Deterministic single soliton generation and compression in microring resonators avoiding the chaotic region

2.4 MB·English

by  Jose A. Jaramillo-Villegas

Deterministic single soliton generation and
compression in microring resonators avoiding the
chaotic region
JoseA.Jaramillo-Villegas,1,3,∗XiaoxiaoXue,1Pei-HsunWang,1DanielE.Leaird,1and
AndrewM.Weiner1,2
1SchoolofElectricalandComputerEngineering,PurdueUniversity,WestLafayette,Indiana47907,USA
5 2BirckNanotechnologyCenter,PurdueUniversity,WestLafayette,Indiana47907,USA
1 3FacultaddeIngenier´ıas,UniversidadTecnolo´gicadePereira,Pereira,Risaralda660003,Colombia
0 ∗[email protected]
2

y Abstract: A path within the parameter space of detuning and pump
a power is demonstrated in order to obtain a single cavity soliton (CS) with
M
certainty in SiN microring resonators in the anomalous dispersion regime.
Once the single CS state is reached, it is possible to continue a path to
9
compressit,broadeningthecorrespondingsinglefreespectralrange(FSR)
1
Kerr frequency comb. The first step to achieve this goal is to identify the

] stable regions in the parameter space via numerical simulations of the
s
Lugiato-Lefever equation (LLE). Later, using this identification, we define
c
i a path from the stable modulation instability (SMI) region to the stable
t
p cavitysolitons(SCS)regionavoidingthechaoticandunstableregions.
o
. © 2015 Optical Society of America. One print or electronic copy may be made
s
forpersonaluseonly.Systematicreproductionanddistribution,duplicationofany
c
i materialinthispaperforafeeorforcommercialpurposes,ormodificationsofthe
s
contentofthispaperareprohibited.
y
h
OCIS codes: (230.5750) Resonators; (190.5530) Pulse propagation and temporal solitons;
p (190.4380)Nonlinearoptics,four-wavemixing.
[

2
v Referencesandlinks
4 1. T.Kippenberg,R.Holzwarth,andS.Diddams,“Microresonator-basedopticalfrequencycombs,”Science332,
4 555–559(2011).
2 2. S.B.PappandS.A.Diddams,“Spectralandtemporalcharacterizationofafused-quartz-microresonatoroptical
3 frequencycomb,” Phys.Rev.A84,053833(2011).
0 3. P.Del’Haye,A.Schliesser,O.Arcizet,T.Wilken,R.Holzwarth,andT.Kippenberg,“Opticalfrequencycomb
. generationfromamonolithicmicroresonator,”Nature450,1214–1217(2007).
1
4. I.S.Grudinin,N.Yu,andL.Maleki,“GenerationofopticalfrequencycombswithaCaF2resonator,” Opt.Lett.
0 34,878–880(2009).
5 5. J.S.Levy,A.Gondarenko,M.A.Foster,A.C.Turner-Foster,A.L.Gaeta,andM.Lipson,“CMOS-compatible
1 multiple-wavelengthoscillatorforon-chipopticalinterconnects,”Nat.Photonics4,37–40(2009).
: 6. L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible
v
integratedopticalhyper-parametricoscillator,”Nat.Photonics4,41–45(2009).
i
X 7. F.Ferdous,H.Miao,D.E.Leaird,K.Srinivasan,J.Wang,L.Chen,L.T.Varghese,andA.M.Weiner,“Spectral
line-by-linepulseshapingofon-chipmicroresonatorfrequencycombs,”Nat.Photonics5,770–776(2011).
r
8. P.Del’Haye,T.Herr,E.Gavartin,M.Gorodetsky,R.Holzwarth,andT.Kippenberg,“Octavespanningtunable
a
frequencycombfromamicroresonator,” Phys.Rev.Lett.107,063901(2011).
9. M.A.Foster,J.S.Levy,O.Kuzucu,K.Saha,M.Lipson,andA.L.Gaeta,“Silicon-basedmonolithicoptical
frequencycombsource,” Opt.Express19,14233–14239(2011).
