinear form B : H 1 p,T (Ω) 3 × H 1 p,T (Ω) 3 → C satisfies a Gårding inequality (this goes back to [1]), i.e. there exist strictly positive constants c 1 and c 2 such that for all H ∈ H 1 p,T (Ω) 3 . Theorem 3.1. Assume that ε −1 r ∈ L ∞ (Ω) is positive and bounded away from zero. Set Re ρ = inf Ω ε −1 r > 0 and choose Im ρ < 0. Then B satisfies (15).

Proof. As in [16, proof of Theorem 1] one shows that
The following identity follows from integrations by parts, the periodicity, and the vanishing normal component of H on Γ 0 , In consequence, Precisely as in [16] one shows now by a Fourier series argument that for a finite-dimensional operator K on H 1/2 p (Γ h ) 3 . Note that the last inequality follows from | Γ h K(H) · H ds | ≤ C Ω |H| 2 dx due to the finite-dimensional range of K and the fact that on finite-dimensional spaces all norms are equivalent. The last inequality implies a Gårding inequality for B.
For simplicity we write from now on H for the searched-for scattered field H s α in (14) and replace the source function curl H ir α by a G ∈ H 1 p (Ω) 3 . The last theorem implies the following corollary.
Corollary 3.2. The variational problem to find H ∈ H 1 p,T (Ω) 3 such that satisfies the Fredholm alternative, i.e., uniqueness of solution implies existence of solution.
Note that this formulation corresponds to the usual variational formulation of the Maxwell equations with perfectly conducting magnetic boundary conditions in smooth bounded domains, see, e.g., [9,Section 4.5(b)]. For special material parameters ε −1 r in we will in the sequel of the paper establish a uniqueness result via a Rellich identity. The next lemma will be useful when proving this identity.
Multiplying (17) by F ∈ H 1 p,T (Ω) 3 , using the Stokes formula, and subtracting the resulting expression from the variational formulation (16), we find that Remark 3.4. Instead of the above variational formulation in H 1 p,T (Ω) 3 , one can also consider formulations in H p (curl α , Ω) 3 , the natural energy space for the second-order Maxwell equations (10), see, e.g., [1]. In H p (curl α , Ω) 3 there is no bounded trace operator for the normal component of the field, and in consequence, the formulation (16) needs to be adapted. Usually, one replaces F → e 3 × (R α × F ) × e 3 by Q(e 3 × H), where Q is a bounded operator between the natural trace spaces H see, e.g., [1]. Obviously this definition only makes sense if all β n are non-zero. If this is the case, then the variational formulation (16) is equivalent to the formulation in H p (curl α , Ω) 3 obtained using Q. Under the assumption that β n = 0, all subsequent results could also be obtained via the formulation in H p (curl α , Ω) 3 .

Integral Identities
This section is concerned with technical lemmas that will be used to derive the Rellich identity and solution bounds subsequently. Roughly speaking, for deriving the Rellich identity, we will multiply the Maxwell equations (17) by x 3 ∂H/∂x 3 and integrate by parts. Therefore, it is the aim of the technical lemmas in this section to analyze the term Re Ω x 3 ∂H/∂x 3 · curl α (ε −1 r curl α H) dx for a solution H ∈ H 1 p,T (Ω) 3 to the problem (16). Note that the first two lemmas need the function H to be in H 2 p (Ω) 3 . These lemmas for the magnetic field formulation actually correspond to the ones for the electric field formulation in [12,Section 3].
We need to introduce some notation. For a vector field F = (F 1 , F 2 , F 3 ) ⊤ we denote by F T = (F 1 , F 2 , 0) ⊤ its transverse part. Recall that ∂ α f /∂x j = ∂f /∂x j + iα j f for a scalar function f and j = 1, 2, 3. Further, we introduce and, for a vector field F = (F 1 , It is straightforwards to show that div α,T − − → curl α,T = 0 as well as curl α,T ∇ α,T = 0. Moreover, a tedious computation shows that and further (Ω) is positive and bounded away from zero and that H ∈ H 2 p (Ω) 3 . Then Proof. Denote by ν the outward unit normal to Ω. Using integration by parts and noting that ν = e 3 on Γ h , and that the boundary term on Γ 0 vanishes since x 3 = 0 on Γ 0 , we find that The next lemma continues the analysis of the term Re Ω ε −1 r (e 3 × ∂H/∂x 3 ) · curl α H dx in the right hand side of (24).
It remains to show that H also satisfies (27). The function H δ satisfies (25) and we consider the limit of this identity as δ → 0. It is easily seen that div α H δ → div α H = 0 in L 2 (Ω). Thus, we have Plugging in these limits into (25) shows that (27) holds.

