Upscaling of Helmholtz Equation Originating in Transmission through Metallic Gratings in Metamaterials

We investigate the transmission properties of a metallic layer with narrow slits. We consider (time-harmonic) Maxwell's equations in the H-parallel case with a fixed incident wavelength. We denote η > 0 as the typical size of the complex structure and obtain the effective equations by letting η → 0. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the slits. For the waves to be in resonance with the height of the metallic layer, the corresponding results can be perfect transmission through the layer.


Introduction
Negative refraction of electromagnetic waves in metamaterials has become of major interest in recent years, compare [1,2], especially to construct small scale optical devices for technical applications in the fields of micro-and nanooptics. Metamaterials are the materials that are not found in nature; instead they are created by the composition of several metals or plastics or both. Due to their precise shape, size, geometry, and arrangement of metals, these metamaterials are capable of influencing the electromagnetic waves by absorbing, bending, or refracting. To create the metamaterials, the composite materials are arranged in repeated (periodic) fashion with periodicity scales smaller than the wavelength of waves. Negative index metamaterial or negative index material (NIM) is a metamaterial where the refractive index (in optics theory, the refractive index of a material is a dimensionless number which describes how light propagates through that medium and is defined as the ratio / , where is the speed of light in vacuum and is the phase velocity of light in the medium) has a negative value over some frequency range when an electromagnetic wave passes through it. Negative index materials are extensively studied in the fields on optics, electromagnetics, microwave engineering, material sciences, semiconductor engineering, and several others.
In this work, we study the phenomena of light wave passing through the subwavelength metallic structure; that is, we investigate the high transmission of light wave through a metamaterial with thin holes inside it. We consider a thin metallic structure (inside a medium) with holes smaller than the wavelength of incident photon which shows the high transmission of light waves through this metallic structure. This high transmission contradicts the classical aperture theory and shows an important feature of metamaterials. To demonstrate the geometry assumed in this work, let us consider Figure 1 where the light wave emerging from a source (l.h.s of the figure) is passing through a metamaterial with negative refractive index and its image is given on the r.h.s. (cf. this figure to that of [3]).
The holes inside the metallic layer are periodically distributed with period > 0 smaller than the wavelength of incident light wave. This layer can be considered as a heterogeneous or perforated media and our goal is to give a physically consistent approach to transmission properties of heterogeneous media using the techniques from homogenization theory and applied analysis. We obtain an effective (upscaled) scaterring problem where the metallic layer with holes is replaced by a homogenized structure with effective permittivity eff and permeability eff . We also obtain the tranmission coefficient in terms of incident wave number  Figure 1: The light wave is originating from the left side and is getting refracted through the metallic plate and its image is on the right (this image is taken from [2]). and incident angle . We will see that, for lossless materials with (real) negative permittivity , perfect transmission = 1 can be obtained for every and suitable value for . In the recent times, several significant investigations for metamaterials have been done. In [4] the connection between the high transmission and the excitation of surface plasmon polaritons has been established. The photonic band structure of the surface plasmons is evaluated numerically, In [5] the authors have calculated the transmission coefficients for the lamellar gratings, while the effect of surface plasmons on the upper and lower boundary of the layer is investigated in [6]. In [7], the effect of finite conductivity is studied. In [8], the relation between the high transmission effect and the negative index material is obtained with a fishnet like structure. A homogenization method is proposed in [9] where the author accentuates the connection between the skin depth of evanescent modes in the metallaic structure and the period of the gratings. Some results in this direction can be found in [10][11][12][13].
