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© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
SupplementaryMaterialofStokesSolitonsinOpticalMicrocavities
Qi-FanYang
,XuYi
,KiYoulYangandKerryVahala
T.J.WatsonLaboratoryofAppliedPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125,USA.
Theseauthorscontributedequallytothiswork.
Correspondingauthor:vahala@caltech.edu
Stokes solitons in optical microcavities
SUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHYS3875
NATURE PHYSICS
| www.nature.com/naturephysics
1
© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
2
I.NUMERICALSIMULATION
Masterequations.
TheinteractionbetweentwosolitonsviacrossphasemodulationandtheRamanprocesshas
beenstudiedinopticalfibersusingcoupledpulsepropagationequations
1,2
.Thisapproachisextendedtostudythe
Stokessolitongenerationprocessinmicroresonators.Apairofcoupledequationsdescribingtheintracavityslowly-
varyingfieldamplitudesfortheprimaryandStokessolitonsystemcanbeadaptedfromtheLugiato-Lefeverequation
(LLequation)
3–6
augmentedbyRamaninteractions
2
,whichreads:
∂E
p
(
φ
)
∂t
=
i
D
2
p
2
2
E
p
∂φ
2
+
i
(1
f
R
)(
g
p
|
E
p
|
2
+2
G
p
|
E
s
|
2
)
E
p
+
i
f
R
E
p
D
1
p
h
R
(
φ
φ
D
1
p
)(
g
p
|
E
p
(
φ
)
|
2
+
G
p
|
E
s
(
φ
)
|
2
)
+
i
f
R
E
s
D
1
p
h
R
(
φ
φ
D
1
p
)
G
p
E
p
(
φ
)
E
s
(
φ
)exp(
i
φ
φ
D
1
p
)
(
κ
p
2
+
i
ω
p
)
E
p
+
κ
ext
p
P
in
,
(S1)
∂E
s
(
φ
)
∂t
=
δ
∂E
s
∂φ
+
i
D
2
s
2
2
E
s
∂φ
2
+
i
(1
f
R
)(
g
s
|
E
s
|
2
+2
G
s
|
E
p
|
2
)
E
s
+
i
f
R
E
s
D
1
p
h
R
(
φ
φ
D
1
p
)(
g
s
|
E
s
(
φ
)
|
2
+
G
s
|
E
p
(
φ
)
|
2
)
+
i
f
R
E
p
D
1
p
h
R
(
φ
φ
D
1
p
)
G
s
E
s
(
φ
)
E
p
(
φ
)exp(
i
φ
φ
D
1
p
)
(
κ
s
2
+
i
ω
s
)
E
s
.
(S2)
Theslowlyvaryingfields
E
j
(subscript
j
=(
p,s
)forprimaryorStokessoliton)arenormalizedtoopticalenergy.
Tosecondorder,thefrequencyofmodenumber
μ
inmodefamily
j
=(
p,s
)isgivenbytheTaylorexpansion
ω
μj
=
ω
0
j
+
D
1
j
μ
+
1
2
D
2
j
μ
2
where
ω
0
j
isthefrequencyofmode
μ
=0,while
D
1
j
and
D
2
j
aretheFSRandthe
second-orderdispersionat
μ
=0.Ωisthecarrierfrequencydifference
ω
0
p
ω
0
s
.Also,
δ
=
D
1
s
D
1
p
isthe
FSR
differencebetweenprimaryandStokessolitonsatmode
μ
=0.
κ
j
isthecavitylossrateand∆
ω
j
isthedetuningof
modezeroofthesolitonspectrumrelativetothecoldcavityresonance.
h
R
(
t
)istheRamanresponsefunction
2
.For
theprimarysoliton,whichisadissipativeKerrsoliton(DKS),thepumpfieldislockedtooneofthesolitonspectral
linesandthis“pump”lineistakenasmode
μ
=0.
κ
ext
p
istheexternalcouplingcoefficientand
P
in
isthepump
power.
g
j
and
G
j
areselfandcrossphasemodulationcoefficients,definedas,
g
j
=
n
2
ω
j
D
1
j
2
nπA
jj
,G
j
=
n
2
ω
j
D
1
j
2
nπA
ps
.
(S3)
wherethenonlinearmodearea
A
jk
isdefinedas
2
A
jk
=
∫∫
−∞
|
u
j
(
x,y
)
|
2
dxdy
∫∫
−∞
|
u
k
(
x,y
)
|
2
dxdy
∫∫
−∞
|
u
j
(
x,y
)
|
2
|
u
k
(
x,y
)
|
2
dxdy
,
(S4)
where
u
j
isthetransversedistributionofthemode.
f
R
=0
.
