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.
Angle of Incidence Magnetic Fraction Metamaterial Microwave Band Reflection Transmission Superlattices Visible Band1. Introduction
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 [1] -[3] . Antireflection coatings require a particular refractive index and quarter wavelength thickness where it operates by overcoming the mis-match between intrinsic impedances of two me- dia. This approach is scalable over a wide spectral range from microwave to far infrared [4] . There have been a few efforts to develop antireflection coatings at THz frequencies by using dielectric meta-materials or left- handed materials (LHM). LHM have both negative permittivity and permeability and consequently have nega- tive index of refraction [5] [6] and are able to match media impedances. The theory of antireflection coating is examined by many authors [7] [8] . The matrix method [9] [10] is usually employed for calculation of reflection coefficient. Chen et al. [4] have presented approach of metamaterial antireflection.
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. [3] have made a theoretical study of the antiref- lection coatings on silicon solar cells. Cory et al. [11] have analyzed the reflection and transmission characteris- tics of a multilayered structure consisting of metamaterials and dielectric slabs. In this paper, I investigate the reflection and transmission properties of a superlattices (LANS)-metamaterial (LHM)-dielectric multilayered structure. Since antireflection coating is formed by two slabs of the same thickness and of opposite refractive in- dices, a pair of LHM and LANS materials is situated between two semi-infinite dielectric media which are con- sidered right handed materials (RHM) of positive refractive index. The superlattice is a lateral anti-ferromag- netic/nonmagnetic (LANS) such as. LANS are described with an effective medium theory. Such description is valid when the wave lengths of the excitations are much longer than the superlattice period where, where k is the magnitude of the wave vector and L = L1 + L2, is the period of the superlattice, L1 and L2 are the thickness of the anti-ferromagnetic layers and non-magnetic layers ,respectively [5] [12] . and. In the theory the electric and magnetic fields of the incident waves are determined in each region by Maxwell’s equations. Then Snell’s law is applied and the boundary conditions are imposed at each interface to obtain the reflection and transmission coefficients. The reflected and transmitted powers of the structure are presented in terms of these coefficients. In the numerical analysis, a recursive method [13] is used to calculate the reflected, transmitted powers as a function of incident wavelength, angle of incidence, layer thickness and magnetic fraction of (LANS). The calculations are performed for electromagnetic radiations in both the visible and microwave bands for a single wavelength by selecting the optimum refractive indices of LHM and RHM in both bands. The suitable LHM in each band is selected. The conservation law of energy is checked and satisfied.
2. Theory
Consider LHM and LANS of electric permittivity and magnetic permeability and respec- tively embedded between two semi-infinite dielectric media of permeability and permittivity as and. A perpendicular polarized wave is incident on the structure at Y = 0 at angle relative to the normal to the boundary as depicted in Figure 1.
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 [5] [12] which is described as a single effective medium, can be written as:
with
where the expressions of and are [5] [12] :
with
represents an isotropy field, the exchange field, and the gyromagnetic ratio. m0 is the sublattice magnetization. The magnetic field of the superlattice is H0. The effective dielectric function of (LANS) is ex- pressed by [5] [13] as:
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 [14] , [15] :
where and are the amplitude of forward and backward travelling waves in the region of order
, is the wave vector inside the material and is the refractive index of it.
In Superlattice (LANS) Region
The curl Maxwell’s equations are [12] :
By these equations
where is the Voigt permeability.
By Maxwell’s equation, the magnetic field in the other regions of order is:
Matching the boundary conditions at each layer interface, where at, and and so on yields six equations with six unknown parameters as:
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 [16] :
For the other interfaces the Fresnel coefficients are:
where,
The reflection and transmission coefficients and respectively of the structure are [17] :
The reflectance and transmittance of the structure are given by:
where and are the conjugate of and respectively.
The law of conservation of energy is [17] :
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 [18]
where is the angular frequency, is the lattice permittivity, is the effective plasma frequency and is the electric damping factor.
For microwave region, I employ a dispersive LHM with and appeared in [15] [19] as:
where and are the electric and magnetic plasma frequencies. and are the electric and magnetic resonance frequencies. and are the scaling filling parameters, and are the electric and magnetic dissipation factors.
3. Numerical Results and Discussion
The parameters were used in carrying out the numerical calculations are [12] : the applied field H0 = 0.8 kG, , , , and for antiferromagnetic layers, for the non-magnetic layers. The relative permeability of the dielectrics is, the re- fractive index of the dielectrics and.
3.1. In Visible Spectral Band
The parameters were used are [18] :, ,.
