of 10
SUPPLEMENTARY INFORMATION
1
www.nature.com/nature
doi: 10.1038/nature08061
Supplementary Information for “A picogram and nanometer scale photonic crystal
opto-mechanical cavity”
Matt Eichenfield,
1
Ryan Camacho,
1
Jasper Chan,
1
Kerry J. Vahala,
1
and Oskar Painter
1,
1
Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, CA 91125
(Dated: March 29, 2009)
These set of notes describe cavity optomechanics in the presence of additional thermo-optic tuning of the
cavity resonance. We find that thermo-optic tuning results in correction factors to both the optical spring and
optomechanical gain. In addition there is an overall saturation of the optomechanical coupling. These effects
can be large for systems with large static thermo-optic tuning and fast thermal decay relative to the mechan-
ical frequency (i.e., small heat capacity). Analysis of the zipper optomechanical cavity indicates that optical
damping can be realized with blue detuned light, in direct opposition to the bare optomechanical effect. Several,
supporting measurements/calculations are also presented, including estimates of the quantum back-action noise
and thermo-mechanical effects, both found to be negligible on the scale of the measured properties of the zipper
cavity system, and measurement of the zipper cavity mechanical
Q
-factor in vacuum (
Q
M
,
vac
>
10
4
). Finally,
methods and parameters used in fitting a steady-state nonlinear optical model, including the gradient optical
force and thermo-optic tuning, to the measured zipper optomechanical cavity response are provided at the end
of the notes.
PACS numbers:
I. THERMO-OPTIC EFFECTS
A. Coupled mode theory
We begin with the set of coupled equations describing mechanical and optical motion,
̇
a
=
(
i
(
o
g
OM
x
)+
Γ
/
2
)
a
+
κ
s
,
(S-1)
̈
x
=
γ
M
̇
x
2
M
x
|
a
|
2
g
OM
ω
o
m
x
,
(S-2)
where
o
(
ω
l
ω
o
)
is the bare laser detuning from the optical cavity resonance (
ω
o
),
Γ
is the optical cavity (energy) decay
rate,
κ
(=
1
/
τ
e
)
is the input coupling rate of the laser into the cavity,
|
s
|
2
is the optical input power,
g
OM
d
ω
/
d
x
is the
optomechanical factor,
γ
M
is the bare mechanical (energy) damping factor,
M
(=
k
/
m
x
)
is the bare mechanical frequency,
m
x
is the bare motional mass of the mechanical resonator, and
a
is the amplitude of the optical cavity field normalized so that
|
a
|
2
represents the stored optical cavity energy. The equation for
a
is written in a slowly varying basis in which the laser frequency,
ω
l
, has been removed from both
a
and
s
.
In order to include the effects of thermo-optic tuning of the cavity resonance, we include a third equation for the cavity
temperature increase,
T
:
̇
a
=
(
i
(
o
(
g
OM
x
+
g
th
T
))+
Γ
/
2
)
a
+
κ
s
,
(S-3)
̈
x
=
γ
M
̇
x
2
M
x
|
a
|
2
g
OM
ω
o
m
x
,
(S-4)
̇
T
=
γ
th
T
+
Γ
abs
|
a
|
2
c
th
,
(S-5)
where
g
th
=
(
d
n
/
d
T
)(
ω
o
/
n
o
)
is the thermo-optic tuning coefficient, d
n
/
d
T
is the thermo-optic coefficient of the optical and
mechanical cavity material,
Γ
abs
is the component of the optical energy decay which is due to material absorption,
c
th
is the
thermal heat capacity of the cavity, and
γ
th
is temperature decay rate.
In order to solve these coupled equations we proceed using a perturbation approach
1
. We assume that the mechanical motion
is harmonic in time with small amplitude parameter
ε
x
,
x
(
t
)=
x
o
+
ε
x
cos
(
M
t
)
. The optical cavity mode amplitude and the
cavity temperature increase can be expanded in terms of the small parameter
ε
x
,
2
www.nature.com/nature
doi: 10.1038/nature08061
SUPPLEMENTARY INFORMATION
2
a
(
x
,
t
)=
n
=
0
ε
n
x
a
n
(
x
,
t
)
,
(S-6)
T
(
x
,
t
)=
n
=
0
ε
n
x
T
n
(
x
,
t
)
.
(S-7)
(S-8)
Keeping terms only to first order in
ε
x
yields the following sets of coupled equations,
0
=
(
i
o
+
Γ
/
2
)
a
0
+
κ
s
,
(S-9)
0
=
2
M
x
o
|
a
0
|
2
g
OM
ω
o
m
x
,
(S-10)
0
=
γ
th
T
o
+
Γ
abs
|
a
0
|
2
c
th
,
(S-11)
and
̇
a
1
=+
i
(
g
OM
x
1
+
g
th
T
1
))
a
0
(
i
o
+
Γ
/
2
)
a
1
,
(S-12)
̈
x
1
=
γ
M
̇
x
1
2
M
x
1
(
a
0
a
1
+
a
0
a
1
)
g
OM
ω
o
m
x
,
(S-13)
̇
T
1
=
γ
th
T
1
+
Γ
abs
(
a
0
a
1
+
a
0
a
1
)
c
th
,
(S-14)
where
x
1
cos
(
M
t
)
and
o
=
o
(
g
OM
x
o
+
g
th
T
o
)
is the time averaged laser-cavity detuning. Fourier transforming the
first-order perturbation equations to convert them from differential to algebraic ones yields,
(
i
(
ω
+
o
)+
Γ
/
2
)
̃
a
1
=+
i
(
g
OM
̃
x
1
+
g
th
̃
T
1
)
a
0
,
