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The characteristics of electromagnetic wave reflection and transmission by multilayered structures consisting of a pair of left-handed material (LHM) and superlattices (LANS) slabs inserted between two semi-infinite dielectric media are investigated for photovoltaic and solar energy applications. Maxwell’s equations are used to determine the electric and magnetic fields of the perpendicular polarized wave incident at each layer. Snell’s law is applied and the boundary conditions are imposed at each layer interface to calculate the reflected and transmitted coefficients of the structure. The reflected, transmitted powers are determined using these coefficients by a recursive method. The reflected and transmitted powers are computed in both visible and microwave spectral band with the appropriate LHM for each band and appropriate location of LANS in the structure. They are illustrated as a function of the incident wavelength, angle of incidence, magnetic fraction of LANS and thickness of the slabs with the emphasis on the appropriate refractive indices. I found that, zero reflectance and maximum transmittance of the incident powers are achieved for visible spectral band at a single frequency if LHM and LANS have the same refractive index of opposite signs with the same width and more magnetic material of LANS while the reflected power is zero for less magnetic material of LANS in the microwave spectral band which realizes antireflection coating in this structure.

In many applications, reflection is undesirable and causes insertion losses, for example. It is well known that the application of one or more antireflection coating (ARC) layer on the front surface of the photovoltaic cells and optoelectronic devices (Lasers, IR diodes, etc.) reduces the amounted reflection of the incident light, which im- proves the device performance [

It reduces the reflection and enhances transmission near a specifically frequency over a wide range of inci- dence angles for both TE and TM polarizations, Bouhafs et al. [_{1} + L_{2}, is the period of the superlattice, L_{1 }and L_{2 }are the thickness of the anti-ferromagnetic layers and non-magnetic layers ,respectively [

Consider LHM and LANS of electric permittivity and magnetic permeability

Wave propagation through a structure consisting of LHM and LANS materials inserted between two semi-infinite dielectric media

Introducing the effective medium theory, the magnetic permeability of the LANS [

with

where the expressions of

with

_{0} is the sublattice magnetization. The magnetic field of the superlattice is H_{0}. The effective dielectric function of (LANS) is ex- pressed by [

The electric and magnetic field vectors for TE waves propagating along x-axis with angular frequency ω are defined as:

The electric field in each region is [

where

The curl Maxwell’s equations are [

By these equations

where

By Maxwell’s equation

Matching the boundary conditions at each layer interface, where at

According to Snell’s law

with

For TE polarized light, at the first interface, the Fresnel coefficient (interface reflection and transmission (r, t) respectively are given by [

For the other interfaces the Fresnel coefficients are:

where

The reflection and transmission coefficients

The reflectance

where

The law of conservation of energy is [

In this work, two cases of LHM are considered. The first when the incident electromagnetic waves in the visi- ble spectral band and other one in microwave band. The frequency―dependent permittivity of LHM in the visi- ble band is described by Drude medium model as [

where

For microwave region, I employ a dispersive LHM with

where

The parameters were used in carrying out the numerical calculations are [_{0} = 0.8 kG,

The parameters were used are [

The relative permeability of LHM is assumed to be −1. The thickness of each slab is assumed to be one-half long of the central wavelength. The reflected, transmitted power of the structure is calculated as a function of wavelength of the incident waves, angle of incidence, layer thickness and magnetic fraction of LANS. Accord- ing to (16) the real part of refractive index of LHM is negative in the wavelength range of (500, 600, 700, 1000) nm where the real part of n_{2} of values (−1.023, −2.34, −3.27, to −5.596) where the damping factor of LHM in this region is ignored and no energy loss is displayed. The central wavelength is assumed to be 600 nm. This choice is based on the spectral stability of the coating and for low reflectance. Stability means that the low-ref- lectance spectrum changes very slightly with refractive index variations as shown by _{d} changes to the values of (2.34, 4.86, 6.58). It shows maximum reflectance R of value <0.25 and minimum transmittance T of value >0.75 at n_{d} of value 2.34 over a wide wavelength range (λ = 500 - 1000 nm). The re- fractive indices of LHM are (−2.34, −4.86, −6.58) at incident wavelength λ of values (600, 900, 1140) nm re- spectively and that of LANS is (n_{3} = 2.49 at λ = 600 nm and magnetic fraction

(a) The reflected, (b) transmitted power as a function of the normal incident wavelength when the dielectric refractive index n_{d} changes as n_{d} =2.34, 4.86, 6.58, f_{1} = 0.7, d = 250 nm

leads to r = 0 and T = 1 around λ = 600 nm are:_{1} = 0.9. This is because

For dispersive LHM with

where

The reflected power as a function of the angle of incidence for dif- ferent wavelength λ = 600, 700, 800 nm, n_{d} =4.86, f_{1} = 0.7, d = 250 nm

(a) The reflected, (b) transmitted power versus the normal incident wavelength when the magnetic fraction f_{1} changes as f_{1} = 0.1, 0.5 ,0.9, n_{d} = 4.86, d = 250 nm

dielectric which will be in Region 3. The reflection, transmission coefficients are rewritten as:

The operating wavelength is assumed to be 0.027 m which is included in the frequency range in which and are simultaneously negative. As increases to the values of (0.1, 0.5, 0.9) of LANS decreases to the values of (2.78, 2.59, 2.39) respectively, of LHM changes to the values () to () in the wavelength range of 0.026 to 0.272 m. R = 0 and t = 1 and zero power dissipation are attained at wavelength range λ = 0.026 - 0.272 m and f_{1} = 0.1 at θ = 30˚

The transmission and reflection of perpindicular polarized waves by a multilayered structure consisting of a pair of LHM and LANS materials embedded between two semi-infinite dielectrics media have been studied in both visible and microwave spectral bands with the appropiate LHM and appropiate location of LANS in the structure. The frequency dependence of

(a) The reflected, (b) transmitted power versus the layer thickness d at normal incident wavelength of λ = 600 nm, 800 nm, n_{d} =4.86, f_{1} = 0.7

(a) The reflected, (b) transmitted, (c) loss power versus the incident wavelength when the magnetic fraction f_{1} changes as f_{1} = 0.1, 0.5, 0.9, n_{d} = 4.86, θ = 30˚, d = 14 mm, γ_{e} = γ_{m} =0.1

λ = 0.026 to 0.272 m with f_{1} = 0.1 at θ = 30˚. The implementation of LANS adjacent to LHM layer dramatically reduces the reflection and greatly enhances the transmission near a specifically frequency. The law of conserva- tion of energy has been satisfied by the obtained results. The obtained results may be used to refine the un- derstanding of any related applications that may be modeled requiring controlling of reflected and transmitted powers as photovoltaic cells and optoelectronic devices.