PT Symmetry Breaking and Nonlinear Optical Isolation in Coupled Microcavities

0.78 MB·

by  Xin Zhou

PT symmetry breaking and nonlinear
optical isolation in coupled microcavities
XinZhou1andY.D.Chong1,2,∗
1DivisionofPhysicsandAppliedPhysics,SchoolofPhysicalandMathematicalSciences,
NanyangTechnologicalUniversity,Singapore637371,Singapore
2CentreforDisruptivePhotonicTechnologies,NanyangTechnologicalUniversity,Singapore
6 637371,Singapore
1 ∗[email protected]
0
2

r Abstract: We perform a theoretical study of the nonlinear dynamics of
a
M nonlinearopticalisolatordevicesbasedoncoupledmicrocavitieswithgain
andloss.Thisrevealsacorrespondencebetweentheboundaryofasymptotic

4 stability in the nonlinear regime, where gain saturation is present, and the
2 PT-breaking transition in the underlying linear system. For zero detuning

and weak input intensity, the onset of optical isolation can be rigorously
]
s derived, and corresponds precisely to the transition into the PT-broken
c
phase of the linear system. When the couplings to the external ports are
i
t unequal, the isolation ratio exhibits an abrupt jump at the transition point,
p
o whose magnitude is given by the ratio of the couplings. This phenomenon
. couldbeexploitedtorealizeanactivelycontrollednonlinearopticalisolator,
s
c inwhichstrongopticalisolationcanbeturnedonandoffbytinyvariations
i intheinter-resonatorseparation.
s
y
© 2016 OpticalSocietyofAmerica
h
p
OCIScodes:(230.4555)Coupledresonators;(230.3240)Isolators;(130.4310)Nonlinear.
[

3
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1. Introduction
For many years, the implementation of compact optical isolators has been a major research
goal in the field of integrated optics [1,2]. Optical isolation requires the breaking of Lorentz
reciprocity;thisistraditionallyachievedusingmagneto-opticmaterials,butsuchmaterialsare
challenging to incorporate into integrated optics devices [3,4]. The most commonly-pursued
alternative method for breaking reciprocity is to exploit optical nonlinearity [1,5–11]. Two
recent demonstrations of nonlinearity-based on-chip optical isolators, by Peng et al. [9] and
Changetal.[10],havedrawnparticularattention.Theseexperimentsfeaturedapairofcoupled
whispering-gallerymicrocavities,onecontaininglossandtheothersaturable(nonlinear)gain.
Lighttransmissionacrossthestructurewasfoundtobestronglynonreciprocal,dependingon
whether it first passed through the gain or loss resonator. Aided by the high Q factors of the
resonators,isolationwasobservedforrecord-lowpowersof∼1µW[9].
Theuseofdualresonatorscontaininggainandlossin[9,10]wasinspiredby“PT symmetric
optics”,whichconcernsopticalstructuresthatareinvariantundersimultaneousparity-flip(P)
and time-reversal (T) operations [12–23]. The concept originated from the observation that
PT symmetricHamiltonians,despitebeingnon-Hermitian,canexhibitrealeigenvaluespectra
[24,25], as well as “PT-breaking transitions” between real and complex eigenvalue regimes.
The PT-breaking transition point is an “exceptional point”, where two eigenstates coalesce
and the effective Hamiltonian becomes defective [26,27]. Near the transition, the dynamical
behavior of the opticalfields can exhibit highly interesting features[28–31]; for instance, the
presence of gain saturation has been found to stabilize PT-symmetric steady states past the
usualPT transitionpoint[29,30].
Despite these intriguing conceptual links, it was not clear from [9,10] how PT symmetry
relatestotheworkingofthenonlinearopticalisolatorsinquestion.Strictlyspeaking,PT sym-
metry holds in the dual-resonator structures only in the linear limit; in the nonlinear regime,
thegainsaturatesandnolongermatchestheloss,sothestructuresarenotPT symmetricand
do not possess distinct “PT-symmetric” or “PT-broken” phases. Peng et al., in [9], indicated
thatopticalisolationoccurs(inthenonlinearregime)ifthesystemistunedsothatitwouldbe
PT-broken in the linear regime; however, the actual correspondence was not shown theoreti-
callynorexperimentally.Thedynamicalbehaviorofthesystem,includingtheuniquenessand
stabilityofthesteady-statesolution(s),wasalsounexplored.