(S-15)
(
i
(
ω
o
)+
Γ
/
2
)
̃
a
1
=
i
(
g
OM
̃
x
1
+
g
th
̃
T
1
)
a
0
,
(S-16)
ω
2
̃
x
1
=
i
ωγ
M
̃
x
1
2
M
̃
x
1
(
a
0
̃
a
1
+
a
0
̃
a
1
)
g
OM
ω
o
m
x
,
(S-17)
(
i
ω
+
γ
th
)
̃
T
1
=
Γ
abs
(
a
0
̃
a
1
+
a
0
̃
a
1
)
c
th
,
(S-18)
Solving for the time-dependent part of the optical cavity energy,
(
a
0
̃
a
1
+
a
0
̃
a
1
)=
f
(
ω
,
o
)
i
|
a
0
|
2
(
g
OM
̃
x
1
+
g
th
̃
T
1
)
,
(S-19)
where we have defined the transfer function
f
as,
f
(
ω
,
o
)=
1
(
i
(
ω
+
o
)+
Γ
/
2
)
1
(
i
(
ω
o
)+
Γ
/
2
)
.
(S-20)
Substituting for
̃
T
1
of eq. (S-18) allows us to solve for the optical cavity energy solely in terms of the mechanical motion,
f
(
ω
,
o
)
1
i
g
th
Γ
abs
c
th
|
a
0
|
2
i
ω
+
γ
th
(
a
0
̃
a
1
+
a
0
̃
a
1
)=
i
|
a
0
|
2
g
OM
̃
x
1
.
(S-21)
Defining
f
(
ω
,
|
a
0
|
2
)
and
g
(
ω
,
o
,
|
a
0
|
2
)
as,
f
(
ω
,
|
a
0
|
2
)=
i
g
th
Γ
abs
c
th
|
a
0
|
2
i
ω
+
γ
th
,
(S-22)
g
(
ω
,
o
,
|
a
0
|
2
)=
f
1
+(
f
)
f
|
1
+
f
f
|
2
,
(S-23)
3
www.nature.com/nature
SUPPLEMENTARY INFORMATION
doi: 10.1038/nature08061
3
allows us to write for the Fourier transform of the time varying component of the cavity energy,
(
a
0
̃
a
1
+
a
0
̃
a
1
)=
ig
(
ω
,
o
,
|
a
0
|
2
)
|
a
0
|
2
g
OM
̃
x
1
.
(S-24)
All of the transfer functions
f
,
f
, and
g
have the property that
h
(
ω
)=
h
(
ω
)
. With ̃
x
1
=(
δ
(
ω
M
)+
δ
(
ω
M
)
/
2, we
have for the cavity energy,
(
a
0
̃
a
1
+
a
0
̃
a
1
)=
i
|
a
0
|
2
g
OM
(
g
(
M
)
δ
(
ω
M
)+
g
(
M
)
δ
(
ω
+
M
))
/
2
(S-25)
=
|
a
0
|
2
g
OM
(
g
(
M
)
δ
(
ω
M
)
g
(
M
)
δ
(
ω
+
M
))
2
i
.
(S-26)
Further symplifying this result yields,
(
a
0
̃
a
1
+
a
0
̃
a
1
)=
−|
a
0
|
2
g
OM
Re
(
g
(
M
))(
δ
(
ω
M
)
δ
(
ω
+
M
))
2
i
+
Im
(
g
(
M
))(
δ
(
ω
M
)+
δ
(
ω
+
M
))
2
.
(S-27)
Finally this gives in the time-domain,
(
a
0
a
1
(
t
)+
a
0
a
1
(
t
))=
|
a
0
|
2
g
OM
Re
(
g
(
M
))
M
(
M
sin
M
t
)
Im
(
g
(
M
))
cos
M
t
,
(S-28)
=
|
a
0
|
2
g
OM
Re
(
g
(
M
))
M
̇
x
1
Im
(
g
(
M
))
x
1
(S-29)
Substituting this result into the equation of motion for
x
1
(
t
)
in eq. (S-13) allows one to identify renormalized mechanical
frequency (
M
) and damping (
γ
M
) terms due to optomechanical and thermo-optic interactions,
(
M
)
2
=
2
M
|
a
0
|
2
g
2
OM
Im
(
g
(
M
))
ω
o
m
x
,
(S-30)
γ
M
=
γ
M
+
|
a
0
|
2
g
2
OM
Re
(
g
(
M
))
M
ω
o
m
x
.
(S-31)
The effects of the thermo-optic tuning of the cavity are manifest in the correction to the pure optomechanical transfer function
(
f
) in the equation for
g
given in eq. (S-23). This correction factor is simply 1
/
(
1
+
f
f
)
. For
|
f
f
|
1 the thermo-optic
correction is small, and can be neglected. In order to make connection with previously derived results for the optomechanical
spring and gain coefficient, we now consider this correction in the sideband unresolved limit, relevant for the current zipper
cavities.
B. Sideband unresolved limit (
M
Γ
)
We begin by evaluating
f
(
M
)
in the limit that
M
Γ
,
f
(
M
)
≈−
2
o
ΓΩ
M
+
i
2
4
,
(S-32)
where we have defined
2
=(
o
)
2
+(
Γ
/
2
)
2
. In the absence of thermo-optic tuning this results in the usual equations for the
sideband unresolved optical spring effect and optomechanical gain,
(
M
)
2
|
T
1
=
0
=
2
M
+
2
|
a
0
|
2
g
2
OM
2
ω
o
m
x
o
,
(S-33)
γ
M
|
T
1
=
0
=
γ
M
2
|
a
0
|
2
g
2
OM
Γ
4
ω
o
m
x
o
.
(S-34)
4
www.nature.com/nature
doi: 10.1038/nature08061
SUPPLEMENTARY INFORMATION
4
As can be seen, this results in an increase in the mechanical frequency and negative damping (positive
gain
) of the mechanical
motion for blue detuned laser light (relative to the steady-state cavity resonance frequency).
We now consider the thermo-optic response in the case where the mechanical frequency is much larger than the thermal decay
rate, a situation commonly found in optomechanical microsystems. We begin with
f
,
f
(
M
)=(
M
γ
th
i
γ
2
th
)
th
2
M
+
γ
2
th
(
M
γ
th
i
γ
2
th
)
th
2
M
,
(S-35)
where we have assumed that the mechanical frequency is much larger than the thermal decay rate (
M
γ
th
) and we have
associated
g
th
Γ
abs
c
th
|
a
0
|
2
/
γ
th
with the static thermo-optic tuning of the cavity resonance,
th
. In order to evaluate
g
, we need
the unresolved sideband limit of
|
f
|
2
,
|
f
|
2
, and 2Re
(
f
f
)
,
|
f
(
M
)
|
2
4
(
o
)
2
4
,
(S-36)
|
f
(
M
)
|
2
γ
th
th
M
2
,
(S-37)
2Re
(
f
(
M
)
f
(
M
))
4
o
th
γ
th
(
2
γ
th
2
M
Γ
)
2
M
4
.
(S-38)
This yields for the transfer function
g
(
M
)
in the sideband unresolved limit and for
M
γ
th
,
g
(
M
)
1
1
+
s
(
|
a
o
|
2
)

