260. R. J. Weiblen, C. R. Menyuk, L. E. Busse, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal

16 Pages·2016·2.87 MB·English

Optimized moth-eye anti-reflective
structures for As S chalcogenide optical
2 3
fibers
R.J.Weiblen,1,∗C.R.Menyuk,1L.E.Busse,2L.B.Shaw,2
J. S.Sanghera,2andI.D.Aggarwal3
1DepartmentofComputerScienceandElectricalEngineering,TRC203,Universityof
MarylandBaltimoreCounty,1000HilltopCircle,Baltimore,MD21250,USA
2NavalResearchLaboratory,Code5620,Washington,D.C.20375,USA
3SoteraDefenseSolutions,Crofton,MD21114,USA
∗[email protected]
Abstract: We computationally investigate moth-eye anti-reflective
nanostructures imprinted on the endfaces of As S chalcogenide optical
2 3
fibers.Withagoalofmaximizingthetransmissionthroughtheendfaces,we
investigatetheeffectofchangingtheparametersofthestructure,including
theheight,width,period,shape,andangle-of-incidence.Usingtheseresults,
we design two different moth-eye structures that can theoretically achieve
almost99.9%averagetransmisisonthroughanAs S surface.
2 3
© 2016 OpticalSocietyofAmerica
OCIScodes:(050.6624)Subwavelengthstructures;(060.2390)Fiberoptics,infrared.
Referencesandlinks
1. LordRayleigh,“Onreflectionofvibrationsattheconfinesoftwomediabetweenwhichthetransitionisgradual,”
Proc.LondonMath.Soc.S1–11,51–56(1879).
2. C.G.BernhardandW.H.Miller,“Acornealnipplepatternininsectcompoundeyes,”ActaPhysiol.Scand.56,
385–386(1962).
3. D. G. Stavenga, S. Foletti, G. Palasantzas, and K. Arikawa, “Light on the moth-eye corneal nipple array of
butterflies,”Proc.Biol.Sci.273,661–667(2006).
4. S.Chattopadhyay,Y.F.Huang,Y.J.Jen,A.Ganguly,K.H.Chen,andL.C.Chen,“Anti-reflectingandphotonic
nanostructures,”MaterialsSci.EngRev.69,1–35(2010).
5. J.Kulakofsky,W.Lewis,M.Robertson,T.Moore,andG.Krishnan,”Designinghigh-powercomponentsfor
opticaltelecommunications,”Proc.SPIE4679,198(2002).
6. S.A.BodenandD.M.Bagnall,“Optimizationofmoth-eyeantireflectionschemesforsiliconsolarcells,”Prog.
Photovolt:Res.Appl.18,195–203(2010).
7. Y.Ou,D.Corell,C.Dam-Hansen,P.Petersen,andH.Ou,”Antireflectivesub-wavelengthstructuresforim-
provementoftheextractionefficiencyandcolorrenderingindexofmonolithicwhitelight-emittingdiode,” Opt.
Express19,A166–A172(2011).
8. D. S. Hobbs, B. D. MacLeod, and J. R. Riccobono, “Update on the development of high performance anti-
reflectingsurfacereliefmicro-structures,”Proc.SPIE6545,65450Y(2007).
9. T.Hoshino,M.Itoh,andT.Yatagai,“Antireflectivegratingintheresonancedomainfordisplays,” Appl.Opt.
46,648–656(2007).
10. J.Sanghera,C.Florea,L.Busse,L.B.Shaw,F.Miklos,andI.Aggarwal,“ReducedFresnellossesinchalcogenide
fibersbyusinganti-reflectivesurfacestructuresonfiberendfaces,” Opt.Express18,26760–26768(2010).
11. N.H.Finkel,B.G.Prevo,O.D.Velev,andL.He,“Orderedsiliconnanocavityarraysinsurface-assisteddesorp-
tion/ionizationmassspectrometry,”Anal.Chem.77,1088–1095(2005).
12. S.Y.Chou,P.R.Krauss,andP.J.Renstrom,“Nanoimprintlithography,”J.Vac.Sci.Technol.B14,4129–4133
(1996).
13. M.Kowalczyk,J.Haberko,andP.Wasylczyk,“Microstructuredgradient-indexantireflectivecoatingfabricated
onafibertipwithdirectlaserwriting,” Opt.Express22,12545–12550(2014).