In this paper, we present a theoretical analysis of the dual-resonator structure, aiming to
clarify the relationships between the PT phase, the performance of the nonlinear optical iso-
lator,andtheuniquenessandstabilityofthesteady-stateopticalmodes.Usingcoupled-mode
theory[32–35],westudytheconditionsforsteady-statesolutionstoexist,andtheasymptotic
stability of those solution(s). We find that stability in the nonlinear system has a close corre-
spondencewiththePT transitionboundaryoftheunderlyinglinearsystem.
In the “weak-input limit”, where the input intensity is low relative to the gain saturation
threshold within the amplifying resonator, we show that the nonlinear solutions at non-zero
frequency detunings are asymptotically stable in the PT-symmetric phase. In the PT-broken
phase, the solutions become unstable at sufficiently large frequency detunings, and the non-
linearsystemexhibitslimit-cycleoscillations,whichmightbeusefulforfrequencygeneration
applications(suchasfrequencycombs).
For small frequency detunings, multiple steady-state solutions can exist in the PT-broken
phase,butonlythehighest-intensitysolutionisasymptoticallystable.Specificallyatzerode-
tuning, there is always one stable steady-state solution, and the nonlinear system exhibits a
sharptransitionbetweenisolatingbehavior(correspondingtothePT-brokenphase)andrecip-
rocalbehavior(correspondingtothePT-symmetricphase).Althoughthistransitioncoincides
exactly with the PT transition point, it is an inherently nonlinear effect, arising from a jump
betweendifferentsolutionbranchesofthetransmissionintensityequations.However,theper-
formance of the isolator can be significantly limited by the contributions to the nonlinearity
causedbyareflectedwave[11].
Wealsoshowthattheperformanceofthenonlinearopticalisolatorisalsomodifiedinause-
fulwaywhenthetworesonator-to-waveguidecouplingratesareunequal.Inthiscase,asmall
shiftacrossthetransitionpointcausestheisolationratio(theratiobetweenforwardandback-
wardtransmissionintensities)toundergoanabruptjump,whichapproachesadiscontinuityin
theweak-inputlimit.Themagnitudeofthisjumpisgivenbytheratioofthecouplingrates.This
phenomenoncanbeusedtorealizeanonlinearopticalisolatorthatexhibitsverylargechanges
intheisolationratio,activelycontrolledbytinyshiftsin(e.g.)theinter-resonatorseparation.
2. Coupled-modeequations
Thedual-resonator structureisshown schematicallyin Fig.1(a).The setupisidentical tothe
experimentsreportedin[9,10],consistingoftwoevanescentlycoupledmicrocavitieswithres-
onant frequencies ω and ω . One resonator contains saturable gain, and the other is lossy.
1 2
Theresonatorsarecoupledtoseparateopticalfiberwaveguides,whichactasinput/outputports
(labeled 1–4), with couplings κ and κ . The direct inter-resonator coupling rate is µ. In the
1 2
“forward transmission” configuration, light is injected from port 1 at a fixed operating fre-
quencyω,exitingatports2and4.Alternatively,inthe“backwardtransmission”configuration,
light is injected at port 4 and exit at ports 1 and 3. We are interested in the level of isolation
betweenports1and4,whichserveastheoperationalinputandoutputportsforthedevice.
