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SupplementaryInformationtoCounter-PropagatingSolitonsinMicroresonators
Qi-FanYang
∗
,XuYi
∗
,KiYoulYang,andKerryVahala
†
T.J.WatsonLaboratoryofAppliedPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125,USA.
∗
Theseauthorscontributedequallytothiswork.
†
Correspondingauthor:vahala@caltech.edu
I.TUNINGOFRELATIVERATEANDOFFSET
Toachieveindependentcontrolofthecounter-propagating(CP)solitonrepetitionratedifferenceandthepump
frequencyoffset,weemployatwo-steptuningprotocol.First,thefrequencydifferenceofthetwoAOMs(seefig.
1b),whichisexactlythesolitonfrequencyoffset∆
ν
,issettoadesiredvalue.Second,theclockwisepumptocavity
resonancedetuningfrequencyistunedbyadjustingthepumplaserlockingsetpointusingtheservocontrol.This
allowsthecontinuoustuningofrepetitionratedifference∆
f
.Aplotofthisprocessbasedoneq.(5)intheMethods
sectionisprovidedinfig.S1.
II.LOCKINGOFCOUNTERPROPAGATINGSOLITONSATTHESAMEREPETITIONRATE
Inthissectionthemechanismofsolitonrepetitionratelockingtoanidenticalrateisdiscussed(Fig.2withinmain
text).Backscatteringofpumplightfromthetaperedfibreusedtocouplepowertotheresonatorisbelievedtoinduce
thislocking.Forexample,considertaperbackscatteringoftheclockwise(CW)pump.Thebackscatteredpumplight
propagateswiththecounter-clockwise(CCW)solitonintheresonator.TheCCWsolitonandthebackscatteredCW
pumpexperiencefour-wave-mixingthatcreatessidebandsontheCCWsoliton.Thesesidebandslieatfrequencies
whichareveryclosetotheCWsoliton.Backscattercouplingofthesesidebands(andsimilarsidebandsproducedby
taperbackscatteringoftheCCWpump)causesthesolitonrepetitionratestolockthroughinjectionlocking.This
processisdescribedschematicallyinfig.S2(a).
012345
0
20
40
60
80
100
120
Pump frequency offset
∆ν
(MHz)
Repetition rate difference
∆
f (kHz)
δω
cw
=30 MHz
δω
cw
=25 MHz
δω
cw
=20 MHz
δω
cw
=15 MHz
δω
cw
=10 MHz
FIG.S1:
IndependenttuningofCPsolitonrepetitionrateandpumpfrequencyoffset.
CalculatedCP
solitonrepetitionratedifferenceversuspumpfrequencyoffsetatdifferentpump-cavitydetuningfrequenciesin
clockwisedirectionbasedoneq.(5)intheMethodsSection.
SupplementaryInformationtoCounter-PropagatingSolitonsinMicroresonators
Qi-FanYang
∗
,XuYi
∗
,KiYoulYang,andKerryVahala
†
T.J.WatsonLaboratoryofAppliedPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125,USA.
∗
Theseauthorscontributedequallytothiswork.
†
Correspondingauthor:vahala@caltech.edu
I.TUNINGOFRELATIVERATEANDOFFSET
Toachieveindependentcontrolofthecounter-propagating(CP)solitonrepetitionratedifferenceandthepump
frequencyoffset,weemployatwo-steptuningprotocol.First,thefrequencydifferenceofthetwoAOMs(seefig.
1b),whichisexactlythesolitonfrequencyoffset∆
ν
,issettoadesiredvalue.Second,theclockwisepumptocavity
resonancedetuningfrequencyistunedbyadjustingthepumplaserlockingsetpointusingtheservocontrol.This
allowsthecontinuoustuningofrepetitionratedifference∆
f
.Aplotofthisprocessbasedoneq.(5)intheMethods
sectionisprovidedinfig.S1.