18istheRamancontributionparameterinsilica.
Adiabaticapproximation.
Asthepulsewidthofthesolitonsinoursystemisseveralhundredfemtosecond,which
ismuchlongerthantheRamanresponsetime(
10fs),thepulsefields
E
p
and
E
s
canbeconsideredslowlyvarying
variableswhencomparedwith
h
R
(
t
).Therefore,theintegralsof
E
p
and
E
s
ineqns.(S1)and(S2)associatedwith
RamanresponsefunctioncanbeexpandedintoaTaylorseriesbywriting
φ
=
φ
(
φ
φ
):
h
R
(
φ
φ
D
1
p
)
|
E
j
(
φ
(
φ
φ
))
|
2
≈|
E
j
(
φ
)
|
2
D
1
p
|
E
j
(
φ
)
|
2
∂φ
τh
R
(
τ
)
(S5)
h
R
(
φ
φ
D
1
p
)
E
j
(
φ
(
φ
φ
))
E
k
(
φ
(
φ
φ
))
e
i
φ
φ
D
1
p
E
j
(
φ
)
E
k
(
φ
)
h
R
(
τ
)
e
i
τ
(S6)
where
τ
=(
φ
φ
)
/D
1
p
.Thenexthighestordertermineqn.(S6)isfoundtohaveanegliibleeffectonsimulation
andisneglectedhere.Usingtheseapproximateforms,thecoupledequationssimplifyasfollows:
3
∂E
p
∂t
=
i
D
2
p
2
2
E
p
∂φ
2
+
i
[
g
p
|
E
p
|
2
+(2
f
R
)
G
p
|
E
s
|
2
]
E
p
iD
1
p
τ
R
E
p
(
g
p
|
E
p
|
2
+
G
p
|
E
s
|
2
)
∂φ
(
κ
p
2
+
i
ω
p
)
E
p
ω
p
ω
s
R
|
E
s
|
2
E
p
+
κ
ext
p
P
in
,
(S7)
∂E
s
∂t
=
δ
∂E
s
∂φ
+
i
D
2
s
2
2
E
s
∂φ
2
+
i
[
g
s
|
E
s
|
2
+(2
f
R
)
G
s
|
E
p
|
2
]
E
s
iD
1
p
τ
R
E
s
(
g
s
|
E
s
|
2
+
G
s
|
E
p
|
2
)
∂φ
(
κ
s
2
+
i
ω
s
)
E
s
+
R
|
E
p
|
2
E
s
.
(S8)
where
R
=
f
R
G
s
Im[
h
R
(
τ
)exp(
i
τ
)
]=
cD
1
p
g
R
(
ω
s
p
)
/
4
nπA
ps
and
g
R
(
ω
s
p
)istheRamangaininsilica.The
Ramanshocktimeisdefinedby
τ
R
=
f
R
τh
R
(
τ
)
andis
2-3fsinsilica.ForsolitonswithafewTHzbandwidth,
othereffectsarenegligible(e.g.,higherorderdispersion,theself-steepeningeffectandRamaninducedrefractiveindex
change
2
).Ifthesolitonpulsewidthiswellbelow100fs,i.e.,ithasabroadbandspectrum,thenthecoupledequations
withhigher-orderRamancorrectionmightberequired.
Phaselockingterms.
Phase-sensitive,four-wave-mixingtermshavealsobeenomittedinEqn.(S7)and(S8).In
principle,thesetermscouldintroducelockingoftheStokesandprimarysolitonfields(inadditiontotheirrepetition
rates).However,forthistooccurtheunderlyingspatialmodefamilieswouldneedtofeaturemodefrequenciesthat
alignreasonablywell(bothinFSRandoffsetfrequency)withinthesameband.Inthiswork,themodefrequences
wereobservedtonotoverlapusingdevicesthatfeaturedspectrallyoverlappingsolitons.
