Parity-Time Symmetric Coupled Microresonators with a Dispersive Gain/Loss

1.5 MB·English

by  Sendy Phang

Parity-Time Symmetric Coupled
Microresonators with a Dispersive
Gain/Loss
SendyPhang1,∗,AnaVukovic1,StephenCreagh2,TrevorM.Benson1,
5
PhillipD.Sewell1,GabrieleGradoni2
1
0 1GeorgeGreenInstituteforElectromagneticsResearch,FacultyofEngineering,
2 UniversityofNottingham,Nottingham,NG72RD,UK
2SchoolofMathematicalSciences,UniversityofNottingham,
n
Nottingham.NG72RD,UK
a
J ∗[email protected]

9
2
Abstract: The paper reports on the coupling of Parity-Time (PT)-

] symmetric whispering gallery resonators with realistic material and
s
c gain/lossmodels.ResponseofthePTsystemisanalyzedforthecaseoflow
i andhighmaterialand gaindispersion,andalso fortwo practicalscenarios
t
p when the pump frequency is not aligned with the resonant frequency of
o
the desired whispering gallery mode and when there is imbalance in the
.
s gain/loss profile. The results show that the presence of dispersion and
c
frequency misalignment causes skewness in frequency bifurcation and
i
s significant reduction of the PT breaking point, respectively. Finally, as
y
coupled WGM resonators are inherently lossy structures, we show that
h
p unbalancing the gain/loss in resonators is required to compensate for
[ inherentlossofthestructureandachieveimprovedPTproperties.

1 © 2015 OpticalSocietyofAmerica
v
5 OCIS codes: (080.6755) Systems with special symmetry;(140.3945) Microcavities;
5 (230.4555)Coupledresonators
4
7
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1. Introduction
PhotonicsisemergingasapopularpracticalplatformfortheexplorationofParity-Time(PT)-
symmetric systems characterized by balanced loss and gain and having a threshold point at
whichrealeigenfrequenciesofthesystemcoalescetobecomecomplexconjugates[1–4].The
existenceofthisthresholdpointisessentialtotheuniquepropertiesofPT-structures,suchas
unidirectionalinvisibilityandsimultaneouslasingandabsorption[2,3].Thisphenomenaopens
new avenues for the realization of practical devices such as lasers, optical memory, optical
switchesandlogic-gates[5–11].Todate,PT-symmetricstructuresbasedonBragggratingsand
coupled optical systems have been investigated both theoretically [2–4,8,12–19] and exper-
imentally [20–25]. Recently, a PT-symmetric system based on two coupled microresonators
and two fiber-taper waveguides has been experimentally demonstrated and shown to exhibit
direction-dependent behavior at a record low power of 1m W [21,25]. This is primarily at-
tributed to strong field localization andbuild up of energyin the resonantwhisperinggallery
modes[21,25],andhasfurtherstrengthenedtheargumentforusingresonantstructuresrather
thanwaveguidesasbuildingblocksofPT-symmetricsystems.IncontrasttoPTsymmetriccou-
pledwaveguidesystemswheretheeigenmodesarepurelyrealbelowthethresholdpoint,thePT
symmetriccoupledmicroresonatorshavecomplexeigenfrequenciesbelowthethresholdpoint
duetoinherentradiationlosses[3,4,13,17].
In this paper we investigate the fundamental properties of the PT resonant system based
on two coupled whispering gallery resonators within the context of both realistic material
properties and practical operating constraints. In particular we discuss how practical disper-
sivepropertiesofmaterialgainandlossthatsatisfytheKramers-Kronigrelationshipaffectthe
performanceofmicrocavity-basedPTresonantstructures.Oursurprisingconclusionisthatac-
countingforlarge,yetrealistic,levelsofdispersionpreservestheessentialthreshold-behaviour
predictedby completelyPT-symmetric dispersionlessmodels, while more moderatelevelsof
dispersioncancompletelychangethecharacteroftheresponseofthesystemtoincreasinggain
andloss. In particular,whenthere ismoderatedispersionthegainandlossmaterialsrespond
differently to frequencyshifts in such a way that sharp threshold points give way to gradual
changesoverarangeofparameters.Whendispersionisincreasedfurther,theresponsereverts
to threshold behaviour of the type seen in non-dispersivePT-symmetric systems, albeit with
somebreakingofdetailedquantitativesymmetry.