10. Y.Okawachi,K.Saha,J.S.Levy,Y.H.Wen,M.Lipson,andA.L.Gaeta,“Octave-spanningfrequencycomb
generationinasiliconnitridechip,” Opt.Lett.36,3398–3400(2011).
11. P.-H.Wang,F.Ferdous,H.Miao,J.Wang,D.E.Leaird,K.Srinivasan,L.Chen,V.Aksyuk,andA.M.Weiner,
“Observationofcorrelationbetweenroutetoformation,coherence,noise,andcommunicationperformanceof
Kerrcombs,” Opt.Express20,29284–29295(2012).
12. Y.K.ChemboandN.Yu,“Modalexpansionapproachtooptical-frequency-combgenerationwithmonolithic
whispering-gallery-moderesonators,” Phys.Rev.A82,033801(2010).
13. L.A.LugiatoandR.Lefever,“Spatialdissipativestructuresinpassiveopticalsystems,” Phys.Rev.Lett.58,
2209(1987).
14. S.Coen,H.G.Randle,T.Sylvestre,andM.Erkintalo,“Modelingofoctave-spanningKerrfrequencycombs
usingageneralizedmean-fieldLugiato-Lefevermodel,” Opt.Lett.38,37–39(2013).
15. Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in
whispering-gallery-moderesonators,” Phys.Rev.A87,053852(2013).
16. S.CoenandM.Erkintalo,“UniversalscalinglawsofKerrfrequencycombs,” Opt.Lett.38,1790–1792(2013).
17. T.Hansson,D.Modotto,andS.Wabnitz,“Dynamicsofthemodulationalinstabilityinmicroresonatorfrequency
combs,” Phys.Rev.A88,023819(2013).
18. C.Godey,I.Balakireva,A.Coillet,andY.K.Chembo,“StabilityanalysisoftheLugiato-LefevermodelforKerr
opticalfrequencycombs.PartI:caseofnormaldispersion,”arXivpreprintarXiv:1308.2539(2013).
19. I.Balakireva,A.Coillet,C.Godey,andY.K.Chembo,“StabilityanalysisoftheLugiato-LefevermodelforKerr
opticalfrequencycombs.PartII:caseofanomalousdispersion,”arXivpreprintarXiv:1308.2542(2013).
20. M. R. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based
frequencycombs,” Opt.Lett.38,3478–3481(2013).
21. A. Coillet, I. Balakireva, R. Henriet, K. Saleh, L. Larger, J. M. Dudley, C. R. Menyuk, and Y. K.
Chembo, “Azimuthal Turing patterns, bright and dark cavity solitons in Kerr combs generated with
whispering-gallery-moderesonators,”IEEEPhoton.J.5,6100409–6100409(2013).
22. M.ErkintaloandS.Coen,“CoherencepropertiesofKerrfrequencycombs,” Opt.Lett.39,283–286(2014).
23. P.Parra-Rivas,D.Gomila,M.Matias,S.Coen,andL.Gelens,“Dynamicsoflocalizedandpatternedstructures
intheLugiato-Lefeverequationdeterminethestabilityandshapeofopticalfrequencycombs,” Phys.Rev.A89,
043813(2014).
24. A.CoilletandY.Chembo,“OntherobustnessofphaselockinginKerropticalfrequencycombs,”OpticsLetters
39,1529–1532(2014).
25. T.Hansson,D.Modotto,andS.Wabnitz,“OnthenumericalsimulationofKerrfrequencycombsusingcoupled
modeequations,” Opt.Commun.312,134–136(2014).
26. T.Herr,V.Brasch,J.Jost,C.Wang,N.Kondratiev,M.Gorodetsky,andT.Kippenberg,“Temporalsolitonsin
opticalmicroresonators,”Nat.Photonics8,145–152(2014).