II.THEORETICALANALYSIS(WEAKSTOKESLIMIT)
Althoughanexactsolutionofthecoupledsolitonscannotbeobtained,thenearthresholdbehavioroftheStokes
solitoncanstillbestudiedanalytically.Inthislimit,theprimarysolitonisunperturbatedbytheStokessolitonsince
theStokessolitonisweak(i.e.,nearthreshold).ItssolutionisthereforegivenbytheSech
2
DKSsolution
6
.The
Stokessolitonequationthenusesthissolutionfortheprimarysoliton.ByselectingthecarrierfrequencyoftheStokes
solitonsuchthat
δ
=0,theequationfortheStokessolitoncanbesimplifiedtothefollowing,
∂E
s
∂t
=
i
D
2
s
2
2
E
s
∂φ
2
+
i
(2
f
R
)
G
s
|
E
s
|
2
E
s
(
κ
s
2
+
i
ω
s
)
E
s
+
R
|
E
p
|
2
E
s
,
(S9)
where
E
p
=
A
sech
istheuncoupledprimarysolitonsolution
6
with
|
A
|
2
=2∆
ω
p
/g
p
and
B
=
2∆
ω
p
/D
2
p
.The
Ramantermscontainingderivativescausesolitonselffrequencyshiftandsubsequentlyaphasechange
7
.However,
theyminimallyaffectpulsewidthandpeakpower
8,9
.AstheprimarysolitonandStokessolitonhavenoabolute
phasecoherence,thesetermshavebeenomitted.Accordingly,theStokessolitonistreatedasawavetrappedina
sech
2
-shapepotentialwellcreatedbytheprimarysoliton,andalsoderivingopticalgainfromtheprimarysoliton.
Theboundedsolutionofthewavefunctioninsuchapotentialhasthefollowingform,
E
s
=
V
sech
γ
Bφ,
(S10)
where,consistentwiththenearthresholdassumption,
V
isasmallamplitudesatisfying
|
V
|
2
|
A
|
2
.Underthese
assumptions,theexponent
γ
isarootofthefollowingequation,
γ
(
γ
+1)=2(2
f
R
)
G
s
D
2
p
/g
p
D
2
s
.
(S11)
Asaphysicalcheckofthisequation,wenotethatundercircumstancesof
G
s
=
g
p
/
2thepotentialwellcreatedbythe
primarysolitonfortheStokessolitonisidenticaltotheprimarysolitonpotentialwell(note:thefactorof1/2comes
aboutfromcrossphasemodulation).Inthiscase,assumingidenticalsecond-orderdispersion(
D
2
s
=
D
2
p
)andalso
f
R
=0,thesolutiontoeqn.(S11)is
γ
=1whichshowsthattheStokessolitonaquiresthesameenvelopeasthe
primarysoliton.
Thresholdcalculation
.OncethepeakpoweroftheprimarysolitonreachesapointthatprovidessufficientRaman
gaintoovercomeStokessolitonroundtriploss,theStokessolitonwillbegintooscillate.Thethresholdcondition
emergesastheconditionforsteady-stateStokessolitonpowerbalance.ThisisreadilyderivedfromtheStokessoliton
2
NATURE PHYSICS
| www.nature.com/naturephysics
SUPPLEMENTARY INFORMATION
DOI: 10.1038/
NPHYS3875
© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
2
I.NUMERICALSIMULATION
Masterequations.
TheinteractionbetweentwosolitonsviacrossphasemodulationandtheRamanprocesshas
beenstudiedinopticalfibersusingcoupledpulsepropagationequations
1,2
.Thisapproachisextendedtostudythe
Stokessolitongenerationprocessinmicroresonators.Apairofcoupledequationsdescribingtheintracavityslowly-
varyingfieldamplitudesfortheprimaryandStokessolitonsystemcanbeadaptedfromtheLugiato-Lefeverequation
(LLequation)
3–6
augmentedbyRamaninteractions
2
,whichreads:
∂E
p
(
φ
)
∂t
=
i
D
2
p
2
2
E
p
∂φ
2
+
i
(1
f
R
)(
g
p
|
E
p
|
2
+2
G
p
|
E
s
|
2
)
E
p
+
i
f
R
E
p
D
1
p
h
R
(
φ
φ
D
1
p
)(
g
p
|
E
p
(
φ
)
|
2
+
G
p
|
E
s
(
φ
)
|
2
)
+
i
f
R
E
s
D
1
p
h
R
(
φ
φ
D
1
p
)
G
p
E
p
(
φ
)
E
s
(
φ
)exp(
i
φ
φ
D
1
p
)
(
κ
p
2
+
i
ω
p
)
E
p
+
κ
ext
p
P
in
,
(S1)
∂E
s
(
φ
)
∂t
=
δ
∂E
s
∂φ
+
i
D
2
s
2
2
E
s
∂φ
2
+
i
(1
f
R
)(
g
s
|
E
s
|
2
+2
G
s
|
E
p
|
2
)
E
s
+
i
f
R
E
s
D
1
p
h
R
(
φ
φ
D
1
p
)(
g
s
|
E
s
(
φ
)
|
2
+
G
s
|
E
p
(
φ
)
|
2
)
+
i
f
R
E
p
D
1
p
h
R
(
φ
φ
D
1
p
)
G
s
E
s
(
φ
)
E
p
(
φ
)exp(
i
φ
φ
D
1
p
)
(
κ
s
2
+
i
ω
s
)
E
s
.
