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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 G.N. Pandey Department of Applied Physics, Amity Institute of Applied Sciences, Amity University, Noida (U.P.), India. Abstract In this present communication, I have calculated the dispersion relation and cosine wave of the structure containing negative index materials to verify transmittance of the considered structure. Such study may be applicable in telecommunication as filters. To calculate the dispersion and cosines wave of the materials we have taken Kronig-Penney model of electronic structure In this addition to this, I have also studied the optical properties of the defect structure containing negative index materials. 1. Introduction Photonic crystals are also known as the electromagnetic wave band gap materials because the electromagnetic wave cannot propagate through the photonic crystal if the incident wavelength is equivalent to the thickness of the unit cell of the crystals. Photonic crystals are the artificial periodic composite materials which exhibit electromagnetic band gaps [1] In 1968, Veselago [2] was the 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 the 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 G.N. Pan ndey withh negative refractive index, douuble negativ ve (DNG) metamaterrials, and other o simiilar names [4]. [ The photonnic crystal (PC) ( is a material m in which w perioddic inclusioons inhibit wave w proppagation due d to destructive innterference from scatttering from m the periiodic repeetition. Thee photonic bandgap b prroperty of PCs P makes them the eelectromagn netic anallog of the electronic e semi-conducctor crystalss [5]. Intendded materiaal fabricatio on of elecctromagneticc band gaaps (EBG) has the goal g of creeating periodic, dieleectric struuctures, withh low loss, and a that are of high quaality. An EB BG affects tthe propertiees of the photon in the same way semicconductor materials m afffect the prroperties off the elecctron. So, it happens thhat the PC is i the perfecct bandgap material, beecause it alllows no propagation p n of light. 2. Theory and a Methoodology By choosing a linearly periodic p reffractive ind dex profile in the perriodic dieleectric matterial one obbtains a given set of wavelength raanges that are a allowed or forbiddeen to passs through thhe periodic dielectric material. m Selecting a paarticular x-aaxis through h the matterial, I shalll assume a periodic p stepp function for f the indexx of the form m; (1)) Where n(x)) =n(x+md)) and m is the translattion factor, which can take the vaalues m=00, ±1, ±2, ±3,…., ± and d=d d 1+d2 is the t period of o the latticee with d1andd d2 is being g the widdth of the tw wo regions having h refracctive indicess n1 and n2 respectively r y. The refracctive indiices profile of the matterials in thee form of rectangular r symmetry iis shown in n the figuure (1).[6] t periodicc structure the one dim mensional wave w If θ is the angle of inncident on this equation for thee spatial parrt of the elecctromagnetiic eigen mode is given as, (2)) Figgure 1: Perioodic refractiive index prrofile of dielectric mateerial. 967 Disppersion Rellation of Def efect Structuure Containing Negativve Index Maaterials where n(x) is given byy equation (1). ( Therefo ore, equationn (2) for waave equation n for twoo media mayy be written as, (3)) (4)) where a and are ray angle in the lay yer of refr fractive inddex n1 and d n2 resppectively. e (3 3) ) and (4) can c be writtten as; By using Bloch’s theorrem, these equations (5)) whrre (6)) On solving above equaation, we obbtain, (7)) Now, abbreeviating the LHS as equatio on (7) may be b written ass, (8)) By using equation e (8)), we may write the dispersion d r relation whhich is obtaained throough the sam me process calculate c thee Kronig-Peenney modeel in periodic potentialss [7]. 3. Result an nd Discussion In thhis present research r paaper I have studied s the cosine c wavee, band struucture propeerties of considered c s structure by KP model and transmiittance of thhe structure is calculateed by TM MM. For the calculationns of the opptical properrties of the defect, we have the deefect matterials of NIM, N for thhe considerred structurre containinng differentt values off the 968 G.N. Pandey cos(Kd) refractive index of the NIM [6, 8 and 9]. Firstly, we introduce the NIM having refractive index nd=-1.05. Figure (2) shows the graph between cos(Kd), Band structure and transmittance versus wavelength (nm) respectively. In this figure cosine wave is crossing the -1 value for the wavelength range 397nm-403nm. This is exactly verified the dispersion relation of the considered structure is calculated by using K.P. Model and the bands structure is the forbidden band within the range of 397nm-403nm which satisfies the 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. -0.8 -1 -1.2 392 394 396 398 400 402 404 406 408 404 406 408 404 406 408 Bandstructure Wavelength[nm] 6 2.5 x 10 2 1.5 392 394 396 398 400 402 Transmittance Wavelength[nm] 1 0.5 0 392 394 396 398 400 402 Wavelength[nm] 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 the NIM as a defect material having the refractive index nd = -1.55 for the considered structure which is shown in figure (3) gives the same graph for cos wave and band structure as given in previous case but this figure gives the transmission peak at 398nm of same intensity but its value is shifted toward lower wavelength. From first and second values of the NIM (i.e n = -1.05 and -1.55). We have the special properties where the tunneling properties is changing as the decreasing the value of NIM. Such properties show that the periodic structure containing NIM has tunable properties which can be changed or shifted the wavelength on the value of the NIM is changed. Dispersion Relation of Defect Structure Containing Negative Index Materials 969 cos(Kd) -0.8 -1 -1.2 392 394 396 398 400 402 404 406 408 404 406 408 404 406 408 Bandstructure Wavelength[nm] 6 2.5 x 10 2 1.5 392 394 396 398 400 402 Transmittance Wavelength[nm] 1 0.5 0 392 394 396 398 400 402 Wavelength[nm] Figure 3: Cos wave, bandsstructure and transmittance of structure for (AB)6DB(AB)2D(BA)6 for nd=-1.55. References [1] [2] [3] [4] [5] [6] [7] [8] [9] E. Yablonovitch, Phys. Rev .Lett., 58, 2059 (1987). V.G.Veselago.Sov.Phys.Usp., 10,.509 (1968). R. A. Shelby, D. R. Smith, S. Shultz, Science, 292, 7779 (2001). N. Engheta, R. W. Ziolkowski, Physics and Engineering Explorations, Wiley & Sons, (2006). W. Chappell, leads the IDEA laboratory at Purdue University "Metamaterials". Research in various technologies. (2009) P. Yeh, “Optical Waves in Layered Media”, John Wiley and Sons, USA (1988) C. Kittel, “Introduction to Solid State Physics” John Wiley and Sons, USA (1953). P.Markos and C.M.Soukoulis, “Wave Propagation from Electron to Photonic Crystal and Left-Handed materials” (2008) X. Li, K. Xie and H .M. Jiang Optics Communications 282, 4295 (2009). 970 G.N. Pandey