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 n2 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 Figure 2. It displays the reflected, transmitted power as a function of the normal incident wavelength when the dielectric refractive index nd 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 nd 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 (n3 = 2.49 at λ = 600 nm and magnetic fraction). changes very slightly with frequency. For these indices, one minimum appears around λ = 600 nm at which the refractive in- dices of LHM, and LANS, dielectrics are approximately closed to each other and opposite in signs which con- firm that high transmitted power can be achieved if, , [11] . These conditions which
Figure 2
(a) The reflected, (b) transmitted power as a function of the normal incident wavelength when the dielectric refractive index nd changes as nd =2.34, 4.86, 6.58, f1 = 0.7, d = 250 nm
leads to r = 0 and T = 1 around λ = 600 nm are:, , , , , ,. Such AR coating systems are used in photodiodes (LASER) and other optoelectronic devices which need a minimum reflectance at a single wave- length. Figure 3 illustrates the reflectance spectra over a wide range of incidence angle for λ of values (600, 700, 800) nm. In the range (θ = 0˚, 27˚) the minimum reflectance is achieved at λ = 600 nm. The implementation of LANS layer adjacent to LHM layer dramatically reduces the reflection and greatly enhances the transmission near a specifically frequency at incidence angles (θ = 0˚, 27˚). By wavelength increase to the value of 800 nm, minimum reflectance is observed at higher incidence angle of value 57˚. The effect of the magnetic fraction on the reflectance and transmittance is described in Figure 4. As increases to the values of (0.1, 0.5, 0.9), the refractive index of LANS decreases to the values of (2.78, 2.59, 2.39) while of LHM increases to the value of (−2.83 to −4.486) in the wavelength range (600 - 850) nm which leads to reflectance increase. It is worth to note that, R = 0 and T = 1 at λ = 600 nm, f1 = 0.9. This is because, ,. Figure 5 demonstrates the effect of thickness d of LHM on the reflectance and transmittance at normal incidence of λ = 600 nm, 800 nm. The slab thickness is changed from 50 nm to 500 nm. It is noticed that reflec- tance and transmittance changes periodically with thickness. Besides that, more transmittance is realized at λ = 800 nm where maximum R is 0.03 while minimum T = 0.97 of the incident power. It is shown that the selected thickness (d = 250 nm) is appropriate for achieving zero reflectance at λ = 600 nm.
3.2. In Microwave Spectral Band
For dispersive LHM with and have parameters appeared in [15] as:, , , , ,. For this LHM, the frequency range in which and are negative extends from 10.4 up to 11.5 GHz with corresponding wavelength extends from 26 up to 28 mm. The thickness of each of LHM and dielectrics slabs is equal to one half-wavelength long at the operating frequency. Reflectance and transmittance are calculated nu- merically as stated above. The law of conservation of energy is [20] :
where is the loss power due to losses in LHM. Since in the structure arrangement shown in Figure 1, there is no effect of magnetic fraction of LANS on the reflected power and there is very small power loss of LHM which can’t be displayed in a figure, I rearranged the structure to be LANS in Region 1 instead of the
The reflected power as a function of the angle of incidence for dif- ferent wavelength λ = 600, 700, 800 nm, nd =4.86, f1 = 0.7, d = 250 nm
Figure 4
(a) The reflected, (b) transmitted power versus the normal incident wavelength when the magnetic fraction f1 changes as f1 = 0.1, 0.5 ,0.9, nd = 4.86, d = 250 nm
dielectric which will be in Region 3. The reflection, transmission coefficients are rewritten as:
Figure 6 displays the reflectance, transmittance and power loss as a function of the incident wavelength for many values of magnetic fraction
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 f1 = 0.1 at θ = 30˚
4. Conclusions
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 and of LHM and that of LANS is taken into account. It has been shown that, the frequency-dependent refractive index of both LHM and LANS plays an important role in the variation of the reflection coefficients of the structure. Low reflection can be achieved for both visible and microwave rays by choosing the proper indicies of the materials constiuted the structure. For incident visible rays, LANS is located as shown in Figure 1 r = 0 and then T = 1 is attained at a sigle wavelength value of 600 nm with magnetic fraction of LANS of value 0.9, incidence angle of value 0˚ or 27˚. For incident microwave rays, LANS is located in Region 1 and the dielectric in Region 3. r = 0 and then T = 1 are achieved at wavelength
Figure 5
(a) The reflected, (b) transmitted power versus the layer thickness d at normal incident wavelength of λ = 600 nm, 800 nm, nd =4.86, f1 = 0.7
Figure 6
(a) The reflected, (b) transmitted, (c) loss power versus the incident wavelength when the magnetic fraction f1 changes as f1 = 0.1, 0.5, 0.9, nd = 4.86, θ = 30˚, d = 14 mm, γe = γm =0.1
λ = 0.026 to 0.272 m with f1 = 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.
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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 f1 = 0.1 at θ = 30˚