of 2
SupplementalSection:Broadbanddispersionengineeredmicroresonatoron-a-chip
I.DISPERSIONPARAMETERSFITTING
0250500750
-250
-500
-750
mode number
22.035
22.040
22.045
21.395
21.400
21.405
FSR (GHz)
FSR (GHz)
EOM-assisted spectroscopy
Sideband spectroscopy
D
2
= 4.17 kHz
D
3
= -9.74 Hz
O
0
= 1550 nm
D
2
= 1.06 kHz
D
3
= 1.89 Hz
O
0
= 1550 nm
a
b
21.355
21.360
21.365
FSR (GHz)
c
D
2
= 0.93 kHz
D
3
= -4.87 Hz
O
0
= 1550 nm
1550
wavelength (nm)
1450
1500
1600
1650
1700
FIG.S1:
Dispersionparametersfitting
(
a
)Mea-
suredFSRofsingle-wedgediskandtheTaylorseries
fit(solidline).Bluedotsrepresentthemeasurement
valueusingtheEOMcombmethod(Fig.3a),andred
dotsaremeasuredbysidebandspectroscopy
S1,S2
for
comparison.TheFSRis22.042GHzat1550nm.(
b-c
)
Sameaspanel
a
,butforquadruple-wedgedisks.FSR:
21.36GHzfor(
b
),21.40GHzfor(
c
)at1550nm.
Theresonancefrequenciesofonemodefamilycanbe
describedasaTaylorseriesshownineqn.(2),andthe
coefficientsoftheseriescorrespondtodispersionparam-
etersat
μ
=0.TheFSRcanalsobeexpressedasa
Taylorserieswiththesamecoefficientsineqn.(2).
FSR
(
μ
)=
D
1
+
D
2
·
μ
+
D
3
2!
μ
2
+
···
(S1)
Fig.S1showsthemeasuredFSRofsingle-and
quadruple-wedgestructures(Fig.3),andthefittedpoly-
nomialcurves(eqn.(S1))totheexperimentalresults.
Here,FSRwasmeasuredusingbothEOM-assisted
(Fig.3a,300-nm-bandwidth)andsideband
S1
(70-nm-
bandwidth)spectroscopymethods.∆FSRpermode
(
D
2
)isrelativelysmall(
<
10kHz/mode),thusitisnec-
essarytoaccumulatethechangeofFSRthroughoutmul-
tiplemodenumbers(
μ
)inordertoperformthefitting.
S1
.
II.CONFIDENCEINSEM/AFM-BASED
DISPERSIONCALCULATION
b
a
c
22.06
22.04
22.02
22.00
21.98
21.92
21.90
21.88
21.86
21.84
21.44
21.42
21.40
21.38
21.36
9001200150018002100
Wavelength (nm)
-25
0
25
50
75
-25
0
25
50
75
-25
0
25
50
75
D
2
(kHz)
D
2
(kHz)
D
2
(kHz)
FSR (GHz)
FSR (GHz)
FSR (GHz)
FIG.S2:
Dispersioninhigher-ordermode
TM1
modeFSRand
D
2
for
(a)
single-,
(b)
double-,
(c)
quadruple-wedgedisksshowninFig.3.Thedotsare
fromthemeasurementandsolidlinesarenumerically
calculatedFSR(Red)and
D
2
(Orange)fromthesame
resonatorstructure.InsetsshowSEMimagesofres-
onatorandthesimulatedTM1modeprofile.
Thefinite-element-simulationcalculatestheresonance
frequencies(eqn.(2))basedonAFMandSEMmea-
surements.Generally,thestructuralparametersfrom
themicroscopemeasurementarefinelytunedwithina
rangeof
±
0.5%ofthemeasuredparametersinorder
toobtainagoodfitting,andthetuningmightcorrect
theuncertaintyofresonatorprofilemeasurements.
