Antireflection Coating at Metamaterial Waveguide Structures for Solar Energy Applications

Available online at www.sciencedirect.com ScienceDirect

Energy Procedia 50 ( 2014 ) 314 – 321

The International Conference on Technologies and Materials for Renewable Energy, Environment and Sustainability, TMREES14 Antireflection Coating at Metamaterial Waveguide Structures for Solar Energy Applications

Mohammed M Shabata* , Muin F Ubeida

aDepartment of Physics, Faculty of Science, Islamic University of Gaza P.O. 108, Gaza, Gaza Strip, Palestinian Authority

Abstract

The optical reflection properties of a periodic metamaterial-dielectric multilayered structure are investigated theoretically. The structure is an antireflection coating if it is placed between two semi-infinite media of the same kind and the two slabs constituted each period have the same width and opposite refractive indices. Maxwell's equations are used to determine the electric and magnetic fields of the incident waves 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 and transmitted powers are determined using these coefficients by a recursive method. In the numerical results, the reflected powers are computed and illustrated as a function of wavelength of the incident waves, angle of incidence and number of periods with the emphasis on the appropriate refractive indices. ©© 20142014 Elsevier Mohammed Ltd. This M Shabat,is an open Muin access F Ubeid.article under Published the CC by BY-NC-ND Elsevier Ltd license. (Selectionhttp://creativecommons.org/licenses/by-nc-nd/3.0/ and peer-review under responsibility). of the Euro-Mediterranean Institute for Sustainable Development (EUMISD). Selection and peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) Key words: Angle of incidence, electromagnetic waves, metamaterial, reflection, refractive index, wavelengths, solar energy.

Introduction

Antireflection coating (ARC) is a type of optical coating applied to the surface of photovoltaic cells and optoelectronic devices (Lasers, IR diodes, etc.) to reduce reflection of the incident light which improves the

* Corresponding author. Tel.: 0097059960004; fax: 00970-082823311 E-mail address: [email protected]

1876-6102 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) doi: 10.1016/j.egypro.2014.06.038 Mohammed M. Shabat and Muin F. Ubeid / Energy Procedia 50 ( 2014 ) 314 – 321 315 efficiency of the system since less light is lost [1-5]. Characteristics of antireflection coating are reported by many authors [6-8] for determining the proper thickness and materials to be used as ARCs. The transfer matrix method [8] and the recursive method [9] are usually employed for calculation of reflection coefficients. Metamaterials (sometimes termed left-handed (LH) materials) are materials whose permittivity İ and permeability ȝ are both negative and consequently have negative index of refraction. These materials are artificial and theoretically discussed first by Veselago [10] over 45 years ago. The first realization of such materials, consisting of split-ring resonators (SRRs) and continuous wires, was first introduced by Pendry [11, 12]. Regular materials are materials whose İ and ȝ are both positive and termed right handed (RH) materials. R. A. Shelby et al [13] have studied negative refraction in LH materials. Kong [14] has provided a general formulation for the electromagnetic wave interaction with stratified metamaterial structures. M F Ubeid et al [15] have presented a numerical study of a structure containing left-handed material waveguide. R. W. Ziolkowski et al [16] haves studied wave propagation in media having negative permittivity and permeability. N. Enghheta et al [17] have shown some interesting applications of LH materials. C. Sabah [18] has investigated the effects of loss factor on plane wave propagation through a left-handed material slab. H. Oraiz et al [19] have formulated a theory for zero reflection of electromagnetic waves from multilayered metamaterial structures. J. Yang et al [20] have studied the cancellation of reflection and transmission at metamaterial surfaces. C. Sabah et al [21] have obtained theoretically high reflection coating with negative and positive refractive indices. H. Cory et al [22] have analyzed the reflection and transmission characteristics of a multilayered structure consisting of left-handed materials and dielectric slabs.

