Optical coupling to nanoscale
optomechanical cavities for near
quantum-limited motion transduction
Justin D. Cohen,
1
,
2
,
3
Se
́
an M. Meenehan,
1
,
2
,
3
and Oskar Painter
1
,
2
,
∗
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr. Laboratory of Applied Physics,
California Institute of Technology, Pasadena, CA 91125, USA.
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena,
CA 91125, USA.
3
These authors contributed equally to this work.
∗
opainter@caltech.edu
Abstract:
A significant challenge in the development of chip-scale cavity-
optomechanical devices as testbeds for quantum experiments and classical
metrology lies in the coupling of light from nanoscale optical mode volumes
to conventional optical components such as lenses and fibers. In this work
we demonstrate a high-efficiency, single-sided fiber-optic coupling platform
for optomechanical cavities. By utilizing an adiabatic waveguide taper to
transform a single optical mode between a photonic crystal zipper cavity
and a permanently mounted fiber, we achieve a collection efficiency for
intracavity photons of 52% at the cavity resonance wavelength of
λ
≈
1538
nm. An optical balanced homodyne measurement of the displacement
fluctuations of the fundamental in-plane mechanical resonance at 3
.
3
MHz reveals that the imprecision noise floor lies a factor of 2
.
8 above the
standard quantum limit (SQL) for continuous position measurement, with
a predicted total added noise of 1
.
4 phonons at the optimal probe power.
The combination of extremely low measurement noise and robust fiber
alignment presents significant progress towards single-phonon sensitivity
for these sorts of integrated micro-optomechanical cavities.
© 2013 Optical Society of America
OCIS codes:
(230.5298) Photonic crystals; (230.3120) Integrated optics devices;
(120.4880) Optomechanics; (280.4788) Optical sensing and sensors; (350.4238) Nanophoton-
ics and photonic crystals.
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OSA
6 May
2013
| Vol.
21,
No.
9 | DOI:10.1364/OE.21.011227
| OPTICS
EXPRESS
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width, correspondingly shorter tapering region, and the ability to eliminate the support tether. However such an
approach would be less robust to slight fiber misalignments arising from fabrication variability of the V-groove
size on the order of 100 - 200 nm. These misalignments are small relative to the 10
μ
m mode diameter used in
the reported device, but would cause substantial mode-mismatch in any scheme with a short waveguide taper.
The optimal mode-matching of the junction would also likely be diminished, as increased mode confinement in
the waveguide tip would render the profile less circularly symmetric.
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mode, for an optically resonant measurement of position in the sideband unresolved regime the phase noise does
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a small component to the imprecision noise floor. Measurements of the phase and intensity noise of our laser
reveal no excess intensity noise and a flat frequency NPSD of
S
ωω
=
5
×
10
3
rad
2
Hz in the frequency range of
interest. Consequently, for the probe powers used in this work, the excess back-action due to technical laser noise
is negligible, and the phase noise contribution to the noise floor lies roughly 60 dB below the shot noise.
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1. Introduction
Nanoscale structures in the form of photonic and phononic crystals have recently been shown
to feature significant radiation pressure interactions between localized optical cavity modes
and internal nanomechanical resonances [1]. Alongside similar advances in the microwave do-
main [2], optomechanical crystals have recently been used to laser cool a nanomechanical os-
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| OPTICS
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cillator to its quantum ground state [3]. The ability to measure and control the quantum state
of such an object ultimately hinges on the quantum efficiency of the optical transduction of
motion. Here we demonstrate high-efficiency optical coupling between an optomechanical zip-
per cavity [4] and a permanently mounted optical fiber through adiabatic mode conversion.
This optical coupling technique greatly improves the collection efficiency of light from these
types of optomechanical cavities over existing methods, and brings the minimum total added
measurement noise to within a factor of 3 of the standard-quantum-limit of continuous position
measurement.
In a weak measurement of position through a parametrically coupled optical cavity there are
two intrinsic sources of measurement noise. Shot noise of the probe laser and excess quantum
vacuum noise due to optical signal loss set the fundamental noise floor of the measurement.
