Post on 13-May-2023
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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