c4.1 omni-directional reflection properties of one directional ...

ICOP 2009-International Conference on Optics and Photonics CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009

OMNI-DIRECTIONAL REFLECTION PROPERTIES OF ONE DIMENSIONAL SUPERCONDUCTOR PHOTONIC CRYSTAL G N Pandey1, Arun Kumar2, Khem B Thapa3 & S. P. Ojha4 1 Department of Applied physics, Amity school of Engg & Tech., Amity University Noida, [email protected] 2 Amity Institute of Telecom Technology & Management, Amity University Noida, [email protected] 3 Department of Physics, UIET, CSIM University Kanpur, [email protected] 4 Department of Applied Physics, IT BHU Varanasi, [email protected] Abstract: We have studied theoretically the ominidirectional reflection (ODR) in one dimensional Photonic Crystal (PC) structure consisting of alternate layers of super conductor-dielectric for TE-mode. The band structure and the reflectance of superconductor-dielectric photonic crystal are calculated by using transfer matrix method and Bloch’s Theorem [1, 2]. The proposed structures give 100% reflection within a very wide range of wavelength of the Electromagnetic Spectrum.

1. INTRODUCTION

2. THEORY

Periodic dielectric structures that exhibit electromagnetic stop bands or PBGs, have attracted much attention because of their ability to control the propagation of electromagnetic waves. The study of photonic band gap consisting of a superconducting material and a dielectric has been reported [2-6]. The gap size is characterized by polarization, penetration depth, and highly dependent on temperature at the vicinity of superconducting transition temperature. The analysis shows that the property of thin photonic crystals may have application in optical region if extremely low relaxation time superconductor is used. This may be asset for superconducting electronics-photonic integration. The electromagnetic properties of Abrkosov vertex lattice as a photonic crystal were investigated by changing Ginzburg-Landau parameter and static magnetic field [5]. In addition to a low frequency band gap below the first band, they also obtained the PBGs for superconductor in the presence of vortices. The photonic crystals is to incorporate a superconductor constituent [3-6], because photonic band structures in PCs determine the optical routine of light transmitting in them, it is also possible to obtain tunable negative refraction in PCs by modifying the band structures. The issue of a super conducting photonic crystal was first investigated by a group in Singapore [2] and reflection properties studied by Ojha et. al. [7]. We have considered one-dimensional superconductor/ dielectric materials, by making use of the transfer matrix method and accompanied by the Bloch theorem. Coffey and Clem also studied microwave transmission and reflection in type–II superconductor in the mixed state [8].

A one-dimensional non-magnetic of refractive index n1 = ε1 and n 2 = ε2 photonic crystal will

be modeled as a periodic dielectric ( ε1 ) and dielectric ( ε 2 ) multilayer structure with a period of d = d1 + d2. Such structure is shown in Fig. 1.

Figure 1. Schematic presentation of superconductor / dielectric periodic structure.

Where d is the spatial periodicity with d1 and d2 are the thickness of the refractive indices n1 and n2 respectively. We consider that a TE mode wave is incident at an angle θ0 from the air medium with a refractive index 1. The index of refraction of the loss less dielectric is given by n i and i = 1, 2. The refractive index profile of the structure given by  nd , n(x) =   ns ,

d2 < x < d 0 < x < d2

(1) with n(x) = n(x + d) where, nd, ns are the refractive indices of the dielectric and superconductor materials. The electric field E(x) within each homogeneous layer is a combination of right-travelling waves and left-travelling waves and so it can be expressed as

1


the sum of an incident plane wave and reflected plane wave. Thus, the electric field in the mth unit

cell can be written as

(md − d1 ) < x < md a m e − ik1 ( x−md) + b m eik 2 ( x −md) ; E(x) =  − ik 2 ( x −md +d1 ) ik 2 ( x −md + d1 ) ;(m − 1)d < x < (md − d1 ) + dme  c m e

M 2,1

(2) In order to calculate the transmission and reflectance for a periodic multilayered structure is calculated using TMM. The elegant ABELES theory will be employed. According to this theory, we must set up the characteristic matrix corresponding to one period, with the result  m1,1 m1,2  m(d) =  (3)   m 2,1 m 2,2  We can obtain dispersion relation as 1 1  K(ω) = cos −1  ( m1,1 + m 2,2 )  d 2    1 1 p p K(ω) = cos −1  cos(k1x d1 ) cos (k 2x d 2 ) −  1 + 2 d 2  p 2 p1 

M 2,1

2

RN = 2

(7)

2

 sin K(ω)d  +   sin NK(ω)d 

2

where N is the number of unit cells (i.e. number of pairs of layers). 3. RESULTS AND DISCUSSIONS Generally the refractive index of the superconductor is temperature dependent i.e. ns = [Cos2θ – (c2 / ω2λL2)]1/2, λL(T) = λL (0) / [1 – (T/Tc)4]1/2 [3-6] The super conductor of Tc=10K with penetration depth 60nm and dielectric material of reflective index 3.87 are considered. In the proposed structure we have calculated the band structure and the reflectance, and assuming that the regions to be linear, homogeneous, nonabsorbing and with no optical activity. From study of the figures, we observed that the band structures and reflectance of the given structure are same at the incident angle zero. When angle of incidence increases, we observed that the range of band gaps and reflectance of the given structure increases. But the band structures and reflectance of the considered structure are shifted towards the higher frequency range. From the Figure (3) it is observed the omnidirectional reflection band in TE – mode of the considered structure. This shows that it can be used as a omnidirectional reflector.

(4)    sin(k1x d1 ) sin(k 2x d 2 )   

(5) The reflection coefficients can be determined and are given by ( M1,1 + M1,2 ps ) p0 − ( M 2,1 + M 2,2 ps ) r= ( M1,1 + M1,2 ps ) p0 + ( M 2,1 + M 2,2 ps ) (6) Here p0 and ps are the first and last medium of the structure, which are given as n 0 cos θ0 n s cos θs p0 = , ps = (TE) Z0 Z0 In our case we have taken n0 = ns = 1.0 for the free space. The expressions for Nth layers, we obtain the reflectivity of the structure, as shown below

Figure 2: Dispersion relation, of the considered structure at angle of incidence 00

2


Figure 3: Omnidirectional Reflection Band for TE - mode of the considered structure.

3


5.

ACKNOWLEDGEMENT I, Dr. G N Pandey with Arun Kumar, thankful to Dr. Ashok K. Chauhan, Founder President, Amity University, Uttar Pradesh, for his interest in research and constant encouragement.

6.

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7.

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C. J. Wu, “Transmission and reflection in a periodic superconductor/dielative film multilayer structure”, J. Electromagnetic wave and applications, vol. 19. pp. 1191-1996, 2005. C. J. Wu, “Transmission and reflection in a periodic superconductor/dielative film multilayer structure”, PIER Symposium-2005, China, Aug. 2005. G N Pandey, Arun Kumar, K S Singh, S P Ojha “Design of Optical reflector using Superconductor – Dielectric Photonic Crystal”, IConTOP 2009, Calcutta, India, March 2009. Mark W. Coffey and John R. Clem, ‘Theory of microwave transmission and reflection in type – II superconductors in the mixed state’, Phys. Rev. B, vol. 48, pp. 342 -350, 1993.