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A chip-scale integrated cavity-electro-optomechanics platform
Martin Winger,
1,
Tim Blasius,
1
Thiago P. Mayer Alegre,
1, 2
Amir H. Safavi-Naeini,
1
Se
́
an Meenehan,
1
Justin Cohen,
1
Søren Stobbe,
3, 4
and Oskar Painter
1, †
1
Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, CA 91125, USA
2
Current address: Instituto de F
́
ısica “Gleb Wataghin”,
Universidade Estadual de Campinas, UNICAMP 13083-859, Campinas, SP, Brazil
3
DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Ørsteds Plads 343, DK-2800 Kgs. Lyngby, Denmark
4
Current address: Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
We present an integrated optomechanical and electromechanical nanocavity, in which a common mechanical
degree of freedom is coupled to an ultrahigh-
Q
photonic crystal defect cavity and an electrical circuit. The sys-
tem allows for wide-range, fast electrical tuning of the optical nanocavity resonances, and for electrical control
of optical radiation pressure back-action effects such as mechanical amplification (phonon lasing), cooling, and
stiffening. These sort of integrated devices offer a new means to efficiently interconvert weak microwave and
optical signals, and are expected to pave the way for a new class of micro-sensors utilizing optomechanical
back-action for thermal noise reduction and low-noise optical read-out.
I. INTRODUCTION
The usually feeble force associated with radiation pressure
[1], a manifestation of the mechanical momentum carried by
all electromagnetic waves, has recently proven to be quite ef-
fective in manipulating and detecting the motion of micro- and
nanomechanical objects embedded within a resonant cavity
[2–4]. The simplest form of a cavity-mechanical system con-
sists of a resonant electromagnetic cavity with its resonance
frequency dispersively coupled to the position of a mechani-
cal object. In such a cavity-based scheme, a narrowband elec-
tromagnetic source is used to pump the cavity. Mechanical
motion translates into modulation in the stored intra-cavity
electromagnetic field, and through the filtering properties of
the cavity, results in an imprinting of the mechanical motion
on the electromagnetic signal. The resonant enhancement
of the pump’s radiation pressure in turn, yields strong back-
action effects which modify the dynamic mechanical and op-
tical properties of the coupled system. Dynamical back-action
effects can include optical stiffening of the mechanical struc-
ture [4–8], damping or amplification of the mechanical mo-
tion [6, 9–11], or electromagnetically induced transparency
[12–14].
Cavity-mechanical systems demonstrating near quantum-
limited position read-out and strong radiation pressure back-
action have been realized both in the optical [15, 16] and the
microwave frequency domains [17, 18]. In the optical do-
main one has the advantage of shot-noise limited read-out
(even at room temperature) and large radiation pressure cou-
pling due to the relatively large operating frequency, whereas
in the microwave domain one has the distinct benefit of
simple electrical interfacing and compact, robust packaging.
Here we present a chip-scale platform for integrating cavity-
optomechanics with conventional micro-electromechanical
Electronic address: winger@caltech.edu
Electronic address: opainter@caltech.edu
systems (MEMS) in which the mechanical degree of freedom
is strongly coupled via radiation pressure to both an electri-
cal circuit as well as a high-
Q
optical cavity [19]. Using an
integrated photonic crystal device we demonstrate wide-band
(
19 nm) electromechanical tuning of the optical cavity reso-
nance, near shot-noise-limited optical read-out of mechanical
motion, and electromechanical locking of the optical cavity to
a fixed laser source. By combining these device attributes, a
series of key optomechanical back-action effects are also re-
alized, including optical stiffening, back-action cooling, and
phonon lasing. It is envisioned that these coupled electro-
and optomechanical systems, driven by radiation pressure and
packaged in a chip-scale form factor, may be used to create
sensors of electrical [20], force [15, 17], acceleration, or mass
[21] operating at the quantum limits of sensitivity and band-
width.
II. A TUNABLE SLOTTED-WAVEGUIDE PHOTONIC
CRYSTAL CAVITY
As discussed above, in this work we seek to develop a
common platform for cavity electro- and optomechanics, in
which both electrical and optical signals are coupled to a com-
mon mechanical degree of freedom [19]. Planar photonic
crystals (PCs) are particularly promising to this end, since
they provide the potential for on-chip integration with well-
established microwave and micro-electromechanical systems
(MEMS) technologies, and large radiation pressure coupling
due to their nanoscale optical mode volumes [8, 22, 23]. Elec-
tromechanical control of microcavities has been shown previ-
ously in one-dimensional zipper and double-membrane cav-
ities [24–26]. These approaches, however, were either lim-
ited by low tuning speed, high leakage-currents, or the use of
low-
Q
cavities, which prohibited the observation of radiation
back-action effects.
