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14 Pages·2009·0.6 MB·English

Extremely short-length surface plasmon
resonance devices
MaximL.Nesterov1∗,AlexandreV.Kats1andSergeiK.Turitsyn2
1A.Ya.UsikovInstituteforRadiophysicsandElectronicsNASofUkraine,
12AcademicianProskuraStreet,61085Kharkov,Ukraine.
2PhotonicsResearchGroup,SchoolofEngineeringandAppliedScience,AstonUniversity,
BirminghamB47ET,UnitedKingdom.
∗Correspondingauthor:
[email protected]
Abstract: The impact of the system design on the control of coupling
betweenplanarwaveguidemodesandsurfaceplasmonpolaritons(SPP)is
analyzed.Weexaminehowtheefficiencyofthecouplingcanbeenhanced
by an appropriate dimensioning of a multi-layer device structure without
usingadditionalgratings.Wedemonstratethatbyproperdesignthelength
of the device can be dramatically reduced through fabrication a surface
plasmon resonance sensor based on the SPP-photon transformation rather
thenonSPPdissipation.
© 2008 OpticalSocietyofAmerica
OCIScodes:(240.6680)Surfaceplasmons;(060.2370)Fiberopticssensors
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1. Introduction
Recentadvancesintheapplicationofsurfaceplasmonpolaritonsinthefieldsofchemicaland
bio-sensorsandnano-sciencehasstimulatedboomingresearchanddevelopment.Surfaceplas-
mon resonance(SPR) based optical bio-sensorsexploita special type of the electromagnetic
surfacewavescoupledtothecollectiveelectronexcitation—surfaceplasmonpolaritons(see
e.g.[1–6])toprobeinteractionsbetweenananalyteunderstudyandabio-molecularrecogni-
tionelementimmobilizedontheSPRsensorsurface.Oneofthemostimportantcharacteristics
oftheSPPs is theirabilitytodetectsmallscale effectsandfieldinteractionsat theinterfaces
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betweenametalandanon-metal.Thesurfaceplasmonpolaritonsattractagreatdealofatten-
tion as a quite unique possibility of the electromagneticfield localization and corresponding
substantial enhancementof the electric field nearthe surface. Sucha high levelof a field lo-
calizationresultsintotheenhancedsensitivityoftheSPRsensors.Specifically,SPPsinsand-
wichedstructureshaveattractedmuchinterestasapromisingwaytoimplementsurfaceplas-
monresonanceopticalsensors[7–11];waveguide-coupledSPRsensorwasfirstdemonstrated
byHarrieKreuwelin[12].Manyofthekeypropertiesofsuchsensors,usingmulti-layersand-
wichedstructures,aredeterminedbytheinteractionbetweenawaveguideandSPPmodes.The
progressintheareaisveryfastandsomeproductsarealreadycommerciallyavailable.There
isbroadrangeofpossibleschemesandconfigurations,anddespitesuccessfulfirstdemonstra-
tionsofsuchdevices,manydesignproblemsarestillopenandinthispaperweexaminesome
oftheseissues.
WewouldliketopointoutthatSPP-basedtechnologyhasaverybroadspectrumofpotential
applications. A feasibility of exploiting the high spectral dispersion achievable in plasmonic
devicesinopticalcommunicationhasbeenrecentlydiscussedin[13].Surfacewavesplayan
importantroleinmanyfundamentalresonantphenomena,suchase.g.the”Woodanomalies”
in thereflectivityandtransmissivity[1,14,15]ofperiodicallycorrugatedmetalsamples, and
are already practically exploited in a wide range of practical devices. A very important area
of the SPP applicationsis in miniaturizationand integrationof optical circuits. A possibility
of high localization of electromagnetic fields using SPP and subwavelength metal structures
makes it possible to overcomethe diffractionlimit of light waves whichis a critical issue in
thedevelopmentofphotoniccircuits.Considerableeffortshavebeenrecentlyplacedintothe
developmentofplasmonsubwavelengthguidingandcouplingdevices[16–20].