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10172
14. B.D.MacLeod,D.S.Hobbs,andE.Sabatino,“MoldableARmicrostructuresforimprovedlasertransmission
anddamageresistanceinCIRCMfiberopticbeamdeliverysystems,”Proc.SPIE8016,80160Q(2011).
15. R. J. Weiblen, C. Florea, A. Docherty, C. R. Menyuk, L. B. Shaw, J. Sanghera, L. Busse, and I. Aggarwal,
“OptimizingmotheyeantireflectivestructuresformaximumcouplingthroughAs2S3 opticalfibers,”inIEEE
PhotonicsConference(IEEE,2012),paperThP3.
16. K.HanandC.-H.Chang“Numericalmodelingofsub-wavelengthanti-reflectivestructuresforsolarmodule
applications,”Nanomaterials4,87–128(2014).
17. J.Yamauchi,M.Mita,S.Aoki,andH.Nakano,“AnalysisofantireflectioncoatingsusingtheFD-TDmethod
withthePMLabsorbingboundarycondition,”IEEEPhotonicsTechnol.Lett.8,239–241(1996).
18. Z.Y.Yang,D.Q.Zhu,M.Zhao,andM.C.Cao,“Thestudyofanano-porousopticalfilmwiththefinitedifference
timedomainmethod,”J.Opt.A:PureAppl.Opt.6,564–568(2004).
19. M.G.MoharamandT.K.Gaylord,“Rigorouscoupled-waveanalysisofplanar-gratingdiffraction,” J.Opt.Soc.
Am. 71,811–818(1981).
20. L.Li,“NewformulationoftheFouriermodalmethodforcrossedsurface-reliefgratings,” J.Opt.Soc.Am.A
14,2758–2767(1997).
21. D.S.Hobbs,B.D.MacLeod,E.SabatinoIII,T.M.Hartnett,andR.L.Gentilman,“Laserdamageresistantanti-
reflectionmicrostructuresinRaytheonceramicYAG,sapphire,ALON,andquartz,”Proc.SPIE8016,80160T
(2011).
22. D.S.Hobbs,“Laserdamagethresholdmeasurementsofanti-reflectionmicrostructuresoperatinginthenearUV
andmid-IR,”Proc.SPIE7842,78421Z(2010).
23. C.Florea,L.Busse,S.Bayyab,B.Shaw,I.Aggarwal,andJ.Sanghera,“Anti-reflectivesurfacestructuresin
spinelceramicwindows,”inThe10thPacificRimConferenceonCeramicandGlassTechnology(2013),paper
S10-014.
24. L.Busse,C.Florea,L.B.Shaw,S.Bayya,G.Villalobos,I.Aggarwal,andJ.Sanghera,“Anti-reflectivesurface
structuresforhighenergylaserapplications,”inAnnualDirectedEnergySymposium(2013),paper13-Symp-
053.
25. R.J.Weiblen,C.Florea,L.Busse,L.B.Shaw,C.R.Menyuk,I.Aggarwal,andJ.Sanghera,“Irradianceen-
hancment and increased laser damage threshold in As2S3 moth-eye antireflective structures,” Opt. Lett. 40,
4799-4802(2015).
26. R.J.Weiblen,C.Florea,C.R.Menyuk,I.Aggarwal,L.Busse,L.B.Shaw,andJ.Sanghera,“Idealcusp-like
motheyeantireflectivestructuresforchalcogenideopticalfibers,”inIEEEPhotonicsConference(2015),paper
WI1.4.
27. A.F.Oskooi,D.Roundy,M.Ibanescu,P.Bermel,J.D.Joannopoulos,andS.G.Johnson,“MEEP:aflexible
free-softwarepackageforelectromagneticsimulationsbytheFDTDmethod,”Comp.Phys.Comm.181,687–
702(2010)
28. D.RaguinandG.Morris,“Antireflectionstructuredsurfacesfortheinfraredspectralregion,” Appl.Opt.32,
1154–1167(1993).
29. M.Steel,T.White,C.M.deSterke,R.McPhedran,andL.Botten,“Symmetryanddegeneracyinmicrostructured
opticalfibers,” Opt.Lett.26,488–490(2001).
30. W.H.Southwell,“Pyramid-arraysurface-reliefstructuresproducingantireflectionindexmatchingonoptical
surfaces,” J.Opt.Soc.Am.A8,549–553(1991).