(S2)
Theslowlyvaryingfields
E
j
(subscript
j
=(
p,s
)forprimaryorStokessoliton)arenormalizedtoopticalenergy.
Tosecondorder,thefrequencyofmodenumber
μ
inmodefamily
j
=(
p,s
)isgivenbytheTaylorexpansion
ω
μj
=
ω
0
j
+
D
1
j
μ
+
1
2
D
2
j
μ
2
where
ω
0
j
isthefrequencyofmode
μ
=0,while
D
1
j
and
D
2
j
aretheFSRandthe
second-orderdispersionat
μ
=0.Ωisthecarrierfrequencydifference
ω
0
p
ω
0
s
.Also,
δ
=
D
1
s
D
1
p
isthe
FSR
differencebetweenprimaryandStokessolitonsatmode
μ
=0.
κ
j
isthecavitylossrateand∆
ω
j
isthedetuningof
modezeroofthesolitonspectrumrelativetothecoldcavityresonance.
h
R
(
t
)istheRamanresponsefunction
2
.For
theprimarysoliton,whichisadissipativeKerrsoliton(DKS),thepumpfieldislockedtooneofthesolitonspectral
linesandthis“pump”lineistakenasmode
μ
=0.
κ
ext
p
istheexternalcouplingcoefficientand
P
in
isthepump
power.
g
j
and
G
j
areselfandcrossphasemodulationcoefficients,definedas,
g
j
=
n
2
ω
j
D
1
j
2
nπA
jj
,G
j
=
n
2
ω
j
D
1
j
2
nπA
ps
.
(S3)
wherethenonlinearmodearea
A
jk
isdefinedas
2
A
jk
=
∫∫
−∞
|
u
j
(
x,y
)
|
2
dxdy
∫∫
−∞
|
u
k
(
x,y
)
|
2
dxdy
∫∫
−∞
|
u
j
(
x,y
)
|
2
|
u
k
(
x,y
)
|
2
dxdy
,
(S4)
where
u
j
isthetransversedistributionofthemode.
f
R
=0
.
18istheRamancontributionparameterinsilica.
Adiabaticapproximation.
Asthepulsewidthofthesolitonsinoursystemisseveralhundredfemtosecond,which
ismuchlongerthantheRamanresponsetime(
10fs),thepulsefields
E
p
and
E
s
canbeconsideredslowlyvarying
variableswhencomparedwith
h
R
(
t
).Therefore,theintegralsof
E
p
and
E
s
ineqns.(S1)and(S2)associatedwith
RamanresponsefunctioncanbeexpandedintoaTaylorseriesbywriting
φ
=
φ
(
φ
φ
):
h
R
(
φ
φ
D
1
p
)
|
E
j
(
φ
(
φ
φ
))
|
2
≈|
E
j
(
φ
)
|
2
D
1
p
|
E
j
(
φ
)
|
2
∂φ
τh
R
(
τ
)
(S5)
h
R
(
φ
φ
D
1
p
)
E
j
(
φ
(
φ
φ
))
E
k
(
φ
(
φ
φ
))
e
i
φ
φ
D
1
p
E
j
(
φ
)
E
k
(
φ
)
h
R
(
τ
)
e
i
τ
(S6)
where
τ
=(
φ
φ
)
/D
1
p
.Thenexthighestordertermineqn.(S6)isfoundtohaveanegliibleeffectonsimulation
andisneglectedhere.Usingtheseapproximateforms,thecoupledequationssimplifyasfollows:
3
∂E
p
∂t
=
i
D
2
p
2
2
E
p
∂φ
2
+
i
[
g
p
|
E
p
|
2
+(2
f
R
)
G
p
|
E
s
|
2
]
E
p
iD
1
p
τ
R
E
p
(
g
p
|
E
p
|
2
+
G
p
|
E
s
|
2
)
∂φ
(
κ
p
2
+
i
ω
p
)
E
p
ω
p
ω
s
R
|
E
s
|
2
E
p
+
κ
ext
p
P
in
,
(S7)
∂E
s
∂t
=
δ
∂E
s
∂φ
+
i
D
2
s
2
2
E
s
∂φ
2
+
i
[
g
s
|
E
s
|
2
+(2
f
R
)
G
s
|
E
p
|
2
]
E
s
iD
1
p
τ
R
E
s
(
g
s
|
E
s
|
2
+
G
s
|
E
p
|
2
)
∂φ
(
κ
s
2
+
i
ω
s
)
E
s
+
R
|
E
p
|
2
E
s
.
