5/3/12

1

420nm

[Notomietal.(2005).]

Resonanceanoscilla:ngmodetrappedforalong:meinsomevolume

(oflight,sound,…)frequencyω0

life:meτ>>2π/ω0qualityfactorQ=ω0τ/2

energy~e–ω0t/Q

modalvolumeV

[Schliesseretal.,PRL97,243905(2006)]

[Eichenfieldetal.NaturePhotonics1,416(2007)]

[C.‐W.Wong,APL84,1242(2004).]

WhyResonance?anoscilla:ngmodetrappedforalong:meinsomevolume

•long:me=narrowbandwidth…filters(WDM,etc.)—1/Q=frac:onalbandwidth

•resonantprocessesallowoneto“impedancematch”hard‐to‐coupleinputs/outputs

•long:me,smallV…enhancedwave/ma^erinterac:on—lasers,nonlinearop:cs,opto‐mechanicalcoupling,sensors,LEDs,thermalsources,…

HowResonance?needmechanismtotraplightforlong:me

[llnl.gov]

metalliccavi:es:goodformicrowave,dissipa:veforinfrared

ring/disc/sphereresonators:awaveguidebentincircle,bendingloss~exp(–radius)

[Xu&Lipson(2005)]

10µm

[Akahane,Nature425,944(2003)]

photonicbandgaps(completeorpar:al+index‐guiding)

VCSEL[fotonik.dtu.dk]

(planarSislab)

UnderstandingResonantSystems

[Schliesseretal.,PRL97,243905(2006)]

•Op:on1:Simulatethewholethingexactly—manypowerfulnumericaltools—limitedinsightintoasinglesystem—canbedifficult,especiallyfor

weakeffects(nonlineari:es,etc.)

•Op:on2:Solveeachcomponentseparately,couplewithexplicitperturba:vemethod(onekindof“coupled‐mode”theory)

•Op:on3:abstractthegeometryintoitsmostgenericform …writedownthemostgeneralpossibleequa:ons…constrainbyfundamentallaws(conserva:onofenergy)…solveforuniversalbehaviorsofawholeclassofdevices

…characterizedviaspecificparametersfromop:on2

“Temporalcoupled‐modetheory”

•  GenericformdevelopedbyHaus,Louisell,&othersin1960s&early1970s–  Haus,Waves&FieldsinOptoelectronics(1984)–  ReviewedinourPhotonicCrystals:MoldingtheFlowofLight,2nd

ed.,ab‐ini:o.mit.edu/book

•  Equa:onsaregeneric⇒reappearinmanyformsinmanysystems,rederivedinmanyways(e.g.Breit–Wignersca^eringtheory)–  fullgeneralityisnotalwaysapparent

(modernnamecoinedbyS.Fan@Stanford)

TCMTexample:alinearfilter

420nm

[Notomietal.(2005).]

[C.‐W.Wong,APL84,1242(2004).]

[Takanoetal.(2006)]

[Ou&Kimble(1993)]

=abstractly:twosingle‐modei/oports+oneresonance

resonantcavityfrequencyω0,life:meτ

port1 port2

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2

TemporalCoupled‐ModeTheoryforalinearfilter

ainput outputs1+s1– s2–

resonantcavityfrequencyω0,life:meτ |s|2=power

|a|2=energy

dadt

= −iω0a −2τa + 2

τs1+

s1− = −s1+ +2τa, s2− =

2τa

assumesonly:•exponen:aldecay(strongconfinement)•linearity•conserva:onofenergy•:me‐reversalsymmetry

canberelaxed

TemporalCoupled‐ModeTheoryforalinearfilter

ainput outputs1+s1– s2–

resonantcavityfrequencyω0,life:meτ |s|2=flux

|a|2=energy

transmissionT

=|s2–|2/|s1+|2

1

ω0

T=Lorentzianfilter

=

4τ 2

ω −ω0( )2 + 4τ 2

ω

ResonantFilterExample

Lorentzianpeak,aspredicted.

Anapparentmiracle:

~100%transmissionattheresonantfrequency

cavitydecaystoinput/outputwithequalrates⇒Atresonance,reflectedwave

destruc:velyinterfereswithbackwards‐decayfromcavity

&thetwoexactlycancel.

Someinteres:ngresonanttransmissionprocesses

Wirelessresonantpowertransfer[M.Soljacic,MIT(2007)]

witricity.com

ResonantLEDemissionluminus.com

(narrow‐band)resonantabsorp:oninathin‐filmphotovoltaic

[e.g.Ghebrebrhan(2009)]

inputpower

outputpower~40%eff.