Thedual-resonatorsystemcanbedescribedbycoupled-modeequations[9,10],formulated
using the standard framework of coupled-mode theory [32–35]. In this section and the next,
we briefly summarize these equations, which have previously been presented in [9,10]. For
forwardtransmission,thecoupled-modeequationsare
da
1 =(i∆ω +g)a −iµa (1)
1 1 2
dt
da √
2 =(i∆ω −γ)a −iµa + κ s (2)
dt 2 2 1 2 in
I =κ |a |2. (3)
F 1 1
Here,a anda denotethecomplexamplitudesfortheslowly-varyingfieldamplitudesinthe
1 2
gain resonator and loss resonators, respectively; ∆ω ≡ω−ω denote the operating fre-
1,2 1,2
quency’s detuning from each resonator’s natural frequency; g>0 and γ >0 are the net gain
rateinresonator1andthenetlossrateinresonator2;s istheamplitudeoftheincominglight
in
inport1;andI isthepowertransmittedforwardintoport4.Forthemoment,weassumethat
F
there is no reflected wave re-entering the system from port 4; the effects of such a reflected
wavewillbediscussedinSection7.
Forbackwardtransmission,adifferentsetofcoupled-modeequationsholds:
da √
1 =(i∆ω +g)a −iµa + κ s (4)
dt 1 1 2 1 in
da
2 =(i∆ω −γ)a −iµa (5)
2 2 1
dt
I =κ |a |2, (6)
B 2 2
whereI isthepowertransmittedintoport1.
B
Thegain/lossratesgandγ consistofseveralradiativeandnon-radiativeterms[9]:
g= 1(cid:0)g(cid:48)−γ −κ (cid:1) (7)
1 1
2
1
γ = (γ +κ ), (8)
2 2
2
whereg(cid:48) istheintrinsicamplificationrateinresonator1,andγ aretheintrinsiclossratesin
1,2
104
103
102
101
100
Backward 10-1
Transmission -2.0 -1.0 0.0 1.0 2.0
106 Forward
104 Backward
102
100
10-2
-2.0 -1.0 0.0 1.0 2.0
10-2
10-3
Forward
Transmission 10-4
-2.0 -1.0 0.0 1.0 2.0
Fig.1.(a)Schematicofaresonatorwithsaturablegaincoupledtoalossyresonator,with
bothresonatorscoupledtoopticalfiberports.Solidarrowsindicateforwardtransmission
(port 1→4), and dashed arrows indicate backward transmission (port 4→1). (b)–(d)
Transmissioncharacteristicsinthelinear(non-gain-saturated)regime,whenthegainand
loss are PT symmetric (g=γ =0.4). Here, we plot the intensity in the active resonator
(|a |2)underforwardtransmission(solidlines),andinthepassiveresonator(|a |2)under
1 2
backward transmission (dashes), versus the frequency detuning. These resonator intensi-
tiesareproportionaltotheforwardandbackwardtransmissionintensitiesviaEqs.(3)and
(6).InthePT-symmetricphaseµ>γ,therearetwotransmissionpeaks;inthePT-broken
phaseµ<γ,thesemergeintoasinglepeak.
theresonators.Untilstatedotherwise,wewillimposethefollowingsimplifyingrestrictions:
κ =κ =γ =γ , (9)
1 2 1 2
∆ω =∆ω ≡∆ω, (10)
1 2
g
g(cid:48)= 0 . (11)
1+|a /a |2
1 s
Equation(9)correspondstoa“criticalcoupling”criterionwithrespecttotheindividualcavity-
waveguidecouplings.Theintrinsiclossandoutcouplingratesarealltunedtothesamevalue;
note also that g = g(cid:48)/2−γ. Equation (10) states that the resonators have the same natural
frequency.Equations(9)–(10)serveassimplifyingassumptions,toavoiddealingwithaprolif-
erationoffreeparameters;later,wewilldiscusstheimplicationsofrelaxingtheseassumptions.
Anotherimportantconstraint,PT symmetry,willbeimposedinthenextsection.Equation(11)
describessaturablegain,whereg istheunsaturatedamplificationrate,anda ∈R+ isagain
0 s
saturationthreshold.
Theexperimentallyrealizedsystemsreportedin[9,10]operatedinthe1550nmwavelength
band,withrateparametersµ,g ,γandκ ontheorderof10MHzin[9],and100MHzin[10].