Rellich Identity and Solution Estimate
For establishing uniqueness of solution to the variational problem (16), we derive in this section the so-called Rellich identity relating | curl α H| 2 and |∂H/∂x 3 | 2 where H is a solution to the homogeneous variational problem corresponding to (16). Then, under suitable non-trapping and smoothness conditions on the material parameter, integral inequality resulting from this identity allow us to obtain estimate for a solution to the homogeneous problem. As mentioned in the introduction, the Rellich identity and solution estimate obtained in this section are much simpler than the ones in [12,Section 4]. It turns out also that the non-trapping assumptions on the parameter material are weaker than the ones in the latter paper. The proof of the Rellich identity is based on an integration-by-parts technique that goes back to Rellich [15]. Typically, this technique requires more regularity of a solution than just to belong to the energy space. In our case we will roughly speaking multiply the Maxwell equations (17), for G = 0 in the right hand side, by x 3 ∂H/∂x 3 and integrate by parts. (Ω) is positive and bounded away from zero. Then the following identity holds for all solutions H ∈ H 1 p,T (Ω) 3 to the homogeneous problem corresponding to (16), Proof. Let H ∈ H 1 p,T (Ω) 3 be a solution to the homogeneous problem corresponding to (16). First, using integration by parts we have Recall equality (23), Combining the last two equations yields where L(H) is the left hand side of (28). It remains now to prove that the right hand side of the latter equation vanishes. First, we recall from Lemma 3.3 that ∂H/∂x Using the latter formulas and replacing k 2 by |n + α| 2 + β 2 n in the first boundary term in (28) yields On the other hand, due to the divergence-free condition, we have n∈Λ |n + α| 2 |Ĥ T,n | 2 − |(n + α) ×Ĥ T,n | 2 = n∈Λ |(n 1 + α 1 )Ĥ 1,n + (n 2 + α 2 )Ĥ 2,n | 2

Now substituting the latter equation into (30) leads to
where we exploited that β n is either a non-negative or a purely imaginary number. The proof is hence finished if we show that βn≥0 β 2 n |Ĥ n | 2 = 0 (since then L(H) = 0, which is the claim of the theorem). First, we compute that Since Re (β n ) ≥ 0 this implies that (The second equation will be exploited later on.) Taking the imaginary part of the variational formulation (16) with G = 0 and F = H, and exploiting Lemma 3.3, we obtain that This implies that |Ĥ n | 2 = 0 for all n such that Re (β n ) > 0. Since β n is either purely imaginary or non-negative, we conclude that βn≥0 β 2 n |Ĥ n | 2 = 0.
The following assumptions on ε −1 r will guarantee a stability estimate and a uniqueness statement for a solution to the variational problem (16): Remark 5.3. Note that (34)(a) implies that ε −1 r ≥ 1, since, by construction, ε −1 r = 1 in {h − η < x 3 < h} for some small η > 0. For the case of periodic non-absorbing structures, the main difference between these non-trapping conditions and the ones for the scalar case in [7] is the additional condition (34)(c). This condition arises from estimating the term 2Re Ω (∇ε −1 r · ∂H/∂x 3 H 3 ) dx in the Rellich identity (28) using the Poincaré-like result above. This is natural since the Rellich identity resulting from a similar technique for the scalar case [7] does not have a corresponding term.
Lemma 5.4. Assume that ε −1 r satisfies the three assumptions in (34). Then there exists C > 0 (independent of k > 0) such that for all solutions H ∈ H 1 p,T (Ω) 3 to the homogeneous problem corresponding to (16). Proof. We first estimate the two boundary terms in (28). We find that −2Re for arbitrary δ, γ > 0. Poincaré's inequality from Lemma 5.2 and the binomial formula imply that Again, we exploit Poincaré's inequality, to find that γ −1 ∂H 3 ∂x .