Themeasurementofhigher-ordermodedispersion
canprovideadditionalconfidenceintheconnecting
resonatorstructureanddispersion.Fig.S2showsthe
measuredFSR,
D
2
oftheTM1mode(seeinsetsfor
modeprofile),andtheresultsofsimulationsfromthe
geometryinFig.3.Here,thestructuralparameters
areexactlysamewiththoseforfundamentalmode
calculations(i.e.,noextratuningofthestructuretofit
Broadband dispersion-engineered microresonator on a chip
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DOI: 10.1038/NPHOTON.2016.36
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201
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2
0
Number of
disks
0
5
10
15
0.5
-0.5
00.5
-0.5
error in
T
1
(
ȗ
)error in
T
2
(
ȗ
)
75
-75
0
error in t (
nm)
D
2
(kHz)
6
-3
0
abc
3
FIG.S3:
Reproducibilityofresonatorfabrica-
tion
Structuralparametervariationsandsimulated
impacton
D
2
for
(a)
θ
1
(outerwedgeangle),
(b)
θ
2
(innerwedgeangle),and
(c)
t(outerwedgeheight)
ofthedouble-wedgedisk.Structuralparameterswere
measuredfromfabricateddouble-wedgediskswiththe
samedesignastheoneinFig.3c,and
D
2
iscalculated
asafunctionofoneoftheparameterswhileotherpa-
rametersarefixed.
theTM1dispersion).
Theimprecisionofmicro-fabrication
S3,S4
mightlower
therepeatabilityofdispersioninfabricateddisks.It
canalsoimpacttherepeatabilityofresonatorgeometry
designtoachievespecifeddispersioncoefficients.Asa
preliminarytestofprocessaccuracy,double-wedgedisks
(29EA)wereidenticallyfabricatedfrommultiplewafers
(4EA)andstructuralparametersweremeasuredusing
anAFM(seeFig.S3).Themeasurementsshow30.4
nm,0.21
,and0.15
standarddeviationsint,
θ
1
,and
θ
2
,respectively(actualvaluesare2.92
μ
m,43.5
,and
10.5
fort,
θ
1
,and
θ
2
).Inordertounderstandhow
thisimprecisionintheprocessimpactsthedispersion
oftheresonator,wehaverunasetofsimulationson
double-wedgedisksandindependentlyvariedtand
θ
1
,
2
by
±
150nmand1
,respectively.Basedonnumerical
simulation,thestandarddeviationsoft,
θ
1
,and
θ
2
approximatelycause0.8,0.01,0.3kHzof
D
2
variations
fromthedesigneddispersionparameter,respectively.
Forsimplicity,
D
2
variationwascalculatedasfunction
ofasingleparameter(oneoft,
θ
1
,and
θ
2
)assuming
othergeometricparametersconstant.
[S1]Li,J.,Lee,H.,Yang,K.Y.&Vahala,K.J.Sideband
spectroscopyanddispersionmeasurementinmicrocavi-
ties.
Opt.Express
20
,26337–26344(2012).
[S2]Del’Haye,P.,Beha,K.,Papp,S.B.&Diddams,
S.A.Self-injectionlockingandphase-lockedstatesin
microresonator-basedopticalfrequencycombs.
Phys.
Rev.Lett.
112
,043905(2014).
[S3]
Lee,H.
etal.
Chemicallyetched,ulta-high-Qresonator
onachip.
NaturePhoton.
6
,369–373(2012).
[S4]Kordts,A.,Pfeiffer,M.H.P.,Guo,H.,Brasch,V.&Kip-
penberg,T.J.Higherordermodesuppressioninhigh-q
anomalousdispersionsinmicroresonatorsfortemporal
dissipativekerrsolitonformation.
Opt.Lett.
41
,452–
455(2016).
2
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| www.nature.com/naturephotonics
SUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHOTON.2016.36
©
201
6
Macmillan Publishers Limited.
All rights reserved.