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Qualitative Theory of Differential Equations, Difference Equations, and Dynamic Equations on Time Scales

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Volume 2016 |Article ID 7436136 | https://doi.org/10.1155/2016/7436136

Hari Shankar Mahato, "Upscaling of Helmholtz Equation Originating in Transmission through Metallic Gratings in Metamaterials", The Scientific World Journal, vol. 2016, Article ID 7436136, 14 pages, 2016. https://doi.org/10.1155/2016/7436136

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

Academic Editor: Tongxing Li
Received28 Apr 2016
Accepted26 Jun 2016
Published21 Sep 2016

Abstract

We investigate the transmission properties of a metallic layer with narrow slits. We consider (time-harmonic) Maxwell’s equations in the -parallel case with a fixed incident wavelength. We denote as the typical size of the complex structure and obtain the effective equations by letting . 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.

1. 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 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 and permeability . We also obtain the tranmission coefficient in terms of incident wave number and incident angle . We will see that, for lossless materials with (real) negative permittivity , perfect transmission 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 [1013].

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 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 of (see Section 1.5 for details) and the limit function in [3] are totally different. In [3], the authors worked with a rectangular metallic subpart of type and therefore by defining a suitable test function, they have shown that the first component of vanishes and they obtain . This is clearly not the case in this paper as the metallic subpart is chosen to be sinusoidal along - due to the geometry of the metallic structure given by Figure 1. This will lead to nonvanishing componenets of and we end up having a different compared to that in [3] and hence, we will obtain a different upscaled equation. Also no explicit representation of can be obtained due to the geometry of .

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 . Let the metallic structure remain unchanged towards -direction and the metallic field, denoted by , is parellel to ; that is, , where .

The heterogenoeus domain has a metallic structure of finite length and finite height in , and the slits (vacuum) are repeated periodically with a small period , 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 has large absolute values, we assume that it depends on and consider . We obtain nontrivial effects due to plasmonic resonance for , compare [3, 13]. If denotes the matallic part in , we setwhere . Due to ohmic losses inside the metal, is always assumed to be positive in a physical system which means we always take and . A material is called a lossless material if . Our particular interest is to study a lossless material with negative relative permittivity; that is, and . For such transverse evanescent modes will be generated in the metal. Since and , 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 given in [3].

1.2. Geometry

Let be a small scale parameter and be the domain under investigation which is bounded in . Let be the representative unit cell in and be an open set in such that and . Let us choose in such a way that it follows a sinusoidal profile along -; that is,Keeping physics of the problem in mind, 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 and relative metal volume is , where . We define . We assume that the compact rectangle contains number of small rectangles of type , that is, of width and height , which include the -scaled versions of the metallic part (cf. Figures 2 and 3), where each is of width and height . The collection of these small -scaled versions of the metallic part is the metallic domain and assume that the two-dimensional heterogeneous metallic structure introduced in Figure 1, denoted by and parellel to -axis, is contained in the closure of the set with ; that is;see Figure 3.

As , . Due to nondimensionalization, we are, however, only interested in .

1.3. Function Spaces

Let and be such that . Assume that and ; then as usual and denote the Lebesgue and Sobolev spaces with their usual norms and they are denoted by and . For the sake of clarity if , thenand if , thenwhere is a multi-index, , and . Similarly, , , and are the Hölder, real, and complex interpolation spaces, respectively, endowed with their standard norms; for definition confer [19, 20]. denotes the set of all Y-periodic -times continuously differentiable functions in for . In particular, is the space of all the Y-periodic continuous function in . The -spaces are as usual equipped with their maximum norm whereas the space of all continuous functions is furnished with supremum norm, compare in [19].

1.4. Two-Scale Convergence

Definition 1. A sequence of functions in is said to be two-scale convergent to a limit iffor all .
By ,, and we denote the two-scale, weak, and strong convergence of a sequence, respectively. Finally, denotes the time interval.

Lemma 2 (cf. [21]). For every bounded sequence in there exists a subsequence (still denoted by same symbol) and such that .

Lemma 3 (cf. [21]). Let be strongly convergent to , and then , where .