OurrecentworkondispersivePT-Bragggratingshasshownthatmaterialdispersionlimits
theunidirectionalinvisibilitytoasinglefrequencywhichisinstarkcontrasttopreviousresults
thatassumedidealizedgain/lossprofileinordertodemonstratewidebandunidirectionalbehav-
ior[19,26].Inthispaper,theperformanceofthemicroresonator-basedPTsystemisanalyzed
forpracticalscenariosinvolving:a)frequencymismatchbetweenthecavityresonantfrequency
andthegainpumpfrequencyand,b)imperfectbalanceofthegainandlossinthesystem.The
analysis of the microresonator-basedPT system is achieved using an exact representation of
theproblembasedonboundaryintegralequationsandexplicitanalyticalresultsaregivenfora
weaklycoupledsystemusingperturbationanalysis[27].Weconcentrateontheweakly-coupled
limitinourdetailedcalculationsbecausethatcapturestheessentialpropertiesofthethreshold
behaviorofthePT-symmetrywhileallowingsimpleanalyticalcalculationstobeused.Finally,
real-timefield evolutionin a two microresonatorPT-symmetricsystem isanalyzedfordiffer-
entlevelsofdispersionusingthenumericaltimedomainTransmissionLineModelling(TLM)
method[9,28,29].
2. PTsymmetriccoupledmicroresonators
InthissectionwedescribethetheoreticalbackgroundofaPT-symmetricsystembasedontwo
coupledmicroresonators.Thesystem,inwhichbothmicroresonatorshaveradiusaandaresep-
aratedbyagapg,isillustratedschematicallyinFig.1.Theactiveandpassivemicroresonators
havecomplexrefractiveindicesn andn respectively,thataretypicallychosentosatisfythe
G L
PTconditionn =n∗,where*denotescomplexconjugate,n=(n′+jn′′),andn′ andn′′ rep-
G L
resent the real and imaginary parts of the refractive index. In practice, localized gain might
be achieved by means of erbium doping and optical pumping of the active microresonator,
while maskingthe lossy microresonatorasin [20–22,25]. Both resonatorsare assumedto be
surroundedbyair.
Therefractiveindexofdispersivematerialsisfrequencydependentbutmustalsosatisfythe
Fig.1.Schematicoftwocoupledcylindricalmicroresonatorsorradiusaandseparatedbya
distanceg.Microresonatorswithgainandlossaredenotedbym RGandm RL,respectively.
causalitypropertybetweentherealandimaginarypartsofthematerialrefractiveindex[30,31].
Thematerialpropertiesareconvenientlymodelledbyassumingadielectricconstantthatuses
aLorentzianmodelfordispersionasin[32]
s 1 1
er(w )=e¥ −j2e00w (cid:18)1+j(w +w s )t +1+j(w −w s )t (cid:19). (1)
Here e¥ denotes the permittivity at infinity, w s denotes the atomic transitional angular fre-
quency,t isthedipolerelaxationtimeands isrelatedtotheconductivitypeakvaluethatis
0
set by the pumpinglevel at w s . The time-varyingcomponenthas been assumed to be of the
formejw tandtherefores >0denoteslosswhiles <0denotesgain.Theparametert controls
0 0
thedegreeofdispersion,witht =0correspondingtoadispersion-lesssystem.Throughoutthis
paper,thefrequency-dependentrefractiveindexisexpressedasn= e (w )andthematerial
r
gain/lossparameterisexpressedusingtheimaginarypartofrefractivepindexasg =w n′′.
3. Analysisofinter-resonatorcouplinginthefrequencydomain
Wenowprovideananalysisofcouplingbetweenresonatorsbasedonboundaryintegralmeth-
ods.Thisapproachisparticularlysuitedtoperturbativeapproximationofthecouplingstrength
in the weak coupling limit but also providesan efficient platform for exactcalculation when
couplingisstrong.Thecalculationis basedonanapproachusedin [27] todescribecoupling
betweenfullyboundstatesincoupledresonatorsandopticalfibers,butadaptedheretoallow
forradiationlosses.Itisalsosimilartomethodsusedin[33–35].