Two-scale convergence has proven to be a very efficient tool in homogenization theory while dealing with the problems where the underlying medium is heterogenous. The concept of two-scale convergence is first introduced by Nguetseng in [14]. This convergence criterion and the results related to it have been used extensively in the homogenization of partial differential equations; see Allaire [15,16], Cioranescu and Donato [17], and Mahato and Böhm [18]. In this work too, we have used the two-scale convergence of an oscillating sequence and its gradient; see Section 1.4. At this point we would like to point out that the geometry of metallic structure in this work is generalized compared to that considered in [3]. In [3] due to the rectangular shapes of the metallic gratings, the coefficients of the effective system were determined by the help of a scalar, one-dimensional shape function : R → C given by the hyperbolic functions; however in our work where the considered geometry is more realistic, such nice representation is not possible. To deal with this problem in this work an eigenvalue approach has been proposed and this eigenvalue problem in the unit cell helps us to determine the effective parameters of the problem.
Although this paper can be compared with [3] in some way, the major difference in this work is that our limit function ⃗ 0 of ⃗ = ∇ (see Section 1.5 for details) and the limit function ⃗ 0 in [3] are totally different. In [3], the authors worked with a rectangular metallic subpart Σ of type (− , )× (−1/2, 1/2) and therefore by defining a suitable test function, they have shown that the first component of 0 vanishes and they obtain 0 = (0, −1 2 ) for ∈ and ∈ \ Σ. This is clearly not the case in this paper as the metallic subpart Σ is chosen to be sinusoidal along 2 -axis due to the geometry of the metallic structure given by Figure 1. This will lead to nonvanishing componenets of 0 and we end up having a different 0 compared to that in [3] and hence, we will obtain a different upscaled equation. Also no explicit representation of ⃗ 0 can be obtained due to the geometry of Σ. In Sections 1.1, 1.2, and 1.3, we will outline the model in detail. In Section 2, we gather some mathematical tools required to do the analysis and we state our main results. In Sections 3 and 4 we will prove the main results.
1.1. Model. We investigate the time-harmonic solutions of a Maxwell equations with a fixed wave number and the corresponding wave length = 2 / . Let the metallic structure remain unchanged towards 3 -direction and the metallic field, denoted by ⃗ , is parellel to 3 ; that is, ⃗ = (0, 0, ), where : R 2 → R.
The heterogenoeus domain Ω has a metallic structure of finite length and finite height in R 2 , and the slits (vacuum) are repeated periodically with a small period > 0, compare Figure 1. The period is assumed to be infinitesimally small with respect to the wavelength . The relative permittivity of the metal is denoted by . Since the permittivity of conductors The Scientific World Journal 3 has large absolute values, we assume that it depends on and consider = . We obtain nontrivial effects due to plasmonic resonance for | | ∝ −2 , compare [3,13]. If Σ denotes the matallic part in Ω, we set where ∈ C. Due to ohmic losses inside the metal, Im( ) is always assumed to be positive in a physical system which means we always take Im( ) ≥ 0 and ̸ = 0. A material is called a lossless material if Im( ) = 0. Our particular interest is to study a lossless material with negative relative permittivity; that is, Im( ) = 0 and Re( ) < 0. For such transverse evanescent modes will be generated in the metal. Since Re( ) < 0 and Im( ) = 0, then from (14) we have ; that is, we can obtain wave like solutions and waves cannot penetrate the metallic grating. These evanescent modes can penetrate only in a region which is given by the skin depth of order . The evanascent mode is related to a surface plasmon solution (in this case a solution which is nonvanishing in the grating but which has exponential decay in the metal). The main aspect of the current work is to generalize the geometric structure of the metallic slab inside Ω given in [3].

Geometry.
