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14 Pages·2009·0.48 MB·English

Discontinuous Galerkin time-domain
computations of metallic nanostructures
KaiStannigel1,2,3,MichaelKo¨nig1,2,JensNiegemann1,2,andKurt
Busch1,2
1Institutfu¨rTheoretischeFestko¨rperphysik,Universita¨tKarlsruhe(TH),76128Karlsruhe,
Germany
2DFG-CenterforFunctionalNanostructures(CFN),Universita¨tKarlsruhe(TH),76128
Karlsruhe,Germany
3InstituteforTheoreticalPhysics,UniversityofInnsbruckandInstituteforQuantumOptics
andQuantumInformation,AustrianAcademyofSciences,Technikerstraße25,6020
Innsbruck,Austria
[email protected]
Abstract: We apply the three-dimensional Discontinuous-Galerkin
Time-Domain method to the investigation of the optical properties of bar-
and V-shaped metallic nanostructures on dielectric substrates. A flexible
finite element-like mesh together with an expansion into high-order basis
functions allows for an accurate resolution of complex geometries and
strong field gradients. In turn, this provides accurate results on the optical
response of realistic structures. We study in detail the influence of particle
size and shape on resonance frequencies as well as on scattering and
absorption efficiencies. Beyond a critical size which determines the onset
of the quasi-static limit we find significant deviations from the quasi-static
theory. Furthermore, we investigate the influence of the excitation by
comparing normal illumination and attenuated total internal reflection
setups. Finally, we examine the possibility of coherently controlling the
localfieldenhancementofV-structuresviachirpedpulses.
© 2009 OpticalSocietyofAmerica
OCIScodes: (000.3860)Mathematicalmethodsinphysics;(000.4430)Numericalapproxima-
tionandanalysis;(240.6680)Opticsatsurfaces:Surfaceplasmons(260.3910)Physicaloptics
:Metaloptics;(290.5850)Scattering:Scattering,particles;(310.6628)Thinfilms:Subwave-
lengthstructures,nanostructures
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#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14935
1. Introduction
Therelevantpropertyofmetalswithrespecttotheiropticalresponseisthelargedensityoffree
electrons.Underappropriateexcitationconditions,theseelectronsundergocoherentcollective
oscillationsrelativetoaratherimmobilebackgroundofpositivelychargedions.Theconfined
geometry provided by metallic nanostructures allows for the resonant enhancement of a dis-
cretesetofso-calledparticleplasmonresonances.Theseleadtolargeelectricfieldsandstrong
field gradients which may be exploited for numerous applications. In an optical context, the
excitationmayberealized,forinstance,viaanexternallightsource.Theresultingstronglylo-
calizedfieldsleadtoenhancedfluorescencefrommolecules[1],facilitatetip-enhancedRaman-
scatteringspectroscopytechniques[2]andon-chipgenerationofhighharmonics[3].Theyeven
allow for coherently amplified radiation based on novel principles such as the spaser [4] and
thelasingspaser[5].Similarly,thestrongfieldgradientsassociatedwithplasmonicnanostruc-
tures start to find applications in novel types of optical tweezers [6–8]. Furthermore, it has
recently been demonstrated experimentally that the utility of metallic nanostructures may be
considerablyenhancedthroughcontrolledexcitationwithshapedbeams[9].Infact,thecurrent
state-of-the-art in realizing beams with prescribed temporal profile in phase, amplitude, and
polarizationwillallowafar-reachingcontroloverthespatio-temporallocalizationofradiation
fortheapplicationsdiscussedabove.
This clearly demonstrates an elevated need for quantitative theoretical understanding on
various levels of sophistication. In fact, a large forest of theoretical and computational meth-
ods has been brought to bear on this problem. These include the choice of material models
and approaches to improve the accuracy of representing the nanostructures’ geometries and
associated electromagnetic fields. For instance, on the linear optical level, metals may be de-
scribedasdispersiveand,hence,absorptivedielectricmaterialswithspatiallylocalresponses.