The photonic crystal structure studied in this work, along
with the conceptually similar tunable Fabry-Perot cavity, are
sketched in Figs. 1(a) and (b). The PC structure is based on
a silicon membrane slotted cavity, in which the optical cavity
arXiv:1111.4602v1 [physics.optics] 20 Nov 2011
2
FIG. 1: (a) Scanning Fabry-P
́
erot cavity example of an electro-optomechanical system, in which cavity mirrors are attached to capacitive
actuators. (b) Displacement profile of the PC implementation of an electro-optomechanical cavity. The cavity is formed as a waveguide defect
in between two individual PC membrane halves (region outlined by the black rectangle), the distance between which can be adjusted using an
electrostatic force generated between pairs of metal wires. (c) & (d) Electric field distribution
|
E
|
2
of the first ((c)) and second order (d)) optical
cavity modes. (e) Scanning-electron micrograph of a processed device in a double-capacitor configuration. The PC membrane is suspended
on struts with
w
1
=
250 nm and
w
2
=
80 nm, a zoom-in of which is shown in (d).
field is localized to the air slot between two PC membranes
that are suspended on flexible struts. Two pairs of metal con-
tacts on each membrane act as capacitive MEMS actuators
that provide for electromechanical control of membrane mo-
tion and the cavity air slot width. Localization of light in the
optical cavity is determined by a two-step compression of the
PC lattice constant [27] along the length of the slotted PC
waveguide formed from the two PC membrane halves [23].
In this work, the PC structure was fabricated with a lattice
constant of
a
=
470 nm, a relative hole radius of
r
/
a
=
0
.
285,
and a slot width of
s
/
a
=
0
.
21 nm so as to produce cavity
modes in a wavelength band around 1500 nm with high-
Q
and
large radiation pressure coupling. Theoretical estimates for
the optical cavity mode frequencies and radiation rates were
calculated using a finite-elements method (FEM) solver which
is part of the COMSOL Multiphysics [28] software package.
The cavity is found to support two high-
Q
modes (theoretical
Q
>
10
6
), FEM simulations of which are shown in Fig. 1(c)
and (d).
The strong light confinement in the slot region makes the
optical mode frequency (
ω
c
) highly sensitive to the separa-
tion
s
of the two membranes with a theoretical optomechani-
cal coupling
g
OM
=
∂ ω
c
/
s
=
ω
c
/
L
OM
=
2
π
×
152 GHz
/
nm
obtained from FEM simulations. The electrostatic actuators
are formed by pairs of gold contacts that together with the
underlying silicon form capacitors (capacitance
C
) which cre-
ate an electrostatic force
F
el
= (
1
/
2
)(
d
C
/
d
w
g
)
V
2
a
when ap-
plying a voltage
V
a
across the capacitor gap
w
g
[29].
F
el
leads to contraction of the capacitors, thus increasing
s
and
leading to a blue-shift of the cavity resonances. Figure 1(e)
shows a scanning-electron micrograph of a device fabricated
on a microelectronics SOI wafer. The cavity membranes are
suspended on
l
=
3
μ
m long struts of width
w
1
=
250 nm
and
w
2
=
80
150 nm, respectively, yielding estimated ef-
fective spring constants for in-plane motion on the order of
k
eff
50 N
/
m. For a metal layer thickness of 200 nm and
capacitor gaps of
w
g
=
200
250 nm we estimate
C
0
.
7 fF
and
F
el
1
.
5 nN
/
V
2
.