Apromisingdirectioninthedevelopmentof SPP-baseddevicesandtechniquesistocom-
bine surface plasmon approaches with well developed dielectric waveguide technologies, in
particularwithfiber-optics[21–24].TheconventionalmodernSPRfibersensorsutilizeBragg
gratingsorlongperiodgratingsto couplelightto the SPPs. Recentlythe couplingefficiency
betweenthecoremodeofthewaveguideandSPPmodesofthemetalfilminplanarandcylin-
drical geometry for sensing purposes have been examined in [25–29]. A variety of possible
configurations of multi-layered structures offers a range of potential device designs and this
fieldis still farfrombeenfullyexplored.EfficiencyofthecouplingbetweenSPP modesand
thewaveguideisthekeytounlockingthefullpotentialofsuchsandwichedmulti-layerstruc-
tureddevices.ThereexisttwokeypossibilitiestocoupleSPPandwaveguidemodes.Thefirst
oneistousethegratingwithaproperreciprocalvectortofulfillthephasematchingconditions.
The second one is through design, using appropriate materials and layers thicknesses at the
givenoperationalwavelength,whenthephasematchingconditionsaresatisfiedduetothelight
tunnelingthroughthebufferlayer.InthisworkweanalyzethebasicpropertiesoftheSPPand
waveguide mode coupling and consider a possibility to optimize the SPR-sensors by system
designthroughtuningdeviceparametersratherthanthroughadditionalgratingswritteninthe
waveguide.Understandingof the details of the interactionof the SPP and waveguide modes
isacriticalpointinthedevelopmentofthehighsensitivityreal-timeSPR-sensorandourgoal
hereistoanalyzesuchinteractionforrathergeneralconfigurationthatcanberelevanttomany
practicalsituations.
2. Fieldsrepresentationandboundaryconditions
ConsidereigenmodesofthemultilayerstructuredepictedinFig.1thatistypicalfortheSPP-
baseddevices.Theindicesc,b,w,−and+inwhatfollowsrelatetothecorrespondinglayers
asshowninFig.1.TakingintoaccountthatSPPisaTM-polarizedwaveand,therefore,inthe
geometryunderconsiderationitcaninteractonlywithTM-polarizedwaves,weconsiderTM-
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polarizedeigenmodes(E={E ,0,E },H={0,H,0})ofthestructure.Thetimedependence
x z
ismovedoutoftheequationsbythecorrespondingFouriertransformandanalysisoftheplane
waves having the form exp(−iωt). We are interested here in solutions that present outgoing
ordecayingwavesbothintheupperhalf-planez≤0(τ=−),andintheregionz≥L+(cid:2)+d
(τ=+)foranygivenfrequencyωandtangentialcomponentofthewavevectork ,
x
Hτ(x,z)=Hτexp{ikxx+iτkzτ[z−δτ,+(d+(cid:2)+L)]}, (1)
(cid:2)
wherek=ω/cisthevacuumwavenumber,kzτ= k2ετ−kx2,Im(kzτ),Re(kzτ)≥0.Similarly,
E
z
E E k
sp SR LR x sp
ε H
- y
0 superstrate ε- x
|εc(ω)| |ε| d metalεc(ω)
ε k ε
b ℓ+d sp buffer b
ε E ε
w wg waveguide w
ε+ L+ℓ+d substrate ε+
z z
Fig.1.Geometryoftheproblemandatypicaldielectricpermittivitydistribution.Thestruc-
tureoftheSPPmodesforboththickandthinmetalfilmsareschematicallyshownalong
withawaveguidemode.
HM(x,z)= ∑ HM|σexp{ik x+iσkM[z−z ]}, (2)
x z M
σ=±
inthemetal(M=c,z∈[0,d]),thebuffer(M=b,z∈[d,d+(cid:2)])andthewaveguide(M=w,
z∈[d+(cid:2),d+(cid:2)+L])layer.Herez =0,z =d,z =d+(cid:2)andthebranchofthesquareroots,
c b w
kM,ischosensothatRe(kM)≥0,Im(kM)≥0andthusthesignσ=+(−)correspondstothe
z z z
waves propagatingor decayingin the positive(negative)directionofthe z-axis, respectively.