II.LOCKINGOFCOUNTERPROPAGATINGSOLITONSATTHESAMEREPETITIONRATE
Inthissectionthemechanismofsolitonrepetitionratelockingtoanidenticalrateisdiscussed(Fig.2withinmain
text).Backscatteringofpumplightfromthetaperedfibreusedtocouplepowertotheresonatorisbelievedtoinduce
thislocking.Forexample,considertaperbackscatteringoftheclockwise(CW)pump.Thebackscatteredpumplight
propagateswiththecounter-clockwise(CCW)solitonintheresonator.TheCCWsolitonandthebackscatteredCW
pumpexperiencefour-wave-mixingthatcreatessidebandsontheCCWsoliton.Thesesidebandslieatfrequencies
whichareveryclosetotheCWsoliton.Backscattercouplingofthesesidebands(andsimilarsidebandsproducedby
taperbackscatteringoftheCCWpump)causesthesolitonrepetitionratestolockthroughinjectionlocking.This
processisdescribedschematicallyinfig.S2(a).
012345
0
20
40
60
80
100
120
Pump frequency offset
∆ν
(MHz)
Repetition rate difference
∆
f (kHz)
δω
cw
=30 MHz
δω
cw
=25 MHz
δω
cw
=20 MHz
δω
cw
=15 MHz
δω
cw
=10 MHz
FIG.S1:
IndependenttuningofCPsolitonrepetitionrateandpumpfrequencyoffset.
CalculatedCP
solitonrepetitionratedifferenceversuspumpfrequencyoffsetatdifferentpump-cavitydetuningfrequenciesin
clockwisedirectionbasedoneq.(5)intheMethodsSection.
Counter-propagating solitons in microresonators
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1
2
φ
Bc
-
φ
Ac
(
π
)
0
5
10
Cavity round trips (10
6
)
-0.1
0.1
a
b
∆ω=
0.05
κ
α=
5%
G/2
π
=100 kHz
Locked
15
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
0
Locked
Locked
Wal
king off
∆ω=
0.1
κ
α=
20%
G/2
π
=100 kHz
∆ω=
0.1
κ
α=
5%
G/2
π
=200 kHz
∆ω=
0.1
κ
α=
5%
G/2
π
=100 kHz
Optical power
CCW
CW
Optical power
Backscattering
Optical frequency
CW
Optical power
Locking
FIG.S2:
MechanismofCPsolitonsynchronization.a,
Ratelockingoccurswhentherepetitionratesof
A
p
and
B
p
areinjection-lockedbythebackscatteringof
B
b
and
A
b
,respectively.Theupperpanelshowsfour-wave-
mixingsidebands(dashedbluelines)onthecomblinesoftheCCWsoliton(solidbluelines).Thesearecreatedby
taperbackscatteringoftheCWpump.Thesesidebandsaresubsequentlybackscatteredwithintheresonatorinto
theCWdirection(middlepanel),wherethey(andtheirCWcounter-parts)induceinjectionlockingoftheCWand
CCWsolitons(lowerpanel).
b,
SimulationofCPsolitonrepetitionratelocking.
φ
Ac
and
φ
Bc
arethepeakposition
ofsolitonsinCWandCCWrotationframes,respectively.Seetextfordiscussionoffourpanels.
Suchadual-pumpedmicroresonatorwithbacscatteringcanbedescribedby
∂A
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
A
)
A
+
i
D
2
2
∂
2
A
∂φ
2
+
F
A
+
√
αF
B
e
i
∆
ωt
+
igA
(
φ,t
)
∫
∞
0
R
(
φ
′
/D
1
)
|
A
(
t,φ
+
φ
′
)
|
2
dφ
′
/D
1
,
(S1)
where
F
A
(
F
B
)isthepumpfieldforthesolitondescribedbyamplitude
A
(
B
),
α
denotesthetaperback-reflection
portionofthepumppower,
κ
isthedecayrateofthesolitonfield,
δω
A
isthefrequencydetuningofthepumpfield
relativetothecavitymodebeingpumped,and
D
2
isthesecondorderdispersion.Thelasttermincludesboththe
KerrandRamannonlinearity
1,2
.Thenonlinearresponseterm
R
(
t
)hastheform
1
R
(
t
)=(1
−
f
R
)
δ
(
t
)+
f
R
h
R
(
t
)
,
(S2)
wherethedelayinelectricalresponseisignoredand
h
R
(
t
)accountsfortheRamanresponse.TheRamanfractionfor
silica(
f
R
=0
.