3.1. Notationandassumptions
Weassumeresonatorsofradiusa,uniformrefractiveindexandTMboundaryconditions.Then
themodetakingtheformy =(J (n kr)/J (n ka))ejmq insidetheisolatedlossyresonatoris
L m L m L
suchthatthesolutionanditsnormalderivativeontheboundaryoftheresonatorcanbewritten
as
¶y
a L =FLy , (2)
¶ n m L
where
zJ′ (z)
FL= m and z=n ka, (3)
m J (z) L
m
where, k is the free-space wave number and y and FG being defined similarly for the gain
G m
resonator. The treatment of coupling in the remainder of this section can be used for other
circularly-symmetric resonators such as those with graded refractive index or with different
boundaryconditions,aslongasanappropriatelymodifiedFLissubstitutedin(2).
m
3.2. Exactsolutionusingboundary-integralrepresentation
An exact boundary integral representation of the coupled problem is conveniently achieved
by applying Green’s identities to a region W which excludes the resonators, along with an
infinitesimallysmalllayersurroundingthem(sothattheboundariesB andB oftheresonators
G L
themselvesliejustoutsideW ).InW ,weassumethattherefractiveindextakesthevaluen =1,
0
sothatthefree-spaceGreen’sfunctionis
j
G (x,x′)=− H (k|x−x′|), (4)
0 0
4
whereH (z)=J (z)−jY (z)denotestheHankelfunctionofthesecondkind(andthesolution
0 0 0
is assumed to have time dependenceejw t). Then, applyingGreen’s identities to the region W
andassumingradiatingboundaryconditionsatinfinityleadstotheequation
¶y (x′) ¶ G (x,x′)
0=ZBG+BL(cid:18)G0(x,x′) ¶ n′ − 0¶ n′ y (x′)(cid:19)s.′ (5)
whenxliesoneitherB orB (andthereforejustoutsideofW ).
L G
WenowexpandthesolutionontherespectiveresonatorboundariesasFourierseries,
y G=(cid:229) a mGejmqG and y L=(cid:229) a mLejmqL, (6)
m m
in the polaranglesq andq centeredrespectivelyonthe gainandlossy resonators,running
G L
inoppositesensesineachresonatorandzeroedonthelinejoiningthetwocenters.Thecorre-
spondingnormalderivativescanbewritten
¶y G =(cid:229) 1FGa GejmqG and ¶y G =(cid:229) 1FGa GejmqG. (7)
¶ n a m m ¶ n a m m
m m
UsingGraf’stheorem[36]toexpandtheGreen’sfunctionG (x,x′)analogouslyinpolarcoor-
0
dinatesoneachboundary,theintegralequation(5),evaluatedseparatelyforxonB andonB ,
L G
leadstoapairofmatrixequations
DGa G+CGLa L = 0
CLGa G+DLa L = 0. (8)
Here,
. .
. .
. .
 a G   a L 
a G= m and a L= m (9)
 a G   a L 
 m+1   m+1 
 ..   .. 
 .   . 
   
are Fourierrepresentationsof the solution onthe boundariesof the gainand lossy resonators
respectively.ThematricesDGandDLarediagonalwithentries
uH′(u)
DG,L=J (u)H (u) FG,L− m , whereu=ka, (10)
mm m m (cid:18) m H (u) (cid:19)
m
andprovidethe solutionsoftheisolated resonators.ThematricesCGL andCLG describecou-
plingbetweentheresonators.ThematrixCGL hasentriesoftheform
u J′ (u)
CGL=J(u)H (w)J (u) FL− L m , (11)
lm l l+m m (cid:18) m J (u) (cid:19)
m
whereu=ka,w=kbandbisthecenter-centerdistancebetweenthegainandlossyresonators.
ThematrixCLGisdefinedanalogouslybyexchangingthelabelsGandL.