1. Introduction
IthasbeenknownsincethetimeofLordRayleighthatmicroscalestructuresonthesurfaceof
opticalinterfacesareeffectiveatreducingFresnelreflections[1].Periodicanti-reflective(AR)
microstructuresarecalled“moth-eye”structuresbecauseoftheirsimilaritytothemicrostruc-
turesontheeyesofnocturnalmoths[2,3].Moth-eyestructuresareabiomimeticmicrostruc-
tured AR surface structure that are effective at reducing Fresnel reflections [4]. In the long
wavelengthlimit,theyworkbyprovidingagradualchangeoftheeffectiverefractiveindexas
lightpropagatesacrosstheair-glassinterface.Theyareespeciallyusefulforhigh-indexmate-
rials, which includes most mid-IR materials, such as chalcogenide glasses. Reducing Fresnel
reflections from optical interfaces is important in mid-IR applications, where high power and
low loss are needed. High power laser radiation that reflects from interfaces can damage in-
struments, while insertion or coupling losses can be major contributors to overall losses in a
system[5].Hence,mid-IRsystemscanbenefitgreatlybyusingmoth-eyestructurestoreduce
reflections from and increasing transmission through interfaces. They are useful in a number
ofapplications,includinglasersystems[5],photovoltaics[6],LEDs[7],automotiveglass[8],
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10173
electronicsdisplays[9],andfiberoptics[10].
Moth-eye structures can be imprinted by numerous methods, including wet or dry etch-
ing [11], nanoimprint lithography [12], or direct laser writing [13]. Sanghera et al. [10] and
MacLeodetal.[14]recentlydemonstratedadirectstampingmethodfornanoimprintingmoth-
eyestructuresontheend-facesofchalcogenideopticalfibers,whereARinterfacesarepartic-
ularly useful because of the large refractive index difference between air and As S (∼2.45
2 3
atawavelengthofλ =2µm).Thedirectstampingmethodallowsstructurestobeaccurately
replicated,andthereflectionasafunctionofwavelengthdependssensitivelyonthestructures
andtheirparameters[15],whichmeansthattheymustbeaccuratelymodeled.Becauseinthe
structuresstudiedinthiswork,themicrostructurefeaturedimensionsareonthesameorderas
thewavelengthoftheincomingradiation(∼1µm),lightinadjacentfeaturesinteracts,andthus
neitherthelong-wavelengthaveragerefractiveindexmodelnortheshort-wavelengthrayoptics
modelisappropriatetodescribetransmissionthroughmoth-eyestructures.Hence,theymustbe
modeledusingrigorouscomputationalmethods[16],suchasthefinite-elementmethod(FEM),
thefinite-differencetime-domainmethod(FDTD)[17,18],orrigorouscoupled-waveanalysis
(RCWA) [19,20], where the results become exact in principle as the grid size and step size
tendtozero(FEMandFDTD)orthenumberofharmonicsbecomesinfinite(RCWA).Wehave
simulatedmoth-eyestructuresusingthreemethods,FEM,FDTDandRCWA.Wefindthatthey
agreeperfectly,butthatRCWAisthefastestandmostaccuratemethod.
Moth-eyestructureshaveseveraladvantagesovertraditionalthin-filmARcoatings,includ-
ing environmental tolerance, surface adhesion, single material fabrication, minimal surface
preparation, and self-cleaning via the lotus effect [8,10]. Additionally, in recent years it has
beenshownthatinmanycasesperiodicmoth-eyestructureshaveahigherlaser-induceddam-
agethreshold(LIDT)thandotraditionalAR-coatedsurfaces[21–24].Weexplainedthiseffect
intermsoftheboundaryconditionsforMaxwell’sequations[25].
Inthiswork,weoptimizetheshapeanddimensionsofmoth-eyestructuresformaximumin-
putandoutputcouplingthroughtheendfacesofAs S opticalfibers.Wepreviouslypresented
2 3
some of these results in brief meeting reports [15,26], but here we show an in-depth investi-
gationofallrelevantmoth-eyestructureparameters,includingthestructureshape,dimensions,
period,packing,andtheangle-of-incidence.Therestofthispaperisorganizedasfollows:In
Sec.2, weverifyourtheoretical modelbycomparingour resultstoexperimentally-measured
results. In Secs. 3–5, we report our investigations of the effects of changing the dimensions
and period of the moth-eye structures (Sec. 3), changing the angle-of-incidence of the input
radiation (Sec. 4), and changing the shape of the moth-eye structure (Sec. 5). We have used
the insights of Secs. 2–5 to design and optimize moth-eye structures with nearly ideal 100%
transmission,andwereportthesedesignsinSec.6.Finally,Sec.7containstheconclusions.
