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: December 16, 2008)
Abstract
We describe the design, fabrication, and measurement of a cavity opto-mechanical system consisting of
two nanobeams of silicon nitride in the near-field of each other, forming a so-called “zipper” cavity. A
photonic crystal patterning is applied to the nanobeams to localize optical and mechanical energy to the
same cubic-micron-scale volume. The picrogram-scale mass of the structure, along with the strong per-
photon optical gradient force, results in a giant optical spring effect. In addition, a novel damping regime
is explored in which the small heat capacity of the zipper cavity results in blue-detuned opto-mechanical
damping.
∗
Electronic address: opainter@caltech.edu; URL:
http://copilot.caltech.edu
1
arXiv:0812.2953v1 [physics.optics] 16 Dec 2008
Recently, there has been keen interest
1
in dynamic back-action caused by electromagnetic
forces in optical
2,3,4,5,6,7
and microwave
8
cavities. Back-action cooling, for example, is being
pursued as a means to achieve quantum ground-state cooling of a macro-scale mechanical oscil-
lator. Work in the optical domain has revolved around milli- or micro-scale structures utilizing
the radiation pressure force. By comparison, in microwave devices, low-loss superconducting
structures have been used for gradient-force mediated coupling to a nanomechanical oscillator of
picogram mass
8
. Here we describe measurements of an optical system consisting of a pair of
specially patterned nanoscale beams in which optical and mechanical energy are simultaneously
localized to a cubic-micron-scale volume, and for which large per-photon optical gradient forces
are realized. The resulting scale of the per-photon force and the mass of the structure enable new
cavity-optomechanical regimes to be explored, where for example, the mechanical rigidity of the
structure is dominantly provided by the internal light field itself. In addition to precision measure-
ment and sensitive force detection
9
, nano-optomechanics may find application in reconfigurable
and tunable photonic systems
10
, RF-over-optical communication
11
, and to generate giant optical
nonlinearities for wavelength conversion and optical buffering
12
.
Optical forces arising from near-field effects in guided-wave structures have been proposed
13
,
and recently demonstrated
14,15
, as a means of providing large optomechanical coupling between
the field being guided and the dielectric mechanical structure providing the guiding. The resulting
optical force can be viewed as an intensity gradient force much like that used to
tweeze
dielectric
particles or to trap cold gases of atoms
16
. In the devices studied in this work, doubly-clamped
silicon nitride nanobeams are converted into optical resonant cavities through the patterning of a
linear array of etched holes (Fig. 1(b)). Bringing two such cavities into the near-field of each other
forms a super cavity supporting even and odd superpositions of the individual beams modes. This
“zipper” cavity, so-named due to its resemblence to the mechanical fastner, allows for sensitive
probing and actuation of the differential motion of the beams through the internal, optical, cavity
field.
A figure of merit for cavity-optomechanical systems is the coupling constant
g
OM
≡
d
ω
c
/
d
x
,
which represents the differential frequency shift of the cavity resonance (
ω
c
) with mechanical dis-
placement of the beams (
x
). For the commonly studied Fabry-Perot cavity structure (Fig. 1(a)),
momentum transfer between the circulating light field and the mechanically-compliant end mir-
ror(s) occurs at a rate of 2
~
k
ph
per round trip time, resulting in an optomechanical coupling con-
stant that scales with the inverse of the cavity length (
L
c
),
g
OM
=
ω
c
/
L
c
. Similarly for whispering-
2
FIG. 1:
Zipper cavity opto-mechanical system and experimental set-up
.
a
, Fabry-Perot and
b
, pho-
tonic crystal optomecachanical systems. Finite-element-method simulated
c
, bonded and
d
, anti-bonded
supermodes of the zipper optical cavity, shown in cross-section.
e,f
, Scanning-electron-microscope (SEM)
images of a typical zipper cavity, indicating the slot width (
s
), the cantilever width (
w
), and the photonic
crystal lattice constant (
a
).
g
, Experimental set-up used to probe the optical and mechanial properties of the
zipper cavity. Acronyms are: erbium-doped fiber amplifier (EDFA), variable optical attenuator (VOA), fiber
polarization controller (FPC), fiber Mach-Zender interferometer (MZI), and photodetected Mach-Zender
transmission (PD
MZI
) and zipper cavity transmission (PD
T
).