3.3. PT-symmetryintheexactsolution
Thesystem(8)canbepresentedmoresymmetricallybyusingthescaledFouriercoefficients
uJ′ (u)
a˜L =J (u) FL− m a L (12)
m m (cid:18) m J (u) (cid:19) m
m
(alongwithananalogousdefinitionofa˜G).Then(8)canberewritten
m
D˜Ga˜G+C˜a˜L = 0
C˜a˜G+D˜La˜L = 0, (13)
wherethediagonalmatricesD˜G,Lhaveentries
H (u)FG,L−uH′(u)
D˜G,L=−j m m m , whereu=ka, (14)
mm J (u)FG,L−uJ′(u)
m m m
andthesame(symmetric)matrixC˜,withentries
C˜ =−jH (w), (15)
lm l+m
couplessolutionsinbothdirections.
Wehaveincludedanoverallfactorof−jintheseequationstoemphasiseanapproximatePT-
symmetrythatoccurswhenn =n∗.Then,inthelimitofhigh-Qwhisperinggalleryresonances
G L
forwhichwemayapproximate
jH (u)≃Y (u) and jH (u)≃Y (u), (16)
m m l+m l+m
thematricesin(13)satisfytheconditions
D˜L ∗≃D˜G and C˜∗≃C˜ (17)
(cid:0) (cid:1)
which are a manifestation of PT symmetry of the system as a whole: deviation from these
conditionsisduetoradiationlosses.
3.4. Perturbativeweak-couplingapproximation
The system of equations (13) can be used as the basis of an efficient numerical method for
determiningtheresonancesofthecoupledsystemwitharbitraryaccuracy.Inpractice,oncethe
gap g=b−2a between the resonators is wavelength-sized or larger, a truncation of the full
systemtoarelativelysmallnumberofmodessufficestodescribethefullsolution.
Inthelimitofveryweakcouplingwemayachieveaneffectiveperturbativeapproximation
byrestrictingourcalculationtoasinglemodeineachresonator.Weconsiderinparticularthe
caseofnearleft-rightsymmetryinwhich
n ≈n . (18)
G L
PTsymmetryisachievedbyfurtherimposingn =n∗,butfornowweallowfortheeffectsof
G L
dispersion by notassuming that this is the case. We build the full solution aroundmodesfor
which
y ≈y ±y , (19)
± G L
wherey andy arethesolutionsoftheisolatedresonatorsdescribedatthebeginningofthis
G L
section.Weuseasinglevalueofmforbothy andy andinparticularapproximatetheglobal
G L
modeusingachiralstateinwhichthewavecirculatesinoppositesensesineachresonator.That
is,weneglectthecouplingbetweenmand−mthatoccursintheexactsolution.
Thenasimpleperturbativeapproximationisachievedbytruncatingthefullsystemofequa-
tions(13)tothe2×2system
a˜G D˜G C˜
M mm =0, where M= mm mm . (20)
(cid:18) a˜mLm (cid:19) (cid:18) C˜mm D˜Lmm (cid:19)
Resonantfrequenciesofthecoupledproblemarethenrealisedwhen
0=detM=D˜G D˜L −C˜2 . (21)
mm mm mm
In the general, dispersiveand non-PT-symmetric,case this reducesthe calculationto a semi-
analytic solutionin which we search for the (complex)rootsof the known2×2 determinant
above,inwhichmatrixelementsdependonfrequencythroughbothk=w /candn=n(w ).
3.5. Furtheranalyticdevelopmentoftheperturbativesolution
Todevelopaperturbativeexpansionwelet
1 1
D0 = D˜G +D˜L and DI = D˜G −D˜L (22)
mm 2 mm mm mm 2j mm mm
(cid:0) (cid:1) (cid:0) (cid:1)
(and note that in the high-Q-factorPT-symmetric case, D˜G ≃(D˜L)∗, both D0 and DI are
mm mm
approximatelyreal).WeassumethatbothDI andC aresmallandcomparableinmagnitude,
mm mm
andexpandangularfrequency
D w
w =w ± 0 +··· (23)
1,2 0
2
aboutarealresonantangularfrequencyofanaveragedisolatedresonatorsatisfying
D0 (w )=0. (24)
mm 0
Thentofirstorderthecoupledresonanceconditionbecomes
0=detM=D w 2D0 ′(w )2+DI (w )2−C˜ (w )2+··· (25)
0 mm 0 mm 0 mm 0
fromwhichtheangularfrequencyshiftscanbewritten
D w C˜ (w )2−DI (w )2
0 = mm 0 mm 0 , (26)
2 p D0 ′(w )
mm 0
whereD0 ′(w )denotesaderivativeofD0 (w )withrespecttofrequency.
mm mm
Wethenarriveatasimplecondition
C˜ (w )2=DI (w )2
mm 0 mm 0
forthreshold(atwhichD w andthetworesonantfrequenciesofthecoupledsystemcollide).In
0
thePT-symmetriccase,whereC˜ andDI areapproximatelyreal(andwhosesmallimaginary
mm mm
parts represent corrections due to radiation losses), we therefore have a prediction for a real
thresholdfrequency.