Let > 0 be a small scale parameter and Ω be the domain under investigation which is bounded in R 2 . Let fl (−1/2, 1/2) × (−1/2, 1/2) be the representative unit cell in R 2 and Σ be an open set in such that Σ ⊂ and fl Σ ∪ ( \ Σ). Let us choose Σ in such a way that it follows a sinusoidal profile along 2 -axis; that is, Keeping physics of the problem in mind, Σ denotes the mettalic part which lies between the two columns of holes in the metallic structure of type introduced in Figure 1. The relative aperture volume = 1 − 3 /2 ∈ (1/4, 1) and relative metal volume is (3/2) , where ∈ (0, 1/2). We define 2 + 1 fl 2 /2 . We assume that the compact rectangle contains (2 + 1) number of small rectangles of type ( − , + ) × (−ℎ, 0), that is, of width 2 and height ℎ, which include the -scaled versions of the metallic part Σ (cf. Figures 2 and 3), where each Σ is of width (3/2) and height ℎ. The collection of these small -scaled versions of the metallic part is the metallic domain Σ and assume that the two-dimensional heterogeneous metallic structure introduced in Figure 1, denoted by Σ and parellel to 1 -axis, is contained in the closure of the set fl (− , ) × (−ℎ, 0) ⊂ Ω with ⊂ Ω; that is; Figure 3.

Mathematical Formulation and Statement of the Main
Results. We study the Maxwell equations in a complex geometry with highly oscillating permittivities. By we denote (i) the dimensionless positive scale parameter which represents the small length scale in the geometry Σ ⊂ R 2 and (ii) the oscillations of large absolute values of the permittivity. We follow the standard nondimenionsalization techniques; for instance, see [3,18,22], and so forth and from here on all the quantities considered in this work are dimensionless unless stated otherwise. For the electric field ⃗ and magnetic field ⃗ , the time-harmonic Maxwell equations are with fixed positive real constants , 0 , and 0 denoting the frequency of the incident waves and the permeability and the permittivity of vacuum, respectively. We postulate that all the quantities are 3 -independent and the polarized magnetic field is given by By orthogonal property of ⃗ and ⃗ , we have ⃗ = ( , , , , 0). Then (12) reduce to By (13), a straightforward calculation yields where we have set 2 = 2 0 0 . We define the coefficient ( ) fl −1 ( ) which can have a negative real part and that it vanishes in the metal as → 0. Thus we have the desired Helmholtz equation which we will study in this paper and is given below. We study solutions ∈ 1 loc (Ω) of where the coefficient is given by The set Σ ⊂⊂ ⊂ Ω describes the complex geometry of the metallic inclusion in Ω; see Figure 3.
Remark 6 (scattering problem). We will investigate the effective behavior of solutions of (15) in two different cases. In the first case we will study an arbitrary bounded sequence of solutions on a bounded domain Ω while the second one concerns the scattering problem. In other words we consider (15) in whole of R 2 . For a given incident wave , which solves ∇ 2 = − 2 in R 2 , we take the Sommerfeld condition as the boundary condition which says that the scattered field = − satisfies for = | | → ∞, uniformly in the angle variable.

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The Scientific World Journal Remark 7. Note that for (15) we have not given any boundary conditions; instead we have considered an arbitrary sequence of solutions; however, the uniqueness of solution of the scattering problem will be proven for every . To state the main results, we rewrite (15) as a system: Comparing with (12), we see that represents (up to a factor and perhaps a rotation) the horizontal electric field ⃗ and since the magnetic field ⃗ ( ) = (0, 0, ( )), system (18) is nothing but (12) itself.