27. P.-H.Wang,Y.Xuan,J.Wang,X.Xue,D.E.Leaird,M.Qi,andA.M.Weiner,“Coherentfrequencycomb
generationinasiliconnitridemicroresonatorwithanomalousdispersion,”in“CLEO:ScienceandInnovations,”
(OpticalSocietyofAmerica,2014),pp.SF2E–3.
1. Introduction
An optical frequency comb is a light source with a number of highly resolved and nearly
equidistant spectral lines. Since its introduction, multiple important applications have been
demonstrated in areas such as communications, metrology, spectroscopy, astronomy and
optical clocks. Optical frequency combs can be generated using mode-locked lasers or
electro-optic modulation of continuous-wave light. Since 2007, multiple experiments have
reported optical frequency comb generation by means of Kerr nonlinearity (wave mixing
process) in microresonators, which offer potential for highly compact and portable solutions.
SuchcombsaretermedKerropticalfrequencycombsorsimplyKerrcombs[1–11].
The understanding of underlying processes and dynamics in Kerr comb generation is
critical to move this technology further to industry and applications. Chembo and Yu in [12]
introduced one of the first simulation approaches using modal expansion. More recently, the
Lugiato-Lefeverequation(LLE)[13]hasbeenwidelyadopted[14–24].Furthermore,Chembo
and Menyuk in [15] demonstrated the equivalence between the LLE and mode coupling
equationsmodelsandHanssonetal.in[25]showedthatbothmodelscanbenumericallysolved
insimilarcomputationaltimes.
Thegeneralizedmean-fieldLLEequationis:
∂E(t,τ) (cid:34) β (cid:18) ∂ (cid:19)k (cid:35) √
t = −α−iδ +iL∑ k i +iγL|E|2 E+ θE (1)
R ∂t 0 k! ∂τ in
k≥2
which is the nonlinear Schro¨dinger equation (NLSE) with damping, detuning and external
pumping, which accurately describes Kerr comb generation. In this equation E(t,τ) is the
complexenvelopeofthetotalintracavityfieldandhereaftersimplythefield,t isthesocalled
slowtimevariable,τ isthefasttimevariable,t istheroundtriptime,α ishalfofthetotalloss
R
perroundtripwhichincludestheinternallinearabsorptionandcouplingloss,δ isthephase
0
detuning,Listhecavitylength,β isthek-orderdispersioncoefficient,γ istheKerrcoefficient,
k
θ isthecouplingcoefficientbetweenthewaveguideandthemicroresonatorandE isthepump
in
field(normalizedsuchthatP =|E |2).Tofacilitateouranalysesinamoregeneralframework
in in
wewilladditionallyusethenormalizeddetuningandpumpfieldaccordingtoEq.2and3.
δ
∆= 0 (2)
α
(cid:114)
γLθ
S=E (3)
in α3
Recently, some authors explored and identified regions corresponding to different types
of operation in the detuning and pump power (∆,|S|2) parameter space in the anomalous
dispersion regime [16,19,23]. The characterized regions are: stable modulation instability
(SMI), unstable modulation instability (UMI), stable cavity solitons (SCS), unstable cavity
solitons(UCS)andcontinuouswave(CW).Additionally,ErkintaloandCoenin[22]explored
thefirst-ordercoherencepropertiesofeachregion.
SingleCSgenerationisadesiredgoalinmuchresearchbecauseityieldsahighcoherence,
singlefreespectralrange(FSR)Kerrcomb.Ithasbeensuggestedthatthisstyleofcombcan
be used as information carriers in optical communications and optical memories. Although
the first experimental observation of a single CS in microresonators was reported in [26], the
difficultytocontrolthenumberofCSsintheSCSregionwasalsoemphasized.Inparticular,in
theusualapproachinwhichdetuningisswepttoapredeterminedfinalvaluewithinputpower
held constant, the number of CSs generated is probabilistic. In the current work, we show a
methodnotonlytoobtainasingleCSwithahighdegreeofcertaintyavoidingthechaoticand
unstableregionssuchasUMIandUCS,butalsotocompressit,movingthesystemthroughthe
SCStohighpowervalues.