(S8)
where
R
=
f
R
G
s
Im[
h
R
(
τ
)exp(
i
τ
)
]=
cD
1
p
g
R
(
ω
s
p
)
/
4
nπA
ps
and
g
R
(
ω
s
p
)istheRamangaininsilica.The
Ramanshocktimeisdefinedby
τ
R
=
f
R
τh
R
(
τ
)
andis
2-3fsinsilica.ForsolitonswithafewTHzbandwidth,
othereffectsarenegligible(e.g.,higherorderdispersion,theself-steepeningeffectandRamaninducedrefractiveindex
change
2
).Ifthesolitonpulsewidthiswellbelow100fs,i.e.,ithasabroadbandspectrum,thenthecoupledequations
withhigher-orderRamancorrectionmightberequired.
Phaselockingterms.
Phase-sensitive,four-wave-mixingtermshavealsobeenomittedinEqn.(S7)and(S8).In
principle,thesetermscouldintroducelockingoftheStokesandprimarysolitonfields(inadditiontotheirrepetition
rates).However,forthistooccurtheunderlyingspatialmodefamilieswouldneedtofeaturemodefrequenciesthat
alignreasonablywell(bothinFSRandoffsetfrequency)withinthesameband.Inthiswork,themodefrequences
wereobservedtonotoverlapusingdevicesthatfeaturedspectrallyoverlappingsolitons.
II.THEORETICALANALYSIS(WEAKSTOKESLIMIT)
Althoughanexactsolutionofthecoupledsolitonscannotbeobtained,thenearthresholdbehavioroftheStokes
solitoncanstillbestudiedanalytically.Inthislimit,theprimarysolitonisunperturbatedbytheStokessolitonsince
theStokessolitonisweak(i.e.,nearthreshold).ItssolutionisthereforegivenbytheSech
2
DKSsolution
6
.The
Stokessolitonequationthenusesthissolutionfortheprimarysoliton.ByselectingthecarrierfrequencyoftheStokes
solitonsuchthat
δ
=0,theequationfortheStokessolitoncanbesimplifiedtothefollowing,
∂E
s
∂t
=
i
D
2
s
2
2
E
s
∂φ
2
+
i
(2
f
R
)
G
s
|
E
s
|
2
E
s
(
κ
s
2
+
i
ω
s
)
E
s
+
R
|
E
p
|
2
E
s
,
(S9)
where
E
p
=
A
sech
istheuncoupledprimarysolitonsolution
6
with
|
A
|
2
=2∆
ω
p
/g
p
and
B
=
2∆
ω
p
/D
2
p
.The
Ramantermscontainingderivativescausesolitonselffrequencyshiftandsubsequentlyaphasechange
7
.However,
theyminimallyaffectpulsewidthandpeakpower
8,9
.AstheprimarysolitonandStokessolitonhavenoabolute
phasecoherence,thesetermshavebeenomitted.Accordingly,theStokessolitonistreatedasawavetrappedina
sech
2
-shapepotentialwellcreatedbytheprimarysoliton,andalsoderivingopticalgainfromtheprimarysoliton.
Theboundedsolutionofthewavefunctioninsuchapotentialhasthefollowingform,
E
s
=
V
sech
γ
Bφ,
(S10)
where,consistentwiththenearthresholdassumption,
V
isasmallamplitudesatisfying
|
V
|
2
|
A
|
2
.Underthese
assumptions,theexponent
γ
isarootofthefollowingequation,
γ
(
γ
+1)=2(2
f
R
)
G
s
D
2
p
/g
p
D
2
s
.
(S11)
Asaphysicalcheckofthisequation,wenotethatundercircumstancesof
G
s
=
g
p
/
2thepotentialwellcreatedbythe
primarysolitonfortheStokessolitonisidenticaltotheprimarysolitonpotentialwell(note:thefactorof1/2comes
aboutfromcrossphasemodulation).Inthiscase,assumingidenticalsecond-orderdispersion(
D
2
s
=
D
2
p
)andalso
f
R
=0,thesolutiontoeqn.(S11)is
γ
=1whichshowsthattheStokessolitonaquiresthesameenvelopeasthe
primarysoliton.