gallery-mode structures, such as the recently studied microtoroid
17
,
g
OM
scales with the perime-
ter length through the radius of the cavity
R
,
g
OM
=
ω
c
/
R
. In the case of the zipper cavity the
optomechanical coupling is exponentially proportional to the slot gap (
s
) between the beams,
g
OM
=
ω
c
/
L
OM
with
L
OM
∼
w
o
e
α
s
. The minimum value of
L
OM
is set by
w
o
which is approx-
imately equal to the beam width, while the decay constant
α
is set by the wavelength of light (
λ
)
and the refractive index contrast of the nanobeam system. Thus, for beam widths on the order of
3
the wavelength of light and for a sub-wavelength slot gap,
L
OM
∼
λ
, independent of the length
of the nanobeams (see Fig. 2(a,b)). This yields an optomechanical coupling more than an or-
der of magnitude larger than can be accomplished in high-Finesse Fabry-Perot cavities
18
or glass
microtoriod structures
17
. In addition, this large optomechanical coupling is realized in a versatile
geometry in which motional mass and mechanical stiffness can be greatly varied, and for which the
mechanical displacement energy density and optical energy density can be efficiently co-localized
at optical wavelengths in the visible-NIR and for mechanical frequencies in the MHz-GHz fre-
quency range.
For the devices studied in this work, optically thin (
t
=
400 nm) stoichiometric silicon nitride
(Si
3
N
4
) is deposited using low-pressure-chemical-vapor-deposition on a silicon wafer in order to
form the optical guiding layer and the mechanical beams. Electron-beam lithography is used to
pattern the zipper cavity consisting of beams of length
l
=
25
−
40
μ
m, widths of
w
=
0
.
6-1
.
4
μ
m, and with an inter-beam gap of
s
=
60-250 nm (Fig. 1(f)). The optical cavity is created in the
nanobeams by patterning holes to form a quasi-1D photonic bandgap for light (see Figs. 1(e-g)).
A C
4
F
8
/SF
6
-based plasma etch is then used to transfer the nanobeam and photonic crystal pattern
into the Si
3
N
4
. This is followed by a wet chemical etch of KOH which selectively etches the
underlying Si subtrate and releases the patterned beams.
As shown in Figure 1(g), optical excitation and probing of the zipper cavity is performed using
a high-efficiency optical fiber taper coupler
19
in conjunction with a bank of tunable external-cavity
diode lasers. A fiber polarization controller is used to adjust the polarization to selectively excite
the transverse electric (TE) polarization modes of the zipper cavity. The zipper optical cavity
design is based upon a graded lattice concept
19,20,21
in which the lattice period is varied harmoni-
cally from the center to the ends of the nanobeam. This results in an optical potential for photons
which increases harmonically as one approaches the cavity center. Localized modes form from
photonic bands near the zone boundary with negative group velocity dispersion
22
, with the funda-
mental mode of the cavity having the
highest
frequency and higher-order cavity modes
decreasing
in frequency (see inset of Fig. 2(c)). Owing to the strong optical coupling between the pair of
nanobeams, the photonic bands in the zipper cavity break into pairs of positive and negative parity
super-mode bands. The positive-parity superposition, designated TE
+
, corresponds to a manifold
of modes which have an even mode profile for the TE electric field polarization and a peak elec-
tric field intensity in the center of slot gap between the beams (Fig. 1(c)). These we term
bonded
modes
13
. The negative-parity TE
−
manifold of modes (the
anti-bonded
modes) have an odd parity
4
FIG. 2:
Optical Spectroscopy of the Zipper Cavity
. Finite-element-method simulation of the
a
, wave-
length tuning versus nanobeam slot gap and
b
, effective optomechanical coupling length parameter (
L
OM
)
for the bonded and anti-bonded fundamental zipper cavity optical modes (
w
=
650 nm).
c
, Measured optical
transmission of a zipper cavity with
w
=
650 nm and
s
=
120 nm showing four orders of the bonded (TE
+
)
resonant modes. Inset: schematic of the graded photonic crystal lattice design and resulting bonded and
anti-bonded resonance manifolds.
d
, Measured optical transmission of a zipper cavity with a larger beam
width and gap (
w
=
1400 nm,
s
=
250 nm) showing the bonded and anti-bonded lowest order optical reso-
nances.
e,f
, Optical transmission versus fiber taper lateral position for each of the bonded and anti-bonded
resonant modes of
d
, indicating the even and odd parity of the modes. Black outline corresponds to position
of zipper cavity.
mode profile and a node at the slot gap center (Fig. 1(d)).