Theorem 8 (upscaled equations). Let the matallic geometry be given by Σ (Figure 3) on a domain Ω ⊂ R 2 and let the coefficient fl −1 be as in (16). On ∈ C we assume that either Im( ) > 0 or < 0 = Im( ). Let ( ) >0 be the sequence of solutions of (15) where is defined by (36). Then the function ∇ ∈ 2 (Ω). The field = ∇ converges weakly to some in 2 (Ω; C 2 ) which is given by Moreover, the limit functions satisfy the system where fl ( 0 By applying Theorem 8 for Ω fl 0 (0) with a large radius 0 > 0, we can treat the scattering problem with an incoming wave generated at infinity. We obtain the strong convergence of the scattered field outside the metallic obstacle and we identify the limit ( ) as the solution of the effective diffraction problem. We define the exterior domain outside of as ext fl R 2 \ . Theorem 9 (effective scattering problem). Let the metallic gratings be given by Σ (Figure 1) and the coefficient ( ) fl −1 ( ) be as in (16). Assume further that is an incident wave solving the free space equation ∇ 2 = − 2 on R 2 and is the unique sequence of solutions to (15) such that = ( − ) satisfies (17) and that the solution sequence satisfies the uniform bound Then → strongly in 2 ( ) with uniform convergence for all derivatives on any compact subset of . The effective field : R 2 → C is determined as the unique solution of the upscaled equation with (17) for the scattered field ( − ). (24) should be understood in the sense of distributions on the whole of R 2 . The exterior field ∈ 1 ( (0) \ ) for every large radius ; hence its trace on from outside, denoted by + , is a well-defined element of 1/2 ( ). Note that as ∇ belongs to 2 ( ), the function (⋅, ⋅) is an element of 1 loc (R). This helps us to define traces of on the horizontal boundary parts from the inside. Moreover, we have the information that the distributional divergence of the vector field = eff ∇ is of class 2 loc (R 2 ). We define the transmission condition on the boundary of with using traces from inside and outside of . We denote by superscript + (resp., by −) traces from outside (resp., by inside); then problem (24) can be rewritten as

Interface Conditions. The homogenized equation
with the transmission (interface) conditions where is defined in (36).

A Priori Estimates
Lemma 10. For an ∈ C with Im( ) > 0, let be defined as in (16). Then there exists a ∈ C such that where 0 > 0 is independent of .
Proof. Since ⊂ Ω, there exists a subdomain Ω ⊂⊂ Ω such that ⊂ Ω ⊂⊂ Ω. Without loss of generality, let us assume that Σ ⊂ Ω and take a cut-off function Θ ∈ ∞ 0 (Ω; [0, 1]), where Θ( ) = 1 on Ω . We test (15) with Θ 2 (⋅) (⋅), where (⋅) is the complex conjugate of (⋅). This gives We employ Lemma 10. For a ∈ C, we multiply (31) by and equate its imaginary part and rearrange the factors of the second integrand which will yield where in the second step we used Young's inequality. We see that the first integral on the r.h.s. of (32) is bounded by the 2boundedness assumption on whereas the second integral on the r.h.s. is bounded by the boundedness of | | and ‖ ‖ 2 (Ω) < ∞. Using the fact that Θ( ) = 1 on Ω , we have where is independent of and .

An Eigenvalue Problem in the Unit Cell . Let us consider the eigenvalues
and we denote { } the associated normalized eigenfunctions in 1 0 (Σ), so that { } is an orthonormal basis of 2 (Σ). Since ∈ C with Im( ) > 0, 2 satisfies the condition We set Let us consider the following boundary value problem: By [23, theorem 8.22], it follows that (i) ( ) is a solution of (37) and this solution is unique if 2 ̸ = for all and (ii) if condition (34) is not fulfilled then (37) has no solution.
In the next theorem we will analyze the behavior of as → 0 in the sense of two-scale convergence, compare [15]. We notice that the geometry is not only periodic in the 1direction but it is also periodic with respect to to the cell fl (−1/2, 1/2) × (−1/2, 1/2). The metal part in the cell is given by Σ ⊂ ; see Figure 2. We recall that the sequnce ( ) >0 is weakly convergent to ∈ 2 (Ω). We define a function 0 ( , ) fl 0 ( 1 , 2 , 1 , 2 ) as where ( ) is a -periodic function defined in (35). We have defined 0 in such a way that, for every ∈ Ω, there holds ( ) = ∫ 0 ( , ) . We will show in next theorem that 2 ⇀ as → 0.
Proof. We divide the proof into three steps.