Thex-dependenceinallthelayersisthesameduetoconservationofthetangentialmomentum
(or,equivalently,as imposedby the boundaryconditions).The electric field componentscan
be derivedfrom the Maxwell equations.Assuming all the media shown in Fig. 1 to be non-
magneticwehave
i ∂
Em(x,z)=− Hm(x,z), (3)
x kε ∂z
m
i ∂
Em(x,z)= Hm(x,z), (4)
z kε ∂x
m
wherem=τ,c,b,w,andε denotesdielectricpermittivityinthem-thlayer(medium),and
m
Euτ(x,z)=Euτexp{ikxx+iτkzτ[z−δτ,+(d+(cid:2)+L)]}, (5)
EM(x,z)= ∑ EM|σexp{ik x+iσkM[z−z ]}. (6)
u u x z M
σ=±
Hereu=x,y stays forthe correspondingcomponentsoftheelectricfield.Theamplitudesof
τ M|σ
theelectricfield,E ,E ,canbeexpressedbymeansofEqs.(3),(4)and(1),(2)as
u u
τkτ k
Eτ= zHτ, Eτ=− x Hτ, (7)
x kετ z kετ
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(C) 2008 OSA 8 December 2008 / Vol. 16, No. 25 / OPTICS EXPRESS 20230
EM|σ= σkzMHM|σ, EM|σ=− kx HM|σ. (8)
x kε z kε
M M
Introducingdimensionlessvariables,
km
α=k /k, βm= z , (9)
x kε
m
Eqs.(7),(8)canbepresentedintheform
α
Eτ=τβτHτ, Eτ=− Hτ, (10)
x z ε
τ
α
EM|σ=σβMHM|σ, EM|σ=− HM|σ. (11)
x z ε
M
Continuityofthetangentialcomponentsofthefieldcrossingtheinterfacesresultsinthesys-
temofeightlinearhomogeneousalgebraicequationsfortheeightamplitudesofthemagnetic
field, HM|σ, Hτ. The nontrivial solutions of these system (eigenmodes of the layered struc-
ture)existwhenthedeterminantisequaltozero.Thisleadsafterstraightforwardalgebratothe
dispersionrelation:
−βw[iawtanφ−bw]+β+[aw−ibwtanφ]=0, (12)
where
βb
aw=ab+bbtanhF, bw= [abtanhF+bb],
βw
(cid:3) (cid:4) (13)
β− ξ β−
ab=1+ tanhΦ, bb= tanhΦ+ ,
ξ βb ξ
ξ≡βcisthesurfaceimpedanceofthemetalfilm,and
Φ=−ikcd, φ=kwL, F=−ikb(cid:2). (14)
z z z
ResolvingEq.(12)yieldsthedispersioncurves,ω=ω(k )[ork =K(ω)] correspondingto
x x
differentmodesof the system. For the sake ofclarity, but withoutloss of generality,in what
followsweneglectfrequencydependenceofalldielectricpermittivitiesexceptthatoneofthe
metallayer,εc=εc(ω).ForthelatterwewillusetheDrudemodel,εc(ω)=ε∞−ωp2/ω(ω+
iγ), where ε∞ is high-frequencyvalue of the permittivity εc(ω), ωp is the plasma frequency,
andγisthedampingrate.InTable1thetypicalparametersforthecommonlyusedmaterials
√
(fordesignofthesensors)areshown,heren = ε ,m=−,b,w,+,denoterefractiveindices
m m
ofthecorrespondingdielectriclayers.
∗
Table1.TableofcommonmaterialparametersfortheSPRsensingdevices
metal silver[7] gold[7,25,26] aluminium[30]
dielectriccore n=1.585[31] n=1.476[30] n=1.47[25]
dielectriccladding n=1.439[31] n=1.473[30] n=1.45[25]
∗
Foropticalconstantsofsilver,goldandaluminiumsee[32–34].