By systematically varying the lattice constant of the devices, and measuring the parity of the
cavity modes using the fiber taper as a near-field probe
19
, one can identify the various zipper cavity
modes. For example, for a zipper cavity with
a
=
640 nm, beam width
w
=
650 nm, and slot gap
s
=
120 nm, the measured transmission scan across the
λ
=
1420-1625 nm range is shown in Fig.
2(c). From shortest to longest wavelength, the resonance peaks all have an even mode profile and
5
are associated with the TE
+
,
0
through TE
+
,
4
modes of the
bonded
manifold of modes. Wavelength
scans of a different zipper cavity, with larger beam width
w
=
1
.
4
μ
m and slot gap
s
=
250 nm,
exhibits a spectrum in which the bonded and anti-bonded mode manifolds overlap (Fig. 2(d)). The
measured on-resonance transmission contrast versus lateral taper position for each of the modes
is shown in Fig. 2(e), indicating their even (bonded) and odd (anti-bonded) mode character. The
optical
Q
-factor of the zipper cavity TE
+
,
0
mode can theoretically reach a value well above 10
6
even in the modest refractive index afforded by the silicon nitride
20,23
. Experimentally we have
measured zipper cavity modes with
Q
-factors in the range of
Q
=
10
4
−
10
5
(Finesse
F
∼
10
4
),
depending largely upon the fill-fraction of the air holes and their scattering of light transverse to
the axis of the quasi-1D photonic bandgap. For devices at the high end of the measured
Q
range
(
Q
∼
3
×
10
5
), we find a significant contribution to optical loss from absorption (see Appendix).
Mechanical motion of the zipper cavity nanobeams is imprinted on the transmitted optical
intensity through the phase modulation of the internal cavity field
17
. Figure 3(a) shows the high-
temporal-resolution (blue curve) and low-pass filtered (red curve) transmitted signal as the input
laser wavelength is swept across the TE
1
,
+
mode of the zipper cavity of Fig. 2(c) at low optical
input power (
P
i
=
12
μ
W). The zoomed-in temporal response of the transmitted intensity for a
detuning on the side of the Lorentzian lineshape (Fig. 3(b)) shows an oscillating signal of fre-
quency
∼
8 MHz and peak-to-peak amplitude of roughly a third of the transmission contrast of the
resonance. Finite-element-method (FEM) simulations (Fig. 3(c,d)) indicate that the lowest order
in-plane common (
h
1
c
) and differential (
h
1
d
) mechanical modes of the pair of coupled nanobeams
have frequencies of 8
.
19 and 8
.
16 MHz (mass,
m
x
≈
43 picograms, and spring constant
k
h
1
≈
110
N/m; see Appendix), respectively, when accounting for
∼
0
.
75 GPa of internal tensile stress in the
nitride film
24
. The corresponding mechanical amplitude of oscillation is calibrated by fitting
g
OM
from the optical spring effect as discussed below and in the Appendix, yielding
L
OM
=
1
.
58
μ
m
(
g
OM
/
2
π
=
123 GHz/nm) and an inferred rms amplitude of motion of approximately
x
rms
∼
5
.
8
pm. This is in good correspondence with both the FEM-simulated optomechanical coupling con-
stant for this device (
L
OM
=
2
.
1
μ
m for
s
=
120 nm in Fig. 2(b)) and the expected thermal ampli-
tude for the
h
1
d
mode (
〈
x
2
th
〉
1
/
2
=
√
k
B
T
/
k
h
1
=
6
.
2 pm).
The RF spectrum of the transmitted optical intensity out to 150 MHz is shown in Fig. 3(e).
Comparison to FEM mechanical simulations
20
allows us to identify many of the resonances in the
RF spectrum, with in-plane mechanical resonances up to 9th-order being visible. The strength of
the corresponding spectral peaks oscillates for odd and even orders of in-plane motion, consistent
6
FIG. 3:
Mechanical Motion Transduction
.
a
, Optical transmission through the zipper cavity of Fig.
