2024年5月18日发(作者：移动硬盘怎么修复)

Transmissionresonancesonmetallicgratingswithverynarrowslits

CondensedMatterTheoryGroup,TheBlackettLaboratory,ImperialCollege,LondonSW72BZ,UnitedKingdom

DepartamentodeF´ısicaTe´oricadelaMateriaCondensada,FacultaddeCiencias(C-V),UniversidadAut´onomadeMadrid,

E-28049Madrid,Spain

Inthisletterweshowhowtransmissionmetallicgratingswithverynarrowanddeepenoughslits

canexhibittrang

atransfermatrixformalismandaquasi-analyticalmodelbasedonamodalexpansion,weshowthat

therearetwopossiblewaysoftransferringlightfromtheuppersurfacetothelowerone:bythe

excitationofcoupledsurfaceplasmonpolaritonsonbothsurfacesofthemetallicgratingorbythe

couplchanisms

canleadtoalmostperfecttransmittanceforthoseparticularresonances.

PACSnumbers:,,,71.36.+c

1

1

,´ıa-Vidal

2

,

1

2

a

r

X

i

v

:

c

o

n

d

-

m

a

t

/

9

9

0

4

3

6

5

v

1

[

c

o

n

d

-

m

a

t

.

m

t

r

l

-

s

c

i

]

2

6

A

p

r

1

9

9

9

Absorptionanomaliesinmetallicgratingshaveat-

tractedmuchattentionsincetheirdiscoverybyWood

[1]heseanomaliesisonlyobservedfor

p-polarizedlight(Hparalleltothegratinggrooves)and

appearsasaminimumonthespecularrefl

itiswellknown[2]thatthisanomalystemsfromthe

excitationofsurfaceplasmonpolaritons(SPPs)bythe

endenceof

theseSPPmodesonthegratingshapeandthepossi-

bleexistenceofmodeslocalizedinthegrooveshavebeen

studiedforanumberofyearsandarestillofinterest

[2–8].Ontheotherhand,theactivityofthelastdecade

inthefieldofphotoniccrystals[9]hasoriginatedare-

newedinterestinthepropertiesofSPPs,astheycanbe

viewedassurfaceelectromagneticmodespropagatingin

1Dperiodicdielectricmedia[10].

Besides,experimentalevidenceoftheexcitationofop-

ticalwaveguidemodesinsidethenarrowgroovesofzero-

orderreflectiongratingshasbeenrecentlygiven[11].

Bothfromfundamentalandpracticalpointsofviewit

wouldbeveryinterestingtoanalyzethetransmission

ryrecently

someexperimentscarriedoutinarraysofsubmicrome-

trecylindricalholesinmetallicfilmshaveshownanex-

traordinaryopticaltransmissionatwavelengthsuptoten

timeslargerthanthediameteroftheholes[12].Thesim-

ilaritiesbetweenthislaststructureandmetalgratings

suggestthepossibilityofequivalentresonanteffectsin

transmissionmetalgratingsofverynarrowslits.

Inthisletterwetesttheoreticallythispossibilityby

analyzingtheresponseoftransmissionmetallicgratings

show

how,forverynarrowslits,thecouplingoftheincident

lightwithsurfaceelectromagneticmodesofthegrating

canleadtoalmostperfecttransmissionresonancesap-

pearingatwavelengthslargerthantheperiodofthegrat-

ingandhencemuchlargerthanthelateraldimensionsof

theslits.

InsetofFig.1showsaschematicviewofthestruc-

turesunderstudywiththedefinitionofthedifferentpa-

rameters:theperiodofthegrating(d),thewidth(a)

andheight(h)strateischaracter-

izedbyadielectricconstant,ǫ.Advancesinmaterial

technologyhaveallowedtheproductionoftransmission

gratingswithwellcontrolledprofiles,whichhavealready

beenusedindifferentinterestingapplications,suchas

polarizersorx-rayspectrometers[13].Fromthetheoret-

icalpointofview,therehavebeensomestudiesofthese

structuresinthelastfewyears[14,15].However,upto

ourknowledge,transmissionpropertiesofverynarrow

slitsthatareperiodicallystructuredremainunstudied.

Inthisletterweconsidermetalgratingsmadeofgold

andweusefixedvaluesforthegratingperiod(d=

3.5µm)andthewidthoftheslits(a=0.5µm),although

thicknessofthemetallicgrating(h)willbevariedbe-

tween0and4µiceofthesegeometricalvalues

ismotivatedbytheexperimentalfindingsofwaveguide

resonancesreportedinRef.[11]forreflectionmetalgrat-

heless,it

shouldbepointedoutthattheeffectsdiscussedinthis

letterdoappearforanyotherrangeprovidedaisvery

smallincomparisontodandthefrequencyofthein-

cidentlightiswellbelowtheplasmafrequencyofthe

lectricfunctionofgoldisdescribedusing

thetablesreportedinRef.[16].

Wehaveanalyzedtheelectromagneticpropertiesof

thesegratingsbymeansofatransfermatrixformalism

[17].Withinthisformalismitispossibletocalculate

transmissionandreflectioncoefficientsforanincoming

uently,thetransmittanceandre-

flectanceofthegratingaswellasreal-spaceelectromag-

neticfi.1showszero-order

transmittancefornormalincidentradiationonmetallic

gratingsinvacuumasafunctionofthewavelengthofthe

tingheight(h)isvariedin

thesecalculationsfrom0.2to4µeseenin

Fig.1a,fordeepenoughgratings(h≥0.6µm)aremark-

abletransmissionpeakappearsforawavelengthslightly

largerthanthegratingperiod(inthiscase3.5µm).This

1

transmissionpeakmovestolargerwavelengthsasthe

gratingheightincreaseswhereasitslinewidthisbroad-

,asillustratedinFig.1b,subsequenttrans-

av-

iorofthetransmittancespectrumasafunctionofthe

metalthicknessseemstosuggestthatthesepeakscould

belinkedtothecouplingofincidentplanewaveswith

waveguideresonancesoftheslits.