Thefieldamplitudescanbeexpressedintermsoftheuppermediummagneticfieldampli-
−
tude,H :
(cid:5) (cid:6)
H− β− H−
Hc|σ= 1−σ , Hb|σ= {ab−σbb},
2 βc 2
(15)
−
H
Hw|σ= {aw−σbw}, H+=H−[awcosφ−ibwsinφ].
2
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(C) 2008 OSA 8 December 2008 / Vol. 16, No. 25 / OPTICS EXPRESS 20231
Thedispersionrelationincludesanumberoflimitingcasescorrespondingtodifferentcom-
binations of thick or thin limits of the building layers. For instance, in the limit of a large
thicknessofthebufferlayer,(cid:2)→∞(F (cid:9)1),thedispersionrelationsplitsintotwodispersion
branches, (cid:5) (cid:6)
β+βb
βb+β+−i βw+ tanφ=0, (16)
βw
ab+bb=0, (17)
whichcorrespondtothewaveguideTMmodesandSPPmodesofthemetalfilm,respectively.
Inturn,thederivedSPPdispersionrelation,Eq.(17),foraratherthickmetalfilm(Φ(cid:9)1)limit
leadstostandardSPPmodes,ab+bb=0⇒(βb+ξ)(β−+ξ)=0.Inthecaseofafinitevalue
ofΦthedispersionrelationdescribesthehybridizeddouble-interface-localizedSPPmodes,
(βbtanhΦ+ξ)(β−tanhΦ+ξ)−ξ2cosh−2Φ=0. (18)
Morespecifically,whenthedielectricpermittivitiesofthesuperstrateandthebuffercoincide,
εb=ε−,thenEq.(18)splitsintotwobranchesandyieldsthedispersionrelationsfortheshort-
range(symmetric)SPPandlong-range(antisymmetric)SPPmodes[1],seeFig.1,
βb=β+b ≡−ξtanh(Φ/2), βb=β−b ≡−ξcoth(Φ/2), (19)
wheresubscripts“+”and“−”correspondtothelong-range(LR),andshort-range(SR)mode,
respectively.
Similartothat,thesolitarywaveguidemodes((cid:2)→∞)dependonthewaveguidelayerthick-
nessandpermittivity.Namely,inthislimitingcasethesolutionofEq.(16)reads
(cid:3) (cid:4)
1 (βb+β+)βw nπ
ε kL= arctan −i + , n=0,±1,... (20)
w βw (βw)2+βbβ+ βw
At the crossingpoints ofthe dispersioncurvesthe phase matchingconditionsaresatisfied
andaninteractionbetweenwavesofdifferentbranchesismostefficient.However,inorderto
stress thedifferencebetweenthesituationswhenthephasematchingconditionsareprovided
by an additional grating and the problem considered here we will use in what follows term
hybridizationofmodes.Intheproblemconsideredherethefinitethicknessofthebufferlayer
resultsinhybridization(combination)ofthewaveguideandSPPmodes.Theresultinghybrid
eigenmodes of the system might have spatial distributions with energy both in the dielectric
waveguideandintheSPPmodes.Wewouldliketoemphasizethatthemostinterestingsitua-
tionsforthedesignofthesensorstakesplacewhenthelevelofthemodehybridization(strong
couplingofthewaveguideandSPPmodes)isratherhigh.Thisoccurswhenthebufferlayeris
relativelythinandinthevicinityofthepointsintheω−k planewherethedispersioncurves
x
ofthefreeSPPandwaveguidemodesintersect.Inthevicinityofsuchpointsthehybridization
ishighandcontrolofoperationnearthispointcanbeprovidedbytheadjustmentofthebuffer
layer.Thisconsiderationoutlinesthedesignrulesandexplainsthechoiceoftheparametersin
theexamplespresentedbelow.
The solutions of the dispersion relation of the multi-layer system, Eq. (12), are plotted in
Fig.2whereweusethenormalizedfrequencyparameterkL=Lω/c,versusthesquarednor-
malizedtangentialcomponentofthewavenumber(i.e.,thesquaredeffectiverefractionindex,
n2 ≡α2 =c2k2/ω2). The most interestingarea forthe parameterslies is the vicinityof the
eff x
intersectionbetweenthesilver-superstrateSPPbranchandthatofthelowest-frequency(funda-
mental)waveguidemode,seeinsetinFig.2.