2(c). The red curve corresponds to the low-pass-filtered (bandwidth 10 kHz) transmission signal showing
the underlying Lorentzian-like cavity resonance.
b
, Temporal oscillations in optical transmission for fixed
detuning, showing large-scale optical power oscillations of frequency
∼
8 MHz. FEM-modeled lowest
order
c
, common and
d
, differential mechanical resonances. Mechanical deformation and color indicates
displacement amplitude, while arrows indicate direction.
e
, Detected RF spectrum with horizontal (
h
) and
vertical (
v
) cantilever modes of motion indicated. Grey colored labels indicate either “missing” resonances
or modes of questionable description. Grey colored curve is the electronic detector noise floor. Insets:
zoomed-in RF spectrum of the
f
, hybridized fundamental
h
1
mechanical modes and
g
, distinct left and right
cantilever modes of third-order in-plane motion (
h
3
). In
f
and
g
the red curves correspond to RF spectra
taken for red detuning and the blue curves correspond to blue-detuned spectra.
with the odd-order mechanical modes having an anti-node of displacement at the center of the
zipper cavity and the even-order modes having a node. The mechanical
Q
-factor of the resonances
are measured to vary between
Q
M
∼
50
−
150, limited by gas-damping
25
in the nitrogen test en-
vironment used in this work. For the
h
1
mechanical resonances (Fig. 3(f)) at 8 MHz, the RF
spectrum shows two other interesting features. The first is the interference between the two reso-
nances as evidenced by the asymmetry in each peak and the narrow central dip. As will be detailed
7
elsewhere, this is due to coupling between the common and differential modes of mechanical os-
cillation for which the common-mode motion is
dark
with respect to our optical read-out method.
The second feature of interest is the slight shift of the
h
1
resonance peaks to lower (higher) res-
onance frequencies for red (blue) laser-cavity detuning. Both of these features are absent for the
h
3
resonance peaks centered around 28 MHz (Fig. 3(g)) for which the optomechanical coupling
is weaker and the frequency-splitting between independent nanobeam motion is much larger than
for the
h
1
modes.
The optically driven zipper cavity not only allows for sensitive mechanical displacement detec-
tion, but can also strongly modify the mechanical motion in two distinct ways. Optical stiffening
of the mechanical resonant structure
5,17
(the so-called “optical spring”) results from the compo-
nent of optical cavity energy (and gradient force) oscillating in-phase with the mechanical motion.
On the otherhand, the finite cavity photon lifetime introduces a non adiabatic, time-delayed, com-
ponent of optical force acting in-quadrature with the mechanical motion. This velocity dependent
force results in detuning-dependent amplification or damping of the mechanical motion. A per-
turbative analysis shows that in the sideband unresolved limit (
Ω
M
Γ
) the effective mechanical
frequency (
Ω
′
M
) and damping rate (
γ
′
M
) are given by the following relations (see Ref. [17]):
(
Ω
′
M
)
2
=
Ω
2
M
+
(
2
|
a
0
|
2
g
2
OM
∆
2
ω
c
m
x
)
∆
′
o
,
(1)
γ
′
M
=
γ
M
−
(
2
|
a
0
|
2
g
2
OM
Γ
∆
4
ω
c
m
x
)
∆
′
o
,
(2)
where
Ω
M
and
γ
M
are the bare mechanical properties of the zipper cavity,
|
a
0
|
2
is the time-averaged
stored optical cavity energy,
∆
′
o
≡
ω
l
−
ω
c
is the laser-cavity detuning,
Γ
is the waveguide-loaded
optical cavity energy decay rate, and
∆
2
≡
(
∆
′
o
)
2
+
Γ
2
.
As is shown in Fig. 4, for higher optical input powers (
P
i
=
5 mW) the internal optical cavity
field provides significant stiffness to mechanical motion of modes in the zipper optomechanical
cavity. The large optomechanical coupling (
L
OM
=
1
.
