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SupplementaryInformationfor:”Opticaltransductionandroutingofmicrowave
phononsincavity-optomechanicalcircuits”
KejieFang,
1,2
MatthewH.Matheny,
1,2
XingshengLuan,
1,2
andOskarPainter
1,2,
∗
1
KavliNanoscienceInstituteandThomasJ.Watson,Sr.,LaboratoryofAppliedPhysics,
CaliforniaInstituteofTechnology,Pasadena,California91125,USA
2
InstituteforQuantumInformationandMatter,
CaliforniaInstituteofTechnology,Pasadena,California91125,USA
(Dated:April7,2016)
I.DESIGNOFMECHANICALCAVITYAND
WAVEGUIDECOUPLING
Intheoptomechanicalcavity-waveguidecoupledde-
vices,wecanchangethecavity-waveguidecoupling(i.e.
γ
e
)onpurpose.Thisisachievedbythedesignofalow-
Q
mechanicalcavitymodeandvaryingthenumberof
mirrorcells(Fig.S-1a).AsshowninFig.S-1b,theblue
curvesarethemechanicalbandstructureofthemirror
unitcell(bluerectangleinFig.S-1a).Wedesignthecav-
itysuchthatthemechanicalcavityfrequency(reddashed
lineinFig.S-1b)overlapswiththebandofmirrorunit
cell,suchthatthemechanicalcavitymodecantunnel
throughthemirrorcellsintowaveguide.Meanwhile,the
opticalcavityfrequency(reddashedlineinFig.S-1c)lies
withintheopticalbandgapofmirrorunitcell,suchthat
theopticalcavitymodekeepshigh-
Q
.Byvaryingthe
numberofunitcells,wefindthesimulatedradiationme-
chanicalcouplingrateintowaveguide(
γ
e
/
2
π
)oscillates
betweenafewMHztoashighas30MHz,duetothe
interferencewithinthemirrorunitcells.
II.SIMULATIONOFPHONONPULSE
PROPAGATION
Inthissection,weshowpropagationandbouncingof
phononpulsesinthecavity-waveguidesystem(Fig.3a)
canbewellsimulatedbyagroupofcoupledmodeequa-
tionsusinginput-outputformalism.Thedynamicscap-
turedbythecoupledmodeequationsisaphononpulse
travellinginawaveguideterminatedbytwocavitieswith
baremechanicalfrequency
ω
mL,R
andwaveguidecou-
plingrate
γ
eL,R
.Weapproximate
ω
mL,R
tobethefre-
quencyofcavity-dominatedmodes
L
1
and
R
1
inthesim-
ulation.Sincetheresponsetimeoftheopticalcavityis
muchshorterthanthatofthemechanicalcavity,wecan
excludethedynamicsofopticalmodesfromtheseequa-
tions.Thus,thecoupledmodeequationscanbewritten
asfollows,
db
L
(
t
)
dt
=
−
(
iω
mL
+
γ
+
γ
eL
2
)
b
L
(
t
)
−
ig
0
L
α
0
L
α
+
L
e
−
iω
s
t
Θ(
t
)Θ(
τ
−
t
)+
√
γ
eL
b
in
,L
(
t
)
,
(S-1)
db
R
(
t
)
dt
=
−
(
iω
mR
+
γ
+
γ
eR
2
)
b
R
(
t
)+
√
γ
eR
b
in
,R
(
t
)
,
(S-2)
b
in
,L
(
t
)=
e
−
αl
(
√
γ
eR
b
R
(
t
−
t
w
)
−
b
in
,R
(
t
−
t
w
)
)
,
(S-3)
b
in
,R
(
t
)=
e
−
αl
(
√
γ
eL
b
L
(
t
−
t
w
)
−
b
in
,L
(
t
−
t
w
)
)
,
(S-4)
where
α
0
L
and
α
+
L
aretheamplitudesofopticalpump
anditsredsidebandintheleftcavity,
τ
isthedurationof
excitationpulse,
ω
s
isthefrequencyofpulse,Θ(
t
)isthe
Heavisidestepfunction,
γ
istheeffectivedecayrateof
theexcitedmechanicalmode,
α
≈
γ/v
g
isthewaveguide
lossrate,and
t
w
=1
/
(2
f
FSR
)
−
1
/
(
γ
eL
+
γ
)
−
1
/
(
γ
eR
+
γ
)
isthesingletriptimethepulsespentinthewaveguide.
Fromthemechanicalspectrumwefind
γ
=2
π
×
2
.