2. Comparisonofcomputersimulationsandexperiment
The goal of the work described in this section is to create and validate a theoretical model
thatagreeswiththeresultsforaparticularmoth-eyestructurethatwasexperimentallystudied
by Sanghera et al. [10] and is shown in Fig. 1(c). We calculate the transmission of the struc-
turedsurfacebysimulatingaplane-waveincidentuponanAs S structuredsurfacetransmit-
2 3
tingmediumusingtheopen-sourcefinite-differencetime-domain(FDTD)computersoftware
MEEP [27]. The simulation propagates a wave-packet of specific bandwidth from a vacuum
incidentmediumthroughthemicrostructureandthencalculatesthetransmissionandreflection
spectrabytakingaharmonictransformofthetime-domainfluxthroughmeasurementsurfaces
thataresituatedaboveandbelowthemicrostructuredsurface.Weuseaspatialresolutionof10
nm for all simulations, and we have verified that increasing the resolution does not affect the
results.Weutilizetheperiodicityofthestructureandonlysolveforthefieldsinasingleunit
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10174
(c)
Fig.1.Schematicviewofamoth-eyeconestructure:(a)Sidecross-sectionview;(b)top
viewshowingaunitcell;(c)SEMwith1-µmscale-bar.
cell, using Floquet-Bloch boundary conditions. Additionally, for circularly symmetric struc-
turesinasquareorhexagonalpackingschemewithnormalincidence,itissufficienttostudy
a single polarization of incoming light, since the results do not depend on the polarization in
thatcase[28,29].Thecalculatedtransmissionspectrahavebeenverifiedbycomparisonwith
thecommercialsoftwarepackageDiffractMOD®byRSOFT,whichusestherigorouscoupled
wavealgorithm(RCWA).
TheexperimentaltransmissionspectrumwasmeasuredattheNavalResearchLaboratory.In
theexperiment,thefiberendfacewasembossedwithamoth-eyestructureusingashimwith
the negative of the desired moth-eye pattern. The transmission through the fiber before and
afterthemoth-eyestructurewasappliedweretestedusinganFTIRspectrometer.Detailsofthe
experimentalsetuptofabricatethemoth-eyestructureaswellasthemeasurementsaregiven
indetailin[10].
Theexperimentalshapeismodeledbyatruncatedconewithahemisphericcapthatisfully
determined by specifying the cone height, h; the base diameter, w ; the tip diameter, w ; and
2 1√
the lattice constant of the packing, s . Since the features are hexagonally packed, s = 3s .
x y x
ThesegeometricparametersareshowndiagrammaticallyinFigs.1(a)and1(b).Theconeap-
proximation to the experimentally-implemented feature has the following parameters, which
canbedeterminedfromtheSEMimage.w =0.2µm,w =0.7µm,h=0.9µm,s =0.92µm,
1 2 x
ands =1.59µm.Foralltheoreticalmodels,wesettherefractiveindexnofAs S tobeequal
y 2 3
to n=2.45, which is a good approximation to the experimentally-measured value over the
wavelengthrangeofinterest.
The calculated transmission spectra are shown in Fig. 2 for light that is coupled into and
outofanopticalfiber.Thespectraareverysimilaratlongerwavelengthsanddifferforshorter
wavelengths for reasons that will be explained shortly. The output coupling exhibits a sharp
falloffatshortwavelengths,whiletheinputcouplingdoesnot.Thetransitionfromhightolow
transmissionasthewavelengthdecreasesiscalledthediffractionedge,issharp,andoccursat
1.9µmforthisstructure.
Figure2alsoshowsthattheexperimentally-measuredtransmissionforlightcoupledoutof
thefiberandthedrop-offintransmissionforshorterwavelengthsmatchthetheoreticalresult.
The measured transmission spectrum is somewhat lower at longer wavelengths, which could
becausedbyaslightlydifferentconeshapeintheexperimentalsystemthaninourtheoretical
modeland/orbymateriallosses,whicharenotincludedinourtheoreticalmodel.