InmanyrelevantcasesthisamountstoemployingthecelebratedDrudeorDrude-Lorentzmod-
els [10]. On the other hand, if non-radiative processes such as the charge-transfer between
molecules and metallic nanostructures in surface- or tip-enhanced Raman scattering become
relevant,fullyquantummechanicalcalculations,e.g.,viaTime-DependentDensity-Functional
Theory (TDDFT) can be employed [11]. Clearly, more complex material models quickly re-
quirerathersignificantcomputationalresourcessothattypicallyonlyverysmallmetalclusters
canbetreatedviaTDDFT.
Even within a linear material model, there exists a large variety of approaches that range
fromthequasi-staticapproximationforsufficientlysmallparticles[12–14]allthewaytofull
Maxwellsolvers(forarecentreview,see[15]).Broadlyspeaking,thelattermaybedividedfur-
therintofrequency-domainandtime-domainmethods.Mostfrequency-domainmethodssuch
asBoundaryElementMethods[16],MultipleMultipoleMethods[17],andtheFiniteElement
Methods[16]lendthemselvestohighlyaccuraterepresentationsofcomplexgeometries.Many
different systems have been studied—mostly driven by what can be realized experimentally
and often guided by antenna design principles [7,15,18–21]. However, certain applications,
for instance those that employ shaped beams [9] and/or coherent control principles [14,22],
are more naturally treated within a time-domain framework. Surprisingly and in contrast to
the numerous frequency-domain methods, the choice of time-domain methods for the treat-
mentofnano-photonicproblemsisratherlimited,theFinite-DifferenceTime-Domainmethod
undoubtedlybeingthemostpopularone[23].
Inthispaper,wepresentouranalysisoftheopticalpropertiesofexperimentallyfeasiblebar-
andV-shapedmetallicnanostructuresusingourimplementation[24]oftherecentlydeveloped
three-dimensionalDiscontinuous-GalerkinTime-Domain(DGTD)method[25].Inessence,the
DGTDapproachcombinesafinite-element-likerepresentationofcomplexgeometrieswithan
explicit time-stepping capability: An adaptive high-order spatial discretization allows for the
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14936
useofacorrespondinghigh-ordertime-integrationschemesothattheresultingsolverisvery
wellsuitedforthequantitativeanalysisofcomplexnano-photonicsystems.
Theremainderofthispaperisstructuredasfollows:Insection2,wepresentabriefintro-
ductiontotheDGTDmethod,whileavalidationofourimplementationisprovidedinsection3,
wherewecomparenumericalresultsformetallicsphereswithanalyticMietheory.Insections
4and 5,we analyze, respectively, nanoscopic silverbarsand V-shaped structureswithregard
totheirsize-dependentpropertiessuchasresonancefrequenciesandattainablefieldenhance-
mentsundervariousexcitationconditions.
2. ThediscontinuousGalerkintime-domainmethod
Inthissection,weprovideabriefoutlineofthebasicideasunderlyingtheDGTDmethod.For
moredetailsandtherelevantformulas,wereferthereadertoRefs.[24–26].
The key feature of the DGTD method is that it allows to use an explicit time-stepping
schemeincombinationwithafinite-elementdiscretization.Thestartingpointofeachcalcula-
tionisthusthetessellationofthecomputationaldomainbyfiniteelements,whichinourcase
are straight-sided tetrahedra. Then, on each element the components of the electromagnetic
fieldareexpandedintopolynomialsofadjustableorder p.Incontrasttotheclassicalfiniteele-
mentmethod[16],thisexpansionispurelylocalandthefieldsareallowedtobediscontinuous
across element boundaries. In order to facilitate the coupling between neighboring elements,
oneintroducesanadditionalpenaltyterm,whichweaklyenforcesthephysicalboundarycon-
ditions.
As a consequence of this procedure, we obtain a large system of coupled ordinary differ-
entialequations(ODEs)fortheexpansioncoefficientsoftheelectromagneticfield.Sincethis
systemconsistsofsmallindependentblocksrelatedtotheindividualelements,itcanberead-
ily integrated using an explicit time-stepping scheme. For the calculations presented in this
work, we employed a standard 4th-order low-storage Runge-Kutta algorithm as described in
Ref.[27].However,sincewithinDGTDthetime-steppingisessentiallydisentangledfromthe
spatialdiscretization(asopposedtothesituationinFDTD),moresophisticatedmethodssuch
as Krylov-subspace based operator exponential techniques [28] could provide an even more
efficientsolver.