4. Resultsanddiscussions
Inthissection,theimpactofdispersionontheperformanceofthePTcoupledmicroresonators
is analyzed. Frequency mismatch between the resonant frequency of the microresonator and
gainpumpfrequencyis investigatedfor practicallevelsofdispersionandthe practicalimpli-
cations of a slight unbalance between the gain and loss in the system are investigated. We
concludethesectionwithaninvestigationofhowcouplingbetweenresonatorsmanifestsitself
inthetimedevelopmentofsolutions.
4.1. Effectsofdispersiononthresholdbehaviorinthefrequencydomain.
For all cases, weakly coupled microresonators are considered, the coupled resonators with
a dielectric constant e¥ = 3.5 [32] have radius a= 0.54m m and are separated by distance
g=0.24m m. Transverse-magnetic(TM) polarization is considered and operation at two dif-
ferent whispering-gallery modes is analysed, namely a low Q-factor mode (7,2) and a high
Q-factor mode (10,1). The correspondingisolated resonator resonant frequenciesare respec-
tively f(7,2) =341.59THz and f(10,1) =336.85THz, with Q-factors Q(7,2) =2.73×103 and
0 0
Q(10,1)=1.05×107.
Figure 2 shows the real and imaginary part of the eigenfrequenciesw and w of the PT-
1 2
symmetriccoupledmicroresonatorswithbalancedgainandloss,g =−g =g ,andisdepicted
0 G L
as a function of the gain/loss parameter g =|w n′′(w )| for both the low and high Q-factor
0 0 0
modes. The gain and loss are assumed to be tuned to the resonant frequency of an isolated
microresonator, i.e. w s =w 0 ≡2p f0. Three different levels of dispersion, controlled by the
parameter t defined in Sec. 2, are considered. These are w s t =0 correspondingto the case
of no dispersion, w s t =212 taken from [32] to exemplify the case of high dispersion and
w s t =0.7toexemplifythecaseoflowdispersion.
Figure 2(a,b) shows the frequency splitting of the real and imaginary part of the complex
eigenfrequenciesfor the case of nodispersion.In the absence ofgain/loss, whereg =0, the
0
supermodesbeatata rate correspondingtothe frequencydifferencesw −w =3.823rad/ps
1 2
and1.164rad/psforthe(7,2)and(10,1)modesrespectively.Figure2(a)indicatesthatoperation
inahigherQ-factormoderesultsinweakercouplingbetweenthemicroresonatorscomparedto
thecaseofoperationinthelowerQ-factormode.Increasingthegainandlossinthesystem,de-
creasesthebeatingrateandthesupermodescoalesceatthethresholdpointsofg =6.86rad/ps
0
and2.1rad/psforthelowandhighQ-factormodesofoperationrespectively,confirmingthat
the high-Q factor mode has a lower threshold point [21]. In the case of operation in the low
Q-factor mode,the eigenfrequenciesshownin Fig. 2(b) have a significant constantand posi-
tive imaginarypartbeforethe thresholdpoint, which is a consequenceof the higherintrinsic
losses due to radiation in that case. The corresponding imaginary part is insignificant in the
caseofthehighQ-factormode,forwhichradiationlossesaremuchsmaller.Furthermoreitis
notedherethatthecoupledsystemfirststartstolase,i.e.oneoftheeigenfrequenciessatisfies
Im(w −w )<0,onlywhenoperatedsignificantlybeyondthethresholdg =7rad/psforthe
1,2 0 0
lowQ-factoroperationwhilethisonsetoccursimmediatelyafterthethresholdpointinthehigh
Q-factorcase.
