Light transmission behaviour as a function of the homogeneity in one dimensional photonic crystals

Transcription

1 Light transmission behaviour as a function of the homogeneity in one dimensional photoniccrystals MicheleBellingeri 1,FrancescoScotognella*,2 1 Dipartimento di Scienze Ambientali, Università di Parma, Parco Area delle Scienze, 33/A 43100Parma,Italy 2 Dipartimento di Fisica, Politecnico di Milano, piazza Leonardo da Vinci 32, Milano, Italy *Correspondingauthor:francesco.scotognella@polimi.it Abstract The average light transmission of one dimensional photonic media has been studied as a function of the medium homogeneity, quantified by the Shannon Wiener index. We have found a decrease in the average light transmission by increasing the Shannon Wiener index up to minimum (corresponding to H'=0.9375): from this point, the transmission increases followingtheshannon Wienerindex.Thebehaviourhasbeenconfirmedfordifferentpairsof materials forming the photonic structure. Nevertheless, we have observed that the trend slopeisproportionaltotherefractiveindexratiobetweenthetwomaterials(nhi/nlow).

2 1.Introduction Thestudyofelectromagneticwavepropagationincomplexandstructuredphotonicmediais a highly relevant research field since it can improve the understanding of some general propertiesoftransportphenomena.[1 3]Complexdielectricstructuresshowvariationsofthe refractiveindexonalengthscalecomparabletothewavelengthoflight.instructureswithan ordered dielectric periodicity, namely photonic crystals, for a certain range of energies and certain wave vectors, light is not allowed to propagate through the medium [4 6]. Such behaviourisverysimilartotheoneofelectronsinasemiconductormaterial,whereenergy gapsariseowingtotheperiodiccrystalpotentialintheatomicscale.photoniccrystalsexistin natureorcanbefabricatedusingawiderangeoftechniques,withthedielectricperiodicityin one, two and three dimensions [7 9]. In the one dimension case, simple and low cost fabrication techniques can be used, as for instance spin coating or co extrusion [7,10] Nowadays,thesematerialsareextensivelystudiedsincetheyfindapplicationinseveralfields, including photonics for low threshold laser action, high bending angle waveguide, superprismeffect,sensorsandopticalswitches.[11 16]Theopticalpropertiesofphotoniccrystals, asforexamplethetransmissionoflight,canbepredictedbyseveralmathematicalmethods [6,17 20].Yet,thesecalculationscanbecomeverycumbersomeasregardslesshomogeneous structures. Simple and not time consuming methods can be very useful for a better comprehensionoftheopticalpropertiesofsuchcomplicatedsystems.recently,conceptsand methods widely used in statistics have been successfully applied to explain light transport phenomenainlévyglasses[21,22]. Herein,wehavestudiedthelighttransmissionpropertiesofonedimensionalphotonicmedia, demonstratingascalinglawbetweentheaveragetransmissionoflightoverawiderangeof wavelengths and the distribution of the diffractive elements in the photonic lattice, i.e. the homogeneity grade of the structure being quantified by the Shannon index, commonly employedinstatisticsandinformationtheory[23].wehavecalculatedlighttransmissionin suchstructuresbyusingafiniteelementmethod.inparticular,wehaveshownthatthelight transmission decreases linearly by increasing the Shannon index, i.e. by increasing the homogeneity of the pillars distribution in the crystals. Interestingly, the result is inverse to whathasbeenobservedintwo dimensionalphotonicmedia[24,25]. 2.OutlineoftheMethod Inthisstudy,weconsideraone dimensionalphotoniccrystalmadeofalternatedlayersoftwo

3 different materials[6]. The two layers are made of Titanium dioxide(nt = 2.45) and Silicon dioxide(ns=1.46).inordertohavealatticeconstanta=200nm,andinordertosatisfya geometricalsettingnzdz~nsds,thethicknessoftitaniumdioxideisdt=75nm,whereasthe thicknessofsilicondioxideisds=225nm[6].infigure1,sio2layersarerepresentedasthree layerswithathicknessof75nmeach:itisthusevidentthatthecrystalunitcellcanbedivided infourequallayers,threeofthemmadeofsio2andoneoftio2. Figure1.One dimensionalphotoniccrystalmadeofalternatedtitaniumdioxideandsilicon dioxidelayers. It is possible to correlate the distribution of the layers in the photonic structure to the Shannon Wienerindex[23].TheShannon WienerH indexisadiversityindexwidelyusedin statisticsandininformationtheory,anditisdefinedas (1) wherepjistheproportionofthej foldspeciesandsisthenumberofthespecies.dividingh' bylog(s)wecannormalizetheindexconstrainingitwithintherange(0,1).inpreviousworks, weusedthenormalizedshannonindex(i.e.0 H' 1)asameasurementofthehomogeneity oftwo dimensionalmedia[24,25].inthoseexperimentswehavedistributedtio2layers,i.e. thej foldspecies,intheslatticecells.intheone dimensionalstructurewecancomputeh by dividingthecrystallengthinacertainnumberofslinearsub units.furthermore,weconsider thetitaniumdioxidelayersasthej foldspecies.thefractionofthelayersbelongingtoeach sub unit represents the proportion pi in Equation (1). Ideally, H is the maximum when all