Forpracticalcomputations,thebasicalgorithmsketchedaboveneedstobecomplemented
by various extensions. For instance, Lu et al. [26] have shown how to implement dispersive
materialsandperfectlymatchedlayer(PML)absorbingboundaryconditionsviaauxiliarydif-
ferentialequations.TheperformanceofthePMLshasbeenoptimizedinRef.[24].Finally,the
well-knowntotal-field/scattered-fieldapproach[23]fortheinjectionofpulsesintoascattering
configurationcandirectlybeusedinDGTD.
3. Scatteringbyametallicsphere
In order to validate our implementation and as a benchmark problem we calculate the total
scattering and absorption cross sections of a metallic sphere. The dispersion of the metal is
modeledasastandardDruderesponsegivenby
ω2
ε(ω)=1− d ,
ω(ω−iγ)
d
wheretheplasmafrequencyω =1.39·1016s−1 andthecollisionrateγ =3.23·1013s−1 cor-
d d
respondtothevaluesgivenbyJohnsonandChristy[10]forsilver.However,itshouldbenoted
thatthesevaluesunderestimatethelossesintheultravioletregime.Wetakethespheretohave
aradiusof50nmandtobeembeddedinvacuum.
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14937
a) b)
0.08 0.014
2μScattering cross section [m] 000...000246 MDGieT tDheory 2μAbsorption cross section [m] 000000.....00000.000001124682 MDGieT tDheory
0 0
100 200 300 400 500 100 200 300 400 500
Wavelength [nm] Wavelength [nm]
Fig.1.(a)Scatteringand(b)absorptioncrosssectionsofasilverspherewithradius50nm
embeddedinvacuum.Thecomputationsinvolve2750tetrahedraforthesphereitsselfand
havebeenperformedwithinterpolationpolynomialsoforderp=3.Theresultsagreewell
withanalyticMietheory.Seethetextforfurtherdetails.
A sketch of the computational setup is shown in Fig.2(a): In order to simulate an open
system,thecomputationaldomainisterminatedbyuniaxialperfectlymatchedlayers(PMLs)
[24].Furthermore,itisdividedintoatotal-fieldandascattered-field(TF/SF)region,whereat
theinterfacebetweentheseregionsabroadbandpulseisinjectedintotheTFregion[23].At
the same interface we record the Fourier components of the Poynting vector both inside and
outside the TF region. An integration and subsequent normalization to the incident spectrum
thenyieldsscatteringandabsorptioncrosssections,respectively.
When carrying out such computations one has considerable freedom in choosing various
parameters.Wehaveperformedourcomputationswithvaryingordersofthespatialdiscretiza-
tion(p=3,4),differentnumbersoftetrahedraforthesphere(infivestepsfrom160to2750)
and different sizes of the PML region (either one or two cells). Although we find that the in-
terpolation order and the size of the PML do have a slight influence on the accuracy of the
result,theerrorisstronglydominatedbythetetrahedrizationofthesphere.Thisresulthasbeen
anticipated, since we are using straight-sided elements which essentially lead to a polygonal
approximationofthesphere’sshape.InFig.1,wedisplaytheresultsforthescatteringandab-
sorptioncrosssectionsforacomputationwhenthesphereismeshedwith2750tetrahedraand
whereathird-orderspatialdiscretizationisused.AcomparisonwithanalyticMietheory[29]
showsthattherelativepointwiseerrorofthespectrumisbelow2%exceptintheregionofthe
highestresonance,wheretheerrorissignificant(upto10%)evenforthefinestdiscretization.
Thisindicatesthatforthereproductionoftheexactpositionandstrengthofhigh-Qresonances,
oneshouldresorttocurvedelementsormuchfinerdiscretizations.Forlower-Qresonancesas
wellasforthegeneralstructureofthespectrumthepresentedcalculationsyieldwell-converged
results.