This paper is interested in reflection properties of a periodic metamaterial-dielectric multilayered structure with a number of periods N. Low reflection of the structure is demonstrated on the rule that, LH material has negative index of refraction while RH material has positive index of refraction. Antireflection coating is formed by two slabs of the same thickness and opposite refractive indices, one is LH and the other is RH material. The slabs are situated between two media of the same kind like vacuum. The distinctive feature of this structure is the weakening of the influence of the angle of incidence. In the theory the electric and magnetic fields of the incident waves are determined in each region using 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 [9, 25] is used to calculate the reflected powers as a function of wavelength, angle of incidence and the number of periods. To check the results of the analysis used in these calculations, the conservation law of energy given in [24, 25] is checked and it is clear that it is satisfied for all examples. The calculations are performed for electromagnetic radiations in the visible regions for a single wavelength by selecting the optimum refractive indices of the LH and dielectric materials. Such AR coating systems are used in photodiodes (LASER) and other optoelectronic devices which need a minimum reflection at specified wavelengths.

Theory

Consider a pair of LH ( 2 , PH $ ) and RH ( 3 , PH $ ) materials is situated between two semi-finite dielectric media. A perpendicular polarized plane wave in region 1 is incident on the plane z = 0 at some angle ș relative to the normal to the boundary (see Fig. 1). 316 Mohammed M. Shabat and Muin F. Ubeid / Energy Procedia 50 ( 2014 ) 314 – 321

Region 1 Region 2 Region 3 Region 4

İ1ȝo – İ 2, – ȝo İ3ȝo İ4ȝo

B3

B2 A4

A3 B1

A2 X ș

A1 Y Ɣ Z

Z = 0 Z = d2 Z = d2 + d3

Fig. 1. Wave propagation through a structure consisting of a pair of dielectric and metamaterial embedded between two dielectric semi- infinite media. The electric field in each region is [26, 27]: & ik"z z ik"z "x Ztxkiz )(  ""  " yeeBeAE ˆ (1) & ** * To find the corresponding magnetic field H , we start with Maxwell's equation u’ E  wB , substituting " " wt && & P HB "" and solving for H " yield: & 1 ik"z z ik"z z ik"z z ik"z "x Ztxkiz )( H " >@( "" x  "" x ) ˆ ( "" z  "" z ) ˆ exekBekAzekBekA (2) "ZP Where A and B are the amplitude of forward and backward traveling waves ( " = 1, 2, 3, 4) , nk Z is " " "" c the wave vector inside the material and n is the refractive index of it. " & & Matching the boundary conditions for E and H fields at each layer interface, that is at z = 0, EE 21 yy and

1x HH 2x and so on. This yields six equations with six unknown parameters [26]:

A1 + B1 = A2 + B2 (3)

k1z k2z  BA 11  BA 22 (4) P1 P2

ik zd22 ik zd22 ik zd23 ik zd23 2  2 3  3eBeAeBeA (5) k k 2z ik zd22 ik zd22 3z ik zd23 ik zd23 2  2eBeA 3  3eBeA (6) P2 P3

ik z dd 323 ik z  dd 323 ik z dd 324 3eA  3eB 4eA (7) k k 3z ik z dd 323 ik z  dd 323 4z ik z dd 324 3eA  3eB 4eA (8) P3 P4 Mohammed M. Shabat and Muin F. Ubeid / Energy Procedia 50 ( 2014 ) 314 – 321 317

Where kkkk 4321 xxxx { Snell's law and:

Z 2 2 k  nn sin 2 T (9) "z c " 1 Fresnel coefficients (interface reflection and transmission coefficients r, t respectively) for perpendicular polarized light are given by [28]:

j iz  PP i kk jz rij (10) j iz  PP i kk jz

2P j kiz tij (11) j iz  PP i kk jz

Where i, j correspond to any two adjacent media, "  PP $ , "z  kk "z for LH medium and "  PP $ ,

"z  kk "z for RH medium ( " = i, j) [10, 26, 28]. The reflection and transmission coefficients r' and t' respectively of the structure are given by [14]: (  dkdk ) B  2 z dki 33  2 z dki 22  erererrrr i2 zz 3322 rc 1 34231212 23 34 (12) 2 zdki 33 2 zdki 22 (2  zz dkdki 3322 ) A1 1 3423  2312  3412 errerrerr (  dkdk ) A ettt i zz 3322 tc 4 342312 (13) 2 zdki 33 2 zdki 22 (2  zz dkdki 3322 ) A1 1 3423  2312  3412 errerrerr The reflectance R and transmittance T of the structure are given by: R = r'r'*, (14)