Thresholdcalculation
.OncethepeakpoweroftheprimarysolitonreachesapointthatprovidessufficientRaman
gaintoovercomeStokessolitonroundtriploss,theStokessolitonwillbegintooscillate.Thethresholdcondition
emergesastheconditionforsteady-stateStokessolitonpowerbalance.ThisisreadilyderivedfromtheStokessoliton
NATURE PHYSICS
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4
Scan time (
μ
s)
0
60
Comb power (a.u.)
0
1
Primary
Stokes
Above
threshold
(a)
Below
threshold
Comb power (a.u.)
0
12
6
Normalized detuning
0
10
20
30
Normalized detuning
10
20
30
(b)
Normalized detuning
0
10
20
30
(d)
Angular coordinate
φ
0
2
π
π
0
2
π
π
(c)
Intracavity power (a.u.)
Primary soliton
Stokes soliton
Angular coordinate
φ
0
2
π
π
Above
threshold
Below
threshold
Primary
Stokes
Mode number
-50
0
50 -50
0
50
Power (10 dB/div)
(e)
(f)
Primary
soliton
Stokes
soliton
Primary
Stokes
0
1
FIG.S1:
Stokessolitonformationinamicroresonator.a)
Simulatedintracavitycombpowerduringalaser
scanovertheprimarysolitonpumpingresonancefromtheblue(left)tothered(right)oftheresonance.Thede-
tuningisnormalizedtotheresonancelinewidth.Theinitialstepcorrespondstotheprimarysolitonformation,
andthesubsequentdecreaseinpowercorrespondstotheonsetoftheStokessoliton.TheStokessolitonpoweris
showninred.
b)
Zoomed-inviewoftheindicatedregionfromfigureS1(a).
c)
Experimentallymeasuredprimary
andStokessolitonpowerduringalaserscanshowingthefeaturessimulatedinS1(a)andS1(b).
d)
Simulationof
theintracavityfieldinthemovingframeofthesolitonsplottedversusthepumplaserdetuning.Thedetuningaxis
isscaledidenticallytofigureS1(a).Thefigureshowstheprimarysolitonstepregion(belowthreshold)aswellas
theonsetoftheStokessoliton(abovethreshold).
e)
TemporaloverlapoftheprimaryandStokessolitonsisnumer-
icallyconfirmedintheplotofnormalizedpowerversuslocationanglewithintheresonator.Theoverlapconfirms
trappingandco-propagation.
f)
IntracavityopticalspectraoftheprimaryandStokessolitons.
equationandtakestheform,
2
π
0
t
|
E
s
|
2
=
2
π
0
(
κ
s
2
R
|
E
p
|
2
)
|
E
s
|
2
=0
(S12)
5
BysubstitutingthesolutionsfortheprimaryandStokessolitonsintoeqn.(S12),theresultingthresholdinprimary
solitonpeakoutputpowerisfoundtobegivenbyeqn.(1)inthemaintext.
Backgroundfield.
Acontinuousbackgroundfieldexistsaspartoftheprimarysoliton(DKS)solution.Inprinciple,
thisconstantbackgoundcouldinducelaseroscillationthroughtheRamanprocess.However,thethresholdpower
forthistooccurcanbeshowntobeclosetothepeakpowerrequiredforStokessolitonoscillation.Moreover,the
backgroundfieldhasapowerlevelthatismanyordersweakerthanthepeakpoweroftheprimarysoliton
6
.This
resultsbecausethepumplaserisfarreddetunedfromthemicrocavityresonance.Asaresultoftheseconsiderations,
theobservedStokesoscillationresultsfrompumpingbytheprimarysolitonand,specifically,spatio-temporaloverlap
oftheStokessolitonwiththeprimary(pump)soliton.Thegoodagreementbetweenmeasurementwiththetheoretical
threshold(eqn.(1)inthemaintext)providesadditionalconfirmation.