0 1,2
The coupling rates µ and κ can be tuned via the inter-resonator and resonator-waveguide
1,2
separations.Theinputpower|s |2rangedfromzerotoaround10–100µW[9,10].
in
Thesuitabilityofthesystemasanopticalisolatorischaracterizedusingthe“isolationratio”,
whichistheratioofforwardtobackwardtransmittanceatfixedinputpower:
T I (I )
R≡ F = F in , (12)
T I (I )
B B in
whereI isobtainedbysolvingEqs.(1)–(3)witha˙ =a˙ =0(steadystate),andI isobtained
F 1 2 B
from Eqs. (4)–(6). When the system is reciprocal, R=1. The isolation ratio was also used
in[9,10]asthefigureofmeritforopticalisolation.However,itisworthnotingthattheforward
andbackwardtransmissionsarebeingcomparedundertheassumptionthat,ineithercase,no
reflectedwaveispresent.WewilldiscussthislimitationingreaterdetailinSection7.
3. Linearoperation
Wenowimposetheimportantconstraintg=γ.Thismeansthatinthelinearregime,a →∞,
s
the gain and loss resonators become PT symmetric. To understand the implications, consider
the“closed”systemwithoutresonator-fibercouplings.Itsdetuningeigenfrequenciesare
(cid:115)
g−γ (cid:18)g−γ(cid:19)2
∆ω =i ± µ2−γg− . (13)
2 2
(cid:112)
When g=γ, these reduce to ∆ω =± µ2−γ2. As µ and γ are varied while keeping g=γ,
thesystemhasaPT symmetry-breakingtransitionatµ =γ.Forµ >γ,thedetuningsarereal
(PT-symmetricphase),andforµ <γ theyarepurelyimaginary(PT-brokenphase).
With the resonator-fiber couplings introduced, the eigenmodes become transmission reso-
nances. In the PT-symmetric phase µ >γ, the resonator modes and transmission amplitudes
(cid:112)
exhibittwointensitypeaks,at∆ω =± µ2−γ2,correspondingtothe(real)detuningsofthe
closed system, as shown in Fig. 1(b). In the PT-broken phase µ <γ, there is a single peak
at zero detuning, as shown in Fig. 1(d). As noted by Peng et al. [9], the PT-symmetric and
PT-brokenphaseswillgiveverydifferentbehaviorsoncegainsaturationisintroduced.
Inthelinearregime,Eqs.(1)–(3)andEqs.(4)–(6)obeyopticalreciprocitybyexplicitcon-
struction [33]. For fixed s , the forward and backward transmission amplitudes are exactly
in
equal,I =I ;theisolationratioisR=1,asshowninFig.1(b)–(d).
F B
4. Nonlinearoperation:multiplesolutionsandstability
Weturnnowtothenonlinear,gain-saturatedregime,settingg =4γ,sothatg→γ asa →∞.
0 s
This means the system would be PT symmetric in the absence of gain saturation. If we use
a asthenaturalintensityscaleforthecoupled-modeequations,thenonlinearsystemhasfour
s
remainingindependentparameters:∆ω,µ,γ,and|s |2.
in
Forfinitea ,opticalreciprocityisbroken.However,thesystemisnolongerPT symmetric,
s
since g(cid:54)=γ, and thus we can no longer rigorously define “PT symmetric” or “PT broken”
phases. Still, we can relate the nonlinear system’s behavior to the PT symmetric phases as
definedinthelinearlimit.