Therefore, to sum up, we obtain the two-scale limit as The Scientific World Journal 9 provided (34) holds. Consequently, for ∈ , the weak limit satisfies (44) Therefore the two-scale limit is given by We know that 0 ( , ) = ( ) = ( ) holds for a.e. ∈ Ω \ and for all ∈ . Moreover, by the assumption on and estimate (30), we have ‖ ‖ 2 (Ω\ ) + ‖∇ ‖ 2 (Ω\ ) ≤ < ∞. This then implies that , up to a subsequence, is strongly convergent to in 2 (Ω \ ) by Aubin-Lion's Lemma, compare [24]. The uniform convergence on compact subsets of and of all its derivatives is a consequence of the fact that Helmholtz equation Δ + 2 = 0.
With the help of Lemma 12, we can completely determine the two-scale limit of the sequence ( ) >0 if we know the function ( ) which is defined in (39). Now we collect the properties of ⃗ , its weak limit ⃗ , and its two-scale limit ⃗ 0 .
Proposition 13. Let ⇀ be as in Lemma 12 and be given by (39). For ⃗ = ∇ , we suppose that ⃗ ⇀ ⃗ fl ( 1 , 2 ) in 2 (Ω : R 2 ). Then ⃗ is characterized as follows: (i) The sequence ⃗ converges in the sense of two scales to ⃗ 0 which is given by (46) (ii) The limit ∇ ∈ 2 (Ω) and it holds: Remark 14. We would like to point out a major difference in our 0 and the limit function 0 in [3]. In [3], the authors worked with a rectangular metallic subpart Σ of type (− , )× (−1/2, 1/2) and therefore by defining a suitable test function, they have shown that the first component of 0 vanishes and they obtain 0 = (0, −1 2 ) for ∈ and ∈ \ Σ. This is clearly not the case in this paper as the metallic subpart Σ is chosen to be sinusoidal along 2 -axis due to geometry of the metallic structure given by Figure 1. This will lead to nonvanishing componenets of 0 and we end up having a different 0 compared to that in [3] and hence, we will obtain a different upscaled equation.
Proof. By (30), it follows that ⃗ is bounded in 2 loc (Ω) which implies that up to a subsequence ⃗ two-scale converges to some ⃗ 0 . The weak limit would then be given as ⃗ ( ) = ∫ ⃗ 0 ( , ) .

Proofs of Theorems 8 and 9
Proof of Theorem 8. The proof of Theorem 8 is a straightforward consequence of Lemmas 11 and 12 and Proposition 13. It is being shown that if, for any subdomain Ω with ⊂ Ω ⊂⊂ Ω, ⃗ = ∇ is bounded in 2 (Ω ), then, up to a subsequence, ⃗ is weakly convergent to some ⃗ in 2 (Ω ).

By Proposition 13, we have the relation between and ⃗
; that is, the weak and the two-scale limits of ⃗ are given in terms of ; see (46) and (47). Since Ω is arbitrary, the results of Proposition 13 hold good in all Ω. Now we obatin the limit problem by dividing the proof into two following cases.
The Scientific World Journal 11 Case 2. Let ∈ ; then again for ∈ ∞ 0 (Ω) from (15) we have The combination of (54) and (55) gives the limit problem as Here we can compare our upscaled equation with the limit problem obtained in [3], especially for ∈ . Due to their rectangular metallic gratings inside R, the component along 1 direction vanishes; that is, the first component of ⃗ = 0 and thus the authors obtained their upscaled equation Proof of Theorem 9. The proof is devided into three steps which are demonstrated below.