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10175
Fig. 2. The theoretical transmission spectra of light coupled into and coupled out of the
fiber. The experimental transmission spectrum of light coupled out of the fiber is also
shown. The Fresnel limit is the transmission spectrum from a plane-wave coupling into
oroutofafiberwithaflatend-face.
Fig.3.(a)Couplingoflightintoandoutofthefiberrepresentedgraphically.Theray-optics
picturesoflightcoupling(b)outofthestructure,and(c)intothestructure.
Finally, Fig. 2 shows the transmission through a bare fiber end-face with no AR structure
forcomparison.TheFresnel-limitedtransmissiondependsonlyontherefractiveindicesofthe
incidentandtransmittingmediumand,inthiscase,isapproximately83%.
2.1. Couplingintoandoutofafiber
Thecouplingoflightintothefiberandthetransmissionoutofthefiberaretwodifferentprob-
lems,asschematicallyshowninFig.3(a).Forlongwavelengths,thebehaviorofbothinputand
output coupling can be expected to be similar, as the electro-magnetic field will average over
the moth-eye surface, effectively homogenizing the refractive index transition and leading to
similartransmissionpropertiesinbothdirections.
For shorter wavelengths, we expect the transmission properties to be very different. In the
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10176
ray-optics limit, normally incident light will be totally internally reflected inside the cones in
the output coupling case, whereas light will be refracted directly into the fiber in the input
couplingcase,asshowninFigs.3(b)and3(c).Totalinternalreflectionoccurswhentheangle
thattheincidentlightraymakeswiththeconesurface,θ,isgreaterthanthecriticalangle,24◦,
asisthecasewithallofthepracticalconedesignsconsidered.
This argument gives an intuitive picture of how a structured surface can yield very differ-
ent behavior for light coupled into and out of the high index material. However, the coupling
at wavelengths close to the size of the structure is complicated due to the interaction of the
electromagneticfieldinadjacentconestructures.Thewavelengthatwhichtheinputandout-
putcouplingdifferiscalledthediffractionedge,becauseforwavelengthsbelowthediffraction
edge,diffractedordersgreaterthanzeroexistinthetransmittedspectrum.Fullnumericalsim-
ulations of Maxwell’s equations are therefore required to obtain a qualitative picture of the
transmissionspectraaswellasaquantitativeresult.
3. Changingthedimensions
Next,weinvestigatechangingthedimensionsofthetruncatedconetodeterminetheeffecton
thetransmissionspectrum.Wevaryeachparameterindependently,startingwiththetipwidth.
3.1. Tipwidth
The calculated transmission spectra for input and output coupling for different values of the
conetipdiameter,w ,areshowninFig.4.Allotherparametersarekeptfixedatw =0.7µm,
1 2
h=0.9µm,s =0.92µm,ands =1.59µm.
x y
As the cone tip diameter is increased from w =0.1 µm to w =0.7 µm, the transmission
1 1
forwavelengthsabove3.5µmincreases.However,forwavelengthsaround2.8µm,thetrans-
mission correspondingly decreases. For output coupling, the transmission drop-off near 2 µm
does not change significantly for different values of w . For input coupling, the transmission
1
decreasesbelow2µmasthetipdiameterincreases.
(a) (b)
Fig.4.Thetransmissionspectraoflightcoupled(a)intoand(b)outofthefiberasthecone
tipdiameterw varies.Theothergeometricparametersarew =0.7µm,h=0.9µm,and
1 2
sx=0.92µm.Theexperimentaltransmissionspectrumoflightcoupledoutofthefiberis
shownforcomparison(blackdashedline).
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10177
3.2. Basewidth
Inthesecondparameterstudy,wevariedthediameterofthebaseofthecone,keepingtheother
parameters fixed at w =0.2 µm, h=0.9 µm, and s =0.92 µm. The resulting transmission
1 x
spectraversuswavelengthareshowninFig.5.Weseethatforverynarrowcones,withw =
2
0.3 µm, the transmission is only slightly increased above the Fresnel limit of a flat interface,
whichisapproximately83%.Forbothinputandoutputcoupling,theslopesofthetransmission
spectrabecomeincreasinglysteep,decreasingfromalmost100%transmissionnear2.5µmto
around92%transmissionnear5µmatthelargestbasediameterthatweconsider,w =0.9µm,
2
where the base of the cones would be almost touching. For input coupling we see that as w
2
increases,thetransmissionincreasesforallwavelengths.Furthermore,foroutputcoupling,we
findthatasw increases,thecutoffat2µmbecomesstronger.