§k · * T = ¨ 4z ¸ t't' (15) © k1z ¹ Where r'* and t'* are the complex conjugate of r' and t' respectively. The law of conservation of energy is given by [14]: R + T = 1 (16) For n'-layers structure shown in Fig. 2, r' and t' are calculated as follows [14, 9]:

1 2 3 n'-1 n'-2 n' . . .

Fig. 2. n' layers of different thicknesses and refractive indices are embedded between two dielectric media

c rr cc  ,1 nnn c 2 dki  cerr c nzn c1)1( rc c c1,2 nnn c nc1 2 c dki nzn c1)1( 1 c c1,2 nnn ccerr 2 dki  c err c nzn c2)2( rc c c nnn c12,3 nc2 2 c dki nzn c2)2( 1 c c nnn cc12,3 err ʟ ʟ 318 Mohammed M. Shabat and Muin F. Ubeid / Energy Procedia 50 ( 2014 ) 314 – 321

Continue on the same procedure until r2c is reached which is the reflection coefficient of the structure as a whole.  cerr 2 z dki 22 rc 12 3 2 2 z dki 22 1 12 3cerr

The same procedure is performed for t2c :

c tt cc  ,1 nnn c ik d c ett c nzn c1)1( tc c c1,2 nnn c nc1 2 c dki nzn c1)1( 1 c c1,2 nnn ccerr ik d c ett c nzn c2)2( tc c c nnn c12,3 nc2 2 c dki nzn c2)2( 1 c c nnn cc12,3 err ʟ ʟ cett ik z d 22 tc 12 3 2 2 z dki 22 1 12 3cerr

Where d 2 , dnc1 and dnc2 are thicknesses of layers 2, n'-1and n'-2 respectively. Maximum Transmittance: In order to minimize r' and maximize t', a pair of slabs of LH and RH materials of the same width and opposite refractive indices are situated between two semi-infinite dielectric media of the same kind.

(2  zz dkdki 3322 ) By referring to Fig. 1 and under these conditions, TT 32 , 2z  kk 3z , r23 0 , t23 1, e 1,

12  rr 34 , 3412 1 rrtt 3412 . Consider these relations into (12), (13) yields r' = 0, t' = 1 for any frequency and for any angle of incidence. For the LH material in each period, the frequency-dependent permittivity is described by the Drude medium model as [29, 30]: Z 2 HZH  p (17) lattice Z 2  iZJ

:KHUHȦLVWKHangular frequency, Hlattice is the lattice permittivity, Z p LVWKHHIIHFWLYHSODVPDIUHTXHQF\DQGȖLV the electric damping factor.

Numerical Results

In this section the reflected power of the structure is calculated as a function of wavelength of the incident waves, angle of incidence and the number of periods. In calculations the following parameters are used as in [29, 30@İlattice 16 14 Ȧp = 1.2 x 10 UDGVȖ [ rad/s. The central wavelength is assumed to be 600 nm. This wavelength is chosen by arbitrary decision, but it must be in the frequency band where the permittivity of the LH material is negative. The relative permeability of the dielectrics is assumed to be 1 while that of the LH material is assumed to be -1. The initial and final media are assume to be vacuum. The thickness of each slab in each period is assumed to be one quarter long of the central wavelength. The calculation are performed for electromagnetic radiations in the visible regions and the reflectance will be minimized at the wavelengths 600, 700, 800, 900 nm. At these wavelengths the real part of refractive indices of the LH material according to (17) are -3.24, -3.27,-4.1, -4.86 respectively. The corresponding refractive indices of the RH material (dielectric) are recommended to be nd = 3.24, 3.27, 4.1, 4.86.

Figure 3 shows calculations for the real part (Re(n)) of the refractive index of the LH material as a function of wavelength of the incident waves. The wavelength is changed between 400 nm and 1100nm, because the negative Mohammed M. Shabat and Muin F. Ubeid / Energy Procedia 50 ( 2014 ) 314 – 321 319 permittivity of the LH material can be realized in this range according to (17).