f
(
M
)+
4
(
f
(
M
))
(
o
)
2
4
,
(S-39)
where we have defined a saturation parameter,
s
, which is equal to,
s
2
o
γ
th
th
(
|
a
0
|
2
)
2
M
2
1
+
th
(
|
a
0
|
2
)
1
2
o
2
M
Γ
o
γ
th

.
(S-40)
Under most situations in which the thermo-optic correction to the bare optomechanics is significant, the static thermo-optic
tuning of the cavity resonance dominates all other rates and only the first term contributes to
s
,
s
2
o
γ
th
th
(
|
a
0
|
2
)
2
M
2
.
(S-41)
One can usefully relate the thermo-optic correction factor in eq. (S-39) to that of the bare optomechanical factor
f
as,
4
(
f
(
M
))
(
o
)
2
4
Re
(
f
(
M
))
2
th
o
γ
th
2
M
Γ
+
i
Im
(
f
(
M
))
2
th
o
γ
2
th
2
2
M
.
(S-42)
Substituting eqs. (S-35,S-39) into eqs. (S-30,S-31) yields the following thermo-optic corrections to the optical spring and
optomechanical gain coefficients in the sideband unresolved limit and for slow thermal response,
(
M
)
2
2
M
+
2
|
a
0
|
2
g
2
OM
o
2
ω
o
m
x

