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

photoniccrystals

MicheleBellingeri1,FrancescoScotognella*,2

1Dipartimento di Scienze Ambientali, Università di Parma, Parco Area delle Scienze, 33/A

43100Parma,Italy2Dipartimento di Fisica, Politecnico diMilano, piazza Leonardo da Vinci 32, 20133Milano,

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

foundadecrease in theaverage light transmissionby increasingtheShannon‐Wiener index

up tominimum (corresponding toH'=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).

1.Introduction

Thestudyofelectromagneticwavepropagationincomplexandstructuredphotonicmediais

a highly relevant research field since it can improve the understanding of some general

propertiesoftransportphenomena.[1‐3]Complexdielectricstructuresshowvariationsofthe

refractiveindexonalengthscalecomparabletothewavelengthoflight.Instructureswithan

ordereddielectricperiodicity, namelyphotonic crystals, for a certain rangeof energies and

certain wave vectors, light is not allowed to propagate through the medium [4‐6]. Such

behaviour isverysimilartotheoneofelectronsinasemiconductormaterial,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, super‐

prismeffect,sensorsandopticalswitches.[11‐16]Theopticalpropertiesofphotoniccrystals,

as forexamplethetransmissionof light,canbepredictedbyseveralmathematicalmethods

[6,17‐20].Yet,thesecalculationscanbecomeverycumbersomeasregardslesshomogeneous

structures. Simple and not time consuming methods can be very useful for a better

comprehensionoftheopticalpropertiesofsuchcomplicatedsystems.Recently,conceptsand

methodswidely used in statistics have been successfully applied to explain light transport

phenomenainLévyglasses[21,22].

Herein,wehavestudiedthelighttransmissionpropertiesofonedimensionalphotonicmedia,

demonstratingascalinglawbetweentheaveragetransmissionof lightoverawiderangeof

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

homogeneityof thepillarsdistribution in the crystals. Interestingly, the result is inverse to

whathasbeenobservedintwo‐dimensionalphotonicmedia[24,25].

2.OutlineoftheMethod

Inthisstudy,weconsideraone‐dimensionalphotoniccrystalmadeofalternatedlayersoftwo

differentmaterials [6].Thetwo layersaremadeofTitaniumdioxide(nT=2.45)andSilicon

dioxide(nS=1.46). Inordertohavea latticeconstanta=200nm,and inordertosatisfya

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 proportionpi in Equation (1). Ideally,H’ is themaximumwhen all

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.

Sincewesplitthecrystal in16sub‐units,themostuniformcrystalhasaTiO2layerforeach

sub‐unit.ThishomogeneousstructurepresentsH’equalto1.Conversely,sinceeachsubunit

can contain up to four pillars, the most aggregated linear configuration we canmake 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).Figure2representsone‐

dimensionalcrystalswithvariousgradeofhomogeneity.

For this studywehave selected elevendifferent grades of homogeneity, i.e.H’= (1; 0.969;

0.9375;0.906;0.875;0.83;0.76;0.66;0.613;0.55;0.5). For each crystal (i.e. eachShannon

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

elementmethod,weassumedaTM‐polarized fieldandweused the scalarequation for the

transverseelectricfieldcomponentEZ

(2)

wheren is therefractive indexdistributionandk0 is the freespacewavenumber[6,26].As

regards the input field,aplanewavewithwavevectorkdirectedalong thex‐axishasbeen

assumed.Scatteringboundaryconditionsintheydirectionhavebeenused.Foracomparison

with the TiO2‐SiO2 photonic media, we have performed the same light transmission

simulations for other two couples ofmaterials. A photonicmediumhas beenmade of Zinc

Oxide(nZ=2)andSiO2,whiletheotherhasbeenmadeofZnOandPoly(hexafluoropropylene

oxide)(PHFPO,nP=1.301),apolymerwithalowrefractiveindex.Therefore,wehaveanalysed

photonicmedia with three different refractive index ratios (1.678 for TiO2/SiO2, 1.537 for

ZnO/PhFPO,1.37forZnO/SiO2).

3.Resultsanddiscussion

Figure 3 shows the transmission spectra of one‐dimensional photonic media, made of

alternated layersofTiO2andSiO2,with threedifferenthomogeneities, i.e.H’=1(black line),

H’=0.9375 (red line) andH’=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. ,whereλBragg isthecentrewavelengthofthestopband,neff is

theeffectiverefractiveindexofthelatticeandΛthespatialperiod(inthiscase,Λ=a=300

nm).On the contrary, thephotonicmediumwithH’=0.9375hasmore several transmission

featuresthatarenotpresentfortheidealphotoniccrystal.Thereisstillaphotonicbandgap,

evenifblueshifted,aswellastwotransmissionpeaksaround900nm.Yet,anewfeatureat

750nmandanotheroneat1250nmoccur.Finally,thephotonicstructurewithhomogeneity

correspondingtoH’=0.5almostshowtwoweakbandsatabout800nmand1000nm.

Figure3.Transmissionspectraoftwoone‐dimensionalphotonicstructureswithdifferent

homogeneity.

Figure4arepresentstheaveragelighttransmissionintherange650‐1350nmasafunctionof

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

black lineconnectsthemeanvaluesof the lighttransmission.Thefactthatthebehaviour is

consistently dissimilar with respect to the results obtained for two‐dimensional photonic

mediadeservesconsideration.Alinearincreaseoftheaveragelighttransmissionasafunction

of the Shannon indexhasbeenobserved in the two‐dimensional case [23]. Yet, in the one‐

dimensional case, we have observed a decrease of the light transmission in the range

H’=0.5÷0.9375 as a function of the Shannon index. Subsequently, we have observed an

increase of the light transmission in the range H’=0.9375÷1. In this trend, the value of

H’=0.9375isaminimum.

Figure4.NormalizedaveragetransmissionasafunctionoftheShannonindex.

For a better understanding of the results obtainedwith this in silico experiment, we have

analysed the trendof theaverage light transmissionasa functionof theShannon index for

differentpairsofmaterials,whichareZnO/SiO2andZnO/PHFPO.Itisnoteworthythat,forthe

threedifferentpairsofmaterials,thetrendisinvariant,asshowninFigure4b,withthesame

minimumatH’=0.9375.Thus,theaveragelighttransmissionasafunctionofH’isindependent

on the refractive index ratio in the one‐dimensional photonic crystal, whichmeans that is

independent on the materials chosen to make the photonic medium. Moreover, we have

observedthat thedecayingslopeofsuchtrends increaseswiththerefractive indexratioby

normalizingthethreetrendstotheaveragelighttransmissionoftheidealphotoniccrystal.In

otherwords,theaveragelighttransmissionfunctionforTiO2/SiO2structureshowsasharper

derivativewithrespecttotheoneforZnO/SiO2structure.

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 ofmaterials chosen to build the photonic structure, even if the trend

sloperaisesbyincreasingtherefractiveindexratio.

Acknowledgements

TheauthorsacknowledgeEmanuelaTenca,AjayR.SrimathKandada,Prof.GuglielmoLanzani

andProf.StefanoLonghiforhelpfuldiscussions.

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