2. DeterministicSingleCSGeneration
TheLLEissimulatednumericallyusingthesplitstepFouriermethod(SSFM).Thesimulation
parameters used in this work are:t =1/226 GHz, β =−4.7×10−26 s2m−1, α =0.00161,
R 2
γ=1.09W−1m−1,L=2π×100µmandθ =0.00064[27].Theseparameterscorrespondtoa
Si N microringresonatorof100µmradiuswithanomalousdispersion,loadedqualityfactor
3 4
Qof1.67×106 andphotonlifetimet =t /(2α)=1.37ns.Addionally,thesimulationsare
ph R
doneinitializingtheintracavityfieldinthefrequencydomainE(ω)toacircularlysymmetric
complex Gaussian noise field with a standard deviation σ = 10−9 [W1/2], which with
noise
normalizationP=|E|2isequivalenttoameanpowerof-150dBmpercavitymodeanduniform
randomphase.
ToshowtheprobabilisticnatureofthenumberofCSafteradetuningsweptatconstantpump
powerweperformsimulationsaccordingtotheinsetsofFig.1(a).Theseinsetsdepicthowthe
detuningissweptlinearlyfrom∆=0toafinalvalueof∆ overatimeintervalof0.3µs(≈218
f
t ) at a constant |S|2 = 18.9 (P = 180 mW). Fig. 1(a) shows the number of temporal peaks
ph in
(intheSCSregionthiscorrespondstonumberofCSs)versusfinalvalueofdetuning∆.The
f
numberofCSsobtaineddependsverysensitivelyon∆ eventhoughthenoiseinitializationwas
f
thesameforallthesimulationsofFig.1(a).Additionally,thenumberofCSsisdifferentwhen
werepeatthesimulationwiththesamedetuningandpumpparametersbutdifferentrealizations
oftheinitialnoise,asshowninthehistogramofFig.1(b).Thebehaviorofthetotalintracavity
energyU ,whichisproportionaltotheintegraloftheintensityoveraroundtriptime,give
Intra
usanideaoftheKerrcombstability.Fig.1(c)showstotalintracavityenergyduringadetuning
swept at constant input power. The noisy behavior in the yellow segment gives us evidence
that the system is in the UMI region. Then, the oscillatory behavior in the blue segment is a
clearcharacteristicofUCSregion.Finally,thetotalintracavityenergybecomesstablewhenthe
microresonatorapproachestheSCSregion.Ourhypothesisinthisworkisthattheuncertainty
inthenumberofCSsintheSCSregionresultsfromchaoticfieldevolutionwhilethesystemis
crossingtheUMIandUCSregions.
(a) UMI UCS SCS CW
20
aks1126 S2|| 18.900 1 2 ∆0 1∆f 1.3
NPe 8 t[µs] t[µs] (b)
4
400
s
0 t
0 3 6 9 12 15 18 21 24 un
∆ o200
f C
(c)
a.u.a468 UMI UCS SCS 6912∆ 0 0N1Pea2ks3∆4f=512.6427(a8)
UIntr2 3
0 0
1 1.1 1.2 1.3
t [µs]
Fig.1.(a)Thenumberofpeaksasafunctionoffinalvalueofdetuning∆ with∆sweptas
f
shownintheinsets.Thepumppower|S|2issettoaconstantvalueof18.9(P =180mW).
in
Thesesimulationsweredoneusingthesamerealizationofinitialnoise.(b)Histogramof
numberofpeaksfor1000simulationswithdifferentrealizationsofinitialnoise,usingthe
same detuning and pump parameters in the simulation of (a) with ∆ =12.42. (c) Total
f
intracavityenergy(blue)anddetuning(green)asafunctionoftimeforthesimulationof
(a)with∆ =12.42.
f
Although some of the boundaries between the regions can be solved analytically (e.g.