18)isassumed.
Byexpandingtheintracavityfieldattwopumpfrequenciesas
A
=
A
p
+
A
b
e
i
∆
ωt
where
A
p
istheamplitudeforthe
existingsolitonand
A
b
isthefieldthatformsinresponsetothebackscatteredpumpfield,wecanderivethefollowing
coupledamplitudeequations
1,2
∂A
p
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
A
)
A
p
+
i
D
2
2
∂
2
A
p
∂φ
2
+
F
A
+
ig
[
|
A
p
|
2
+(2
−
f
R
)
|
A
b
|
2
]
A
p
+
igτ
R
D
1
A
p
∂
(
|
A
p
|
2
+
|
A
b
|
2
)
∂φ
,
(S3)
∂A
b
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
B
)
A
b
+
i
D
2
2
∂
2
A
b
∂φ
2
+
√
αF
B
+
ig
[
|
A
b
|
2
+(2
−
f
R
)
|
A
p
|
2
]
A
b
+
igτ
R
D
1
A
b
∂
(
|
A
p
|
2
+
|
A
b
|
2
)
∂φ
,
(S4)
wheretheRamangainbetweenthetwofieldshasbeenignoredas∆
ω
ismuchsmallerthanthematerial’sRaman
shift.
τ
R
∼
2
.
4fsistheRamanshocktimeinsilica
1–3
.
As
α
1,weassumethatthebackscatteredfieldisaperturbationtotheexistingsolitonfield(
|
A
b
||
A
p
|
).
Theexistingsolitonthereforemaintainspropagationasasoliton.Next,theintracavitybackscatteringisadded(see
equationsinMethodsSection)byincludingcouplingfromtheexistingsolitonpropagatingintheopposingdirection.
Theamplitudeofthissolitonisdenotedby
B
p
anditsweakpumpbackscatteringcomponentisdenotedby
B
b
such
that
B
=
B
p
+
B
b
e
−
i
∆
ωt
.Thecompletecoupledequationsaregivenby,
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3
Interferogram intensity (a.u.)
0.5
Time (ms)
1
0
0
10
Interferogram intensity (a.u.)
16.5
μ
s
b
Time (
μ
s)
a
20
Locked
Unlocked
FIG.S3:
a,
InterferogramoftheCPsolitonswhentherepetitionratesarelocked.Thesolitonoffset∆
ν
is77kHz.
b,
InterferogramoftheCPsolitonswhentherepetitionratesareunlocked.Takenfromfig.2cofmaintext.The
solitonoffset∆
ν
is3
.
9MHz.
∂A
p
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
A
)
A
p
+
i
D
2
2
∂
2
A
p
∂φ
2
+
F
A
+
ig
[
|
A
p
|
2
+(2
−
f
R
)
|
A
b
|
2
]
A
p
+
igτ
R
D
1
A
p
∂
(
|
A
p
|
2
+
|
A
b
|
2
)
∂φ
+
iGB
b
,
(S5)
∂A
b
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
B
)
A
b
+
i
D
2
2
∂
2
A
b
∂φ
2
+
√
αF
B
+
ig
[
|
A
b
|
2
+(2
−
f
R
)
|
A
p
|
2
]
A
b
+
igτ
R
D
1
A
b
∂
(
|
A
p
|
2
+
|
A
b
|
2
)
∂φ
+
iGB
p
.
(S6)
∂B
p
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
B
)
B
p
+
i
D
2
2
∂
2
B
p
∂φ
2
+
F
B
+
ig
[
|
B
p
|
2
+(2
−
f
R
)
|
B
b
|
2
]
B
p
+
igτ
R
D
1
B
p
∂
(
|
B
p
|
2
+
|
B
b
|
2
)
∂φ
+
iG
∗
A
b
,
(S7)
∂B
b
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
A
)
B
b
+
i
D
2
2
∂
2
B
b
∂φ
2
+
√
αF
A
+
ig
[
|
B
b
|
2
+(2
−
f
R
)
|
B
p
|
2
]
B
b
+
igτ
R
D
1
B
b
∂
(
|
B
p
|
2
+
|
B
b
|
2
)
∂φ
+
iG
∗
A
p
.