A. Fabrication
Samples were fabricated from silicon-on-insulator material
from SOITEC. A lift-off mask for the metal contacts is de-
fined by electron-beam lithography in ZEP-520A positive e-
beam resist. We then deposit a 5nm/200nm thick Cr/Au layer
in an electron-beam evaporator and strip the resist with the ex-
cess metal on top in Microposit 1165 photoresist remover. A
fresh layer of ZEP-520A is applied, and the etch-pattern for
the PC structures, together with the necessary cut-outs for ca-
pacitor gaps, membrane suspensions, and strain-relief slices is
exposed. The pattern is transferred into the silicon by a radio-
frequency plasma of C
4
F
8
/
SF
6
chemistry. The excess e-beam
resist is removed by cascaded immersion into trichloroethy-
lene, Microposit 1165 remover, and a 10 min etch in Piranha
solution (3:1 H
2
SO
4
:H
2
O
2
) at 120
C. The cavity membranes
are released from the underlying SiO
2
layer by immersion
into 48% Hydrofluoric acid. Cleaning of the sample surface
is finalized by an additional Piranha cleaning step, followed
by a rinse in de-ionized water and a 1 min immersion into
1:10 HF : H
2
O. Finally, samples are glued to a copper sample
holder using GE varnish and electrically contacted with gold
wires by ultrasonic wire-bonding.
3
FIG. 2: (a) Plot of the normalized transmission spectrum of a de-
vice with zero applied voltage showing both the fundamental and
the second-order optical resonance. (b) Cavity resonance wave-
lengths from (a) versus applied voltage (
V
2
a
), indicating quadratic
wavelength-tuning of the cavity modes with tuning parameter
α
=
0
.
051 nm
/
V
2
. (c) RF power spectral density of laser light trans-
mitted through the second order cavity mode. The resonances at
3.18, 3.28, and 3.61 MHz correspond to modes with hybridized in-
and out-of-plane character. The insets show FEM-simulations of the
eigenmodes of a single membrane half in top- and sideview.
III. OPTICAL AND MECHANICAL
CHARACTERIZATION
A. Optical spectroscopy
The PC cavities are optically investigated by resonant trans-
mission spectroscopy using a near-field technique based on
a dimpled tapered optical fiber, the long-range evanescent
field of which is brought into optical contact with the cav-
ity [30]. A swept-wavelength narrow-band telecommunica-
tions test laser allows for obtaining cavity transmission spec-
tra. Figure 2(a) shows the transmission spectrum of a de-
vice with
w
2
=
150 nm. The two resonances at 1545
.
63 nm
and 1554
.
45 nm correspond to the cavity modes depicted
in Fig. 1(c) and (d), respectively. When increasing the ap-
plied voltage
V
a
, these resonances blue-shift, as can be seen
in Fig. 2(b). For a maximum applied voltage of
V
a
=
19 V
the fundamental (second order) mode reaches a total shift of
18
.
3 nm (
19
.
1 nm) or
+
2
.
32 THz (
+
2
.
4 THz) without a
noticeable reduction of the optical
Q
-factor. As expected, cav-
ity tuning follows a quadratic voltage dependence. Defining
the tunability
α
by
λ
c
=
α
·
V
2
a
, this corresponds to a mea-
sured
α
=
0
.
051 nm
/
V
2
, in good correspondence with the
FEM electromechanical simulations of the structure. For de-
vices with
w
2
=
80 nm, we were able to achieve tunabilities up
to
α
=
0
.
088 nm
/
V
2
(see Appendix B). The accessible tuning
range of a given device is limited by electrical arching be-
tween the contacts, which occurs around
V
max
20 V in a Ni-
trogen atmosphere at ambient pressure. Also, due to the large
parallel resistance in excess of 400 G
, current flow in these
structures is negligible, minimizing heating and allowing for
ultralow power operation. Viewed as a wide-range (
>
2 THz)
tunable optical filter (bandwidth
1 GHz) operating in the
telecom C-band, or as a narrowband modulator/switch with
ultra-low switching voltage (
V
π
=
10 mV at a bias voltage of
V
a
=
10 V), the present device performance is impressive due
in large part to the strong optomechanical coupling.
B. Mechanical mode spectrocsopy
In addition to the optical properties of the PC cavity, the me-
chanical mode structure of the presented system can be inves-
tigated by monitoring the radio-frequency (RF) power spec-
tral density (PSD) of laser light transmitted through a cavity
mode. To this end, we launch a tunable external cavity diode
laser into the fundamental or second order cavity mode, and
actively stabilize the cavity frequency to a detuning of half an
optical linewidth from the laser. The transmitted pump light
is detected on a high-speed photodetector (125 MHz band-
width), and the fluctuation power spectral density of the pho-
tocurrent is computed with a high-speed digitizing oscillo-
scope. In Fig. 2(c) this is shown for transmission of a probe
laser through the second order cavity mode. The strongly
transduced resonant features between 3 MHz and 3
.