4. Opticalpropertiesofsilvernano-bars
Asamorepracticalexamplewestudysilvernano-barsthatarepositionedonaglasssubstrate.
Such structures may serve as optical antennas for concentrating light on the sub-wavelength
scaleand,thus,areofconsiderableinterestforapplicationsthatincludenonlinearspectroscopy
oropticaltweezing.
InFig.2(b)wedisplaythebar’sgeometricmodelwhichfeaturesarealisticroundingwith
radius r =w/2 at the endings of the bar. In this context, we would like to point out that
end
thefinite-elementdiscretizationallowsustomodeltheroundingwithoutstaircasingaswould
be the case within standard FDTD. While this might be of limited relevance in the case of a
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14938
a) b)
Fig.2.(a)Sketchofthegeneralcomputationalsetupforcomputingtheopticalpropertiesof
metallicnanostructures.(b)Dimensionsofanano-barthatispositionedonaglasssubstrate.
ThetwopointsAandBmarktherecordingsitesfortheelectricfieldenhancements(notto
scale).
bar-shapedobject,itcertainlyisofhighrelevanceforaV-shapedobjectasdiscussedinsection
5.
For this setup, we compute the field enhancement factors upon plane wave excitation and
alsoinvestigatethequestionoftheonsetofthequasi-staticlimit,i.e.,wedeterminethemax-
imumsizeforwhichthepropertiesofthebararewelldescribedbyapurelyelectrostaticcal-
culation.Infact,localfieldenhancementsnearrod-shapedorspheroidalmetallicnanoparticles
havebeenconsideredbeforebyotherauthors(seee.g.Refs.[18,19,30,31]).However,ourcom-
putationstakeintoaccountthatinmostexperimentalrealizationsnano-particlesaredeposited
onto a dielectric substrate. Furthermore, we determine the dependence of the achievable en-
hancementonparticlesizewhichservesasareferencefortheV-structurethatweconsiderin
section5.
4.1. Localfieldenhancement
For the computation of the field enhancement just before the tips of a nano-bar we proceed
as follows: We determine the (lowest) resonance frequency f of the structure by recording
res
thescatteringandabsorptioncrosssectionsasdescribedinsection3.Then,inasecondcom-
putation, we irradiate the structure by a monochromatic plane wave with the very resonance
frequency,wheretheamplitudeisslowlyrampedupfromzerotoitsfinalvalue.Thistime,we
recordthevaluesofthefieldsinfrontofthetip.Thesimulationisstoppedwhentheenhance-
menthassaturated.Forthesystemsdescribedbelow,thistakesapproximately30opticalcycles
(cf. Fig.3(a)). We have carried out these computations for bars of fixed widths and heights,
w=h=20nm, and whose lengths range between l =100–200nm. Figure3(b) shows the ex-
tractedenhancements10nmabovethesubstrateatsitesAandBthatarelocated1nmand5nm
awayfromthetip,respectively(c.f.Fig.2(b)).Tocheckwhetherourresultsareconverged,we
have performed all calculations with interpolation orders p=2 and p=3. Finally, for com-
pleteness we provide in Table 1 the wavelengths λ corresponding to the lowest resonance
res
frequenciesaswellastherespectivequalityfactorsQfordifferentbars.
4.2. Quasi-staticlimit
Inspired by the popularity of quasi-static calculations of the optical properties of metallic
nanostructures and the numerous exciting predictions that have been made for such systems
[12–14,32], we seek to find the maximum size of a nano-bar for which such a description is
still adequate. We will present the discussion in terms of the experimentally accessible scat-
tering and absorption efficiencies, which we obtain by normalizing the corresponding cross
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14939
a) b)
100 90
Point A
80 Point B
E(t)/E at point Ayy,inc −55000 Field enhancement567000
40
−100 30
0 10 20 30 40 100 125 150 175 200
Time [1/fres] Length l [nm]
Fig.3.Localfieldenhancementsforsilvernano-barsonglasssubstrates:Panel(a)displays
thesaturationoftherecordedtimesignalforthey-componentoftheelectricfieldatpoint
A(barlengthl=150nm).Panel(b)showstheextractedvaluesofthefieldenhancementat
pointsAandBforbarsofdifferentlengthsl(seeFig.2forfurtherdetailsofthesystem).