III.FORMATIONOFSTOKESSOLITON
TorevealfurtherdetailsontheStokessolitonformation,apumpinglaserscanisperformednumericallyasshown
inFig.S1(a)(andzoom-inofscaninFig.S1(b)).128modesareemployedinthesimulation.Theformationofa
stepintheprimarycombpower(blue)indicatesprimarysolitonformation
6,8
.However,adecreaseinpowerofthe
primarysolitonisnextobservedthatoccurswithanincreaseinpoweroftheStokessoliton(red).Thesamefeatures
arealsoobservedexperimentally(seeFig.S1(c)).Bystudyingtheintracavityfieldevolution,thesimulationshows
thecorrespondencebetweenthesefeaturesandthesolitonformation(seeFig.S1(d)).Also,thesimulationshows
thattheprimaryandStokespulsesoverlapinspaceandtime,i.e.,confirmationofopticaltrapping(seefigureS1(e)).
ThecalculatedspectrafortheprimaryandStokessolitonsareprovidedinfigureS1(f).
Thereisalsoasupplementarymovieoftheseresults.
1
Headley,C.&Agrawal,G.P.Unifieddescriptionofultrafaststimulatedramanscatteringinopticalfibers.
JOSAB
13
,
2170–2177(1996).
2
Agrawal,G.P.
Nonlinearfiberoptics
(Academicpress,2007).
3
Lugiato,L.A.&Lefever,R.Spatialdissipativestructuresinpassiveopticalsystems.
Phys.Rev.Lett.
58
,2209(1987).
4
Matsko,A.
etal.
Mode-lockedkerrfrequencycombs.
Opt.Lett.
36
,2845–2847(2011).
5
Chembo,Y.K.&Menyuk,C.R.Spatiotemporallugiato-lefeverformalismforkerr-combgenerationinwhispering-gallery-
moderesonators.
Phys.Rev.A
87
,053852(2013).
6
Herr,T.
etal.
Temporalsolitonsinopticalmicroresonators.
NaturePhoton.
8
,145–152(2014).
7
Karpov,M.
etal.
Ramanself-frequencyshiftofdissipativekerrsolitonsinanopticalmicroresonator.
Physicalreviewletters
116
,103902(2016).
8
Yi,X.,Yang,Q.-F.,Yang,K.Y.,Suh,M.-G.&Vahala,K.Solitonfrequencycombatmicrowaveratesinahigh-qsilica
microresonator.
Optica
2
,1078–1085(2015).
9
Yi,X.,Yang,Q.-F.,Yang,K.Y.&Vahala,K.Theoryandmeasurementofthesolitonself-frequencyshiftandefficiencyin
opticalmicrocavities.
OpticsLetters
41
,3419–3422(2016).
4
NATURE PHYSICS
| www.nature.com/naturephysics
SUPPLEMENTARY INFORMATION
DOI: 10.1038/
NPHYS3875
© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
4
Scan time (
μ
s)
0
60
Comb power (a.u.)
0
1
Primary
Stokes
Above
threshold
(a)
Below
threshold
Comb power (a.u.)
0
12
6
Normalized detuning
0
10
20
30
Normalized detuning
10
20
30
(b)
Normalized detuning
0
10
20
30
(d)
Angular coordinate
φ
0
2
π
π
0
2
π
π
(c)
Intracavity power (a.u.)
Primary soliton
Stokes soliton
Angular coordinate
φ
0
2
π
π
Above
threshold
Below
threshold
Primary
Stokes
Mode number
-50
0
50 -50
0
50
Power (10 dB/div)
(e)
(f)
Primary
soliton
Stokes
soliton
Primary
Stokes
0
1
FIG.S1:
Stokessolitonformationinamicroresonator.a)
Simulatedintracavitycombpowerduringalaser
scanovertheprimarysolitonpumpingresonancefromtheblue(left)tothered(right)oftheresonance.Thede-
tuningisnormalizedtotheresonancelinewidth.Theinitialstepcorrespondstotheprimarysolitonformation,
andthesubsequentdecreaseinpowercorrespondstotheonsetoftheStokessoliton.TheStokessolitonpoweris
showninred.
b)
Zoomed-inviewoftheindicatedregionfromfigureS1(a).
c)
Experimentallymeasuredprimary
andStokessolitonpowerduringalaserscanshowingthefeaturessimulatedinS1(a)andS1(b).
d)
Simulationof
theintracavityfieldinthemovingframeofthesolitonsplottedversusthepumplaserdetuning.Thedetuningaxis
isscaledidenticallytofigureS1(a).Thefigureshowstheprimarysolitonstepregion(belowthreshold)aswellas
theonsetoftheStokessoliton(abovethreshold).
e)
TemporaloverlapoftheprimaryandStokessolitonsisnumer-
icallyconfirmedintheplotofnormalizedpowerversuslocationanglewithintheresonator.Theoverlapconfirms
trappingandco-propagation.
f)
IntracavityopticalspectraoftheprimaryandStokessolitons.
equationandtakestheform,
2
π
0
t
|
E
s
|
2
=
2
π
0
(
κ
s
2
R
|
E
p
|
2
)
|
E
s
|
2
=0
(S12)
5
BysubstitutingthesolutionsfortheprimaryandStokessolitonsintoeqn.(S12),theresultingthresholdinprimary
solitonpeakoutputpowerisfoundtobegivenbyeqn.(1)inthemaintext.