6 MHz
correspond to mechanical modes of the structure that origi-
nate from the in-plane tuning mode (right inset in Fig. 2(c))
of the two individual membranes, split by fabrication asym-
metries. Moreover, hybridization with a near-resonant out-of-
plane (flexural) mode (see left inset in Fig. 2(c)) that originates
from the breaking of out-of-plane symmetry induced by the
presence of the top metal contacts gives rise to the additional
features at 2.9 and 3.15 MHz (see Appendix C). The mechan-
ical modes shown here can also be resonantly addressed by
driving the actuators with a sinusoidal modulation voltage (see
Appendix E). The
Q
-factors of the mechanical modes were
found to be in the range 50–100, limited by air-damping [8],
thus allowing for high-speed tuning of the structure at rates
limited by the mechanical time-constant of 20
μ
s.
Despite its unique benefits for the readout and manipulation
of micromechanical motion, optomechanical back-action has
hitherto not found technological application in large part due
to the need for elaborate tunable laser-sources to control the
4
FIG. 3: Optical spring reduction of thermal noise: (a) Transmission spectrum (blue) of the second order cavity mode. The actuators are driven
by a triangular signal (orange) with an amplitude of 1 V and a frequency of 50 Hz. (b) RF spectra of the pump laser transmission as function
of
n
cav
. The optical spring effect shifts the mechanical mode at 3
.
6 MHz to higher frequencies. (c) Normalized transmission spectrum of the
fundamental cavity mode (blue) of a device with
α
=
0
.
055 nm
/
V
2
together with a Voigt fit (red). (d) False color plots of the transmission
scans of the fundamental cavity mode as function of the intracavity photon number
n
cav
in the second order mode. The horizontal green lines
indicate the intra-cavity photon numbers at which the individual scans in (e)–(g) were taken. (h) Linewidth of the fundamental cavity mode as
function of
n
cav
in the second order mode. Blue dots show the FWHM linewidth extracted from (d) while the red dots show the linewidth as
an effective optical
Q
-factor.
relative cavity-pump laser detuning. In the presented system,
however, frequency tunability is solely afforded by electrome-
chanical actuation, thus allowing for the study of optomechan-
ical effects using simple fixed-frequency laser sources. As an
example of this, Fig. 3(a) shows an oscilloscope trace of the
transmission of a strong pump laser (
P
i
=
270
μ
W) through
the second order cavity mode (blue curve) while applying a
50 Hz triangular wave to the actuators (orange curve). Both
the triangular shape of the transmission curve and the asym-
metry between forward- and backward scans arise from the
well-known thermal bistability of silicon microcavities [31].
Using electromechanical frequency tuning then, we can ac-
tively lock the cavity to a fixed-frequency pump laser. To this
end, we actively control the cavity electrical contacts with
a commercial PI-control loop. The error signal is obtained
from the transmission level of the pump laser which therefore
is proportional to the intracavity photon number
n
cav
. Addi-
tionally, although not performed here, using a feedback loop
of sufficient bandwidth allows for active feedback cooling
(“cold-damping”) and amplification of the mechanical mode
[19, 32].
IV. OPTICAL STIFFENING
As an example of electrically controlled optomechanical
back-action, we study the optical spring effect by tuning the
cavity in resonance with the blue-detuned pump laser. Fig-
ure 3(b) shows a series of RF-modulation spectra while chang-
ing the intracavity photon number
n
cav
in the second order
cavity mode. This is achieved by actively locking the tun-
able cavity to different levels of the pump laser transmission
as described above. The higher frequency mode initially at
3
.
61 MHz is renormalized by the radiation pressure coupling
to the internal cavity field into the in-plane differential mode
of Fig. 1(b) [7], and shifts to
8 MHz for
n
cav
=
7 500 (note
that the lower frequency mode at 3
.
3 MHz shifts very little, as
it is renormalized to the uncoupled common mode of motion
between the membrane halves). The observed frequency shift
is consistent with
g
OM
=
2
π
×
215 GHz
/
nm, in reasonable
agreement with the theoretically expected value.