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(C) 2008 OSA 8 December 2008 / Vol. 16, No. 25 / OPTICS EXPRESS 20232
12
WG WG WG
n=2 n=1 n=0
3 10 kL
2 y,
c
8 en
u
q
e
metal-superstrate SPP 6 d fr
e
z
hilgohss reasd iaarteioan 3 4 mali
a 2 1 resonance or
metal-buffer SPP frequency 2 n
b
ε ε=ε ε 0
1.2 1.4 1.6 1.8 - 2 b + 2.2 2.4 w 2.6 2.8
normalized propagation constant, α2
Fig.2.Dispersionrelationsforthemulti-layersystemofFig.1.Thick(red)curvesrepresent
thesolutionsofthecompletedispersionrelationgiveninEq.(12).Thin(gray)WG-named
curvescorrespondtothedispersionrelationofthesolitarydielectricwaveguide,Eq.(20),
andathicksilvermetalfilm,d→∞.Thedashed(blue)curvesrepresentthehigh-frequency
andlow-frequencysolutionsofthedispersionrelationforasolitarythinsilvermetalfilm.
Theparametersare:n−=1.37,nb=n+=1.439,nw=1.585,thethicknessesofthelayers
ared=58nm,(cid:2)=400nm,L=250nm.
Thecouplingofthesemodesduetothefinite-thicknessofthebufferlayerresultsinarepul-
sionofthedispersionbranches.However,inthevicinityoftheinoculatingcurvesintersection
thedistancebetweenthemodifieddispersionbranchesisrelativelysmall.Theintersectionde-
finesthe“resonancefrequency”,k L=3.37(ω /(2π)=6.44·1014Hz)forL=250nm,see
res res
Fig.2,thatoptimizestheconsideredstructurepropertiesforsensorapplications.Notethatour
approachcanbealsousedinthereverseproblem-whenthedesignofthemulti-layerstructure
isadjustedtosomerequiredoperationalfrequency.
The magnetic field evolution along the x-axis for the solitary metal film SPP modes, i.e.,
when the buffer and the waveguide thicknesses are zero-valued, (cid:2),L→0, could be derived
fromEqs.(1), (2)and(15),wherek and ωobeyEq.(18).Thisdistributionin (z,x) planeis
x
showninFig.3fortheabove“resonancefrequency”,ω .
res
Fieldevolutionfortheeigenmodesofthesolitarydielectricwaveguide(weconsiderherethe
limitorvanishingbufferandthemetallayerthicknesses:d,(cid:2)→0)couldbederivedfromthe
sameequationsbutfork andωobeyingEq.(20).Thecorrespondingmagneticfielddistribu-
x
tionisshowninFig.4,forω=ω .
res
The magnetic field distributions in the plane (z,x) for three eigenmodes of the multilayer
systematthe“resonancefrequency”areshowninFigs.5,6and7.
Itisseenthatbyanappropriatefittingoftheparametersofamulti-layerstructureitispos-
sible to switch between different regimes and to control efficiency of coupling between the
waveguidemodeandSPPmodes.