58
μ
m) and comparitively small motional
mass (
m
x
∼
43 picograms) of the
h
1
d
-zipper-cavity mechanical mode results in a giant optical
spring effect
5
, shifting the mechanical frequency from 8 to 19 MHz (Fig. 4(g)). This corresponds
to an optical stiffness greater than five times that of the intrinsic mechanical stiffness of the silicon
nitride cantilevers. For the
h
3
d
in-plane mode, the frequency shift is smaller due to the slightly
reduced optomechanical coupling factor of this mode and its larger bare frequency. An additional
8
FIG. 4:
Optical Spring and Damping
.
a
, Measured (green curve) and fit model (red curve) normalized
optical transmission versus wavelength sweep in units of sweep time. Dashed blue curve corresponds to
low power model curve.
b
, Conversion between sweep time delay and normalized cavity detuning, fit from
model curve in (a), with zero time delay corresponding to zero laser-cavity detuning.
c
, Intensity image
of the measured RF power spectrum versus cavity detuning (time delay) of the optical transmission signal
from the zipper optical cavity of Fig. 2(c) at an input optical power of 5
.
1 mW (dropped power of 1
.
4 mW).
Measured and modeled
d
,
f
total RF power and
e
,
g
resonance frequency of the
h
1
,
d
mechanical mode, versus
detuning.
d
,
e
correspond to low optical input power (
P
i
=
127
μ
W) while
f
,
g
are for high optical input
power (
P
i
=
5
.
1 mW). Blue (green) curves correspond to a model with (without) optomechanical damping.
Red squares are measured data points.
feature in Fig. 4(c) is the mechanical mode mixing that occurs as the optical spring tunes the
h
1
d
mode through other mechanical resonances. The mode mixing is most prevelant as the
h
1
d
sweeps away from the
h
1
c
mode and through the
h
2
in-plane modes near
Ω
/
2
π
=
16 MHz (Fig.
4(c)). This mixing of mechanical modes is due to the highly anisotropic and motion-dependent
optical stiffness and its renormalizing of the mechanical eigenmodes of the structure, and will be
discussed in more detail elsewhere.
9
Figures 4(d-g) compare the measured integrated RF power in the
h
1
d
mechanical resonance line
and its mechanical frequency to a nonlinear optical model of the zipper cavity system including
the optical gradient force and thermo-optic tuning of the cavity (see Appendix). At low optical
power (Figures 4(d,e)), a single estimate for
g
OM
based upon optical FEM simulations fits both
the total measured RF power (or
〈
x
2
〉
) and optical frequency of the
h
1
d
mode over a large detun-
ing range. At higher powers (Figures 4(f,g)), the same estimated
g
OM
fits the optical frequency
tuning of the
h
1
d
mode, but severely over estimates the total RF power (green curve in Fig. 4(f))
where the optomechanical interaction is strongest. Damping of the mechanical motion is quite
unexpected for blue-detuned laser excitation
17
. FEM numerical simulations of the zipper cavity
indicate thermo-mechanical effects
26,27
produce a response several orders of magnitude too small
to explain the observed damping; however, a theoretical analysis of the cavity dynamics including
the thermo-optic effect
28
shows that the severely phase-lagged and damped thermo-optic tuning
of the cavity introduces a significant correction to eq. (2). Owing to the small heat capacity of the
zipper cavity, thermo-optic tuning reverses the sign of the damping coefficient of the bare optome-
chanical response for blue-detuned pumping (the correction to the optical spring is found to be
small, at the 10
−
4
level). The numerical model including the thermo-optic correction to the spring
and damping terms is shown as a blue curve in Figs. 4(d,f), with the fit to the high power data
now in much better agreement. The model indicates that at a detuning of
∆
′
0
≈
Γ
/
4 (Fig. 4(f)) the
thermal motion of the
h
1
d
resonance is being damped from
x
rms
≈
7 pm down to 1 pm, at a bath
temperature of 360 K.
Beyond the giant optical spring effect afforded by the large optomechanical coupling and
picogram-scale mass of the zipper cavity, thin-film photonic crystals offer a highly flexible, chip-
scale architecture for coupling optical and mechanical degrees of freedom. In the area of quan-
tum cavity-optomechanics, significant improvements in optical
Q
to values approaching 5
×
10
6
(
F
∼
10
6
) can be expected with new processing technqiues
20,29
, which along with increased me-
chanical frequency (
>
100 MHz), will push the system into the important sideband resolved
limit
30,31
. Applications to optical cavity QED
18
also exist, where rapid cavity frequency shift-
ing may be utilized for single-photon generation and quantum-state transfer. Finally, by combin-
ing phononic
32
with photonic crystal concepts, simultaneous routing and localization of acoustic
and optical waves can be envisioned. Such a platform would expand both quantum and classical
applications, and enable integration not possible in current optomechanical microsystems.