(i) Uniqueness of the Limit Problem. With a fixed incident field we will show that the limit problem (24) has a unique solution. On the contrary, let us assume that 1 and 2 are the two solutions of (24) and set = 1 − 2 . We consider the equations satisfied by difference of two solutions as We claim that ( , ) = 0 for ∈ R and ∈ . The main ingredient for this uniqueness result is Rellich's first lemma and the fact that eff is real and eff has positive imaginary part. In fact eff is identity and eff is 1 outside of . Let us denote the surface of a sphere (0) of radius by (fl (0)), where is chosen so large such that ⊂ (0). Let 0 be such ; then by (59), we have This gives Now we multiply (58) by and integrate over 0 (0). Since (58) holds only in the sense of distributions and due to possible jumps on Γ hor , we approximate by smooth functions. By divergence theorem we have The surface integral on r.h.s. of (62) is well defined. This can be argued as follows: outside of , is a solution of the Helmholtz equation ∇ 2 = − 2 and so it is analytic in the exterior of . Therefore the traces of and are well defined in 0 , compare [25]. Comparing the imaginary parts of (62) and investing the knowledge of Im( eff ) > 0, then Therefore from (63) we have = 0 in . Since 0 is chosen arbitrarly, for every from (60) it follows that Thus by Rellich's first lemma (which states that the solutions u of the Helmholtz equation on an exterior domain 12 The Scientific World Journal satisfying property (64) vanish) we obtain = 0 in all of R 2 which concludes the proof of the uniqueness property, compare [25].
(ii) Convergence to the Limit Problem Assuming an 2 -Bound. Let the radius 0 > 0 be such that ⊂ 0 (0) and set Ω fl 0 (0). We begin with the assumption that The proof basically follows as the one for Theorem 8. Using (65), up to a subsequence, passing the limit as → 0, we obtain that ( ) = ( ) ( )| Ω\ + −1 ( ) ( )| solves (24). We only need to verify the radiation condition (17). By Lemma 12 it follows that and ∇ are uniformly convergent on every compact subset of R 2 \ . Let us choose < 0 such that ⊂⊂ (0) ⊂⊂ Ω. By [25, theorem 2.4] and end remark of that theorem, we have from the Sommerfeld radiation condition that the scattered field = − coincides on R 2 \ (0) with its Helmholtz representation through values and derivatives of − on (0). By the similar representation formula, using the values and derivatives of − on (0), we can extend into all of R 2 to a solution of the Helmholtz equation ∇ 2 = − 2 outside of . Thus this construction of shows that − satisfies the Sommerfeld radiation condition. The uniform convergence of → and ∇ → ∇ on (0) implies the uniform convergence of and its derivatives on all compact subsets of exterior of . Finally by uniqueness of the limit from part (i), ⇀ as → 0 for the whole sequence. This shows that the Sommerfeld radiation condition holds for = | | → ∞. which establishes (17).
(iii) Boundedness of . In the previous step the limit problem is obtained assuming (65) is true. We will prove that (65) holds true by the method of contradiction. We suppose that → ∞, up to a subsequence, as → 0. Now we consider the normalized sequence Due to linearity, V solves the original scaterring field problem with incident field V = / → 0 as → ∞. Following the proofs of Lemma 12 and parts (i) and (ii), the function = V ( )| Ω\ + −1 V ( )| is the unique solution of (24) and satisfies the Sommerfeld wave condition. By the construction of V , we obtain = 0 and therefore V → 0 weakly in 2 (Ω).
The Transmission Coefficient. After having the matrix in hand, our next step is to calculate the transfer coefficient . With the help of matrix , we map the values ( , ∇ ⋅ ⃗ ) at 2 = 0+ to the values ( , ∇ ⋅ ⃗ ) at 2 = −ℎ−; that is, (1 + , cos( )(−1 + )) 1 sin( ) will get mapped to ( , cos( )) 1 sin( ) . In other words, Here since we are only interested in the transmission coefficient , we eliminate the unknown . Now we follow a simple elimination technique shown in [3] and introduce two vectors V ∈ C 2 and ∈ C 2 by Since the left hand side of (70) is ⃗ V, multiplying it with V ⊥ will result in the elimination of from (70). This leads to By (73), we have determined the transmission coefficient which depends on wave number , height of the metallic structure ℎ, the aperture volume , effective material parameter , and the angle . We note that = √( / ), where is defined by the help of an eigenvalue problem in the metallic part Σ and we also notice that depends on the wave number by the relation = ( ). For a rather simple , the graph of | | 2 against the wave number is shown in figure 4 in [3].