1
MHzfor
L
1
mode(themaincoherently-drivenmode)
duringthepulsemeasurement;andbyfittingthepulse
tailsdetectedineachcavitywefind
γ
eL
=2
π
×
34
.
7MHz
and
γ
eR
=2
π
×
25
.
5MHz.Usingtheseparameters,
|
b
L
|
and
|
b
R
|
canbenumericallycalculatedfromthecoupled
modeequationsandtheproportionalvoltagesignalsare
showninFig.3a.Thesimulatedresultcapturesthemain
featuresofthemeasuredpulsedata.Inparticular,the
pulsesplittingobservedfromcavity
R
isduetothefact
thatthepulsefrequencyisnotinresonantwithcavity
R
andthusexperiencesdestructiveinterferenceinsidethis
cavity.
Thephonontransferefficiencyfromcavity
L
tocavity
R
isabout
e
−
γ/
(2
f
FSR
)
≈
67%.Thephonontransferef-
ficiencyfromcavitytowaveguideforcavity
L
and
R
is
γ
eL
(
R
)
/
(
γ
eL
(
R
)
+
γ
)
≈
94%(92%).
Wesummarizethemeasuredmechanicalmodeparam-
etersofthedeviceinFig.3inTableI,where
g
0
forthe
Optical transduction and routing of microwave
phonons in cavity-optomechanical circuits
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k (π/a)
0
0.2
0.4
0.6
0.8
1
Mechanical Frequency (GHz)
0
1
2
3
4
5
6
7
8
Mirror cell #
2
4
6
8
10
Γ
X
0
50
100
150
200
250
Optical Frequency (THz)
γ /2π
(MHz)
e
k
0
30
25
20
15
10
5
b
c
d
Mirror cells
Mirror cells
a
Defect center
FIG.S-1:
a
SEMimageofatypicalnanobeamcavity.
b
Mechanicalbandstructureofthemirrorcell(bluecurves)and
mechanicalcavityfrequency(reddashedline).InsetisthemodeprofileofsecondbandatΓpoint.
c
Opticalbandstructureof
themirrorcell(bluecurves)andopticalcavityfrequency(reddashedline).Greyregionislightcone.Insetisthemodeprofile
offirstbandat
X
point.
d
Oscillationofmechanicalcavityandwaveguidecouplingwithvariationofnumberofmirrorcells.
TABLEI:Mechanicalmodeparameters
g
0
/
2
π
(MHz)
γ
i
/
2
π
(MHz)
γ
e
/
2
π
(MHz)
L
1
0.85
3.7
L
2
0.75
3.7
34.7
L
3
0.63
3.9
R
1
1.39
3.6
25.5
L
j
(
R
k
)modesiswithrespectiveto
O
L
(
R
)
opticalcavity
modes.
III.DERIVATIONOFTHEMICROWAVE
S
−
MATRIXFORTHEOPTOMECHANICAL
CAVITY-WAVEGUIDESYSTEM
Here,wederivethe
S
−
matrixforamicrowavesignal
traversingtheoptomechanicalcavity-waveguidesystem.
Weassumethemechanicalamplitudeissmallsuchthat
onlythefirst-orderopticalsidebandneedstobeconsid-
ered.Inthenextsection,wewillverifythesmallme-
chanicalamplitudeassumption.