1
+
W
1
+
s
,
(S-43)
γ
M
γ
M
|
a
0
|
2
g
2
OM
Γ∆
o
4
ω
o
m
x

1
+
V
1
+
s
,
(S-44)
where the correction factors are,
W
=
2
th
o
γ
2
th
2
2
M
=
2
th
Γ

γ
th
M
2
Γ∆
o
2
,
(S-45)
V
=
2
th
o
γ
th
2
M
Γ
=
2
th
Γ

γ
th
M
2
o
γ
th
.
(S-46)
5
www.nature.com/nature
SUPPLEMENTARY INFORMATION
doi: 10.1038/nature08061
5
It should be noted that both
W
and
V
are dependent upon the (time) average stored cavity energy through the static thermo-optic
tuning,
th
. It is also noteworthy that since the thermo-optic tuning is negative for most cavity materials (heat generates a red
shift of the cavity resonance),
W
will be a positive quantity and
V
a negative one for blue detuned laser input (
o
>
0). In this
way the thermo-optic correction tends to increase the bare optical spring effect and reduce the bare optomechanical gain when
one tunes to the blue side of the cavity resonance. This negative correction to the optomechanical gain can then result in an
effective mechanical damping on the stable blue-detuned side of the cavity resonance if
|
V
|
>
1, a case study of which will be
explored below. The situation is reversed for a red detuned laser input, with the optical spring effect tending to be reduced and
the optomechanical damping being enhanced.
Before proceeding to study specific examples, it is useful to estimate the correction factors and the saturation parameter for
detunings close to the maximal bare optomechanical response,
|
o
|≈
Γ
/
2. Substituting this detuning into eqs. (S-41,S-45,S-46)
yields,
|
s
(
|
o
|
=
Γ
/
2
)
|≈
2
th
Γ
2
γ
th
M
2
,
(S-47)
|
W
(
|
o
|
=
Γ
/
2
)
|≈
2
|
th
|
Γ