Backgroundfield.
Acontinuousbackgroundfieldexistsaspartoftheprimarysoliton(DKS)solution.Inprinciple,
thisconstantbackgoundcouldinducelaseroscillationthroughtheRamanprocess.However,thethresholdpower
forthistooccurcanbeshowntobeclosetothepeakpowerrequiredforStokessolitonoscillation.Moreover,the
backgroundfieldhasapowerlevelthatismanyordersweakerthanthepeakpoweroftheprimarysoliton
6
.This
resultsbecausethepumplaserisfarreddetunedfromthemicrocavityresonance.Asaresultoftheseconsiderations,
theobservedStokesoscillationresultsfrompumpingbytheprimarysolitonand,specifically,spatio-temporaloverlap
oftheStokessolitonwiththeprimary(pump)soliton.Thegoodagreementbetweenmeasurementwiththetheoretical
threshold(eqn.(1)inthemaintext)providesadditionalconfirmation.
III.FORMATIONOFSTOKESSOLITON
TorevealfurtherdetailsontheStokessolitonformation,apumpinglaserscanisperformednumericallyasshown
inFig.S1(a)(andzoom-inofscaninFig.S1(b)).128modesareemployedinthesimulation.Theformationofa
stepintheprimarycombpower(blue)indicatesprimarysolitonformation
6,8
.However,adecreaseinpowerofthe
primarysolitonisnextobservedthatoccurswithanincreaseinpoweroftheStokessoliton(red).Thesamefeatures
arealsoobservedexperimentally(seeFig.S1(c)).Bystudyingtheintracavityfieldevolution,thesimulationshows
thecorrespondencebetweenthesefeaturesandthesolitonformation(seeFig.S1(d)).Also,thesimulationshows
thattheprimaryandStokespulsesoverlapinspaceandtime,i.e.,confirmationofopticaltrapping(seefigureS1(e)).
ThecalculatedspectrafortheprimaryandStokessolitonsareprovidedinfigureS1(f).
Thereisalsoasupplementarymovieoftheseresults.
1
Headley,C.&Agrawal,G.P.Unifieddescriptionofultrafaststimulatedramanscatteringinopticalfibers.
JOSAB
13
,
2170–2177(1996).
2
Agrawal,G.P.
Nonlinearfiberoptics
(Academicpress,2007).
3
Lugiato,L.A.&Lefever,R.Spatialdissipativestructuresinpassiveopticalsystems.
Phys.Rev.Lett.
58
,2209(1987).
4
Matsko,A.
etal.
Mode-lockedkerrfrequencycombs.
Opt.Lett.
36
,2845–2847(2011).
5
Chembo,Y.K.&Menyuk,C.R.Spatiotemporallugiato-lefeverformalismforkerr-combgenerationinwhispering-gallery-
moderesonators.
Phys.Rev.A
87
,053852(2013).
6
Herr,T.
etal.
Temporalsolitonsinopticalmicroresonators.
NaturePhoton.
8
,145–152(2014).
7
Karpov,M.
etal.
Ramanself-frequencyshiftofdissipativekerrsolitonsinanopticalmicroresonator.
Physicalreviewletters
116
,103902(2016).
8
Yi,X.,Yang,Q.-F.,Yang,K.Y.,Suh,M.-G.&Vahala,K.Solitonfrequencycombatmicrowaveratesinahigh-qsilica
microresonator.
Optica
2
,1078–1085(2015).
9
Yi,X.,Yang,Q.-F.,Yang,K.Y.&Vahala,K.Theoryandmeasurementofthesolitonself-frequencyshiftandefficiencyin
opticalmicrocavities.
OpticsLetters
41
,3419–3422(2016).
NATURE PHYSICS
| www.nature.com/naturephysics
5
SUPPLEMENTARY INFORMATION
DOI: 10.1038/
NPHYS3875