ABSTRACT 1. INTRODUCTION

Transcription

1 Dispersio Reltio Effetive Iex of Refrtio i Terry Oe Dimesiol Photoi B Gp Struture Khem B.Thp, S. K. Sigh S. P. Ojh * Deprtmet of Applie Physis, Istitute of Tehology, Brs Hiu Uiversity, Vrsi005, INDIA ABSTRACT I this pper, we the stuy ispersio reltio effetive Iex of refrtio properties of Terry Oe Dimesiol Photoi B Gp (PBG) Struture i.e. perioi multilyer otie y trsltig three lyers uit ell osistig of mterils with refrtive iies, of thiesses,, l respetively y reiprol lttie vetors. The ispersio reltio for ifiite terry oe-imesiol PBG is erive o the sis of Kroig-Peey moel. This struture provies itiol egree of freeom i ispersio otrol reltive to ovetiol (iry) oe-imesiol PBG struture. Sie the three lyers re ivolve i the formtio of gp, muh roer rge of ispersio otrol re otie. Moreover, er the gp i PBG mterils the effetive iex of refrtio i terry oeimesiol PBG struture hs ee stuie. Key wors : Terry oe-imesiol PBG strutures, ispersio reltio, group veloity, egtive effetive group iex of refrtio effetive iex of refrtio. INTRODUCTION I reet yers, the propgtio of eletromgeti (EM) wves i perioi ieletri strutures hs reeive muh experimetl theoretil ttetio. The photoi gp strutures of these systems exhiit itervls of frequeies i whih EM wves re forie. The existee of the photoi gps le to vriety of iterestig pheome of oth fumetl prtil iterest hs potetil pplitios i my sietifi tehil res. [,] Oe Dimesiol perioi struture, referre s Photoi B Gp (PBG) rystls hve ee extesively stuie [-4]. Suh rystls re typilly ompose of ltertig mterils of high low refrtive iex. This rrgemet gives rise to series of wvelegth pss- gps, where the light is trsmitte or reflete y the rystl i mer similr to the gps i the perioi tomi rrgemet i semioutors. Oe imesiol multiplyer strutures re wiely employe i ifferet res of optis lser physis. A umerous pplitio of suh strutures re se o the existee of PBG i their trsmissio spetr, risig ue to strog ouplig of forwr wrs wves with the erti frequey regio. [5] A high refletio oeffiiet provie y PBG mterils is effiious i the opertio of suh wiespre optil ompoet s multilyers otigs, Brgg refletors, perioi wve-guies rrow- filters. Reetly, Aumov et.l. [6] exmie the pilities of terry oe-imesiol PBG strutures to simulteously phse-mth severl optil fiels with ifferet frequeies propgtig i ispersive mei. Ojh et.l [7] stuie egtive group veloity effetive refrtive group iex, for oe imesiol PBG strutures. T. Fuy et.l. [8] hee the reliztio of egtive refrtio i sl tht osists of perioi multilyers of two mterils with ifferet refrtive iies. We stuie the ispersio reltio, group veloity effetive iex of refrtio of perioi multilyer strutures for the propgtig of EM wve. For this, we hve hose terry oe-imesiol struture. We hve formulte ispersio reltio for terry oe-imesiol struture o the sis of Kroig-Peey moel [9] s well s Trsfer mtrix metho. I this pper, we hve opte Trsfer mtrix metho for stuyig the refletio trsmissio of EM wve through perioi multilyer meium [0, ]. The terry oe-imesiol PBG struture is perioi multilyer otie y trsltig uit ell osistig of three ifferet mterils of ifferet refrtive iies. O the