Figure2(c,d)showstherealandimaginarypartsoftheeigenfrequenciesforthecaseofstrong
dispersion,correspondingtotheparametervaluesw s t =212takenfrom[32].Theseareagain
shown for both high and low Q-factor modes. It is noted that the threshold pointfor the low
Q-factormodeis reducedfromg =6.86rad/pstog =6.47rad/psin thiscase whileforthe
0 0
highQ-factormodeitremainsunchangedat2.1rad/ps(comparedtothecaseofnodispersion).
Belowthethresholdpointtheimaginarypartsoftheeigenfrequenciesarenotconstant,butare
instead skewed towards a lossy state with positive and increasing imaginary part. Extension
Fig. 2. Frequency bifurcation of PT-coupled microresonator with a balanced gain (gG=
−g0)andloss(gL=g0)asafunctionofgain/lossparameteratthepeakofpumpingbeam
g0=w s n′′(w s )forthreedifferentdispersionparameters,(a,b)wt =0,(c,d)wt =212and
(e,f)wt =0.7
beyondthethresholdpointshowsthattheimaginarypartsoftheeigenfrequenciesdonotsplit
evenlyandarealsoskewedtowardsoverallloss,implyingthatinthehighlydispersivecasethe
eigenfrequenciesbotharecomplexbutnolongercomplexconjugatesafterthethresholdpoint.
Therealandimaginarypartsoftheeigenfrequenciesforthecaseoflowlevelsofdispersion,
forwhichwetakew s t =0.7,areshowninFig.2(e,f).Figure2(e)showsthatthereisnoclear
thresholdpointinthiscase:theimaginarypartssplitforverylowvalueofthegain/lossparam-
eterg ,withnosharppointofonset.Theappearanceofathresholdpointtypicallyassociated
0
withPT-behaviorislostandtheeigenfrequenciesarealwayscomplexvalued.
The key conclusion to be made from Fig. 2 is therefore that PT-like threshold behavior is
observedin the cases of nodispersionand ofhigh dispersion,butnotforcases of intermedi-
atedispersion.Whilethereissomeskewnessinthehigh-dispersioncase,whichamountstoa
quantitativedeviationfromstrictPT-symmetry,thereisanessentialqualitativesimilaritytothe
dispersionlesscasein whichthereappearstobea sharpthreshold.Bycontrast,in thecaseof
intermediate dispersion there is no sharp transition point and the imaginary parts of the two
frequenciesbegintodivergefromthebeginning.
Tofurtherinvestigateandexplainthisphenomenon,weexaminethedependenceofthereal
Fig.3.(a)Impact of dispersiontotherealpart ofmaterialatatomictransitionalangular
frequencyw s duetothepresenceofgainandlossfordifferentdispersionparameters;(b)
ContrastbetweentherealpartofeigenfrequenciesofPT-coupledmicroresonatorsfortwo
different gain/loss parameter, i.e.g0=7.5rad/ps for (7,2) and 2.54 rad/ps for the (10,1)
modeasfunctionofdispersionparametert .
Fig.4.Frequency bifurcationofcoupled microresonatorswithbalancedgainandlossas
functionofgain/lossparametersgs ,fortwodifferentatomictransitionalfrequenciesw s =
2p (f0+d )withd =−0.1and0.1THz.
partofthecomplexrefractiveindexonthedispersionparameter2w s t .Thisdependenceisplot-
tedinFig.3(a)forthecasesofbothgainandloss,forwhichwerespectivelytakes 0=±2e0w s
andw s =w 0.Figure3(a)showsthattherealpartsoftherefractiveindicesbehavedifferently
for the cases of loss and gain in the system, with the maximum difference occurring when
t =1/(2w s ).However,intwolimitingcasest =0(dispersion-lesssystem)andt →¥ (strong
dispersion), the real parts of the refractiveindex converge.This means that the PT condition
n =n∗ canonlybesatisfiedaccuratelyforthecasesofnodispersionandofhighdispersion.
G L
Forthecaseofintermediatedispersionthereisnecessarilysomediscrepancybetweenthereal
partsoftherefractiveindicesoftheresonators.
Figure 3(b) shows the minimum difference in the real parts of the two eigenfrequencies