The optical spring effect is rather unique in that it only af-
fects the dynamic spring constant of the mechanical system
responding to fluctuations around mechanical equilibrium, but
leaves alone the static stiffness of the structure [7]. Increas-
ing the wide-range tunability of a micro-mechanical device
by reducing the spring constant
k
eff
=
m
eff
ω
2
m
naturally leads
to a compromise in which the noise is increased due to ther-
5
FIG. 4: Electrically controlled optomechanical back-action: (a) False-color plot of RF optical transmission spectra as a function of
V
a
. For
blue detuning we observe stiffening and amplification of mechanical motion, whereas we observe softening and damping for red detuning.
(b) RF spectra for a blue-detuned pump laser below (lower) and above the lasing threshold (upper panel). (c) Time trace of the cavity optical
transmission in the phonon lasing regime. (d) Waterfall plot of the RF optical transmission spectra of the mechanical modes in the cooling
regime, with the pump-laser a half-linewidth red-detuned from the cavity. Curves go from red to blue as
n
cav
is increased. (e) Plot of the higher-
frequency 3
.
6 MHz mechanical mode linewidth (blue dots) and effective temperature (red dots) versus
n
cav
under red-detuned pumping.
mal processes. The frequency jitter of the cavity resonance
in the highly flexible structures of this work is estimated to
be
λ
rms
= (
λ
c
/
L
OM
)
k
B
T
/
m
eff
ω
2
m
=
18
.
1 pm, almost the
entire measured optical linewidth of 22 pm. As a result, time-
averaged transmission scans like that of the fundamental op-
tical cavity resonance shown in Fig. 3(c) are predominantly
thermally broadened. The red line here shows a fit assuming a
Voigt line profile (see Appendix D) that allows us to estimate
an intrinsic linewidth of 6–9 pm with thermal Gaussian line
broadening of
18 pm. Using optomechanical back-action
this thermal noise can be overcome, without sacrifice in tun-
ability, by increasing
k
eff
using the optical spring effect.
In order to investigate the effect of the reduction of thermal
membrane motion by increasing
k
eff
via the optomechanical
spring effect, we monitor transmission spectra of the funda-
mental cavity mode as function of
n
cav
stored in the second or-
der (pump) cavity mode. To this end, we use two separate tele-
com external-cavity diode lasers that are combined via a fiber-
based optical beamsplitter before entering the fiber taper and
that are individually detected after being separated by a fiber-
based transmission/reflection bandpass filter at the taper out-
put. One laser (pump laser) is kept at a fixed wavelength close
to the second order optical cavity mode. Again, the detuning
with the pump laser can then be controlled electrostatically.
At the same time, the second laser is swept across the fun-
damental mode, resulting in the transmission spectra shown
in Fig. 3(d) for various values of
n
cav
. As
n
cav
increases, the
cavity modes red-shift due to heating of the structure, which
counter-acts the electrostatically induced blue-shift and re-
sults in the saturation of cavity tuning. At the same time, the
linewidth of the fundamental cavity mode decreases signifi-
cantly, as can be seen from the cuts through Fig. 3(d) shown
in Figs. 3(e)–(g). The linewidths
λ
c
extracted from the trans-
mission curves are shown as the blue bullets in Fig. 3(h),
while the red bullets express the width as an effective
Q
-
factor
Q
eff
=
λ
c
/
λ
c
. While the initial linewidth is 21 pm,
for
n
cav
=
7 500 we observe narrowing to 8 pm, corresponding
to an intrinsic optical
Q
-factor of 200 000 (see SI for details).
This is more consistent with the observed cavity linewidths
of
3 pm (
Q
5
·
10
5
) on nominally identical, mechanically
rigid test cavities.
V. ELECTRICALLY CONTROLLED
RADIATION-PRESSURE BACK-ACTION
Using electromechanical control of the cavity frequency,
we can also realize parametric amplification (phonon lasing)
and back-action cooling. Figure 4(a) shows mechanical spec-
tra of a different device while sweeping the fundamental cav-
ity mode across resonance with a pump laser with
P
i
=
25
μ
W.
For a blue-detuned pump laser (
V
a
<
7
.
23 V) we observe
stiffening of the mechanical modes – similar to Fig. 3(h) –
while for red detuning we observe softening, indicated by a
reduction of the mechanical mode frequencies. In the electro-
optomechanical PC cavity, we can switch between the two
regimes by using a fixed-frequency pump and simply chang-
ing a voltage.