In the linear regime, the solutions to the coupled-mode equations were unique. With non-
linearity, the coupled-mode equations can have multiple steady-state solutions. For forward
transmission,steady-statesolutionsaredeterminedbycombiningEqs.(1)–(2)into:
(cid:16) (cid:17) (cid:16) (cid:17)
|α|2x3+ 2|α−1|2−2−β x2+ |α−2|2−2β x−β =0, (14)
whereα,β,andxarethefollowingdimensionlessvariables:
α = (−i∆ω+γ)2+µ2, β = µ2 1 (cid:12)(cid:12)(cid:12)sin(cid:12)(cid:12)(cid:12)2, x=(cid:12)(cid:12)(cid:12)a1(cid:12)(cid:12)(cid:12)2. (15)
γ(−i∆ω+γ) ∆ω2+γ2 γ (cid:12)as(cid:12) (cid:12)as(cid:12)
Sincex∈R+,thereiseitherone,two,orthreephysicalsteady-statesolutions.Theremustbe
at least one solution, since the polynomial has a positive third-order coefficient and negative
3.0
1.0 Forward Transmission Forward Transmission
0.5 2.0
Unstable
Asymptotically
0.0 Unstable 3-sol stable 1.0
-0.5
-1.0 0.0
1.0 Backward Transmission 0.05 Backward Transmission
0.5 Unstable
0.00
Asymptotically
0.0 3-sol stable
-0.5 Unstable 0.05
-1.0
0.10
0.00 0.25 0.50 0.75 1.00 1.25 0 0.2 0.4 0.6 0.8 1.0
PT critical point
Fig. 2. (a)–(b) Domains in which the nonlinear coupled-mode equations have multiple
steady-statesolutions,forforward(a)andbackward(b)transmission.Here,weshowthe
parameter space defined by the frequency detuning ∆ω and inter-resonator coupling µ,
with fixed γ =0.4, sin =0.5, and as =3; symbols indicate the points in the parameter
spacecorrespondingtothecurvesin(c)and(d).Withinthesmall-∆ω regionboundedby
theredcurves,thehighest-intensity(oronly)solutionisasymptoticallystable.(c)–(d)Plots
showingtheemergenceofmultiplesolutionsatseveralvaluesofµ,fixing∆ω=0.Thehor-
izontalaxisisthenormalizedintensityinthegainresonator,|a1/as|2;theverticalaxisis
theleft-handsideofthecubicEq.(14),anditscounterpartforbackwardtransmission;the
steady-statecoupled-modeequationsaresatisfiedwhenthecurvescrosszero.
zeroth-ordercoefficient.Thebackwardtransmissioncaseishandledsimilarly,usingEqs.(4)–
(5);itgivesthesamecubicequationasEq.(14),butwiththereplacement
β = 1 (cid:12)(cid:12)(cid:12)sin(cid:12)(cid:12)(cid:12)2. (16)
γ (cid:12)as(cid:12)
Solvingthepolynomialrevealsadomaininparameterspacewheretherearethreephysical
steady-statesolutions,outsideofwhichthesolutionisunique.ThisisshowninFig.2(a)–(b).
Thethree-solutiondomainlieswithinthe“PT-broken”phaseofthelinearsystem,µ <γ.
Theboundariesofthethree-solutiondomaindependonγ ands ,asthechoiceofforward
in
or backward transmission. It consists of two sets of curves; the black curves in Fig. 2(a)–(b)
involveadegeneracyoftwolow-intensityrootsofthecubicpolynomial[Fig.2(c)–(d)].Cross-
ing this boundary causes no discontinuity in the intensity of the stable steady-state solution.
TheredcurvesinFig.2(a)–(b)involvethedegeneracyoftwohigh-intensityrootsofthecubic
polynomial(14);crossingthisboundarydestabilizesthesteady-statesolution.
Through numerical stability analysis, detailed in Appendix A, we find that the highest-
intensitysolutioninthethree-solutiondomainisasymptoticallystable(i.e.,theLyapunovex-
ponents are all negative). The two lower-intensity solutions are unstable: small perturbations
from these steady states eventually evolve into the highest-intensity state. In the one-solution
domain,thesolutionisasymptoticallystableforsmalldetuning∆ω,andunstableforlarge∆ω.
Interestingly,theregionofasymptoticstabilityinthenonlinearsystemiscloselyconnected
to the PT symmetry phases of the linear system. For µ <γ, which corresponds to the PT-
brokenphase,thefrequencyrangeofasymptoticstabilityisboundedbythesolidanddashed
red curves shown in Fig. 2(a)–(b). These bounds diverge at µ =γ, which corresponds to the
transitionfromthePT-brokentothePT-symmetricphaseinthelinearsystem.Forµ >γ,the
steadystatesolutionbecomesasymptoticallystableforall∆ω.