γ
th
M
2
,
(S-48)
|
V
(
|
o
|
=
Γ
/
2
)
|≈
2
|
th
|
Γ

γ
th
M
2
Γ
2
γ
th
.
(S-49)
The correction factor to the optomechanical gain (damping) is seen to be
Γ
/
2
γ
th
times larger than that of the correction to the
optical spring effect. For optomechanical systems of micron scale and high optical
Q
,
Γ
/
2
π
10 MHz and
γ
th
/
2
π
10 kHz are
reasonable numbers, which means the gain correction is on the order of a thousand times larger than the spring correction. For
more modest optical
Q
systems (
Q
10
5
), the gain correction is a million times larger than the spring correction. The saturation
parameter scales similarly to the optical spring correction factor, with an extra factor of 2
th
/
Γ
. Thus, for static thermo-optic
tuning greater than the cavity linewidth (thermo-optic bistability) the optical spring correction due to thermo-optic tuning
always
serves to quench the bare optomechanical effect. The optomechanical gain (damping), however, can be enhanced over a useful
parameter regime. We now proceed to analyze the thermo-optic effects on the properties of the
zipper
optomechanical cavity.
C. The
zipper
optomechanical cavity
The zipper cavity studied in the manuscript has an optical
Q
-factor on the order of
Q
3
×
10
4
(
Γ
/
2
π
6 GHz or roughly
a
δλ
50 pm linewidth), a mechanical frequency
M
/
2
π
10 MHz, and a thermal decay rate of roughly
γ
th
/
2
π
8 kHz (see
below). These devices have significant optical absorption at
λ
1550 nm, resulting in a static thermo-optic tuning of roughly
∆λ
th
4 nm (100 cavity linewidths) for a time-averaged stored cavity energy of 3 fJ (
P
i
5 mW). The correction and saturation
parameters for the zipper cavity under this sort of optical input power and at the “optimal” detuning are,
|
s
(
|
o
|
=
Γ
/
2
)
|≈
2
×
10
3
,
(S-50)
|
W
(
|
o
|
=
Γ
/
2
)
|≈
8
×
10
6
,
(S-51)
|
V
(
|
o
|
=
Γ
/
2
)
|≈
10
.
(S-52)
We see that, because of the large thermo-optic tuning and reasonably fast thermal response (a result of the small heat capacity),
for the zipper cavity the optomechanical gain reverses sign at high enough optical input power for blue detuned pumping,
resulting in strong optomechanical damping of the mechanical motion. This is what we see in our measurements. The optical
spring is left unaffected and the overall saturation of the optomechanical coupling is negligible.
II. QUANTUM BACK-ACTION NOISE
Quantum back-action on the mechanical oscillator, due to the quantum fluctuations of the internal optical cavity field, results
in an effective standard quantum limit (SQL) to which the mechanical oscillator’s position can be determined
2
. Most analyses
concerning quantum back-action noise in cavity-optomechanical systems are specific to the scattering radiation pressure force
in which the optical cavity length is intimately related to the optomechanical coupling factor. This results in relations that
depend upon cavity Finesse instead of cavity
Q
. Cavity-optomechanical systems that utilize gradient optical forces have an
optomechanical coupling that scales with the inverse of a length,
L
OM
, related to the transverse geometry of the cavity. As
6
www.nature.com/nature
doi: 10.1038/nature08061
SUPPLEMENTARY INFORMATION
6
such, all optomechanical relations end up more naturally being couched in terms of cavity
Q
and
L
OM
(or
g
OM
). In the case of
quantum back-action noise, for “measurement” times much longer than the mechanical decay time and in the bad cavity limit,
the displacement noise power spectral density due to radiation force fluctuations within the cavity can be written as (following
an analysis similar to Ref. [2]),
S
BA
x
/
2
π
;
o
=
Γ
2
3
=
6
g
2
OM
KQ
2
ω
3
o
χ
2
M
(
)
P
d
,
(S-53)
where
Q
ω
o
/
Γ
is the
loaded
optical cavity
Q
-factor,
K
(
0
.
1 for our measurements) is a coupling parameter describing the
strength of the cavity loading by the fiber taper waveguide
3
, and the mechanical susceptibility is given by,
χ
2
M
(
)=(
m
2
x
(
2
2
M
)
2
+
m
2
x
2
2
M
/
Q
2
M
)
1
. In deriving this relation we have assumed the laser-to-cavity detuning is chosen at the optimal point
(in terms of transducing mechanical motion via detection of the transmitted laser power) of
o
=
Γ
/
2
3. At this detuning the
dropped optical power into the cavity is
P
d
=(
3
/
4
)(
4
K
/
(
1
+
K
)
2
)
P
i
, where
P
i
is the input optical power to the cavity. The
thermal displacement noise power spectral density is given by the well known one-sided spectral density,
S
th
x
(
/
2
π
)=
4
k
B
Tm
x
M
Q
M
χ
2
M
(
)
.
(S-54)
The ratio
S
BA
x
(
/
2
π
)
/
S
th
x
(
/
2
π
)
defines the relevance of the quantum back-action noise in the presence of thermally-induced
Brownian motion, which for our measurement geometry (under optimal detuning) is given as,
S
BA
x
/
2
π
;
o
=
Γ
2
3
S
th
x
(
/
2
π
)
=
6
g
2
OM
KQ
2
Q
M
4
k
B
Tm
x
M
ω
3
o
P
d
.
(S-55)
Evaluating this ratio for the zipper cavity optomechanical system studied in this work, we find
S
BA
/
S
therm
0
.
03 per Watt of
dropped optical cavity power at a bath temperature of
T
=
300 K. The largest optical power used in the experiments described in
this work is
P
i
=
5 mW, roughly four orders of magnitude below that required to produce significant quantum back-action noise
(on the scale of the thermal noise) for the zipper cavity at room temperature.
In addition to quantum back-action and thermal noise, the ultimate displacement sensitivity is also limited by the optical noise
and electrical noise involved with the measurement of the optical signal. For the direct photodetection of the transmitted optical
intensity of the zipper cavity used in this work, one can show that the
resonant
effective displacement noise power spectral
densities for shot-noise-limited (
S
SN
; not performed in our work) and photoreceiver-noise-limited detection (
S
PD
; the NEP of
our detector is 2
.
5 pW/Hz
1
/
2
at the frequency of the
h
1
d
mode) are:
S
SN
x
M
/
2
π
;
o
=
Γ
2
3
=
2
ω
3
o
(
1
+
K
)
2
3
K
1
3
η
g
2
OM
Q
2
P
1
d
,
(S-56)
S
PD
x
M
/
2
π
;
o
=
Γ
2
3
=
2
ω
2
o
3
g
2
OM
Q
2