4 sub unitscontainthesamenumberoflayers.ontheotherhand,whenallthelayersareinone solesub unit,h becomestheminimum:thisstructureisthemostpossibleaggregated.aswe said above, in this study we have made one dimensional crystals with 64 layers, 16 are Titaniumdioxideand48Silicondioxidelayers. Figure2.Schemeofone dimensionalphotonicstructureswithdifferenthomogeneity. Sincewesplitthecrystalin16sub units,themostuniformcrystalhasatio2layerforeach sub unit.thishomogeneousstructurepresentsh equalto1.conversely,sinceeachsubunit can contain up to four pillars, the most aggregated linear configuration we can make is a crystalwherefoursub unitsenclosefourtio2.forthecrystaltopologyselectedinthisstudy, the aforementioned configuration is the most non homogeneous one, with H =0.5. In this work,thecrystalsetiscorrelatedtoh rangingintheinterval(1,0.5).figure2representsonedimensionalcrystalswithvariousgradeofhomogeneity. For this study we have selected eleven different grades of homogeneity, i.e. H = (1; 0.969; ; 0.906; 0.875; 0.83; 0.76; 0.66; 0.613; 0.55; 0.5). For each crystal(i.e. each Shannon index), five different realizations have been considered by allocating the TiO2 layers in a randomfashionwithinthebelongingsub unit.thus,wehavefivedifferentcrystalsandfive different structures with the same global homogeneity and equal Shannon index. In other words, the number of layers in each sub unit is the same in benchmark crystal and in its permutations. For the calculation of the light transmission of the photonic structures through the finite

5 element method, we assumed a TM polarized field and we used the scalar equation for the transverseelectricfieldcomponentez (2) wherenistherefractiveindexdistributionandk0isthefreespacewavenumber[6,26].as regards the input field, a plane wave with wave vector k directed along the x axis has been assumed.scatteringboundaryconditionsintheydirectionhavebeenused.foracomparison with the TiO2 SiO2 photonic media, we have performed the same light transmission simulations for other two couples of materials. A photonic medium has been made of Zinc Oxide(nZ=2)andSiO2,whiletheotherhasbeenmadeofZnOandPoly(hexafluoropropylene oxide)(phfpo,np=1.301),apolymerwithalowrefractiveindex.therefore,wehaveanalysed photonic media with three different refractive index ratios (1.678 for TiO2/SiO2, for ZnO/PhFPO,1.37forZnO/SiO2). 3.Resultsanddiscussion Figure 3 shows the transmission spectra of one dimensional photonic media, made of alternated layers of TiO2 and SiO2, with three different homogeneities, i.e. H =1(black line), H = (red line) and H =0.5 (green line). We selected these three homogeneities since they display the most significant dissimilarities in their transmission spectra between the gradesofhomogeneitychoseninthisstudy.furthermore,theycorrespondtotheextremesin thetrendrepresentedinfigure4(seebelow).forh =1,i.e.fortheidealphotoniccrystal,we havenoticedawidephotonicbandgapataround1020nm,thatisingoodagreementwiththe BraggSnelllaw,i.e.,whereλBraggisthecentrewavelengthofthestopband,neffis theeffectiverefractiveindexofthelatticeandλthespatialperiod(inthiscase,λ=a=300 nm). On the contrary, the photonic medium with H = has more several transmission featuresthatarenotpresentfortheidealphotoniccrystal.thereisstillaphotonicbandgap, evenifblueshifted,aswellastwotransmissionpeaksaround900nm.yet,anewfeatureat 750nmandanotheroneat1250nmoccur.Finally,thephotonicstructurewithhomogeneity correspondingtoh =0.5almostshowtwoweakbandsatabout800nmand1000nm.

6 Figure3.Transmissionspectraoftwoone dimensionalphotonicstructureswithdifferent homogeneity. Figure4arepresentstheaveragelighttransmissionintherange nmasafunctionof the Shannon index for TiO2/SiO2 photonic media, normalized to the average light transmission of the ideal photonic crystal. The five points for each Shannon index value (exceptforh =1,forwhichitisnotpossibletopermutethestructurefortopologicalreasons) correspondtothefivedifferentrealizations,i.e.tothefivecrystalsofequalhomogeneity.the blacklineconnectsthemeanvaluesofthelighttransmission.thefactthatthebehaviouris consistently dissimilar with respect to the results obtained for two dimensional photonic mediadeservesconsideration.alinearincreaseoftheaveragelighttransmissionasafunction of the Shannon index has been observed in the two dimensional case [23]. Yet, in the onedimensional case, we have observed a decrease of the light transmission in the range H = as a function of the Shannon index. Subsequently, we have observed an increase of the light transmission in the range H = In this trend, the value of H =0.9375isaminimum.