2
(a) (b)
Fig.5.Thetransmissionspectraoflightcoupled(a)intoand(b)outofthefiberversusthe
conebottomdiameterw .Theothergeometricparametersarew =0.2µm,h=0.9µm,
2 1
andsx=0.92µm.Theexperimentaltransmissionspectrumoflightcoupledoutofthefiber
isshownforcomparison(blackdashedline).
3.3. Height
Thetransmissionspectraversuswavelengthfordifferentconeheights,h,areshowninFig.6.
All other parameters are kept fixed at w =0.2 µm, w =0.7 µm, s =0.92 µm, and s =
1 2 x y
1.59µm.Thetransmissionabove3.5µmincreasesastheconeheightincreases;however,the
transmission also decreases slightly with increasing h near a wavelength of 3 µm. The same
behavior is seen for both input and output coupling. The height does not greatly affect the
transmission below 2 µm for input coupling; for output coupling the transmission increases,
although it is still considerably below the Fresnel limit of 83%, and the sharp cutoff is still
present.
3.4. Period
Thefinalgeometricparameterthatwechangeinourmodelisthelatticespacingortheperiodof
thecones,s .Thelatticepackingiskepthexagonal;deviationsfromthispackingbychoosing
x
s differently would cause different polarizations of light to have different transmission spec-
y
tra[28,29].Allotherparametersarekeptfixedatw =0.2µm,w =0.7µm,h=0.9µm.The
1 2
resultsareshowninFig.7.Wefindthatthetransmissionincreaseswithdecreasings .Thisre-
x
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10178
(a) (b)
Fig.6.Thetransmissionspectraoflightcoupled(a)intoand(b)outofthefiberversusthe
coneheighth.Theexperimentaltransmissionspectrumoflightcoupledoutofthefiberis
shownforcomparison(blackdashedline).
sultqualitativelyagreeswiththeresultsofchangingtheconebasediameter:Inbothcases,the
packing density of the cones increases, causing the interaction of the fields between cones to
increase,whichinturnincreasestheresultingtransmission.Theresultsforoutputcouplingare
showninFig.7(b).Inthiscase,wefindthatchangingthelatticeconstantchangesthecut-off
ordiffractionedgewavelength,whichincreasesass increases.
x
(a) (b)
Fig. 7. The transmission spectra of light coupled (a) into and (b) out of the fiber versus
theconehexagonalpackingspacingsx.Theexperimentaltransmissionspectrumoflight
coupledoutofthefiberisshownforcomparison(blackdashedline).
4. Angleofincidence
The effect of changing the angle of incidence of the input plane wave is shown in Fig. 8 for
angles up to 30◦ for the same moth-eye structure that was discussed in the previous section.
For input coupling, the transmission spectrum is almost unaffected by changes in the angle
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10179
of incidence. This result makes sense conceptually, as argued previously from the ray picture
in Fig. 3(c). For output coupling, the transmission drops significantly for angles of incidence
above10◦,goingtozeroforwavelengthsabove2µmforanincidenceangleof30◦.
For single mode fibers the low transmission at non-normal incidence for output coupling
wouldnotbeaproblemsincetheangleofincidenceforthefundamentalmodewouldbeclose
tonormal;however,formultimodefibersthisissueshouldbetakenintoconsideration.
(a) (b)
Fig. 8. The transmission spectra of light coupled (a) into and (b) out of the fiber versus
theincidenceangleoftheinputplane-wave,φ.Theexperimentaltransmissionspectrumof
lightcoupledoutofthefiberisshownforcomparison(blackdashedline).