Lemma 4 (cf. [21]). Let be a sequence in such that in . Then and there exists a subsequence , still denoted by same symbol, and such that .

Lemma 5. Let be a bounded sequence of functions in such that and are bounded in . Then there exists some functions such that , , and .

Proof. (i) Since and are bounded sequence of functions in and , respectively, then there exists and such that and as . This means that, for the sequence , we havefor all . We integrate by parts the l.h.s, and which givesAnd it follows from (7) and (8) that .
(ii) To prove the second part of the lemma, let us choose , where and . Note that the boundedness of implies the boundedness of in and hence, by part (i) there exists such that and . Now let us assume that , and then by definitionWe integrate by parts the l.h.s.; thenWe compare (9) and (10) which leads us toSince is independent of and , from (11) it follows that must be the gradient of some function such that ; that is, . This completes the proof.

1.5. 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 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 arewith fixed positive real constants , and denoting the frequency of the incident waves and the permeability and the permittivity of vacuum, respectively. We postulate that all the quantities are -independent and the polarized magnetic field is given by , where . By orthogonal property of and , we have . Then (12) reduce toBy (13), a straightforward calculation yieldswhere we have set . We define the coefficient which can have a negative real part and that it vanishes in the metal as . Thus we have the desired Helmholtz equation which we will study in this paper and is given below. We study solutions ofwhere the coefficient is given byThe set describes the complex geometry of the metallic inclusion in ; 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 while the second one concerns the scattering problem. In other words we consider (15) in whole of . For a given incident wave , which solves in , we take the Sommerfeld condition as the boundary condition which says that the scattered field satisfiesfor , uniformly in the angle variable.

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 , system (18) is nothing but (12) itself.

Theorem 8 (upscaled equations). Let the matallic geometry be given by (Figure 3) on a domain and let the coefficient be as in (16). On we assume that either or . Let be the sequence of solutions of (15) such that in for . We define as the functionwhere is defined by (36). Then the function . The field converges weakly to some in which is given byMoreover, the limit functions satisfy the systemwhere

By applying Theorem 8 for with a large radius , 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 .

Theorem 9 (effective scattering problem). Let the metallic gratings be given by (Figure 1) and the coefficient be as in (16). Assume further that is an incident wave solving the free space equation on and is the unique sequence of solutions to (15) such that satisfies (17) and that the solution sequence satisfies the uniform boundThen strongly in with uniform convergence for all derivatives on any compact subset of . The effective field is determined as the unique solution of the upscaled equationwith (17) for the scattered field .

1.5.1. Interface Conditions

The homogenized equation (24) should be understood in the sense of distributions on the whole of . The exterior field for every large radius ; hence its trace on from outside, denoted by , is a well-defined element of . Note that as belongs to , the function is an element of . 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 is of class .

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 aswith the transmission (interface) conditionswhere is defined in (36).

2. Derivation of the Effective Model

2.1. A Priori Estimates

Lemma 10. For an with , let be defined as in (16). Then there exists a such thatwhere is independent of .

Proof. Let (if with , then ). For an arbitrary small , let us define such that . There exists a constant such that For , we have

Lemma 11 (gradient estimate). Suppose that the solution of (15) is a bounded sequence in ; that is, . Then for every compactly contained subdomain , the following estimate holds:where is independent of the scale parameter .

Proof. Since , there exists a subdomain such that . Without loss of generality, let us assume that and take a cut-off function , where on . We test (15) with , where is the complex conjugate of . This givesWe employ Lemma 10. For a , we multiply (31) by and equate its imaginary part and rearrange the factors of the second integrand which will yieldwhere 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 -boundedness assumption on whereas the second integral on the r.h.s. is bounded by the boundedness of and . Using the fact that on , we have , where is independent of and .

2.2. An Eigenvalue Problem in the Unit Cell

Let us consider the eigenvalues of the problemand we denote the associated normalized eigenfunctions in , so that is an orthonormal basis of . Since with , satisfies the conditionWe setLet 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 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 in the sense of two-scale convergence, compare [15]. We notice that the geometry is not only periodic in the -direction but it is also periodic with respect to to the cell . The metal part in the cell is given by ; see Figure 2.

