Layered system with metamaterials

The layered system composed of metamaterial and vacuum layers is examined. We assume the metamaterial is the isotropic, homogeneous, dispersive and non-absorptive media. The permittivity and permeability of the metamaterial are equal and described by using a single Lorentz contribution. The electric Green's function is obtained for the case, when the permittivity and permeability of the metamaterial are equal -1. For this case we observe the no reflection effect. Also, the photonic band gap structure of the similar infinite layered system is described. For changing layers widths, the absence of the no reflection effect is shown.


Introduction
Metamaterials are known as negative index materials (NIMs) since for certain frequencies they have the negative refractive index [1]. By using NIMs, new covered surfaces and cloaking materials can be created [2], as well as superlenses with the resolving power, exceeding the diffraction limit [3]. Among systems with NIMs (NIM systems), layered NIM systems are widely known. The simplest two-layer NIM system is considered in [4]. The three-layer NIM system, which is the model for the superlens, is studied in [3,5,6]. The studies of the multilayer NIM systems (i.e., systems with the layers count greater than three) are presented in [7,8]. A number of NIM studies are dedicated to obtaining the Green's function which describes an electromagnetic field of a point source [4,6,9]. An electromagnetic field value at any point can be determined with the Green's function. The layered systems are considered also as one-dimensional photonic crystals (1DPCs) [10,11]. Numerous investigations of 1DPC were carried out in recent years [12,13]. But most of these ones consider systems without the frequency dispersion. The first goal of our work is to obtain expressions for the electric Green's function for a multilayer NIM system, composed of arbitrary finite number of layers filled with the metamaterial and vacuum. The magnetic Green's function can as well be obtained by the same way. The second goal of our work is to describe the photonic band gap (PBG) structure for the similar but infinite NIM system (photonic crystal). We assume the metamaterial is the isotropic, homogeneous, dispersive and non-absorptive media, the permittivity and permeability stand equal and are described by using a single Lorentz contribution [4,14]. We are interesting in the electric field in the case (NIM situation), when the permittivity and permeability of the metamaterial are equal -1 [4].

Finite NIM system
We study the NIM system composed of (N+M+1) parallel layers, where N, M ≥ 3 are natural odd integers (figure 1 . All even layers (as well as the zero layer) are Δ1 in width, and filled with a metamaterial. All odd layers are Δ2 in width, and filled with a vacuum. The last-to-left-side (with -M index) and last-to-right-side (with N index) layers are the half spaces unbounded along the direction of the 3 x axis and are empty (vacuum). The point source is located in the zero layer and defined by coordinates of the vector   ,, e e e , i.e., 31 0 y    . We assume the translation invariance along the plane of layer's surfaces. We consider the Maxwell's equations in a differential form ,, e e e ,  is the Hamilton operator,  is a cross product symbol as well as a numerical product symbol,  is an inner product symbol as well as a matrix product symbol. Also, we consider the auxiliary field equations where 0  and 0  are the electric and magnetic constants, which we set 00 1   for brevity [4],    e unit vectors forms the Cartesian basis. We consider the isotropic homogeneous metamaterial in layers. Therefore, the electric and magnetic permeabilities do not depend on coordinates of the x vector, i.e., ( , where U is the 33 unit matrix. Also, we assume that the metamaterial is the dispersive non-absorptive media. In that case, the susceptibilities consist of a sum of Lorentz contributions [14]. We deal with a single dispersive Lorentz contribution [4]. We assume that the permittivity and permeability of the metamaterial stand equal and where  ,   x y and the magnetic field function ( , ) t Hx can be obtained in a similar way. Therefore, below we consider only electric Green's function.

Electric Green's function
By using the Laplace (5) and Fourier (6) transforms for the Helmholtz equation (8) and the boundary conditions (4) we obtain the system of equations for the electric Green's function in every layer with transverse electric (TE) and transverse magnetic (TM) modes. We solve this system by using the recurrence relation approach, and obtain results in the NIM situation with ẑ   . The electric Green's function in the NIM situation is expressed in the following way: for   in the layers filled with the metamaterial, i.e., with  T  T  T  T  3  3  3  3  3  3 T  3  3  1  3  1  3  3  3 1( , , )   22  T  T  T  T  3  3  3  3  3 3 22 in the layers filled with a vacuum, i.e., with , 3, T  T  T  T  3  1  3  3  3  3 3 22 Note that expressions for Green's function have only one term responsible for propagation and no terms responsible for reflection, i.e., there is no reflection for the frequencies   , which correspond to ( ) ( ) 1 zz     . The magnetic Green's function can as well be obtained with reasoning similar to the presented above.
One notices that the Green's function as a resolvent kernel is defined for regular values of the spectral parameter only, i.e., outside the operator spectrum (see, e.g., [15]). It is, of course, valid for any non-real value. As for the real axis, one has the Green's function only at points which do not belong to the spectrum. Correspondingly, the obtained Green's function formula is valid only if the NIM frequencies   do not belong to the spectrum of the electric Helmholtz operator ( , ) e z Lx . In a limiting case of periodic 1DPC (when N and M tend to infinity) the Green's function at the real axis exists only in the spectral gaps.