(S8)
where,forsimplicity,onlyasinglescattererisassumedinthecavitysothatΓ(
θ
)=
Gδ
(
θ
).
Fig.S2(b)numericallystudieslockingoftheCPsolitonrepetitionratesbysolutionoftheabovecoupledsoliton
equations.ByplottingthetimeevolutionofthedifferenceintheCPsolitons’peakposition(
φ
Ac
−
φ
Bc
)withintheir
ownmovingframes,wecanextracttheirrepetitionratedifferencefromtheslopeofthecurves.Theupperpanelin
fig.S2(b)showshowthesolitonsratelockafterafewcavityroundtrips.Backscattercouplingvalues(internaland
taper)areindicatedasisthepumpdetuninginnormalizedunits.Inthesecondpanel,thepumpdetuningisincreased
andthisleadstounlocking.However,inthethirdandfourthpanelslockingisagainrestoredbyeitherincreasingthe
tapercouplingorthebackscattercoupling.
Finally,themeasuredinterferogramintheratelockedconditionisshowninfig.S3.Thetracecontainsnofeatures
otherthannoise.Fig.2cofthemaintextinwhichtheratesaredistinctandunlockedisincludedinfig.S3for
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Frequency offset (kHz)
-600
-600
0
Power (10 dB per division)
CP soliton
Dual resonator
FIG.S4:Zoom-inofRFspectrashowingdualsolitonbeatnotes.ThebluetracedenotestheCPsolitonswithin-
tegrationtime50ms.Theredtracerepresentstheresultsfromsolitonsgeneratedintwodistinctmicroresonators
withintegrationtime200
μ
s.
comparison.
III.COMPARISONBETWEENCPSOLITONDUALCOMBANDTWO-RESONATORDUALCOMB
Microresonatorsolitondual-combspectroscopyhasrecentlybeendemonstratedusingsolitonsgeneratedfromtwo
distinctresonators
4
.Asdiscussedinthemaintext,CPsolitonseliminatetheneedfortworesonatorsandalsoimprove
thestabilityoftheinterferogramspectrumthroughtheCPsolitonlocking.Herewecomparethedual-combspectra
fromCPsolitonsanddualresonators.Fig.S4overlaysazoom-inoftheinterferogramspectraforcasesinwhich
two,distinctresonatorsareused(repetitionratedifferenceoforder1MHz)andphase-lockedCPsolitonsareused
(repetitionratedifferenceoforder10sofkHz).Adramaticdifferenceinthespectralwidthofeachindividualspectral
lineoftheCPsolitoncase(bluespectrum)versusthedualcombspectrallineisapparent.Thesignal-to-noiseratio
isalsogreatlyimprovedfortheCPsolitoncase.Inprinciple,thedualcombsystemcouldbephaselocked,however
thelockingmechanismdemonstratedhereisintrinsictothephysicsofthecounter-propagatingsolitons.Ittherefore
avoidscomplexexternallockingapparatusandtherebyprovidesamajorsimplification(evenbeyondtheelimination
asecondfrequencycomb)tothesesystems.
1
G.P.Agrawal,
Nonlinearfiberoptics
(Academicpress,2007).
2
Q.-F.Yang,X.Yi,K.Y.Yang,andK.Vahala,Nat.Phys.
13
,53(2017).
3
X.Yi,Q.-F.Yang,K.Y.Yang,andK.Vahala,Opt.Lett.
41
,3419(2016).
4
M.-G.Suh,Q.-F.Yang,K.Y.Yang,X.Yi,andK.Vahala,Science,doi:10.1126/science.aah6516(2016).
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DOI: 10.1038/NPHOTON.2017.117