In the one-solution domain, the onset of asymptotic instability (at sufficiently large ∆ω) is
PT critical point
PT critical point
Fig.3.Isolationratioversusµ/γatzerofrequencydetuning(∆ω=0),for(a)weakinputs
sin=0.15andas=3,and(b)stronginputsregimesin=9andas=3,usingseveralchoices
ofγ.Intheweak-inputregime,theisolationratioismainlydeterminedbythePT-breaking
parameter µ/γ. The system becomes reciprocal for µ/γ >1, corresponding to the PT-
symmetric phase of the linear system. (c) Close-up of the isolation ratio behavior in the
weak-inputregime,showingthekinkinthedependenceonµ/γ atthePT transitionpoint
µ/γ=1.Circlesshowexactnumericalsolutionsofthecoupled-modeequations,andthe
solidcurveshowstheanalyticapproximationsofEqs.(22)–(23).
associated with the appearance of sustained time-domain beating in both the resonator inten-
sities and the transmittance. This is a Hopf bifurcation [36] from a stable state to limit cycle
behavior(seeFig.7inAppendixA).
5. Isolationratiosatzerodetuning
Let us now focus on zero detuning, ∆ω =0. In this case, there is always an asymptotically
stablesteady-statesolution,andweshallbeabletoderiveanimportantconnectiontothePT
transitionofthelinearsystem.Thevariablesα andβ,definedinEq.(15),simplifyto
(cid:18) (cid:19)2
µ
α =1+ (17)
γ
 (cid:18) (cid:19)2
β = |sin/as|2 × µγ , (Forward) (18)
γ
 1, (Backward).
Hence,thecubicpolynomialinEq.(14)isentirelydeterminedbytwoquantities:(i) µ/γ and
(ii)|s /a |2/γ.ThefirstquantityisalsothetuningparameterforthePT transition.Thesecond
in s
quantitydeterminesthestrengthoftheinputrelativetothegainsaturationthreshold.Wewill
beparticularlyinterestedinthe“weak-input”limit,definedas
√
s (cid:28) γa . (19)
in s
Whenβ (cid:28)1,thesteadystatebehaviorwillbeprincipallydeterminedbythePT-tuningparam-
eterµ/γ.
Figure3plotstheisolationratioR≡I /I versusµ/γ,forseveraldifferentvaluesofγ and
F B
s . In the weak-input regime, the isolation ratio curves are almost identical for different γ,
in
whichverifiesthatthesystemiscontrolledbythecombinationµ/γ.Forµ/γ <1,correspond-
ing to the PT-broken phase of the linear system, we find that R>1, and hence the system
functions as a good optical isolator. For µ/γ >1, we find that R≈1. This agrees with the
qualitativebehaviorsreportedin[9].
Letusexaminethevicinityofthetransitionpointingreaterdetail.Figure3(c)showsthatin
theweakinputregime,theisolationratiocurveexhibitsakinkatµ/γ =1.Tounderstandthis,
wereturntothedefinitionoftheisolationratio:
I x
R= F =(µ/γ)−2 F, (20)
I x
B B
wherex andx arethesolutionstoEq.(14)fortheforwardandbackwardtransmissioncases.
F B
Forβ →0,Eq.(14)reducesto
(cid:18) 1−(µ/γ)2(cid:19)2
x x− ≈0. (21)
1+(µ/γ)2
For µ/γ <1, the double-root in Eq. (21) is positive. Hence, in this approximation, the three-
solutiondomaindiscussedinSection4extendsovertheentirerange µ/γ <1alongthezero-
detuningline.Theasymptoticallystablesolutioncorrespondstothedouble-root,whichisequal
for forward and backward transmission, to lowest order in β. Hence, we can use Eq. (20) to
showthat
√
R≈(µ/γ)−2 for µ/γ <1, s (cid:28) γa . (22)
in s
For µ/γ >1,thedouble-rootisnegative,sotheonlyvalidrootintheβ →0limitisx=0.