Hamiltonian-Hopf limit between CW and SMI regions [23]), there is still no theoretical
solution for the boundary between the SMI and UMI regions. Therefore, we characterize the
(∆,|S|2)parameterspacevianumericalsimulationstogainknowledgeabouttheseboundaries.
We initialize the microresonator pump at a detuning ∆=0 and power |S|2 =6.3 (60 mW)
with well-defined behavior in the SMI region. This point corresponds to the location of the
red triangle in Fig. 2(a) where a Turing rolls pattern [21,24] of 13 equally spaced peaks
is formed. The intensity and power spectrum obtained at this point are shown in Fig. 2(c)
(left) and (center), respectively. This pattern corresponds to a Kerr comb with frequency
spacing between lines equal to 13 times the microresonator FSR and possesses low intensity
noise and high stability, as shown by its steady state intracavity energy in Fig 2(c) (right).
AccordingtoErkintaloandCoenin[22],theUMIregionexhibitshighfirst-ordercoherence.
CoilletandChemboin[24]demonstratedthatTuringrollspatternsachievephaselockedstates
independently of the initial conditions and can be easily generated by decreasing the optical
frequencyofthepumplasertowardstheresonance,aspresentedintheirexperimentalexamples.
These reasons make Turing rolls pattern a good choice for our initial point. After 200 ns, we
jumpinasinglesteptoapointin(∆,|S|2)parameterspaceandkeepthesystemtherefor1.8µs
(≈1310t ).Simulationswererepeatedusingthesamerealizationofinitialnoisefordifferent
ph
final values of detuning ∆ and pump power |S|2 up to 6.21 and 10.5 (100 mW), respectively.
Fig. 2(a) shows the number of time domain peaks for each final point. Each point (∆,|S|2)
is classified based on intensity, spectrum and total intracavity energy stability at the end of
the simulation in order to determine the boundaries of the different operating regions. The
characteristicsofeachregionareshowninFig.2(c)to(f)andtheresultofthisclassification
isshowninFig.2(b).Notably,U intheUMIandUCSregionsischaracterizedbynoisyand
Intra
oscillatorybehavior,respectively,whileintheSMIandSCSregionsitisconstantasmentioned
before.
(a) 0.1 (b) 0.1
10 10 (d) (e)
0.09 14 0.09
9 9
8 0.08 12 8 0.08
0.07 0.07
7 10 7
S2|f 56 +00|.S4|22Offset00..0056[W]f 8 S2|f 56 (c) (f) 00..0056[W]f
| 4 -0.42 0.04Pin 6 | 4 CW 0.04Pin
SCS
3 S2| |Sf|2 0.03 4 3 UCS 0.03
2 |6.3 ∆ 0.02 2 2 SMI 0.02
1 ∆ 0 f 0.01 1 UMI 0.01
0 0.2µs 0
0 0 0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
∆ ∆
f f
|E(t,τ)|2 a.u. |E˜(t,ω)|2 [dB] U a.u.
Intra
(c)
0 6
8
MI −40 4
S 4 2
−80
0 0
(d)
80 0 6
MI −40 4
40
U 2
−80
0 0
(e)
0 6
40
CS −40 4
U 20 2
−80
0 0
(f)
0 6
30
CS 20 −40 4
S 10 2
−80
0 0
−2 −1 0 1 2 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2
τ [ps] ω [Prad/s] t [µs]
Fig. 2. (a) The number of temporal peaks as a function of the final point in (∆,|S|2)
parameter space at low pump power. The red triangle shows the initial point of the
simulations.Thesesimulationsweredoneusingthesamerealizationofinitialnoise.The
blackcurvescorrespondtoCATsusingEq.4with|S|2 of-0.42,0,and0.42equivalent
Offset
to offsets of -4, 0 and 4 mW respectively. (b) Characterized regions according to the
features of each region shown in (c) to (f). (c-f) Final intensity (left), spectrum (center)
andintracavityenergyversustimet(right)forthepoint(c)(∆ =0,|S|2=6.3)intheSMI
f f
region,(d)(∆ =3.9,|S|2=10.5)intheUMIregion,(e)(∆ =4.8,|S|2=10.5)inthe
f f f f
UCSregion,and(f)(∆ =4.8,|S|2=6.3)intheSCSregion.