Table1.Lowestfrequencyresonancewavelengthsλres andcorrespondingqualityfactors
Qfornano-barswithdifferentlengthslandfixedwidthandheight,w=h=20nm.
l[nm] 100 125 150 175 200
λ [nm] 750 880 1008 1136 1260
res
Q 22.2 20.6 19.4 18.7 18.1
sectionstothegeometriccrosssectionoftheobjectnormaltotheincidentbeam.
Beforeanalyzingthenano-bars,itisinstructivetofirstconsiderthecaseofametallicsphere
withpermittivityε(ω)andunitpermeabilityforwhichtheefficienciesinthequasi-staticlimit
canbedeterminedanalyticallyas[33]
(cid:2) (cid:2) (cid:3) (cid:4)
Qsca= 38x4(cid:2)(cid:2)(cid:2)εε((ωω))−+12(cid:2)(cid:2)(cid:2)2, Qabs=2xIm εε((ωω))−+12 forx=ka(cid:2)1.
Here,kisthewavevectoroftheincidentlightandatheradius,i.e.,thecharacteristiclength,of
theparticle.Theaboveexpressionsinformusthatthespectralprofileoftheefficiencies—andin
particularthepositionoftheresonance—isindependentoftheparticlesize,butratherdepends
ontheparticleshape.Thesizemerelyinfluencesthemagnitudesoftheefficiencies.Infact,this
statementistrueforscatterersofarbitraryshape.Thus,thequasi-staticlimitisreachedifthe
resonancefrequencyoftheparticleissizeindependentandtheefficienciesscaleasQ ∝a4
sca
and Q ∝a with the particle’s characteristic length a. Note that we are not considering the
abs
crosssectionswhichwouldscalewithanadditionalfactorofa2each.
In order to address the question whether the nano-bars are in the quasi-static limit, we
perform a scaling analysis and check for the above criteria: We start with a specific structure
andcomputeitsspectralresponseneartheresonancefrequency.Then,wescalealldimensions
by a common factor s<1, and repeat the computations. The behavior of the efficiencies as
a function of the scaling factor s can be used to detect the onset of the quasi-static limit. For
simplicity, we reduce the shape of the nano-bar to a rectangular box with sharp corners. This
reducesthecomputationaleffortsincesmalltetrahedraintheroundedendingsofthebarslimit
the simulation time-step. At the same time, while details of the individual structures slightly
differ,thescalingbehaviorforroundedbarsandbarswithsharpcornerscanbeassumedtobe
thesame.
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14940
a) b)

Scattering efficiency [a.u.]111000−−042 Absorption efficiency [a.u.]12345 100000000 ........(75432110l=5 5 5((((( 1lllll=====(((0lll===54321070000105n5nnnn5nnmnmmmmnmm)mm))))))))
10−6 0
500 550 600 650 500 550 600 650
Wavelength [nm] Wavelength [nm]
c) Peak scattering efficiency [a.u.]1111100000−−−01321 ∝ s4 FSiitmteudla etixopnosnent = 3.97 d) Peak absorption efficiency [a.u.]0.12485 ∝ sFSiitmteudla etixopnosnent = 0.98
0.05 0.1 0.150.2 0.3 0.40.5 0.75 1 0.05 0.1 0.150.2 0.3 0.40.5 0.75 1
Scale factor s Scale factor s
Fig.4.Scalinganalysisforanano-barwithreferencegeometryparametersl=100nmand
w=h=25nm (see Fig.2). The graphs show the behavior of (a) the scattering and (b)
theabsorptionefficienciesuponscalingofthisreferencegeometrybyacommonfactors,
whichisgivenforbothpanelsinthelegendof(b).Panels(c)and(d)showthepeakvalues
of the respective efficiencies as a function of the scaling factor on a double-logarithmic
scale.Alinearfittothethreedatapointsthatcorrespondtothesmalleststructuresisused
todeterminetheexponentofthepower-lawdependence.Theunitsfortheefficienciesare
thesameinallgraphs.
