Advance in Electronic and Electric Engineering. ISSN 2231-1297, Volume 3, Number 8 (2013), pp. 965-970 Research India Publications http://www.ripublication.com/aeee.htm Dispersion Relation of Defect Structure Containing Negative Index Materials Department of Applied Physics, Amity Institute of Applied Sciences, Amity University, Noida (U.P.), India. Abstract In this present communication, I have calculated dispersion relation and cosine wave of structure containing negative index materials to verify transmittance of considered structure. Such study may be applicable in telecommunication as filters. To calculate dispersion and cosines wave of materials we have taken Kronig-Penney model of electronic structure In this addition to this, I have also studied optical properties of defect structure containing negative index materials. 1. Introduction Photonic crystals are also known as electromagnetic wave band gap materials because electromagnetic wave cannot propagate through photonic crystal if incident wavelength is equivalent to thickness of unit cell of crystals. Photonic crystals are artificial periodic composite materials which exhibit electromagnetic band gaps [1] In 1968, Veselago [2] was first time who proposed a peculiar medium possesses a negative index (NIM) which contain negative of permittivity (ε) and permeability (µ) at certain frequency. A NIM or metamaterial causes light to refract, or bend, differently than in more common positive refractive index materials. Negative index metamaterials or negative index materials (NIM) are artificial photonic crystal structures which has negative refractive index over some frequency range [3]. This is not naturally occurring materials but engineered materials or structures which are known as metamaterials. Metamaterials which exhibit a negative value for refractive index (NIM) are often referred to by any of several names and terminologies: "left-handed media (LHM), backward wave media (BW media), media
9666 with negative refractive index, double negative (DNG) metamaterials, and or similar names [4]. The photonic crystal (PC) is a material in which periodic inclusions inhibit wave propagation due to destructive interference from scattering from periodic repetition. The photonic bandgap property of PCs makes m electromagnetic analog of electronic semi-conductor crystalss [5]. Intended material fabrication of electromagneticc band gaps (EBG) has goal of creating periodic, dielectric structures, with low loss, and that are of high quality. An EBG affects properties of photon in same way semiconductor materials affect properties of electron. So, it happens that PC is perfect bandgap material, because it allows no propagation of light. 2. Theory and Methodology By choosing a linearly periodic refractive index profile in periodic dielectric material one obtains a given set of wavelength ranges that are allowed or forbidden to passs through periodic dielectric material. Selecting a particular x-axis throughh material, I shalll assume a periodic step function for index of form; (1) Where n(x) =n(x+md) and m is translation factor, which can take values m=0, ±1, ±2, ±3,., and d=d 1 +d 2 is period of lattice with d 1 and d 2 is being width of two regions having refractive indices n 1 and n 2 respectively. The refractive indices profile of materials in form of rectangular symmetry is shown in figure (1).[6] If θ is angle of incident on this periodic structure one dimensional wave equation for spatial part of electromagnetic eigen mode is given as, (2) Figure 1: Periodic refractive index profile of dielectric material.
Dispersion Relation of Defect Structure Containing Negative Index Materials 967 where n(x) is given by equation (1). Therefore, equation (2) for wave equation for two media may be written as, (3) (4) where and are ray angle in layer of refractive index n1 and n2 respectively. By using Bloch s orem, se equations (3) ) and (4) can be written as; (5) whre (6) On solving above equation, we obtain, (7) Now, abbreviating LHS as equation (7) may be written as, (8) By using equation (8), we may write dispersion relation which is obtained through same process calculate Kronig-Penney model in periodic potentials [7]. 3. Result and Discussion In this present research paper I have studied cosine wave, band structure properties of considered structure by KP model and transmittance of structure is calculated by TMM. For calculations of optical properties of defect, we have defect materials of NIM, for considered structure containing different values of
968 refractive index of NIM [6, 8 and 9]. Firstly, we introduce NIM having refractive index n d =-1.05. Figure (2) shows graph between cos(kd), Band structure and transmittance versus wavelength (nm) respectively. In this figure cosine wave is crossing -1 value for wavelength range 397nm-403nm. This is exactly verified dispersion relation of considered structure is calculated by using K.P. Model and bands structure is forbidden band within range of 397nm-403nm which satisfies cosine wave. The transmittance graph shows one transmission peak approximately at 400nm within forbidden band and two transmission peak approximately at 396nm and 404nm which is out of forbidden range. cos(kd) Bandstructure -0.8-1 -1.2 2.5 x 106 2 1.5 Transmittance 1 0.5 0 Figure 2: Cosine wave, band structure and transmittance of structure for (AB)6DB(AB)2D(BA)6 for nd = -1.05. Secondly, we are taking NIM as a defect material having refractive index n d = -1.55 for considered structure which is shown in figure (3) gives same graph for cos wave and band structure as given in previous case but this figure gives transmission peak at 398nm of same intensity but its value is shifted toward lower wavelength. From first and second values of NIM (i.e n = -1.05 and -1.55). We have special properties where tunneling properties is changing as decreasing value of NIM. Such properties show that periodic structure containing NIM has tunable properties which can be changed or shifted wavelength on value of NIM is changed.
Dispersion Relation of Defect Structure Containing Negative Index Materials 969 cos(kd) Bandstructure Transmittance -0.8-1 -1.2 2.5 x 106 2 1.5 1 0.5 0 Figure 3: Cos wave, bandsstructure and transmittance of structure for (AB)6DB(AB)2D(BA)6 for nd=-1.55. References [1] E. Yablonovitch, Phys. Rev.Lett., 58, 2059 (1987). [2] V.G.Veselago.Sov.Phys.Usp., 10,.509 (1968). [3] R. A. Shelby, D. R. Smith, S. Shultz, Science, 292, 7779 (2001). [4] N. Engheta, R. W. Ziolkowski, Physics and Engineering Explorations, Wiley & Sons, (2006). [5] W. Chappell, leads IDEA laboratory at Purdue University "Metamaterials". Research in various technologies. (2009) [6] P. Yeh, Optical Waves in Layered Media, John Wiley and Sons, USA (1988) [7] C. Kittel, Introduction to Solid State Physics John Wiley and Sons, USA (1953). [8] P.Markos and C.M.Soukoulis, Wave Propagation from Electron to Photonic Crystal and Left-Handed materials (2008) [9] X. Li, K. Xie and H.M. Jiang Optics Communications 282, 4295 (2009).
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