10
Appendix
A. Optomechanical coupling, effective mass, and spring constant
For complex geometries and motional patterns, one must use a consistent definition of displace-
ment amplitude,
x
, in determing
g
OM
,
m
x
(motional mass), and
k
eff
(effective spring constant). In
this work we use a convention in which
x
(
t
)
represents the amplitude of motion for a normalized
mechanical eigenmode displacement field pattern:
u
n
(
z
,
t
) =
x
n
(
t
)
f
n
(
z
)
√
1
l
R
l
0
|
f
n
(
z
)
|
2
d
z
,
(3)
where
n
is a mode label,
l
is the length of cantilever, and, for the simple cantilever geometry con-
sidered here, the displacement vector is only a function of position along the axis of the cantilevers
(
z
). With this definition of amplitude, the effective motional mass is simply the total mass of the
two cantilevers (
m
x
=
m
c
=
43 picograms) and the effective spring constant is defined by the usual
relation
k
eff
=
m
c
Ω
2
M
, with
Ω
M
the mechanical eigenmode frequency. The amplitude associated
with zero-point motion and used in the equipartition theorem to determine the thermal excita-
tion of the mechanical mode is then
x
n
(
t
)
. In the case of the fundamental differential mechanical
mode of motion for the two cantilevers of the zipper cavity, this normalization prescription yields
u
h
1
d
(
z
,
t
)
≈
x
h
1
d
(
t
)(
sin
(
π
z
/
l
)
ˆ
x
1
+
sin
(
π
z
/
l
)
ˆ
x
2
)
, where ˆ
x
1
and ˆ
x
2
are in-plane unit vectors associ-
ated with the two nanobeams of the zipper cavity and pointing in opposite directions away from
the center of the gap between the nanobeams. Thus, to be consistent,
g
OM
for the
h
1
d
mode must
be defined approximately as the rate of change of cavity frequency with respect to
half
the change
in slot gap (
g
OM
≈
d
ω
c
/
1
2
d
δ
s
), as the amplitude
x
h
1
d
(
t
)
corresponds to a change in slot gap of
2
x
h
1
d
(
t
)
near the center of the cavity.
B. Optical transmission, measured RF spectra, and motional sensitivity
RF spectra are measured by direct detection of the optical power transmitted through the zipper
cavity using a 125 MHz bandwidth photodetector (noise-equivalent-power NEP
=
2
.
5 pW/Hz
1
/
2
from 0-10 MHz and 22
.
5 pW/Hz
1
/
2
from 10-200 MHz, responsivity
R
=
1 A/W, transimpedance
gain
G
=
4
×
10
4
V/A) and a high-speed oscilloscope (2 Gs/s sampling rate and 1 GHz bandwidth).
As shown in Fig. 1(g), a pair of “dueling” calibrated optical attenuators are used before and
11
after the zipper cavity in order to vary the input power to the cavity while keeping the detected
optical power level constant. The measured electrical noise floor is set by the circuit noise of the
photodetector for the optical power levels considered in this work, corresponding to
−
125 dBm/Hz
near 10 MHz. The motional sensitivity of the
h
1
d
mechanical mode is measured at 9
×
10
−
16
m/Hz
1
/
2
for an optical input power of 12
μ
W (corresponding to a dropped power of 3
.
5
μ
W, and
an estimated 660 stored cavity photons). At the power levels considered in this work, optical force
noise contribution to the motional sensitivity is negligible.
C. Calibration of input power and intra-cavity photon number
A fiber-taper optical coupling technique was used to in-couple and out-couple light from the
zipper cavity. The fiber taper, although extremely low-loss on its own (88% transmission efficiency
in this work), was put in contact with the substrate near the zipper cavity in order to mechanically
anchor it during all measurements (thus avoiding power-dependent movement of the taper due to
thermal and/or optical forces). The total fiber taper transmission after mechanical anchoring of
the taper to the substrate is 53%. In order to accurately determine the optical power reaching the
cavity (determined by the optical loss in the taper section before the cavity) we measure the cavity
response at high optical power (resulting in thermo-optic tuning of the cavity and optical bistability
in the transmission response) with the input sent in one direction and then in the other of the taper.