Onthetechnicalside,weproceedasintheprevioussectiontodeterminethecrosssections.
When scaling the system down, we rescale only the bar itsself, while we keep the computa-
tionaldomainunchangedinsize.ThisensuresthatthePMLsdonotbecomedrasticallysmaller
than the resonant wavelength and, therefore, our procedure avoids spurious reflections. Fur-
thermore, we design the mesh in such a way that bars with different scaling factors s consist
of approximately the same number of tetrahedra so that we do not loose any accuracy. As a
result, the simulation timestep (which is proportional to the size of the smallest tetrahedron)
isproportionaltothescalingfactorand,thus,therequiredCPU-timeisinverselyproportional
tothescalingfactor.Inordertosavecomputationalresources,thetwosymmetryplanesofthe
problemwereexploitedtoreducethenumberofelementsbyafactorof1/4.
InFig.4,wedisplaytheresultsforthescalingofabarwithareferencegeometryof100×
25×25nm3 down to 5% of its original size. For simplicity, we omit the dielectric substrate
in these calculations. From the overview plots Fig.4(a) and Fig.4(b), we observe the well-
knownfactthatabsorptioninsmallparticlesdominatesoverscattering.Withregardtothefirst
criterion formulated above, we notice that the position of the resonance saturates for scaling
factorss≤s ≈0.15.Regardingthesecondcriterion,wededucethepowerlawdependence
crit
ofthecrosssectionsfrompanels(c)and(d).Consistentwiththefirstcriterion,wefindthesame
criticalscalingfactors ≈0.15.Therefore,weconcludethatforthesetupconsistingofsilver
crit
nano-barsinvacuum,thequasi-staticlimitinthesensedefinedabovesetsinforstructureswith
spatialextensionsassmallasl(cid:2)15nm.
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14941
a) b)

60 A (1nm in front of tip)
55 B (5nm in front of tip)
nt
me50
e
nc45
a
h
en40
d
Fiel35
30
25
100 125 150 175 200
Arm length [nm]
Fig.5.(a)SketchofthesetupusedfortheV-structurecomputations.Theradiiofthetips
andthearmedgesaretakentobehalfofthearmwidthwunlessotherwisenoted.Forthe
calculationofthefieldenhancementsthestructureisplacedonaglasssubstrateandthe
fieldsarerecordedatpointsAandB.Again,A(B)islocatedhalftheheightofthemetal
filmabovethesubstrateand1nm(5nm)awayfromthetip.(b)Summaryoftheextracted
enhancementsatthetwositesAandBforV-structuresofdifferentlengthsl,fixedwidth
andheight,w=h=20nm,andfixedapexangleα=30◦.
5. OpticalpropertiesofsilverV-shapednanostructures
Asasecondexample,weconsiderV-shapedstructures,whichhavebeenextensivelyanalyzed
by Stockman et. al. [12,13] using a quasi-static approach. In fact, these authors have found
that such systems lend themselves to realizing strong local-field enhancements at their tips.
Inviewofourscalinganalysisforthenano-barsandtheresultingexperimentalchallengesin
fabricating high-quality structures of corresponding size, we perform our analysis for the full
Maxwell equations. This allows us to understand how the results of the quasi-static analysis
manifestthemselvesinexperimentallymoreaccessiblesystems.
InFig.5(a)wedepictthesetupthatweusefortheV-structurecomputationsthroughoutthis
section.Exceptforthenanostructurethecomputationalsetupincludingthematerialparameters
is the same as in the previous section. We regard this setup as quite realistic since it includes
smoothedgeswhichinevitablyarisewhensuchstructuresarefabricatedviafocussedionbeam
facilities or using electron beam lithography. Again, we want to point out that the adaptive
finite-elementdiscretizationallowsustomodelthestructurewithoutintroducingunphysically
sharpedges.Incontrasttothebar-geometry,theroundededgesatthetipsoftheVsdo,however,
makeasignificantdifferencewhencomparedtothecorrespondingcaseswithinfinitelysharp
tips.