2 sis of ispersio reltio it hs ee emostrte tht terry DPBG struture provies itiol egrees of freeom for ispersio otrol, egtive refrtive iex group veloity.. THEORETICAL ANALYSIS The Mxwell s wve equtio for EM propgtig log the x-xis my e writte s, E(x) β E(x) 0 () x i where β si i, is the propgtio ostt ( i,, ), is wve vetor gle (x) e the perioi refrtive iex profiles of the struture give y, 0 < x < (x) < x < () l < x < 0 with (x) (x ), where, re the three refrtive iies of the mterils with thiess, l respetively. Here is the frequey of eletromgeti wve l is the perio of the lttie (Figure ). Now the wve equtios i the three lterte regios my e writte s, E(x) β E(x) 0 ; 0 < x < () x E(x) β E(x) 0 ; < x < () x E(x) β E(x) 0 ; -l< x < 0 () x The eletri fiel E(x) withi eh homogeeous lyer is omitio of right trvelig left trvelig wve so it e expresse s the sum of iiet ple wve reflete ple wve. Therefore solutio of equtio () i e x i e x < < ; 0 x E(x) e ix e ix ; < x < e e ix f e ix ; l < x < 0 (4) where, β os β os

3 os β Here,,,,, e f re ostts,, re the ry gles i the lyers the re relte with the iiet gle s follows, si os, si os Now imposig the otiuity of E(x) x E(x) t the iterfes x 0,,, -l, we get four equtios hvig four uow ostts. By solvig these equtios we get the followig mtrix equtio [0]; D C B A (5) where A,B,C D re the mtrix elemets for uit ell trsltio mtrix tht reltes the omplex mplitue of the iiet wve the reflete ple wve i oe lyer of uit ell to those of the equivlet lyer i the ext uit ell. Usig Bloh s theorem, ix t i e (x)e E t) (x, E β, where ) (x E ) (x E, simplifyig we oti the ispersio reltio for the propose struture give y ( ) D A os ) K( ( ) l) ) os( )si( si( l) ) si( ) os( si( l) ) si( ) si( os( l) )os( ) os( os( os K (6) Here, K is ow s the Bloh wve umer whih is futio of, is lso the futio of. For orml iiet 0 β. Therefore ove equtio eomes l os si si l si os si l si os os l os os os os ) K( (7) The expressio for the group veloity g V e lulte y usig the formul, g K V, (8) eepig β s ostt.

4 Besies the usul effetive refrtive iex [ eff (p)] give y [],.K( ) eff (p), (9) So [] itroue the oept of effetive group iex [ eff (g)] s follows eff (g) (0) Vg This formul for lultig eff (g) ws lso employe y other worers [4-6]. The oeffiiet of refletio of the propose struture e eue y usig the reltio [7], 0 R N, () 0 N 0 where 0 0 represet the omplex mplitues of iiet reflete wves. The oitio N 0 implies the oury oitio tht to the right of perioi struture there is o wve iiet o it. By solvig the equtio, we otie the expressios for the refletivity of the struture, whih is give s C R N, () si(k C si(nk) where N is the umer of the uit ell C is the mtrix elemet of the uit ell of equtio (5).. RESULTS For the se of umeril lultios we hve te three lterte lyers of mterils viz. () ir, MgF, ZS with refrtive iies.0,.8,.5 rtio 75%, 5%, 0% respetively of the totl with of uit ell () glss, MgF, ZS with refrtive iies.5,.8,.5 rtio 5%, 45%, 0% respetively of the totl with of uit ell. I our lultios, we hve use ritrry uits of frequeies. I the se of Air-MgF-ZS struture, fig. epits the vritio of Bloh wve vetor, group veloity, effetive group iex ( eff (g)), refrtive iex ( eff (p)) with. π Here, photoi gps re lerly oserve for frequey rge. -.06, , Ourree of egtive group veloity is worth otig. Moreover, ( eff (g)) eomes t 7.49 to 7.48 t π However, ( eff (p)) oes ot eome egtive t y frequey rge osiere for our lultio. Fig. 4 presets the reflete of the sme struture for ifferet gles of iiee. It is iterestig to ote tht 00% refletivity is otie etwee the frequeies.4 to 8.70 t 75 o. Moreover, the rge of the full refletivity goes o iresig shifts towrs higher frequeies s we irese the iiet gles. Next, for the struture Glss-MgF-ZS, ispersio urve, vritio of group veloity, eff (g) eff (p) with frequey re epite i fig.. I this se we oserve tht umer of photoi gps irese though with of the gp ereses. Here, group veloity ereses for the hge struture resultig i the ehemet of eff (g) whih ttis the vlue up to -54. Moreover, eff (p) slightly irese t ll vlues of frequeies osiere. I this struture, the refletivity is 00% t 75 o for the full rge of frequey. The umer of gps ireses with the erese with.