3. Transmissionpropertiesofthemulti-layerstructure
Considernowtransmittanceofthedescribedabovemulti-layersystem with thelengthofthe
metal layer that differsfromthe dielectric waveguidelength.We calculate here the transmit-
tanceasaratiobetweenthetotaloutputenergyfluxatx=100μ m,andtheinputenergyflux
atx=0.Theinputenergyfluxatthestartingpointx=0correspondstothesolitarywaveguide
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(C) 2008 OSA 8 December 2008 / Vol. 16, No. 25 / OPTICS EXPRESS 20233
high-frequency SPP mode
1000
e
m) ud
nate , (nz 5000 claddingε- 01.75eld amplit
oordi metal filmεc claddingεb etic fi
c -500 -1.75gn
a
m
(a)
-1000
0 500 1000 1500 2000 2500
propagation direction x, (nm)
low-frequency SPP mode
1000
e
m) ud
nate , (nz 5000 metal filmεc claddingε- 01.75eld amplit
oordi claddingεb etic fi
c -500 -1.75gn
a
m
(b)
-1000
0 500 1000 1500 2000 2500
propagation direction x, (nm)
Fig.3.ThemagneticfielddistributionforthetwoSPPeigenmodesofthesolitarymetalfilm
atω=ωres (ωresL/c=3.37).αh2igh=2.268,αl2ow=2.649otherparametersarethesame
asinFig.2.Thea(b)figureisthefielddistributionforthehigh-frequency(low-frequency)
SPPmode of the metal filmcorresponding to the upper (lower) SPPdispersion curve in
Fig.2.
waveguide mode
1000
e
m) 500 ud
zoordinate , (n-1-0500000 wavcclleaagdduddiiidnnggeεεε+bw 01.75etic field amplit
c -1.75gn
-1500 ma
-2000
0 1000 2000 3000 4000 5000
propagation direction x, (nm)
Fig. 4. The magnetic field evolution for the solitary waveguide mode, α=1.56, other
parametersarethesameasinFig.2
fundamentalmode,and the outputenergyflux is obtainedbyintegratingthe x-componentof
the energy flux density across the section x=100 μ m from z=−3 μ m to z=6 μ m. The
electromagneticfieldwithinthestructurewasobtainedbydirectnumericalsimulationsofthe
problembymeansofthefiniteelementmethod(FEM),byusingacommercialtoolCOMSOL.
The calculationlengthof100 μ m was chosenbecauseit was foundto be enoughto observe
theeffect.Thecomputationofthelongerstructuresrequiressignificantlylargercomputerre-
sources.However,it does notaddsignificantlynewinformation— use ofa longercomputa-
tionaldomainleadstoamoreprecisecharacterizationofafinestructureoftheregionnearthe
#100122 - $15.00 USDReceived 14 Aug 2008; revised 12 Nov 2008; accepted 15 Nov 2008; published 24 Nov 2008
(C) 2008 OSA 8 December 2008 / Vol. 16, No. 25 / OPTICS EXPRESS 20234
500
1 e
zoordinate , (nm) --5000 metal filmεc wsbuuaspfvfueeebrrgss luttarriyaadetteeerεεεεw+b- 01.75etic field amplitud
c -1000 -1.75gn
a
m
-1500
0 200 400 600 800 1000
propagation direction x, (nm)
Fig. 5. The magnetic field evolution for the eigenmode of the system corresponding to
the point 1 in Fig. 2. Compare the field distribution with that of the right in Fig. 3. The
parametersarethesameasinFig.2.
500
2 e
zoordinate , (nm) -5000 metal filmεc wsbuuaspfvfueeebrrgss luttarriyaadetteeerεεεεw+b- 01.75etic field amplitud
c-1000 -1.75gn
a
(a) m
-1500
0 200 400 600 800 1000
propagation direction x, (nm)
500
3 e
zoordinate , (nm) -5000 metal filmεc wsbuuaspfvfueeebrrgss luttarriyaadetteeerεεεεw+b- 01.75etic field amplitud
c -1000 -1.75gn
a
m
(b)
-1500
0 200 400 600 800 1000
propagation direction x, (nm)
Fig.6.Themagneticfieldevolutionfortheeigenmodesofthesystemcorrespondingtothe
points2(a)and3(b)inFig.2.
minimumattheresonancefrequency(itbecomesslightlywideranddeeper),butthemainfea-
turesofthefrequencydependenceofthetransmittancearepracticallynotchanged.Thedirect
numericalsimulationisconvenientforshortstructures,becauseitcouldbeeasilyappliedina
widefrequencyrange.Consideringnarrowfrequencyrangeapplicationswithlongerstructures,
analyticalmethods,e.g.suchasapproachesthatwediscussbelow,mightbepreferable.