5.1. Localfieldenhancement
In analogy with the analysis of the nano-bar structures, we first determine the local field en-
hancements at positions A and B that are, respectively, located 1nm and 5nm in front of the
tip and h/2 above the glass substrate. We list the resonance wavelengths and correspond-
ing Q-factors for the lowest frequency resonance of V-structures with different lengths, fixed
w=h=20nm,andfixedapexangleα=30◦onaglasssubstrateinTable2.
InFig.5(b),wedisplaythecorrespondingresultsforthefieldenhancementsatpositionsA
andB.Itisobviousthatthegeneraltendencyisthesameasinthecaseofthenano-bars,namely
thattheenhancementincreasesnotablywithparticlesize.Remarkably,thelocalfieldenhance-
ments attainable for V-structures under plane wave illumination are slightly below those of
nano-barsofsamelengthsandtipradii(c.f.Fig.3(b)).
For the V-structure, the tip radius can be varied independently of the arm width. We have
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14942
Table2.Resonancewavelengthsλres andqualityfactorsQforthelowestfrequencyreso-
nanceofV-structuresofdifferentlengthsonaglasssubstrate.Forallstructures,wehave
fixedparametersw=h=20nmandα=30◦.
l[nm] 100 125 150 175 200
λ [nm] 662 772 884 994 1106
res
Q 14.6 13.1 12.3 11.7 11.4
exemplarily computed the field enhancement for V-structures with fixed w=h=20nm and
l=150nmandtipradiiofr =8nmandr =12nm.Asexpected,theenhancementincreases
tip tip
with decreasing radius and the resulting field enhancements deviate from those with r =
tip
w/2=10nmbyapproximately±5-8%.
InadditiontothefieldenhancementsatpointsAandBinfrontofthetipwealsoextracted
thecompletefielddistributionsonresonance.InFig.6,wedisplaytwocorrespondingintensity
maps:Panel(a)depictstheintensitydistribution|E(x,y)|2 inthecenterplaneofthestructure,
i.e.,10nmabovethesubstrate,andpanel(b)depictstheintensitydistributioninaplane30nm
abovethesubstrate.Thesecomputationsrevealatotalofthreehotspots,twoattheendsofthe
armsandoneattheendofthetipofthenano-V.
Apartfromtheenhancementsdeterminedwithacontinuouswaveexcitation,wehavealso
performed computations with an ultrafast excitation via a narrow-band Gaussian pulse. The
temporal envelope of these pulses was exp(−t2/2σ) and their spectra were centered at the
resonance frequencies of the respective structures. The temporal widths σwere chosen to be
threeopticalcycles.However,accordingtothediscussionofsection4,thisdoesnotsufficefor
the enhancement to saturate. As a result, the recorded peak enhancements have been about a
factorof2/3smallerthanthoseobtainedwiththemonochromaticexcitation.
5.2. ATIR-vs.NIT-spectroscopy
Fromtheexperimentalside,thereexisttwospectroscopicapproachesforanalyzingindividual
nanostructuresthatarepositionedonasubstrate.Themoretraditionalapproachiscommonly
referredtoasattenuatedtotalinternalreflection(ATIR)spectroscopyandexploitsthefactthat
the presence of a nano-particle at the substrate-air interface leads to scattering of radiation
a) b)
Fig. 6. Intensity distributions |E(x,y)|2 normalized to unit illumination for a monochro-
maticon-resonanceexcitationofasilverVwithequalwidthandheightw=h=20nmand
armlengthl=150nm.Panels(a)and(b),respectively,depicttheintensity10nmand30nm
abovethesubstrate.Notethattheintensityisdisplayedonalogarithmicscale.
#110638 - $15.00 USD Received 27 Apr 2009; revised 12 Jun 2009; accepted 26 Jun 2009; published 7 Aug 2009
(C) 2009 OSA 17 August 2009 / Vol. 17, No. 17 / OPTICS EXPRESS 14943