5 4. CONCLUSIONS O the stuy of ifferet refrtive iex profile their rtio of thiesses it is oserve tht Terry oe imesiol PBG strutures provie itiol egree of freeom to otrol ispersio reltios, group veloity, effetive iex of refrtio [oth eff (g) eff (p)] reltive to ovetiol iry oe imesiol PBG strutures. The effiy of the stuy of terry oe imesiol PBG struture lies i the stuy of phse-mthig, egtive refrtive iex superlumil ehvior. ACKNOWLEDGEMENT Oe of the uthors, Mr. S. K. Sigh is thful to the Couil of Sietifi Iustril Reserh (CSIR), INDIA for proviig fiil ssiste. REFERENCES. Speil issue o evelopmet Applitio of mterils exhiitig PBG : C.M. Bowe, J.P. Dowlig H.O. Everitt, J. Opt. So. Am. B., 0(), 79 (99).. Photoi Crystls: puttig ew twist o light, J.D. Joopoulos, P.R. Villeeuve S. F, Nture, 86,4 (997).. Mesuremet of spoteous emissio ehemet er the oe imesiol photoi ege of semioutor heterostruturres, M.D. Toi M. Slor, Phys. Rev. A, 5, 799 (995). 4. Pulse seo hrmoi geertios i o lier oe imesiol perioi strutures, M. Slor, M. J. Bloemer et. l. Phys. Rev. E, 56, 66 (997). 5. Eletromgeti propgtio i perioi strtifie mei, P. Yeh, A. Yriv, Shi-Shw Hog, J. Opt. So. Am., 67, 4 (977). 6. Terry oe-imesiol photoi gp strutures: Dispersio reltio, Extee Phse-mthig Ailities Attoseo Outloo, A.N. Numov A.M. Zheltiov, Lser physis, (7), 879 (00). 7. Group Veloity, Negtive Ultr High iex of Refrtio i Photoi B Gp Mterils, S. P. Ojh, S. K. Srivstv, Mirowve Optil Tehology Letters, 4, 8 (004). 8. Sl les with restrie light propgtio i perioi multilyer, Toshio Fuy J. Tomig, J. Opt. So. Am.,, 80 (004). 9. C. Kittel, Itroutio to soli stte physis Joh Wiley & sos, New Yor, P. Yeh, Optil wves i Lyere Mei, Joh Wiley& Sos, New Yor M. Bor Wolf, Priiples of optis, Pergmo, Oxfor, Aomlous iex of refrtio i photoi B Gp Mterils, J.P. Jotho C.M. Bowe, J. Mo. Opt. 4, 45 (994).. K. So, Optil Properties of Photoi Crystls, Spriger, Germy, Effetive Optil Properties ssoite with wve propgtio i perioi rystls, D.Y. Jeog, Y.H. Ye Q.M. Zhg, J.Appl. Phys., 9 (00). 5. Group veloity, eergy veloity, superlumil propgtio i fiite photoi gp strutures, G. D Aguo, M. Cetii, M. Slor, C. Siili, C. M. Bowe, J. W. Hus, M. Bertolotti, Phys. Rev. E 6, 0660 (00). 6. Negtive group veloity group ely i left-he mei, J. F. Wooley M. Mojhei, Phys. Rev. E, 70, (004). *E-mil: spojh@yhoo.om, spojh@rs.eret.i

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