TheHamiltonianofthesystemundercontinuouswave
operationinvolvestwoopticalcavitymodes
a
L,R
with
frequency
ω
cL,R
parametricallycoupledtoacommonme-
chanicalmode
b
,
ˆ
H
=
∑
k
=
L,R
ω
ck
ˆ
a
†
k
ˆ
a
k
+
ω
m
ˆ
b
†
ˆ
b
+
∑
k
=
L,R
g
0
k
(
ˆ
b
†
+
ˆ
b
)ˆ
a
†
k
ˆ
a
k
+
∑
k
=
L,R
i
√
κ
ek
α
pk
e
−
iω
pk
t
(ˆ
a
k
−
ˆ
a
†
k
)
,
(S-5)
wherethelasttermisthepumpsofthetwoopticalcav-
ities.Forsimplicityweassumethepumpinglasersare
blue-detunedfromthecavityresonanceswhichistrue
forallofourexperiments.Supposethepumpinglaser
forcavity
L
ismodulatedatfrequency
ω
byamicrowave
signal,thentheoperatorsofthesystemcanbedecom-
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posedintocarriersandsidebands,
ˆ
a
k
=
α
0
k
e
−
iω
pk
t
+
α
+
k
e
−
i
(
ω
pk
−
ω
)
t
,
ˆ
b
=
β
−
e
−
iωt
,
(S-6)
whereweonlykeeptheredsidebandofthepumping
lasersbecauseofrotatingwaveapproximation,giventhe
sidebandresolvedcondition
ω
m
κ
k
ofourdevice.Sup-
posethepumpsarestrongenoughsuchthatthecar-
rieroperatorscanbetreatedasstaticvariables,thenthe
equationsofmotionofthesystemcanbederivedafter
substitutingEq.S-6intoEq.S-5,
iωα
+
k
=(
i
∆
k
−
κ
k
2
)
α
+
k
−
ig
0
k
α
0
k
β
−
−
√
κ
ek
α
in
,
k
,
(S-7)
−
iωβ
−
=
−
(
iω
m
+
γ
i
2
)
β
−
−
∑
k
ig
0
k
α
0
k
α
+
k
,
(S-8)
where∆
k
=
ω
pk
−
ω
ck
≈
ω
m
.SolvingEq.S-7andEq.S-8
inthefrequencyrange
|
ω
−
ω
m
|
κ
k
,weobtain
β
−
=
ig
0
L
√
κ
eL
2
κ
L
α
0
L
i
(
ω
m
−
ω
)+
γ
i
2
−
∑
k
2
g
2
0
k
|
α
0
k
|
2
κ
k
α
in
,L
,
(S-9)
α
out
,R
=
−
√
κ
eR
α
+
R
(S-10)
=
−
4
g
0
L
g
0
R
√
κ
eL
κ
eR
/
(
κ
L
κ
R
)
α
0
L
α
0
R
i
(
ω
m
−
ω
)+
γ
i
2
−
∑
k
2
g
2
0
k
|
α
0
k
|
2
κ
k
α
in
,L
.
FromEq.S-10,peakopticalgainat
ω
=
ω
m
is
G
max
=
|
α
out
,R
|
2
|
α
in
,L
|
2
=
4
C
L
C
R
(1
−
C
L
−
C
R
)
2
,
(S-11)
where
C
L
(
R
)
=
|
γ
OM
,L
(
R
)
|
/γ
i
isthecoorperativityofme-
chanicalmode
b
withopticalmodes
a
L
(
R
)
.
UsingtheresultofEq.S-10,themicrowavesignal
transfer
S
−
matrixcanbederived
S
RL
(S-12)
≡
V
NA
,
in
V
NA
,
out
=
η
oL
η
oR
G
e
G
EDFA
(
i
ω
cR
ω
m
/
√
κ
eR
)
α
out
,R
α
0
R
(2
V
π
/π
)
(
α
in
,L
/
(
iω
m
α
0
L
/
√
κ
eL
)
)
=
η
oL
η
oR
G
e
G
EDFA
2
V
π
/π
4
g
0
L
g
0
R
ω
cR
ω
2
m
/
(
κ
L
κ
R
)
|
α
0
L
|
2
|
α
0
R
|
2
i
(
ω
m
−
ω
)+
γ
i
2
−
∑
k
2
g
2
0
k
|
α
0
k
|
2
κ
k
where
V
NA
,
out
and
V
NA
,
in
aretheoutputanddetected
electricalvoltageofthenetworkanalyzerrespectively,
η
oL,R
istheopticallossoftheinputandoutputportsof
thedeviceandfiberrespectively,
G
EDFA
and
G
e
arethe
gaincoefficientsofEDFAandphotodetectorrespectively,
and
V
π
isthevoltagerequiredtoproduceaphaseshift
of
π
oftheelectro-opticmodulator.
IV.ANALYSISOFLINEAROPERATIONAND
NOISECHARACTERISTICSOFTHE
OPTOMECHANICALMICROWAVE
FILTER/DELAYLINE
Inthissection,weexaminetheassumptionofweak
mechanicalamplitudeunderstrongopticalpumpandan-
alyzetheoptomechanicalmicrowavefilter/delaylineper-
formanceintermsoflinearityandnoisecharacteristics.
Wefindthatthermo-opticeffectconstrainsthemechan-
icalamplitudeduetosaturationoftheoptomechanical
gain.Thiseffectsetsthelinearoperationrangeandthe
suppressionofthemechanicalthermalnoise.
