5/3/12 1 420 nm [ Notomi et al. (2005). ] Resonance an oscilla:ng mode trapped for a long :me in some volume (of light, sound, …) frequency ω 0 life:me τ >> 2π/ω 0 quality factor Q = ω 0 τ/2 energy ~ e –ω0t/Q modal volume V [ Schliesser et al., PRL 97, 243905 (2006) ] [ Eichenfield et al. Nature Photonics 1, 416 (2007) ] [ C.‐W. Wong, APL 84, 1242 (2004). ] Why Resonance? an oscilla:ng mode trapped for a long :me in some volume • long :me = narrow bandwidth … filters (WDM, etc.) — 1/Q = frac:onal bandwidth • resonant processes allow one to “impedance match” hard‐to‐couple inputs/outputs • long :me, small V … enhanced wave/ma^er interac:on — lasers, nonlinear op:cs, opto‐mechanical coupling, sensors, LEDs, thermal sources, … How Resonance? need mechanism to trap light for long :me [ llnl.gov ] metallic cavi:es: good for microwave, dissipa:ve for infrared ring/disc/sphere resonators: a waveguide bent in circle, bending loss ~ exp(–radius) [ Xu & Lipson (2005) ] 10µm [ Akahane, Nature 425, 944 (2003) ] photonic bandgaps (complete or par:al + index‐guiding) VCSEL [fotonik.dtu.dk] (planar Si slab) Understanding Resonant Systems [ Schliesser et al., PRL 97, 243905 (2006) ] • Op:on 1: Simulate the whole thing exactly — many powerful numerical tools — limited insight into a single system — can be difficult, especially for weak effects (nonlineari:es, etc.) • Op:on 2: Solve each component separately, couple with explicit perturba:ve method (one kind of “coupled‐mode” theory) • Op:on 3: abstract the geometry into its most generic form …write down the most general possible equa:ons …constrain by fundamental laws (conserva:on of energy) …solve for universal behaviors of a whole class of devices … characterized via specific parameters from op:on 2 “Temporal coupled‐mode theory” • Generic form developed by Haus, Louisell, & others in 1960s & early 1970s – Haus, Waves & Fields in Optoelectronics (1984) – Reviewed in our Photonic Crystals: Molding the Flow of Light, 2nd ed., ab‐ini:o.mit.edu/book • Equa:ons are generic ⇒ reappear in many forms in many systems, rederived in many ways (e.g. Breit–Wigner sca^ering theory) – full generality is not always apparent (modern name coined by S. Fan @ Stanford) TCMT example: a linear filter 420 nm [ Notomi et al. (2005). ] [ C.‐W. Wong, APL 84, 1242 (2004). ] [ Takano et al. (2006) ] [ Ou & Kimble (1993) ] = abstractly: two single‐mode i/o ports + one resonance resonant cavity frequency ω 0 , life:me τ port 1 port 2
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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
5/3/12
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)]
5/3/12
3
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…
5/3/12
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
5/3/12
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).]
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