Inordertoanalyzethephysicaloriginofthesetrans-

missionresonances,wehavealsodevelopedanapproxi-

rporatetwomainsimplifi-

cationstotheexactmodalmethodreportedin[18].First,

asthefrequencyregimeweareinterestedinisbelow

theplasmafrequencyofthemetal,surface-impedance

boundaryconditions(SIBC)[14]areimposedonthe

metallicboundaries,exceptontheverticalwallsofthe

-

ond,weonlyconsiderthefundamentaleigenmodeinthe

modalexpansionoftheelectricandmagneticfieldsinside

theslits,whichisjustifiedinthelimitwherethewave-

lengthoflightismuchlargerthanthewidthoftheslits.

Thevalidityofthesetwoapproximationsisconfirmedby

thegoodagreementbetweentheresultscalculatedbythis

simplifiedmodalmethodandthenumericalsimulations

thissingle-modeapproximationthetwofieldamplitudes

insidetheslits(theoneassociatedwiththee

ik

0

z

wave

andtheotherwiththee

−ik

0

z

one)areproportionalto

1/D,wherethedenominatorDisgivenby:

D=(1−(1+η)φ)(1−(1+η)ψ)e

ık

0

h

−(1+(1−η)φ)(1+(1−η)ψ)e

−ık

0

h

,

withk

0

=2π/λ,η=ǫ

metal

,andψgivenbythesum:

ψ=

a

2

−1/2

(1)

))

2

isassociated

withthem-thdiffntityφisalso

givenbyEq.(2)butwithǫ=1.

Thezero-ordertransmittancespectrumofthegrating

iscompletelygovernedbythebehaviorofthedenomina-

mple,fornormalincidenceandgratings

invacuum,zero-ordertransmittancesimplifiesto:

d

16(

a

T

0

=.

|D|

2

|1+η|

4

(3)

Moreover,wehavefoundthatthereisaclosecorre-

spondencebetweenmaximaofzero-ordertransmittance

andspectralpositionsofthezerosoftheimaginarypart

ofD,ℑ(D).Thisresultallowsustoanalyzethenatureof

theelectromagneticmodesresponsibleforthetransmis-

sionresonancesshowninFig.1justbystudyingthezeros

2

ofℑ(D)asgivenbyEq.(1).Also,byvaryingtheangle

ofincidenceθwecancalculatethephotonicbandstruc-

ture,ω(k

x

),.2awe

showthephotonicbandstructureforthecaseh=0.6µm

(blackdots)and,forcomparison,theenergeticpositions

oftheSPPexcitationinthelimith→0(graydots).

Notethatduetotherangeofphotonenergieswearean-

alyzing,theseSPPfrequenciesalmostcoincidewiththe

energeticpositionsoftheRayleighanomaliesthatare

linkedtozerosoftransmittance[2].Ascanbeseenin

theinsetofFig.2a,averynarrowbandgapbetween

thefi

lowerbranchatk

x

=0isassociatedwiththetransmis-

sionpeakatλseproximity

totheenergyofSPPbandssuggeststhatthistransmis-

sivemodeisassociatedwiththeexcitationofasurface

plasmonwithSPPcharacterineachsurfaceofthegrat-

won,wenamethiskindofresonancesas

ncreased,newbandsthatare

associatedwithwaveguidemodesoftheslitsappearin

nbeseeninFig.2bwhichshows

thatforh=3µmaflatwaveguidebandispresentat

ω=calizedmodeisresponsibleforthe

transmissionpeaklocatedatλ=7.5µm(seeFig.1b).

Theothertransmissionresonanceobtainedforh=3µm

atλ≈5µmcorrespondstothelowerbranchofthefirst

fferentlyfromtheh=0.6µmcase,astrong

hybridizationbetweenSPPbandsandawaveguidemode

ofsimilarenergyresultsintheopeningofaverybroad

,thetransmissivemodeappearingatλ≈5µm

willpresentahybridcharacterbetweencoupledSPPs

ore,bylookingatthe

photonicbandstructureofsurfaceplasmons,wecancon-

cludethattransmissionresonancesappearinginFig.1

aremainlyduetotheexcitationoftwokindsofelectro-

magneticmodes:coupledSPPsforλ≈dandwaveguide

resonancesforλ≫.2canalsogiveusadditional

informationaboutthedependenceofthetransmission

resonancesontheangleofincidence,θ.Atotallydiffer-

entdispersionrelationforthetwokindsoftransmissive

,transmittance

associatedwithcoupledSPPswillshowastrongdepen-

denceonθwhereasforwaveguideresonancestransmis-

sionisalmostindependentofθ.

Usingthesimplifiedmodalmethod,wecanalsostudy

indetailthebehavioroftransmissionresonancesasa

functionofthewidthoftheslits,spurpose,

weshowinFig.3zero-ordertransmittancecurvesfor

gratingsinvacuumofthickness(a)h=0.6µmand(b)

h=3.0µminthewavelengthregionwheretransmission

thoftheslitsisvariedbetween

0and1.5µcaseoftransmissionresonances

linkedtocoupledSPPs(Fig.3a)aminimumvalueof

aisneededinordertocoupleSPPsofeachsurfaceof

histhreshold(whosewavelengthde-

pendsonthedepthoftheslitsandforh=0.6µmis