Wide‐angleSpli^ers

[S.Fanetal.,J.Opt.Soc.Am.B18,162(2001)]

WaveguideCrossings

[S.G.Johnsonetal.,Opt.LeN.23,1855(1998)]

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WaveguideCrossings

empty

5x5

3x3

1x1

Anotherinteres:ngexample:Channel‐DropFilters

[S.Fanetal.,Phys.Rev.LeN.80,960(1998)]

Perfectchannel‐droppingif:

Tworesonantmodeswith:•evenandoddsymmetry•equalfrequency(degenerate)•equaldecayrates

Coupler

waveguide1

waveguide2

(mirrorplane)

DimensionlessLosses:Q

1

ω0

T=Lorentzianfilter

=

4τ 2

ω −ω0( )2 + 4τ 2

ω

FWHM1Q=2ω0τ

…qualityfactorQ

qualityfactorQ=#op:calperiodsforenergytodecaybyexp(–2π)

energy~exp(–ω0t/Q)=exp(–2t/τ)

infrequencydomain:1/Q=bandwidth

fromtemporalcoupled‐modetheory:

Q=ω0τ/2

MorethanoneQ…

Qw

Asimplemodeldevice(filters,bends,…):

Qr

Q1

Qr1

Qw1= +

Q=life:me/period=frequency/bandwidth

Wewant:Qr>>Qw

1–transmission~2Q/Qr

worstcase:high‐Q(narrow‐band)cavi:es

losses(radia:on/absorp:on)

TCMT⇒

Nonlineari:es+Microcavi:es?weakeffects∆n<1%

veryintensefields&sensi:vetosmallchanges

Asimpleidea:forthesameinputpower,nonlineareffectsarestrongerinamicrocavity

That’snotall!nonlineari:es+microcavi:es =qualitaUvelynewphenomena

NonlinearOp:csKerrnonlineari:esχ(3):(polarizaUon~E3)

•Self‐PhaseModula:on(SPM)=changeinrefrac:veindex(ω)~|E(ω)|2

•Cross‐PhaseModula:on(XPM)=changeinrefrac:veindex(ω)~|E(ω 2)|2

•Third‐HarmonicGenera:on(THG)&down‐conversion(FWM)=ω→3ω,andback

•etc…ω

ω

ω

3ω

ω

ω

ω’s

Second‐ordernonlineari:esχ(2):(polarizaUon~E2)•Second‐HarmonicGenera:on(SHG)&down‐conversion

=ω→2ω,andback•Difference‐FrequencyGenera:on(DFG)=ω1, ω2→ω1-ω2

•etc…

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4

Nonlineari:es+Microcavi:es?weakeffects∆n<1%

veryintensefields&sensi:vetosmallchanges

Asimpleidea:forthesameinputpower,nonlineareffectsarestrongerinamicrocavity

That’snotall!nonlineari:es+microcavi:es =qualitaUvelynewphenomena

let’sstartwithawell‐knownexamplefrom1970’s…

ASimpleLinearFilter

in out

Linearresponse:LorenzianTransmisson

Filter+KerrNonlinearity?

in out

Linearresponse:LorenzianTransmisson shi�edpeak?

+nonlinearindexshi�=ωshi�

Kerrnonlinearity:∆n~|E|2

stable

stableunstable

Op:calBistability

Bistable(hysteresis)response(&evenmul:stableformul:modecavity)

Logicgates,switching,recUfiers,amplifiers,

isolators,…

[FelberandMarburger.,Appl.Phys.LeN.28,731(1978).]

Powerthreshold~V/Q2(incavitywithV~(λ/2)3,

forSiandtelecombandwidthpower~mW)

[Soljacicetal.,PRERapid.Comm.66,055601(2002).]

TCMTforBistability[Soljacicetal.,PRERapid.Comm.66,055601(2002).]

ainput outputs1+ s2–

resonantcavityfrequencyω0,life:meτ,SPMcoefficientα ~χ(3)

(fromperturba:ontheory)

|s|2=power

|a|2=energy

dadt

= −i(ω0 −α a 2 )a − 2τa +

2τs1+

s1− = −s1+ +2τa, s2− =

2τa

givescubicequa:onfortransmission

…bistablecurve

TCMT+Perturba:onTheory

SPM=smallchangeinrefrac:veindex…evaluate∆ωby1st‐orderperturba:ontheory

⇒ allrelevantparameters(ω,τorQ,α)canbecomputedfromtheresonantmodeofthelinearsystem

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5

AccuracyofCoupled‐ModeTheory

semi‐analy:cal

numerical

[Soljacicetal.,PRERapid.Comm.66,055601(2002).]

Op:calBistabilityinPrac:ce

420nm

[Notomietal.(2005).][Xu&Lipson,2005]

Q~30,000V~10op:mum

Powerthreshold~40µW

10µm

Q~10,000V~300op:mum

Powerthreshold~10mW

ExperimentalBistableSwitch

Silicon‐on‐insulator

420nm

Q~30,000Powerthreshold~40µWSwitchingenergy~4pJ

[Notomietal.,Opt.Express13(7),2678(2005).]