5. Changingtheshape
Inthissection,westudytheeffectofchangingtheshapeofthemoth-eyestructureelement.The
shapeswestudyareahalf-ellipsoidalshape,asinusoidalshape,andatruncatedpyramid.The
truncatedpyramidelementisaflat-sidedsquarepyramidwiththetopcutoff.Thehalf-ellipsoid
structureistheupperhalfofanellipsoidorientedalongthez-axis,whoseheightisgivenbythe
equation
(cid:34) (cid:18) x (cid:19)2 (cid:18) y (cid:19)2(cid:35)1/2
z(x,y)=h 1− − , (1)
w /2 w /2
2 2
wherexandyaretheaxesparalleltothesurface,zistheheightperpendiculartothesurface,h
istheelementheight,andw isthebasewidth.Thesinusoidalshapeisgivenbytheupperhalf
2
ofacosinerotatedaroundthez-axis,whichhasaheight
(cid:40) π (cid:34)(cid:18) x (cid:19)2 (cid:18) y (cid:19)2(cid:35)1/2(cid:41)
z(x,y)=hcos + , (2)
w w /2 w /2
2 2 2
whereagainxandydenotetheaxesthatareparalleltothesurface,zdenotestheheightperpen-
diculartothesurface,histheelementheight,andw isthebasewidth.Weshowcrosssections
2
ofeachfeatureshapeinFigs.9(a)–(c).
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10180
Fig.9.Cross-sectionof(a)asinusoidalstructure,(b)ahalf-ellipsoidstructure,and(c)a
truncatedpyramidstructure.(d)Thepyramidalsurfacefromthetop,showingthepolariza-
tionanglesusedinthesimulations.
5.1. Positivestructures
First,westudiedpositivestructuresformedfromtheseelementshapes.Positivestructurescor-
respondtothegrayareasinFig.9beingAs S ,withthewhiteareas(background)beingair.We
2 3
compare the transmission spectra for positive elements of three different heights h=0.8 µm,
1.2µm,and1.6µmandtwobasewidthsofw =0.7µmand0.9µm.Thepyramidalelement
2
hasanextraparametercomparedtothesinusoidalandhalf-ellipsoidelements,whichisthetop
widthw ;forthefollowingcalculations,wesetw =0.15µm.Thepyramidalstructureisalso
1 1
notrotationallysymmetricaboutthez-axis,unliketheothertwostructures,andwilltherefore
havedifferenttransmissionpropertiesdependingonthepolarizationoftheincidentlight.
The calculated transmission spectra in the case of output coupling for the sinusoidal and
half-ellipsoidsurfacesareshowninFigs.10(a)and10(b)respectively.Forthesinusoidalsur-
face, the transmission is greater for larger base widths. The transmission is generally flatter
for taller elements, although there is little change between h=1.2 µm and h=1.6 µm. For
the half-ellipsoid surface, the transmission is greater for larger base widths and is flatter for
tallerelements,andthehighesttransmissionisobtainedwhentheelementiswidestandhasa
height h=1.2 µm or h=1.6 µm. However, for the half-ellipsoid surface with w =0.7 µm,
2
as the height is increased, the transmission actually drops below the transmission for shorter
elements.Asimilardipoccurswiththesinusoidsurface,butitislesssignificant.
ThecalculatedtransmissionspectraforthepyramidalsurfaceareshowninFig.11(a).The
highest and widest pyramidal structure of h=1.6 µm and w =0.9 µm performs the best of
2
all the shapes considered and has a transmission of almost unity across the wavelength range
λ =2–5µm.
Figure11(b)showsthetransmissionfordifferentpolarizationstatesoftheincomingelectric
field. A polarization angle of φ =0◦ corresponds to the electric field parallel to the sides of
thebaseofthepyramidalandapolarizationangleofφ =45◦ correspondstotheelectricfield
aligneddiagonallytothebaseofthepyramidal,asshowninFig.9(d).Thetransmissiondrops
slightlywhenφ =45◦,comparedtowhenφ =0◦.Weexpectthatthetransmissionforallother
polarizationsliesinbetweentheseextremecases.
Inmostcases,wefoundthatincreasingthebasewidthandheightoftheelementincreased
the transmission across the wavelength range. However, this behavior was not found for all
element shapes in all height ranges; for example, the transmission for the half-ellipsoid with
basewidthofw =0.7µmisworsefortallerelements.Also,forboththesinusoidalandhalf-
2
ellipsoid elements, there is no improvement in transmission when the height increases from
#258966 Received 11 Feb 2016; revised 20 Apr 2016; accepted 25 Apr 2016; published 2 May 2016
© 2016 OSA 2 May 2016 | Vol. 24, No. 9 | DOI:10.1364/OE.24.010172 | OPTICS EXPRESS 10181