Effect of negative permittivity and permeability on the transmission of electromagnetic waves through a structure containing left-handed material

We investigate the characteristics of electromagnetic wave reflection and transmission by multilayered structures consisting of a pair of left-handed material (LHM) and dielectric slabs inserted between two semi-infinite dielectric media. The theoretical aspect is based on Maxwell's equations and matching the boundary conditions for the electric and magnetic fields of the incident waves at each layer interface. We calculate the reflected and transmitted powers of the multilayered structure taking into account the widths of the slabs and the frequency dependence of permittivity and permeability of the LHM. The obtained results satisfy the law of conservation of energy. We show that if the semi-infinite dielectric media have the same refractive index and the slabs have the same width, then the reflected (and transmitted) powers can be minimized (and maximized) and the powers-frequency curves show no ripple. On the other hand if the semi-infinite dielectric media have different values of refractive indices and the slabs have different widths, then under certain conditions the situation of minimum and maximum values of the mentioned powers will be reversed.


INTRODUCTION
Metamaterials (sometimes termed left-handed materials (LHMs)) are materials whose permittivity  and permeability  are both negative and consequently have negative index of refraction.These materials are artificial and theoretically discussed first by Veselago [1] over 40 years ago.The first realization of such materials, consisting of split-ring resenators (SRRs) and continuous wires, was first introduced by Pendry [2,3].Regular materials are materials whose  and  are both positive and termed right handed materials (RHMs).R. A. Shelby et al. [4] have studied negative refraction in LHMs.I. V. Shadrivov [5] has investigated nonlinear guided waves in LHMs.N. Garcia et al. [6] have shown that LHMs don't make a perfect lens.Kong [7] has provided a general formulation for the electromagnetic wave interaction with stratified metamaterial structures.M. M. Shabat et al [8] have discussed Nonlinear TE surface waves in a left-handed material and magnetic super lattice waveguide structure.I. Kourakis et al. [9] have investigated a nonlinear propagation of electromagnetic waves in negative-refraction index LHM.H. Cory et al. [10] and C. Sabah et al. [11] have estimated high reflection coatings of multilayered structure.Oraizi et al. [12] have obtained a zero reflection from multilayered metamaterial structures.
In this paper we consider a structure consisting of LHM and dielectric slabs inserted between two semiinfinite dielectric media.A plane polarized wave is obliquely incident on it.We use Maxwell's equations and match the boundary conditions for the electric and magnetic fields of the incident waves at each layer interface.Then we solve the obtained equations for the unknown parameters to calculate the reflection and transmission coefficients.We take into account the frequency dependence of permittivity and permeability of the LHM (in contradict with [10,11]), widths of the slabs, refractive indices of the media and angle of incidence of the incident waves.Maximum and minimum transmitted (minimum and maximum reflected) powers of the considered structure are proposed.The numerical results are in agreement with the law of conservation of energy given by [10,13].It is found that the numerical results of Figure 3 (the case a ≠ b) is similar to Figure 4(b) obtained by [14], this is another evidence for validity of the performed computations.
The paper is organized as follows: our theory is formulated in section 2. Numerical results and applications are described in section 3. Our conclusions are presented in section 4.

THEORY
Consider four regions each with permittivity i  and permeability i  , where i represents the region order.
 ).A polarized plane wave in Region 1 incident on the plane z = 0 at some angle  relative to the normal to the boundary (see Figure 1).
The electric field of the incident wave in Region 1 can be written as [7][8][9][10]: To find the corresponding magnetic field 1 H , we start with Maxwell's equation [15]: Figure 1.Wave propagation through a structure consisting of a pair of dielectric and metamaterial embedded between two dielectric semi-infinite media.
The electric and magnetic fields in Regions 2, 3 and 4 can be written in the same manner as follows: where is the wave vector inside the material and Matching the boundary conditions for E and H fields at each layer interface, that is at z = 0, 1 2 Letting A = 1 and solving these six equations for the unknown parameters enable us to calculate the reflection and transmission coefficients B and J respectively [10,15].The reflected power R equal to the reflection coefficient B times its complex conjugate and the transmitted power T equal to the transmission coefficient J times its complex conjugate [10,15], leading to; The law of conservation of energy is given by [10,13]: where:

NUMERICAL RESULTS AND
ploy a non-d

APPLICATIONS
For the metamaterial in region 3 we will em ispersive metamaterial with  and  given by [2,3,12,[15][16][17]:   (10)(11)(12)(13)(14)(15).These equations are solved for the parameters B and J. Then the reflected and transmitted powers R and T can be calculated.The transmitted power is plotted as a function of  under different conditions as follows: The dependence of 3  on  for the metamaterial in Region 3 taken into account [2][3][4][5][15][16][17], it is co is nsidered by [10,11]  In this case the left hand side of (18) is equal to 0.999 999 99 which verifies the law of conservation of energy.
It can be realized from Figure 3 that this law is satisfied by the performed computations.The same procedure can be applied to other computations in this paper.Low reflected and high transmitted powers can be achieved if n n  , ].In this case R = 0 (minimum) and T = 1 (maximum) for any frequency an ny f incidence (Figure 4).If  aterial and dielectric slabs, a high-reflected and low-transmitted powers are achieved, for which the dependence of R and T on frequency and on the angle of incidence is consequently diminished (Figure 7).Note that the maximum value of T is 0.09 at 0 o .This value is smaller by a factor of 11 than that obtained in Figure 4. Figure 7 is different from that obned by [10,11] in the fact that, they had used a LHM with properties invariant with frequency.In our paper, the properties of the LHM in Region 3 depends on frequency (this can be realized from Eqs.20 and 21).

CONCLUSIONS
n this case by a udiciou ination of metam tai ctromagnetic waves through sisting of a pair of LHM and di The propagation of ele multilayered structures con electric slabs inserted between semi-infinite dielectric media has been studied.The followed method has been based on Maxwell's equations and matching the boundary conditions for the electric and magnetic fields at each interface layer.The frequency dependence of  and  of the LHM has been taken into account.The reflected and transmitted powers have been calculated nume cally.The dependence of them on various parameters has been studied.Low and high transmitted powers have been achieved for any frequency and for any angle of incidence.The law of conservation of energy at z = a + d, E 3 = E 4 and 3 4  H H .This yields the following equations [10,12-15]:

3 
 in addition to iz k are used in

3 .
The difference between the two cases can be noticed Figure2.The thickness of the slabs is taken to be the same a = d,  ,  is kept constant.In this case the structure's reflected and transmitted powers variation with frequency are smooth and show no ripples as shown in Figure To check the validity of computations for the case a ≠ b as an example, let ω = 2π11 GHz, then: μ 3 = -0.641635 071, ε 3 = -3.389624 4113, k 4z /k 1z = 1, B = -0.570959 7786 + 2.958 090 86i, J = -0.743521 1938 + 0.184 524 7865i, R = 0.413 126 8376, T = 0.586 873 1625.

n n  and a = d [ 10 d
for a angle o

Figure 2 .
Figure 2. Transmitted power variation with frequency.Two cases are taken into account: μ 3 is a function of ω, μ 3 is a constant (-μ 0 ). ri

Figure 3 .
Figure 3.The reflected and Transm θ is kept constant for two cases with respect to the widths a and d itted powers variation with f ency when n 1 = n 4 , n 2 = |n 3 | and of the slabs: a = d, a ≠ d.