NEP
P
d
2
,
(S-57)
(S-58)
where
η
(
=
0
.
67 for our set-up) is the overall detection efficiency of the transmitted optical power, and again an optimal detuning
point of
o
=
Γ
/
2
3 is assumed in transducing the mechanical motion. Minimizing the sum of
S
SN
x
and
S
BA
x
with respect to
dropped optical power (assuming the two noise sources are independent, which they are strictly not in our measurement scheme),
one finds (to within a factor of 4
/
3) the
resonant
SQL for the displacement noise power spectral density,
S
SQL
(
M
/
2
π
)=
2
Q
M
/
m
x
2
M
. A plot of the calculated different components of displacement sensitivity noise, on-resonance with the
h
1
d
mechanical mode, are given in Fig. S-1 versus dropped optical cavity power for our system. The three differenent dropped cavity
power levels used in the measurements presented in the main text are shown as black, vertical dashed lines (corresponding to 3
.
8,
38, and 1500
μ
W). The cyan horizontal dashed line is the resonant SQL. One can clearly see from this plot that the thermal noise
swamps all other noise sources for the range of our measurements. In particular, at the power level used in the measurements
displayed in Fig. 3 of the main text (
P
i
=
12
.
7
μ
W,
P
d
=
3
.
8
μ
W), the displacement noise sensitivity for our photoreceiver
(5
×
10
17
m/Hz
1
/
2
) is within a factor of four of the SQL (quantum-back action noise is below the SQL at this power level).
7
www.nature.com/nature
SUPPLEMENTARY INFORMATION
doi: 10.1038/nature08061
7
10
-11
10
-12
x
h1
displacement sensitivity (m/Hz
1/2
)
10
-17
10
-13
10
-14
10
-15
10
-16
dropped optical power (W)
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
SN+BA
PD+BA
SN+BA+th
PD+BA+th
SQL
FIG. S-1:
Theoretical plot of the on-resonance displacement noise components for the
h
1
d
mechanical mode
. The various component
labels are: SN=shot noise, PD=photoreceiver noise, th=thermal noise, BA=quantum back-action. The parameters for the detection scheme are
η
=
0
.
67, NEP
=
2
.
5 pW/Hz
1
/
2
,
o
=
Γ
/
2
3, and
K
=
0
.
09. The parameters used for the
h
1
d
mode are
M
/
2
π
=
7
.
9 MHz,
Q
M
=
80, and
g
OM
/
2
π
=
123 GHz/nm. The zipper optical cavity properties are
Q
=
28
,
000,
Q
i
=
30
,
000, and
λ
o
=
1543 nm.
III. THERMO-MECHANICAL EFFECTS
It is also important to consider thermo-mechanical effects (i.e., direct mechanical actuation stemming from thermal effects
such as the pressure rise in the gas between the zipper nanobeams or thermal expansion of the nanobeams and surrounding
supports)
4
. Thermo-mechanical effects can not only produce a temperature dependent shift in the cavity resonance frequency
as described above in the case of the thermo-optic effect, but in addition they can directly produce a force on the nanobeams.
On first blush, one might expect that thermo-mechanical effects are responsible for the blue-detuned damping measured in the
zipper cavities (for instance, the sign of a thermo-mechanical force due to the pressure rise in the gas between the nanobeams
would be opposite that of the optical force). However, even an overly optimistic estimate of the magnitude of thermo-mechanical
effects indicates that this is not the case.
The steady-state temperature rise inside the cavity at the largest optical input powers used in this work (
P
i
=
5 mW) is
roughly
T
0
=
60 K (estimated from the measured thermal tuning rate of the cavity as described below). The optical energy
inside the cavity is being modulated by roughly
β
=
15% of the time-averaged internal cavity energy due to thermal motion
of the nanobeams. The component of the zipper cavity temperature oscillating in-phase with the optical cavity energy at the
mechanical frequency (
M