Thethermo-opticeffectinducedopticalcavityfre-
quencyshiftcanbedescribedbythefollowingequa-
tions[1]
δω
c
=
−
ω
c
n
Si
(
T
0
)
dn
Si
(
T
0
)
dT
AδT,
(S-13)
δT
=
rςc
2
n
Si
(
T
0
)
2
V
TPA
n
2
c
,
(S-14)
where
n
c
iscavityphotonnumber,
n
Si
,
r
,and
ς
isthe
refractiveindex,thermalresistance,andtwo-photonab-
sorptioncoefficientofsiliconrespectively,
c
isthespeed
oflight,
V
TPA
isthecavityvolumefortwo-photonab-
sorption,and
A
isaperturbationtheorycoefficient
A
=
∫
Si
|
E
(
r
)
|
2
d
r
∫
n
Si
(
T
0
)
2
|
E
(
r
)
|
2
d
r
.SubstitutingEq.S-14intoEq.S-13,
andusingtheparametersofsilicongiveninRef.[1],along
with
A
≈
7
.
5
×
10
−
2
,
V
TPA
≈
(
λ/n
Si
(
T
0
))
3
,wehave
δω
c
=
ξn
2
c
,ξ
≈−
33
.
9Hz
.
(S-15)
Weproceedtoincludethetermofthermo-opticeffect
(Eq.S-15)totheequationsofmotion(Eqs.S-7andS-8),
whicharethenmodifiedtobe
iω
̄
α
+
=
(
i
(
∆+
ξ
(
|
α
0
|
2
+
|
̄
α
+
|
2
)
2
)
−
κ
2
)
̄
α
+
−
ig
0
α
0
̄
β
−
,
(S-16)
−
iω
̄
β
−
=
−
(
iω
m
+
γ
i
2
)
̄
β
−
−
ig
0
α
0
̄
α
+
−
√
γ
i
β
in
,β
in
=
√
γ
i
n
th
/
2
,
(S-17)
wherewehavedenoted ̄
α
+
and
̄
β
−
asthestaticvalueofthecorrespondingoperatorswithoutinputopticalside-
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bandsignalandwehaveincludedthemechanicalther-
malnoiseinput.Also,wespecificallyconsidertheoper-
ationwiththe
R
1
mode,andthusignoreopticalcavity
L
whichhasmuchweakercouplingwith
R
1
compared
toopticalcavity
R
.Eqs.S-16andS-17canonlybe
solvednumericallyforagenericpumpcondition.Tore-
vealthethermo-opticeffectonthemechanicalamplitude,
weconsideraspecialpumpconditioncorrespondingto
theoriginalthresholdofmechanicalself-oscillation,i.e.
4
g
2
0
|
α
0
|
2
/
(
κγ
i
)=1.Inthiscase,wecananalyticallysolve
forthedown-convertedphotonnumberandmechanical
amplitudeat
ω
=
ω
m
fromEqs.S-16andS-17,assuming
∆=
ω
m
and
|
̄
α
+
||
α
0
|
,
|
̄
α
+
|
2
=(
κγ
i
n
th
4
ξ
2
)
1
/
5
,
(S-18)
|
̄
β
−
|
2
=
κ
γ
i
(
κγ
i
n
th
4
ξ
2
)
1
/
5
+
n
th
.
(S-19)
For
κ
=2
π
×
0
.
8GHz,
γ
i
=2
π
×
3
.
6MHz,
g
0
=2
π
×
1
.
39
MHz,
n
th
=
k
B
T
0
ω
m
≈
1000,wehave
|
̄
α
+
|
2
≈
1900,
|
α
0
|
2
≈
380,and
|
̄
β
−
|
2
≈
3
.
7
×
10
5
.Itistheoptical
resonanceshiftinducedbythermo-opticeffectthatsat-
uratesoptomechanicalgainandpreventsrunawayofthe
mechanicalamplitudeatthethreshold.
Nowwecanestimatewhetherthemechanicalampli-
tudeislargeenoughtoinducenonlinearitythroughexci-
tationofhigherorderopticalsidebands.Thenonlinear-
ityarisesduetopumpsaturationandoccurswhenthe
amplitudeofthefirstorderopticalsidebandsignificantly
deviatesfrombeinglinearlyproportionaltothemechan-
icalamplitude,i.e.approximation
J
1
(
z
)
≈
z
2
breaks
down[2],where
z
=
g
0
√
4
|
̄
β
−
|
2
+2
/ω
m
isthenormalized
mechanicalamplitude.For
|
̄
β
−
|
2
≈
3
.