f f
Because instantaneously jumping to a single final state in detuning-power parameter space
as in the simulations of Figs. 2(a) is not physically realizable, we next define a path that can
befollowedsmoothlybetweenourstartingpoint(∆=0,|S|2=6.3)andatargetpointchosen
tofallwithinthesingleCSregion(∆=4.35,|S|2=5.04).Inparticular,weselectapaththat
goesaroundthechaoticandunstableregions.Wecallsuchapathachaos-avoidingtrajectory
(CAT).InthisstudyweconsideraCATgivenbythefollowing:
|S|2=4.15exp(−3.09∆)+2.15exp(0.196∆)+|S|2 (4)
Offset
ThisCATisdefinedusingacurve-fittingtoolandtakestheformofatwo-termexponential
function. Additionally, we introduce a power offset parameter |S|2 which allows us to
Offset
describeafamilyofCATs,portrayedbytheblackcurvesinFig.2(a).Totesttheperformance
oftheCATwerunsimulationsinthreestages.Inthefirststagewedwellattheselectedinitial
point (∆=0,|S|2 =6.3) for 1.0 µs (≈728 t ) to produce a steady state Kerr comb in the
ph
SMI regime. Next, we sweep the detuning linearly from ∆=0 to a final value ∆ over a 0.3
f
µsinterval(≈218t )whilevaryingtheinputpoweraccordingtotheCATinEq.4.Wethen
ph
continuethesimulationatfixeddetuningandinputpowerforanadditional2µs(≈1445t ).
ph
(a) (b)
16
aks12 S2||f 6.3 1 1.3∆0 1∆f 1.3 nts1000
NPe 8 t t ou 500
4 Single CS C
0 0
0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8
∆f NPeaks ∆f=7.45 (a)
(c) (d)
0.42 0.42
2|Offset 0.201 2|Offset 0.201
S S
|−0.21 |−0.21
−0.42 −0.42
0 2 4 6 8 10 12 0 2 4 6 8 10 12
∆ ∆
f f
Fig.3.(a)Thenumberofpeaksasafunctionoffinalvalueofdetuning∆ with∆swept
f
asshownintheinsetwithpumppoweradjustedthroughtheCAT,Eq.4,with|S|2 =0
Offset
usingthesamerealizationofinitialnoiseforallsimulations.(b)Histogramofnumberof
peaksfor1000simulationswithdifferentrealizationsofinitialnoisewiththesamepump
power and detuning parameters of the simulation of (a) with ∆ =7.45. (c-d) Region in
f
which a single CS is generated when a uniform offset |S|2 is applied to the CAT for
Offset
differentvaluesoffinaldetuning∆ with(c)constantdetuningintervalof0.3 µsand(d)
f
constantdetuningspeedof25unitsof∆perµs.
Figure3(a)showsthenumberoftemporalpeaksversusthefinalvalue∆ forfixedrealization
f
ofinitialnoise.Theinsetsillustratethecoordinatedvariationof∆and|S|2intimeaccordingto
theCAT.Thegreenareadepictsarangeof∆ values(from4to10.8)forwhichasingleCSis
f
alwaysobtained.Inanothercasewekeep∆ fixedbutrepeatthesimulationfor1000different
f
realizationsofinitialnoise.AsshowninthehistogramofFig.3(b),asingleCSisobtainedevery
time.Fig.3(c)issimilartoFig.3(a),inthatwekeepthenoiseinitializationfixedandvary∆,
f
except that now we use different values of |S|2 from -0.52 to 0.52 equivalent to an offset
Offset
of-5to5mWtotheCAT.Thegreenshadedareaagainshowsthecaseswherewegeneratea
singleCS.Additionally,wetesttheCATperformancekeepingthesweeprateofthedetuning
constantat25unitsof∆perµsfordifferentvaluesof∆.Thisisincontrasttothesimulationof
f
Fig.3(c)inwhichthedetuninginsweptoveraconstantintervalof0.3 µs,implyingdifferent
sweep rates for different final detunings. The simulation results are displayed in Fig. 3(d),
whichagainshowsagreenareainwhichasingleCSstateisreached.Moreover,weperform
simulationsusinghalfandtwicethesweeprateofthedetuningwithverysimilarresultsofFig.