We recall that the sequnce is weakly convergent to . We define a function aswhere is a -periodic function defined in (35). We have defined in such a way that, for every , there holds . We will show in next theorem that as .

Lemma 12 (two-scale limit). Let , weakly converging to in , be a sequence of solutions of (15). Then for the function defined in (38) it holds that .
Outside of , the strong convergence holds in . More precisely, together with all its derivatives converges uniformly on every compact subset .

Proof. We divide the proof into three steps.
(i) From the assumption on and the estimate (30), the sequences and are bounded in . Then there exists such that, up to a subsequence, and as . As a periodic function, and can be extended by periodicity to all . This shows that which implies that belongs to , in particular, in and has a trace on . In other words, does not jump accross by trace theorem (cf. [19, theorem  5.5.1]).
Next, we investigate the coefficient on the set . From (30), it follows that which implies strongly in . Since strong convergence implies the two-scale convergence, by localisation Lemma (cf. [3]) the two-scale limit vanishes a.e. in and in . Due to , it implies that the function is constant in and for ; and it is constant everywhere for . We use this -independence to define a function asWe note that, at this stage of the proof, and are defined as the two-scale limit of and by (39), respectively.
(ii) Characterisation of Two-Scale Limit for . We claim that, for a.e. , the function , which belongs to , solves the linear boundary value problemwhere (40a), (40b), and (40c) hold in the distributional sense in . To verify this, we choose , where and a periodic function on with . Using as the test function in (15), we obtainPassing the two-scale limit as Since was chosen arbitrarily, (40a), (40b), and (40c) hold. For every , we write , where and is -periodic. Clearly, satisfies the equationThen, for , as shown in Section 2.2, we express uniquely in terms of the orthonormal basis . Note that if the condition (34) is violated, the equation in has no solution and we are led to .
Therefore, to sum up, we obtain the two-scale limit as
, provided (34) holds. Consequently, for , the weak limit satisfiesTherefore the two-scale limit is given bywhere .
(iii) Strong Convergence Outside of R. We know that holds for a.e. and for all . Moreover, by the assumption on and estimate (30), we have . This then implies that , up to a subsequence, is strongly convergent to in 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 .

With the help of Lemma 12, we can completely determine the two-scale limit of the sequence if we know the function which is defined in (39). Now we collect the properties of , its weak limit , and its two-scale limit .

Proposition 13. Let be as in Lemma 12 and be given by (39). For , we suppose that in . Then is characterized as follows:(i)The sequence converges in the sense of two scales to which is given by(ii)The limit and it holds:

Remark 14. We would like to point out a major difference in our and the limit function in [3]. In [3], the authors worked with a rectangular metallic subpart of type and therefore by defining a suitable test function, they have shown that the first component of vanishes and they obtain . This is clearly not the case in this paper as the metallic subpart is chosen to be sinusoidal along - due to geometry of the metallic structure given by Figure 1. This will lead to nonvanishing componenets of and we end up having a different compared to that in [3] and hence, we will obtain a different upscaled equation.

Proof. By (30), it follows that is bounded in which implies that up to a subsequence two-scale converges to some . The weak limit would then be given as .
The Field outside of . For , . Then by Lemma 12, uniformly on compact subsets of . This leads toThe Field in the Metal Part of . We note that in and in ; therefore (30) gives and . This implies and by [21, theorem  17], we have a.e. in . Moreover, for a.e. .
Divergence of . Due to boundedness assumption on , by (18) we have . For and , we test (18) by which gives Since is arbitrary, for a.e. which implies for in distributional sense. This shows that is independent of ; that is, , some function in only.
Next we determine the relation between and as shown in [3]. We define , is Y-periodic, and . We choose a test function , where and . We use and as . ThenSince and is nonvanishing in and by (39) it implies that , all these lead toTherefore, for and for reminder for .
Proof of (i). To conclude this part, the arguments rely on that of [3]. We consider . We intend to show that