From the asymmetry in the thermo-optic tuning in the cavity for both directions one can determine
the asymmetry in the optical loss, and thus determine the optical loss before and after the zipper
cavity. Finally, this method along with calibrated measurements of the optical power at the input
and output of the taper, can determine accurately the optical power reaching the zipper cavity
(the
input
power) and dropped by the cavity. From calibration of the wavelength sweep using the
fiber Mach-Zender interferometer one can also accurately measure the cavity linewidth and the
corresponding loaded cavity
Q
. The average stored photon number can then be determined from
the dropped power and the loaded cavity
Q
. The TE
1
,
+
mode is chosen to study in detail, instead
of the TE
0
,
+
fundamental mode, due to its spectral alignment with the EDFA gain bandwidth,
allowing for the higher power measurements presented in Fig. 4.
12
D. Calibration of laser-cavity detuning
The transduction from mechanical motion to modulated intra-cavity power, and consequently
measured RF photodector spectrum, depends sensitively on the detuning point of the laser from
the cavity resonance. Accurate measurement of the laser-cavity detuning, even for large detunings
(
>
5 half-cavity-linewidths), is required to compare the theoretical model with measured data for
the optical spring and damping shown in Fig. 4. Several methods exist to determine the laser-
cavity detuning, including calibration of the transduced modulated photodector signal for a known
mechanical or optical modulation, or simple inversion of the normalized optical transmission sig-
nal using the measured Lorentzian response of the cavity. For the swept measurements presented
in this work, we have opted to calibrate accurately the laser wavelength versus sweep time using
a fiber-based Mach-Zender interferometer (FSR
=
1
.
57 pm at
λ
∼
1480 nm), and to use this to
fit and convert sweep time to laser-cavity detuning by comparing with a nonlinear model of the
cavity system that incorporates thermo-optic and gradient-force tuning. The thermo-optic cavity
tuning versus temperature was measured to be 14
.
9 pm/K by direct measurement of the resonance
wavelength shift over a 20 K temperature range. The optomechanical coupling constant
g
OM
was estimated from both simulation, based upon an FEM model of scanning-electron-microscope
(SEM) images taken of the cavity geometry, and a fit to the peak measured mechanical frequency
shift. The nonlinear cavity model, incorporating the measured thermo-optic effect and the fit
g
OM
,
then provides an accurate conversion between wavelength and detuning from the cavity. The
above method for calibrating laser-cavity detuning is simple to employ with the swept wavelength
method used in this work, and found to be much more accurate than relying on the low-pass-
filtered optical transmission contrast to infer detuning (especially for large detunings where the
transmission contrast is below the percent level).
E. Zipper cavity optical loss
As mentioned above, although the optical force dominates the cavity tuning at MHz frequen-
cies, the static tuning of the cavity is still largely (
∼
80%) provided by the thermo-optic effect
through optical absorption and subsequent heat generation within the zipper cavity. Calculation of
the thermal resistance of the silicon nitride zipper cavity indicates that optical absorption accounts
for approximately 6% of the total optical cavity loss for the device in Fig. 4(a) (an absorption-
13
limited
Q
∼
4
.
8
×
10
5
). We attribute the optical absorption loss in the zipper cavity to surface-
states
29
of the “holey” silicon nitride beams, rather than absorption in the bulk of the silicon ni-
tride film, due to the much larger
Q
values we have measured in less surface-sensitive microdisks
formed from the same silicon nitride material. Properly chosen chemical surface treatments should
enable
Q
-factors approaching the bulk-absorption-limited value of
Q
b
∼
5
×
10
6
at
λ
=
1
.
5
μ
m,
and perhaps even higher at shorter wavelengths where optical absorption from overtones of the
vibrational modes of the N-H bond is reduced.
Acknowledgements
The authors would like to thank Qiang Lin for extensive discussions regarding the mechanical
linewidth broadening of the devices studied in this work and for pointing out the origin of the
mechanical resonance interference. Funding for this work was provided by a DARPA seedling
effort managed by Prof. Henryk Temkin, and by the National Science Foundation.
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