10 MHz) is roughly
T
q

(
γ
th
/
M
)
2
β∆
T
0
, whereas the in-quadrature component of the zipper
cavity temperature is
T
p

(
γ
th
/
M
)
β∆
T
0
. This assumes of course that
γ
th

M
, as is the case for the zipper cavity. Using
some of the numbers estimated below, we find
γ
th
/
M

10
3
, so that the in-phase and in-quadrature modulations in the cavity
temperature are at most
T
q

10
5
K and
T
p

10
2
K, respectively.
We first consider a thermo-mechanical force from the thermal expansion in the nanobeams. The resulting in-plane displace-
ment (which couples to the optical field) is difficult to simply estimate as it sensitivitly depends upon the beam clamping. We
have performed finite-element-method (FEM) simulations of our stuctures, with an accurate representation of our clamping ge-
ometry, and find that the resulting in-plane displacement is
δ
x
=
80 pm for
T
0
=
60 K at the center of the zipper cavity. From
the above estimated in-phase and in-quadrature temperature oscillations for this static temperature shift, we find the correspond-
ing in-phase and in-quadrature thermo-mechanical displacements,
δ
x
q

1
.
3
×
10
17
m and
δ
x
p

1
.
3
×
10
14
m, respectively.
The effective in-plane force producing these in-plane displacements is related to the spring constant of the structure, and given
by,
F

m
x
2
M
δ
x
. Putting this all together, we arrive at in-phase and in-quadrature (relative to the mechanical oscillation)