3(d).AllthesesimulationsdemonstratethattheCATforrepeatablegenerationofasingleCS
is nonunique and can be robust over a finite range of operation. It should be emphasized that
thisCATcouldtakeotherformsdifferentthanatwo-termexponentialfunction,providedthat
itavoidstheunstableandchaoticregions.
We note that thermal nonlinearities play an important role in practical microresonators.
The response time of such thermal nonlinearities is much slower than that of the Kerr
nonlinearity,whichcomplicatescomputationalstudies.Forthisreasonthermalnonlinearityis
notincludedinthesimulationsinthispaper.Furtherresearchisrequiredtodeterminewhether
thecurrentschemefordeterministicgenerationofsinglecavitysolitonsremainseffectivewith
theinclusionofthermalnonlinearity.Ataminimumweanticipatethatitmaybenecessaryto
slowdowntheCATsweepspeedtoatimescaleconsistentwiththethermalnonlinearity.
3. CScompression
(a) 1 (b) 1
100 100
0.9 18 0.9
90 90
0.8 16 0.8
80 80
0.7 14 0.7
70 70
S2|f 5600 00..56[W]f 1102 S2|f 5600 00..56[W]f
| 40 0.4Pin 8 | 40 SCS 0.4Pin
2300 S2||63|S∆f|2 00..23 46 2300 USCMCWSI 00..23
10 ∆ 0 f 0.1 2 10 UMI 0.1
0 0.2 µ s 0
0 0 0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30
∆ ∆
f f
Fig.4.Thenumberoftemporalpeaksasafunctionofthefinalpointin(∆,|S|2)parameter
space at high pump power. The red triangle shows the initial point of the simulations.
Thesesimulationsweredoneusingthesamerealizationofinitialnoise.Theblackcurve
correspondstoacompressionfunctionusingEq.5.Theinitialpointofthiscompression
functionisthefinalpointoftheCATwith|S|2 =0inFig.2(a).(b)Characterizedregions
Offset
usingthesamecriteriaofFig.2(b).
Our final goal is to demonstrate compression behavior when we move the system from
relativelylowtorelativelyhighdetuningandpowerunderatrajectorythatremainswithinthe
SCSregion.AprocesssimilartothecharacterizationofFig.2isperformedtoexplorebehavior
over a larger range of parameter space (final values of detuning ∆ and input power |S|2 up to
31 and 105 (1 W), respectively), starting at initial point of detuning ∆=0 and pump power
|S|2=63(0.6W).ThispointhashigherpowerthantheinitialpointusedinFig.2(a)becausein
thishigherrangeofpower,themicroresonatorneedsmoreinitialenergytoshowallthepossible
regions of behavior without collapsing to the CW state. The results of the simulations in this
range and the classification are shown in Figs. 4(a) and 4(b), respectively. Then, we define a
compression function from (∆=4.35,|S|2 =5.04), the end of a previous CAT trajectory, to
(∆=27.45,|S|2=105)usingatrajectorygivenby
|S|2=2.846exp(0.1316∆) (5)
Again, this function is not unique, and a wide range of end points ∆ and |S|2 lead to similar
results,providedthatthetrajectoryremainswithintheSCSregion.