7
×
10
5
calculated
above,thedeviationisonlyabout1%.Intheexperi-
ment,wefindforthelargestpumppower
P
pL,R
≈
0
.
2
mW,
|
̄
β
−
|
2
≈
2
.
0
×
10
5
,whichgives
z
=0
.
18andalin-
eardeviationof0
.
4%.Asaresult,scatteringintohigher
orderopticalsidebandsdoesnotneedtobeincludedin
Eq.(S-16)and(S-17).
Next,weconsidertheresponseofthemechanicaloscil-
latortoasmallinputopticalsidebandsignalbypertur-
bativeexpansionofEqs.S-16andS-17.Inthiscasethe
coherentmechanicalamplitudeofEq.S-9ismodifiedto
be
β
−
=
ig
0
L
√
κ
eL
2
κ
L
α
0
L
i
(
ω
m
−
ω
)+
γ
i
2
−
∑
k
2
g
2
0
k
|
α
0
k
|
2
−
2
iδ
k
+
κ
k
α
in
,L
,
(S-20)
where
δ
k
=
ξ
(
|
α
0
k
|
2
+
|
̄
α
+
k
|
2
)
2
isthethermo-optic-effect
inducedopticalfrequencyshift.Eqs.S-10andS-12can
bemodifiedcorrespondingly.AccordingtoEq.S-20,the
effectivemechanicallossrateis
γ
eff
=
γ
i
−
∑
k
4
g
2
0
k
|
α
0
k
|
2
κ
k
+
∑
k
4
g
2
0
k
|
α
0
k
|
2
κ
k
(
δ
k
κ
k
/
2
)
2
.Thedeviationfromalinearre-
sponsecanbecausedbytheadditionalcavityphotons
fromtheinputsignal,andischaracterizedbythera-
tio
r
=
|
α
+
k
|
2
/
|
̄
α
+
k
|
2
(inourdevicethecontribution
Frequency oset (kHz)
48
12
16
Phase noise suppression (dB)
0
5
10
15
20
FIG.S-2:Phasenoisesuppressionratiobetweenmicrowave
signalpowerof-15dBmand-30dBmatlargestopticalpump
level.SolidlineistheoreticalvaluecalculatedusingEq.S-23
anddotsareexperimentaldata.
ismainlyfrom
α
+
R
).Atthetheoreticalself-oscillation
threshold,wefindthe1dBcompressionpointofthe
S
−
matrixtobeequaltoamicrowavepowerof-19dBm
(assuming
|
α
0
L
|
2
=
|
α
0
R
|
2
).Intheexperiment,wefind
forthelargestopticalpumppower
P
pL,R
=0
.
2mW
(whichisslightlyabovetheself-oscillationthreshold),
the1dBcompressionpointoccursatamicrowavesignal
powerof-15dBm.Forreversedoperation(cavity
R
as
input),the1dBcompressionpointisreducedbyafactor
of
γ
OM
,L
/γ
OM
,R
(assuming
|
α
0
L
|
2
=
|
α
0
R
|
2
).
Wenowanalyzethenoisecharacteristicsoftheop-
tomechanicalcavity-waveguidesystem.Thedominant
formofnoiseisfromthermallyexcitedphononsinthe
system.Fromthethermally-addedmechanicalnoiseref-
eredtotheinputsignal(
κ
e
/κ
)
−
1
γ
i
n
th
/
|
γ
OM
|
[3],wede-
finethenoise-equivalentopticalsignalpower
P
NE
=
κ
κ
e
ω
c
γ
i
n
b
γ
OM
2
πB,
(S-21)
where
B
isthebandwidthofthecoherentsignalandall
thequantitiesarereferredtotheinputcavity.Thenthe
noise-equivalentmicrowavesignalpoweris
V
2
NE
=
4
π
2
P
NE
P
p
V
2
π
.
(S-22)
Wefindthatforthelargestpumppowerwhenoperating
at
R
1
resonance,iftheinputportiscavity
L
,thenoise
equivalentmicrowavepoweris-30dBm;iftheinputport
iscavity
R
,thenoiseequivalentmicrowavepowerreduces
to-70dBmbecauseofthesignificantlyenhanced
γ
OM
of
cavity
R
with
R
1
mode.
Foraself-oscillatingmechanicaloscillator,theintrinsic
oscillatornoisecanbesuppressedbytheinjectionofan
externalcoherentsignal[4].Thesuppressedphasenoise
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