7 Figure4.NormalizedaveragetransmissionasafunctionoftheShannonindex. For a better understanding of the results obtained with this in silico experiment, we have analysed the trend of the average light transmission as a function of the Shannon index for differentpairsofmaterials,whicharezno/sio2andzno/phfpo.itisnoteworthythat,forthe threedifferentpairsofmaterials,thetrendisinvariant,asshowninfigure4b,withthesame minimumath = Thus,theaveragelighttransmissionasafunctionofH isindependent on the refractive index ratio in the one dimensional photonic crystal, which means that is independent on the materials chosen to make the photonic medium. Moreover, we have observedthatthedecayingslopeofsuchtrendsincreaseswiththerefractiveindexratioby normalizingthethreetrendstotheaveragelighttransmissionoftheidealphotoniccrystal.in otherwords,theaveragelighttransmissionfunctionfortio2/sio2structureshowsasharper derivativewithrespecttotheoneforzno/sio2structure.

8 Conclusions Inthisstudywehaveengineeredone dimensionalphotonicstructureswithdifferentgradeof homogeneity.suchhomogeneityisquantifiedbytheshannonindex,whichiscommonlyused instatisticsandinformationtheory.bymeansofafiniteelementmethod,wehavesimulated the optical properties of the engineered one dimensional structure. Moreover, we have observedthattheaveragelighttransmissionasafunctionoftheshannonindexshowsatrend that is dissimilar from the one already reported for two dimensional structures [24,25]. AveragelighttransmissiondecreasesbyincreasingtheShannonindexuptoH =0.9375,while itisgrowingwithashannonindexintherange(0.9375,1).moreover,thistrendbehaviouris independent of pairs of materials chosen to build the photonic structure, even if the trend sloperaisesbyincreasingtherefractiveindexratio. Acknowledgements TheauthorsacknowledgeEmanuelaTenca,AjayR.SrimathKandada,Prof.GuglielmoLanzani andprof.stefanolonghiforhelpfuldiscussions.

9 References [1] F.Scheffold,G.Maret,Phys.Rev.Lett.81,5800(1998). [2] D.S.Wiersma,P.Bartolini,A.Lagendijk,R.Righini,Nature390,671(1997). [3] D. S. Wiersma, R. Sapienza, S. Mujumdar, M. Colocci, M. Ghulinyan, L. Pavesi, J. Opt. A: PureAppl.Opt.7,S190(2005). [4] E.Yablonovitch,Phys.Rev.Lett.58,2059(1987). [5] S.John,Phys.Rev.Lett.58,2486(1987). [6] J.D.Joannopoulos,R.D.Meade,J.N.Winn,PhotonicCrystals:moldingtheflowoflight (PrincetonUniversityPress,Princeton,NJ,1995). [7] L.D.Bonifacio,B.V.Lotsch,D.P.Puzzo,F.Scotognella,G.A.Ozin,Adv.Mater.21,1641 (2009). [8] C.Lopez,Adv.Mater.15,1679(2003). [9] J.E.G.J.Wijnhoven,W.L.Vos,Science281,802(1998). [10] K. D. Singer, T. Kazmierczak, J. Lott, H. Song, Y. Wu, J. Andrews, E. Baer, A. Hiltner, C. Weder,Opt.Express16,10358(2008). [11] O.Painter.R.K.Lee,A.Scherer,A.Yariv,J.D.O Brien,P.D.Dapkus,I.Kim,Science284, 1819(1999). [12] F.Scotognella,D.P.Puzzo,A.Monguzzi,D.S.Wiersma,D.Maschke,R.Tubino,G.A.Ozin, Small5,2048(2009). [13] A.Mekis,J.C.Chen,J.Kurland,S.Fan,P.R.Villeneuve,J.D.Joannopoulos,Phys.Rev.Lett. 77,3787(1996). [14] J.Serbin,M.Gu,Adv.Mater.18,21(2006). [15] V. Morandi, F. Marabelli, V. Amendola, M. Meneghetti, D. Comoretto, Adv. Func. Mater. 17,2779(2007). [16] K. Busch, S. Lölkes, R. B. Wehrspohn, H. Föll (eds.), Photonic Crystals: Advances in Design,FabricationandCharacterization(Wiley,Weinheim,2004). [17] W.Axmann,P.Kuchment,J.Comput.Phys.150,468(1999). [18] D.C.Dobson,J.Comput.Phys.149,363(1999). [19] S.G.Johnson,J.D.Joannopoulos,Opt.Express8,173(2001). [20] M. Ghulinyan, C. J. Oton, L. Dal Negro, L. Pavesi, R. Sapienza, M. Colocci, D. S. Wiersma, Phys.Rev.B71,094204(2005). [21] J.Bertolotti,K.Vynck,L.Pattelli,P.Barthelemy,S.Lepri,D.S.Wiersma,Adv.Func.Mater. 20,965(2010).

10 [22] P.Barthelemy,J.Bertolotti,D.S.Wiersma,Nature453,495(2008). [23] C.E.Shannon,Amathematicaltheoryofcommunication,BellSystemTechnicalJournal 27,379(1948). [24] M.Bellingeri,S.Longhi,F.Scotognella,JEOSRP5,doi: /jeos (2010). [25] M.Bellingeri,F.Scotognella,Opt.Mater.,doi: /j.optmat [26] C.S.Lien,PhysicsofOptoelectronicDevices(JohnWiley&Sons,NewYork,1995).