ToprovideacompletepictureofthegenerationandcompressionprocessesofasingleCS,
we run a simulation in five stages, as shown in Fig. 5(a), which plots the coordinated change
indetuningandinputpowerversusslowtime.Figure5(e)showstheintensityandspectrumof
thesteadystateinitialpoint(Turingrollspatternof13temporalpeaks).InFig.5(b)theTuring
rollspatternfallintoasingleCSaround1.4µs.Fig5(f)showstheintensityandspectrumofthe
generatedsingleCS.InFig.5(c)we canobservehowtheKerrcombisbroadeningwhilethe
systemisgoingthroughthecompressionfunctionbetween1.6µsand2.6µs.Figures5(f)and
(g)showthattheCSiscompressedfrom73fsto29fsfullwidthathalfmaximum(FWHM)
pulse width and the bandwidth ∆ω/2π goes from 3.98 to 10.2 THz. Figure 5(d) showsU
Intra
along a CAT leading to single CS generation and subsequent compression. Neither noisy nor
oscillatorybehaviorisseen,providingevidencethatthesystemremainsinSMIorSCSregions
forallsimulationtimeandshouldconsequentlybehighlystablewithlowintensitynoise.
Although we are not taking into account high order dispersion terms and other nonlinear
effects(e.g.Raman)thatwillbepresentinrealsystems,webelievetheseresultsmakeclearthat
once a single CS state is reached compression should be possible while maintaining stability
throughappropriatecoordinatedincreasesin∆and|S|2.
(a)150
6 30
100 4 20 |E(t,τ)|2 a.u. |E˜(t,ω)|2 [dB]
2S| 50 2 1 1.3 10∆ (e) 0
| 0 s 9
0 µ −30
0 6
(b) −2 150 .1 −60
ps]−01 91020u. t= 03 −90
τ[ 1 60 a. (f) 24 0
30 s
2 µ6 16 −30
(c)0.8 0 .1 −60
s] 1 −20 = 8
/ ] t −90
d1.2 −40B 0
Pra1.4 −60[d (g)150 0
ω[1.6 −80 sµ6100 −30
(d) 4 .3 −60
u. = 50
a.3 t −90
UIntra12 0−2 −1τ 0[ps]1 2 0.8 ω1 [P1.r2ad1/.4s]1.6
0
0 1 2 3
t [µs]
Fig. 5. Generation and compression processes of a single CS through the CAT and
compressionfunctionsinEq.4andEq.5,respectively.Fromtoptobottom:(a)pumppower
(blue)anddetuning(green),(b)temporalintensity,(c)spectrum,(d)totalintracavityenergy
vs.slowtimet.(e-g)Intensity(right)andspectrum(left)at(e)t=1µs,steadystateinitial
point,(f)t=1.6µs,aftersingleCSgenerationand(g)t=3.6µs,aftercompression.
4. Conclusions
In this work, we have presented a novel method to generate a single CS in anomalous
dispersionmicroresonatorsinahighlydeterministicwaythroughcoordinatedtuningofpump
frequencyandpowerinordertoavoidchaoticandunstableoperatingregimes.Thesimulation
resultspresentedherecouldhelpaccelerateprogresstowardsestablishingmicroringresonators
as highly coherent and stable single FSR Kerr comb sources for practical applications.
Furthermore, we have also demonstrated a way to compress the single CS by further
coordinatedvariationofthepumpparametersintheregionwhereCSsarestable.
Acknowledgments
This work was supported in part by the National Science Foundation under grant ECCS-
1102110, by the Air Force Office of Scientific Research under grant FA9550-12-1-0236,
and by the DARPA PULSE program through grant W31P40- 13-1-0018 from AMRDEC.
JAJ acknowledges support by Colciencias Colombia through the Francisco Jose de Caldas
Conv.529scholarshipandFulbrightColombia.JAJisgratefultoVictorTorres-Companyand
EvgeniiNarimanovforfruitfuldiscussionsandtoJosephLukensforacarefulreadingofthis
manuscript.