Infinite NIM system
Now we examine the system similar to the studied above, but composed of infinite count of parallel layers. 12    is the period of the system and the system is a periodical 1DPC. We consider the , where c is the speed of light in vacuum. We use the similar to (5) Fourier transform with t time and consider the permittivity and permeability of the metamaterial as in (7) (11) By using the Fourier transforms (10) and (6) for the Helmholtz equation (11), the boundary conditions (4) and the Floquet-Bloch theorem [15,16] we obtain the following dispersion equation: where  is a yet undefined wave vector, called the Bloch wave vector,  (7). Therefore, we have the identical PBG structure for TE and TM modes.

Solutions
The Helmholtz equation has non-trivial solutions when the dispersion equation (12) holds true. By using the Fourier transforms (10) and (6), the boundary conditions (4) and the Floquet-Bloch theorem [16,17], we obtain the following solutions in metamaterial layers (subscript 1) and vacuum layers (subscript 2) for TM mode: where ( , ) C  is an unknown function, which can be obtained from some initial conditions. For the TE mode we obtain the same expressions (13)-(14) but with swapped and for   as follows:   where ()     is given by equation (9). Note that the solution (15), as well as the solutions (16)-(18), has only one term responsible for propagation and no terms responsible for reflection, i.e., there is no reflection at the NIM frequency  . But this effect is observed only when the dispersion equation (12) holds true, i.e., only for permitted bands (at the spectrum).

Numerical results
To study the PBG structure of the considered 1DPC we use a numerical approach. We consider the equality (12) and seek for which ω and κ values it holds true with any θ value, which belongs to the    normalized frequency has values from 1.410 6 till 3.210 5 m -1 ) and for κ values from 0 till 0.8 ηm -1 (i.e., from 0 till 810 5 m -1 ). We use four following cases of the 1  and 2  parameters: 1.   In the case 1, we observe the NIM frequency belonging to a permitted band for all κ values. Therefore, there is no reflection for any κ value in the NIM situation. With changing the 1  or 2  parameters, i.e., in the cases 2, 3, and 4, we observe the faster narrowing of permitted bands with the increasing of κ value, than it is in the case 1. In cases 3 and 4 for small κ value permitted bands shift to the zero ω value. The contained the NIM frequency permitted band splits into two bands. As in the case 2, one observes the absence of reflection in the NIM situation only for a part of κ values set.

Conclusion
In this paper, we solved the problem of obtaining the electric Green's function for the layered NIM system with the point source was located inside the certain metamaterial layer. We assumed that the permittivity and permeability of the metamaterial stand equal and are defined with the single Lorentz contribution. In the NIM situation, we obtained relations for the Fourier transformed electric Green's function. The magnetic Green's function can as well be obtained with reasoning similar to the presented above. The obtained formulas are symmetric, relative to the position of the point source. This fact shows the correlation with the physical conception of the electromagnetic field propagated into the system composed of isotropic homogeneous layers. The system can be composed of arbitrary finite number of layers. This fact allows us to use the considered system as a model for simulation or engineering of the real objects, such as superlens systems and multilayer NIM coverings. Besides, the reflection in the NIM situation is absent. This fact was discussed earlier [4,6] and holds true only if the NIM frequency does not belong to the spectrum of the electric Helmholtz operator. For a limiting case, when we deal with infinite periodic NIM system, the obtained formulas correspond to gaps. Also, we solved the problem of describing the photonic band gap structure of the similar infinite layered NIM system (1DPC). In the permitted bands we obtained expressions for the scalar field functions, which contained only one term responsible for propagation and no terms responsible for reflection, i.e., there was no reflection at the NIM frequency in permitted bands. By the numerical approach, we observed the NIM frequency belonging to a permitted band, i.e., in the NIM situation there was no reflection for any direction. With changing the layers widths the permitted band contained the NIM frequency was splitted into two bands, i.e., for the NIM frequency there was the absence of reflection not for any direction.