0 400 500 600 700 800 900 1000 1100 -1

-2

-3

-4

-5

the left-handed materiall left-handed the -6

Real part of reracrive index of of part reracrive Real -7

Wavelength (nm)

Fig. 3. Calculated real part of refractive index versus wavelength for the LH material considered in each period.

Figure 4 shows the reflected power as a function of frequency for normal incidence and for one period when the refractive index of the dielectric changes (nd = 3.24, 3.27, 4.1, 4.86). In this example it can be seen that, the reflected power is minimum at the wavelengths 600, 700, 800, 900 nm for nd = 3.24, 3.27, 4.1, 4.86, respectively. At these wavelengths the refractive indices of the LH material and the dielectric are equal in magnitude and opposite in signs. The reflected power has an oscillation behaviour in the range 400-600 nm (for nd = 3.24), 400-700 (for nd = 3.27), 400-800 nm (for nd = 4.1) and 400-900 (for nd = 4.86), this range increases with nd. 1 nd = 2.34 nd = 3.27 0.8 nd = 4.1 nd = 4.86 0.6

0.4 Reflectance

0.2

0 400 500 600 700 800 900 1000 1100 Wavelength (nm)

Fig. 4. The reflected power as a function of wavelength when the refractive index of the dielectric changes nd = 3.24, 3.27, 4.1, 4.86.

Figure SUHVHQWVWKHUHIOHFWHGSRZHUYHUVXVWKHZDYHOHQJWKIRUYDULRXVDQJOHVRILQFLGHQFH ș ÛÛÛ RQH period and for nd = 3.27. The selected value of nd corresponds to minimum reflected power at the wavelength 700 320 Mohammed M. Shabat and Muin F. Ubeid / Energy Procedia 50 ( 2014 ) 314 – 321

nm. As it can be seen from the figure, the influence of the angle of incidence is very weak for the wavelength 700 nm where the conditions of minimum reflected power are satisfied. 1

0.8

0.6

0.4 Reflectance ș Û 0.2 ș Û ș Û 0 400 500 600 700 800 900 1000 1100

wavelength (nm)

Fig. 5. The reflected power versus wavelength for three value of the angle RILQFLGHQFHș ÛÛÛ

Figure 6 depicts the reflected power against wavelength when the number of periods changes (N = 1, 2, 3), for normal incidence and for nd = 3.27. It is observed that, the reflected power shows increasing, decreasing and oscillating behaviours in different ranges of the wavelengths. These ranges increase with the number of periods. The peak to peak value of each oscillation decreases with the number of periods. 1 N = 1 0.8 N = 2 N = 3 0.6

0.4 Reflectance

0.2

0 400 500 600 700 800 900 1000 1100

wavelength (nm)

Fig. 6. The reflected power against wavelength for three values of the number of periods N = 1, 2, 3. Mohammed M. Shabat and Muin F. Ubeid / Energy Procedia 50 ( 2014 ) 314 – 321 321

Conclusions

The reflection properties of a periodic metamaterial-dielectric multilayered structure are studied in detail. The structure is arranged and the required equations which describe the propagation of electromagnetic waves through it are pointed out. Then a recursive method is described to demonstrate the transmission and reflection powers of the structure. In the numerical results, the reflected powers are calculated as a function of wavelength of the incident waves, angle of incidence and the number of periods. It has been shown that, the frequency-dependent permittivity of the LH material plays an important role in the variation of the reflection coefficients of the structure. Low reflection can be achieved for visible rays by choosing the proper indices of the materials constituted each period of the structure. The distinctive feature of the studied structure is the weakening of the influence of the angle of incidence. The results obtained in this paper can be helpful to improve the performance of the photovoltaic cells and optoelectronic devices (Lasers, IR diodes etc.) which requires a reduction of their reflectance at specified frequencies. Moreover, this study will make a foundation for future works and provides some insight into the potential applications of the LH materials, in particular the control of reflection and transmission.

References

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