Fornon-zeroβ,thisrootbecomesO(β),soEq.(18)impliesthatx /x ≈(µ/γ)2.Thisyields
F B
theisolationratio
√
R≈1 for µ/γ >1, s (cid:28) γa . (23)
in s
Thelimitingexpressions(22)–(23)areplottedinFig.3(c),andagreewellwiththenumerical
solutions.ThishelpsexplainwhythePT phaseofthelinearsystemaffectstheisolationfunc-
tionalityofthenonlinearsystem.Bothphenomenaaredeterminedbytheparameterµ/γ,witha
criticalpointatµ/γ=1.Thekinkintheisolationratioarisesfromswitchingsolutionbranches
atthecriticalpoint.
6. Imbalancedinput/outputcouplings
Thusfar,wehaveassumedthatthewaveguide-resonatorcouplings,κ andκ ,areequal.Ifthe
1 2
couplingsareunequal,theisolationbehaviorofthesystemcanbequitedifferent.Tostudythis,
wereplaceEq.(9)with
γ +κ =γ +κ =2γ. (24)
1 1 2 2
Forg =4γ,thegaininresonator1is
0
2γ
g= −γ, (25)
1+|a /a |2
1 s
whichensuresthatthe decoupledsystemremainsPT symmetricwithcriticalpoint µ =γ,as
before. With this generalization, the steady-state equations (14)–(16) are altered only by the
replacements
κ
β → 2β (Forward)
γ
(26)
κ
β → 1β (Backward).
γ
Fig.4.Isolationratioversusµ/γ fordifferentmicrocavity-waveguidecouplingrates.The
system parameters are ∆ω =0, sin=0.03, as=3, γ =1, and√g0=4γ. Thin solid lines
showtheanalyticapproximationintheweak-inputlimit(sin(cid:28) γas),givenbyEq.(27).
ByvaryingthecouplingsandlossessothatEq.(24)issatisfied,wecanaccessdifferentvalues
ofκ /κ ,subjecttotheconstraint0<κ ,κ <2γ.
1 2 1 2
The discussion of Section 5 generalizes to this case in a straightforward way. Using the
previous zero-detuning and weak-input assumptions, we find that x /x ≈1 for µ/γ <1, as
F B
before;butfor µ/γ >1,Eq.(26)givesx /x ≈β /β =(κ /κ )(µ/γ)2.Theisolationratio
F B F B 2 1
nowbecomes
R=(κ /κ )(µ/γ)−2x /x
1 2 F B
(cid:26) (κ /κ )(µ/γ)−2 for µ/γ <1 (27)
≈ 1 2
1 for µ/γ >1.
Forκ (cid:54)=κ ,thispredictsadiscontinuityintheisolationratioatµ/γ =1.
1 2
Figure4plotsdependenceoftheisolationratiosonµ/γ,atzerodetuning,forthecasesof(i)
κ (cid:28)κ ,(ii)κ (cid:29)κ ,and(iii)κ =κ .Inallthreecases,theisolationratioapproachesunity
1 2 1 2 1 2
for µ/γ >1. However, for µ/γ <1, the unequal-coupling curves exhibit an abrupt change
corresponding to a factor of κ /κ (which is two orders of magnitude for these examples).
1 2
Interestingly, for κ (cid:28)κ , the isolation ratio in fact decreases below unity, before increasing
1 2
againasµ/γ →0.ThenumericalresultsmatchEq.(27)verywell.
Thisphenomenonmaybeexploitedindeviceapplicationsforrealizinganactivelyswitch-
able optical isolator. Using a small variation in the µ/γ parameter (e.g., by varying the inter-
cavityseparation,whichaffectsµ),wecanswitchbetweenstrongopticallyisolatingandrecip-
rocalregimes.
7. Effectofasimultaneousreflectedwave
We have analyzed the nonlinear system and its isolation ratio under the assumption that light
propagates in one direction at a time (i.e., forward or backward). This is a good assumption
if the isolator is part of a optical circuit operating with optical pulses, such that any reflected
pulse re-entering the isolator (due to scattering from other parts of the circuit) does so at a
latertime,aftertheinitialpulsehasalreadydiedaway.Whenforwardandbackwardwavesare
simultaneouslypresent,however,bothcontributetothenonlinearity,causingtheisolatortofail.
Thisisagenerallimitationofopticalisolatorsbasedonnonlinearity[11].