Theresultsofthenumericalmodelingofthetransmissioncoefficientforthelongmetalfilm
lengthL >100 μ m (formally,L →∞) are presentedin Fig. 8. Thefrequencyrangein the
calculationscorrespondstothestripeareaintheFig.2.
One can see rather narrowdip in the transmission coefficientnear the defined above“res-
onance frequency”,shown by the vertical line. Evidently, the transmission falls due to exci-
#100122 - $15.00 USDReceived 14 Aug 2008; revised 12 Nov 2008; accepted 15 Nov 2008; published 24 Nov 2008
(C) 2008 OSA 8 December 2008 / Vol. 16, No. 25 / OPTICS EXPRESS 20235
mode H1
5 mode H2
mode H3
dielectric permittivity
distribution
-400 -200 200 400 600 800 1000
ε
y
vit coordinate z, (nm)
mitti -5
er
p
c
ctri
e
el
di-10
Fig.7.Squaredmagneticfieldevolutionforthreeeigenmodesofthesystemforthepoints
1,2and3ofFig.2,andthedielectricpermittivitydistribution.Theparametersarethesame
asinFig.2.
tationoftheSPP modesaccompaniedbystrongenhancementofthedissipativelosses.Other
localminimaofthetransmissioncoefficientarecausedbyappearanceoftheradiationlosses,
cf.Fig.2.Wewouldliketoemphasizethatintheconsideredstructurethereexists3eigenmodes
ofthesystemsimultaneously(forsomegiveninputfrequency),seeFig.2.Thisfactleadstothe
fieldinterferencepatterneasilyseeninFig.9.Forthe“resonancefrequency”thelargestperiod
ofthepatternisgivenbythefollowingsimpleexpression:
2π
X = , (21)
beat k −k
x2 x3
wherek andk aretherealpartsofthepropagationconstantsofthelong-periodmodes2and
x2 x3
3 of the complete system, and could be foundfrom the dispersionrelation. In this particular
instanceX ≈69.5μ m.Thesmallfeaturesinthepatternareduetointerferenceoftheshort-
beat
periodmode1withthelong-periodmodes2and3.Thatis,correspondingcharacteristiclength
of the pattern is X(cid:12) (cid:13)2π/(k −k )(cid:13)3.9 μ m, cf. Fig. 9. These features are in excellent
beat x1 x2
agreementwiththeresultsoftherecentexperimentalwork[29],whereauthorsconsidersimilar
geometryand discuss the couplinglengthandenergyexchangebetweenwaveguideandSPP
modes.
Attheinputofthesystemmainpartoftheenergywasinthedielectricwaveguideandthen
isgraduallytransferredtothemetalfilm.Itisseenthatthebeathalf-periodX /2givesusa
beat
characteristiclengthforenergyexchangebetweenthecore-andthehigh-frequencymetalfilm-
localized mode(couplinglength)and at the distance x=X /2the maximaltransferof the
beat
energyintothemetal-film-localizedmodeoccurs,seeFig.9.Consequently,lowtransmission
coefficients (corresponding to almost total energy transfer from waveguide to the SPP) can
be achieved by choosing the metal film length of the order L =X /2, see Fig. 8. This
opt beat
distanceis definedasaminimallengthofthemetalfilmthatcanprovideforthetotalenergy
transfer from the waveguidecore to the metal. Since the process of the energytransfer from
the core to the SPP and in the backward direction in the absence of losses is cyclic, there
existanumberofmetalfilmlengthsforwhichtheenergytotalvanishinginthecoreappears:
L =X /2+mX ;m=0,1,2,....However,theminimalmetalfilmlengthhasadvantage
m beat beat
ofa largecontrastbetweenresonantandnon-resonanttransmission(transmissionat resonant
and non-resonant frequencies) due to the small dissipation losses in the shorter metal film.
#100122 - $15.00 USDReceived 14 Aug 2008; revised 12 Nov 2008; accepted 15 Nov 2008; published 24 Nov 2008
(C) 2008 OSA 8 December 2008 / Vol. 16, No. 25 / OPTICS EXPRESS 20236