Tuning the cavity such that the pump laser is blue detuned
from the optical mode results in phonon lasing, while red de-
tuning leads to cooling. In this system we can realize both
regimes, as we will show in the following. Figure 4(b) shows
6
RF-spectra for driving the system with
P
i
=
250
μ
W on the
blue side, with detunings below (lower panel) and above (up-
per panel) the lasing threshold. Above threshold we observe
line-narrowing and an enhancement of the mechanical reso-
nance peak by approximately four orders of magnitude. In the
time domain, this corresponds to a large, periodic modulation
of the cavity transmission signal, as is evident from the time
trace shown in Fig. 4(c). For red detuning, we observe cool-
ing of the membrane motion. To this end, we mechanically
anchor the fiber taper on one of the cavity membranes in or-
der to suppress spurious out-of-plane modes (see Appendix F)
and to reduce temporal drift of the fiber taper. We then lock
the cavity a half-linewidth red from the pump laser and moni-
tor RF-spectra while increasing the power
P
i
launched into the
cavity. Figure 4(d) shows the membrane displacement spec-
tral density
S
xx
for a series of
n
cav
. The resonance at 3.55 MHz
corresponds to motion of a single membrane. Clearly, we ob-
serve optical damping of the mechanical mode with increasing
photon number, evident from the increasing linewidth
ν
m
of
the resonance (blue dots in Fig. 4(e)). Measuring the area
under the mechanical resonance (proportional to the phonon
occupancy and thus the effective temperature) shows that in
addition to optically-induced damping, there is optical cool-
ing of the mechanical motion, with the effective temperature
of the mode (red dots in Fig. 4(e)) reducing to
T
eff
150 K.
VI. CONCLUSION
As the above measurements indicate,
the electro-
optomechanical PC cavity structure demonstrated here pro-
vides a unique level of control of the optical properties of
the device via electromechanical actuation, and, importantly,
maintains the device characteristics necessary for strong radi-
ation pressure coupling and optomechanical back-action. This
device architecture is also adaptable to integration with higher
frequency (GHz) mechanical resonances (in the form of op-
tomechanical crystals for instance [22]), enabling operation
in the sideband-resolved regime critical for quantum applica-
tions [20]. Moreover, the scalability afforded by the CMOS-
compatible, on-chip architecture of this device platform, al-
lows for mass production, integration with on-chip electron-
ics, and the straightforward engineering of coupled-cavity ar-
rays. Applications for such devices range from wavelength
filters and ultralow-power optical modulators, to quantum-
limited force sensors [15, 17]. Moreover, future devices in
which an optical cavity and a radio-frequency resonant circuit
share a common mechanical degree of freedom are foresee-
able, paving the way for experiments at the interface of cavity
opto- and electro-mechanics [19], such as the implementation
of efficient inter-band signal conversion [20].
Acknowledgments
This work was supported by the DARPA/MTO ORCHID
program through a grant from the AFOSR, the DARPA/MTO
MESO program through a grant from SPAWAR, and the NSF
(CIAN grant no. EEC-0812072 through University of Ari-
zona). S.St. gratefully acknowledges The Danish Council for
Independent Research (Project No. FTP 10-080853).
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8
Appendix
Appendix A: Theory of capacitive cavity tuning
The static displacement of a cavity membrane for an ap-
plied voltage
V
a
is given by balancing the capacitive force ex-
erted by the actuator and the spring force created by a dis-
placement of the membrane by
x
from its rest position:
F
cap
(
x
) =
F
spring
(
x
)
,
(A1)
1
2
d
d
x
C
(
x
)
V
2
a
=
k
eff
x
.
(A2)
Here,
C
(
x
)
is the displacement-dependent capacitance and
k
eff
is the effective spring constant of the membrane. A realistic
model for a capacitor including fringing effects is
C
(
x
) =
a
(
w
g
x
)
n
,
(A3)
where
w
g
is the initial separation of the capacitor plates. With
Eq. (A3), Eq. (A2) has solutions for
x
w
g
/
(
n
+
2
)
. For larger
displacements, the linear spring force cannot compensate the
attractive capacitive force, such that the capacitor collapses
[29]. For the fabricated devices with
w
g
=
200
250 nm how-
ever, such sticking occurs for displacements much larger than
those realized by voltages up to 20 V (which are on the order
of 10
20 nm), such that the only limiting factor for tuning
here is given by electrical breakdown of the structure. For
small displacements
δ
x
, an approximate solution of Eq. (A2)
is
δ
x
=
na
2
k
eff
w
n
+
1
g
V
2
a
.
(A4)
For the device studied in the main text with
w
g
=
230 nm and
w
2
=
80 nm we calculated the values
C
0
.
73 fF,
n
=
0
.
68,
and
k
eff
50 N
/
m using FEM simulations, such that with
a
Cw
n
g
we expect
F
cap
1
.
05 nN and
δ
x
/
V
2
a
≈−
21
.
6 pm.
A similar result is found by performing coupled electrostatics-
elasticity FEM simulations that account for motion of the
structure under the influence of the electrostatic force. The
blue bullets in Fig. A1(a) shows the peak membrane displace-
ment as function of
V
a
for the studied device. Clearly, the
membrane displacement follows a quadratic dependence (red
line) with a displacement of
δ
x
/
V
2
a
=
13 pm. The discrep-
ancy between the displacement calculated in Eq. (A4) and the
result of FEM simulations arises from the capacitive force act-
ing in a distributed manner across the entire capacitor, as op-
posed to the assumption of a point force made in Eq. (A1).
Moreover, the intuitive model introduced above assumes one-
dimensional motion of the cavity membranes. From FEM
simulations, however, we find that membrane motion under
the influence of capacitive actuation has a significant perpen-
dicular component due to the different elastic moduli of sili-
con and gold.
In order to calculate the frequency shift of the optical cav-
ity modes induced by the displacement of the cavity mem-
branes, we computed the cavity resonance frequencies for a
series of membrane separations using COMSOL Multiphysics
[28]. Results of such a simulation can be seen in Fig. A1(b).
Clearly, there is a linear relation
ω
c
=
g
i
,
OM
x
between dis-
placement and frequency, with
g
i
,
OM
=
2
π
×
73 GHz
/
nm per
membrane. In the presence of two actuated membranes, the
total frequency shift per nanometer of membrane separation
is given by
g
OM
=
2
g
i
,
OM
=
2
π
×
146 GHz
/
nm or a 1.1 nm
wavelength shift per nanometer of displacement. The linear
relation in Fig. A1(b) persists over the entire tuning range on
the order of 20 nm, consistent with the experimentally ob-
served quadratic tuning. We note that
g
OM
denotes the opti-
cal frequency shift induced by the electrostatically induced
displacement profile which in general is not identical to a
mechanical eigenmode of the structure. The optomechanical
coupling constants of mechanical modes are denoted by
g
OM
.
The estimated value for the voltage-dependent displace-
ment in Eq. (A4) together with
g
OM
calculated above yields
a predicted wavelength tunability of
α
=
g
OM
2
π
λ
2
c
c
nC
2
k
eff
w
g
=
0
.
025 nm
/
V
2
(A5)
around
λ
c
=
1550 nm. From coupled electrostatics-elasticity
FEM simulations we find
α
=
0
.
015 nm
/
V
2
. Experimentally,
we observe a tunability of 0
.
051 nm
/
V
2
, approximately three
times larger than the expected value. We attribute the discrep-
ancy to the limited accuracy with which we can determine the
geometrical parameters of the structure.
Appendix B: Study of fabricated devices
In the design of the presented tunable cavities, the main
figures to maximize tunability are the membrane stiffness
expressed by
k
eff
and the capacitive force proportional to
d
C
(
x
)
/
d
x
. For a plate capacitor with cross-section
A
we have
C
(
w
g
) =
ε
0
A
w
g
and therefore
F
cap
w
2
g
, such that it is bene-
ficial to minimize the capacitor gap size
w
g
. With the process
used for fabrication of the metal mask, we are limited to gaps
with
w
g
200 nm. Further reducing the feature size could be
achieved by reducing the metal layer thickness, however, this
sacrifices stability in the wet chemistry process used for cavity
lithography. The floppiest struts we could reproducibly fabri-
cate had a length of 3
μ
m and a width of
w
2
=
80 nm. The
following table gives an overview of accessible tuning ranges
achievable with typical device geometries.
geometry
tunability
α
w
g
=
200 nm &
w
2
=
80 nm
0.078
w
g
=
200 nm &
w
2
=
130 nm
0.052
w
g
=
250 nm &
w
2
=
80 nm
0.059
In particular for small
w
2
, we observe a variation of
α
up to
50% for devices processed under identical conditions. This
observation justifies our explanation of the discrepancy be-
tween theoretical and experimentally observed values of
α
.