When converted into units of mechanical quanta this imprecision noise
N
imp
decreases with in-
creasing probe power. However, higher laser probe power comes at the cost of radiation pressure
backaction driving an additional occupation noise
N
BA
on top of the thermal mode occupation
N
th
. For an optically resonant measurement of position, the noise terms in units of mechanical
occupation quanta are
N
imp
=
κ
2
γ
64
n
c
g
2
κ
e
η
cpl
η
meas
,
N
BA
=
4
n
c
g
2
κγ
,
(1)
where
g
is the optomechanical interaction rate,
κ
and
γ
are respectively the optical and me-
chanical loss rates,
κ
e
is the extrinsic cavity loss rate,
η
cpl
is the optical efficiency between
the cavity and the detection channel, and
η
meas
accounts for excess technical noise and signal
loss accumulated in experiment-specific optical components. As intracavity photon number
n
c
is varied, an optimal input power
P
min
is reached where the imprecision noise and back-action
noises are equal and the total added noise is minimized to
N
min
=
(
N
imp
+
N
BA
)
min
=
1
2
√
η
cpl
η
meas
κ
e
/
κ
.
(2)
In the ideal case this point, known as the standard-quantum-limit (SQL), adds 1
/
2 quanta of
noise to the measurement, equal to the zero-point fluctuations of the oscillator [5]. The SQL
can only be reached in the limit of noise-free, lossless detection (
η
cpl
η
meas
=
1) and perfect
waveguide loading (
κ
e
=
κ
). Thus,
N
min
is a suitable figure of merit for the ultimate quantum
efficiency of an optomechanical detector of position. Although experiments in both the optical
and microwave domains have brought the imprecision noise level down to below 1
/
4 quanta [6,
7], and recent microwave experiments have achieved total added noise within a factor of 4 of
the SQL [2], current state-of-the-art optical devices have been limited to 14
−
80 times the
SQL [3, 8, 9]. Such experiments are limited partly by technical noise (e.g. added noise from
amplifiers), but a substantial amount of imprecision is introduced by poor quantum efficiency
of the optical readout. Here we identify another figure of merit to allow for cross-platform
comparison of detection methods. The quantum efficiency of a general measurement apparatus
will be limited to
η
CE
=
η
cpl
κ
e
/
κ
, the collection efficiency of intracavity photons into the
detection channel before further signal processing.
To date, most nanoscale optomechanical experiments utilize evanescent coupling between
the optical cavity and an adiabatically tapered optical fiber [10, 11]. While this method offers
low parasitic losses, standing wave resonators such as the optomechanical cavities considered
here radiate symmetrically into two oppositely propagating modes of the fiber, each at a rate
κ
e
.
Thus the fraction of
n
c
routed into the detection channel
κ
e
/
κ
=
κ
e
/
(
κ
i
+
2
κ
e
)
does not exceed
1
/
2 even in the ideal case of negligible intrinsic cavity loss rate
κ
i
. To reach the overcoupled
regime
κ
e
/
κ
>
1
/
2, a single-sided coupling scheme is necessary.
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| OPTICS
EXPRESS
11229
h
overlap
= 95 %
h
tether
= 95 %
5 μm
100 μm
a
h
taper
= 98 %
2
μm
01
c
Q
e
= 15,000
h
mirror
= 95 %
2
μm
-0.2 0 0.2
d
Q
i
= 75,000
2
μm
-1 0
1
25 μm
e
b
f
Fig. 1. Scanning electron microscope (SEM) images illustrating the optical coupling
scheme and mode conversion junctions, with overlayed mode profiles simulated via Finite-
Element-Method (FEM) of optical power in (c) and (d), and electric field in (e) and (f).
(a) Fabricated device after fiber coupling via self-aligned v-groove placement. (b) Detailed
view of the zipper cavity. (c) The optical-fiber/Si
3
N
4
-waveguide junction. (d) The waveg-
uide with supporting tethers after adiabatically widening to 1
.
5
μ
m. (e) Photonic crystal
taper section. (f) Photonic crystal defect cavity.
2. Single-sided fiber coupling
We have implemented a coupling scheme which routes light from a photonic crystal zipper
cavity to a cleaved single-mode optical fiber tip with high efficiency in a fully single-sided
interface. The fiber is self-aligned in a Si V-groove (Fig. 1), secured with epoxy, and butt-
coupled to a mode-matched Si
3
N
4
waveguide, which adiabatically widens [12–15] to match the
width of the zipper cavity nanobeam. The waveguide then end-couples to the cavity through a
truncated photonic crystal mirror. The robust fiber alignment offers another key advantage over
evanescent and grating-coupler techniques that call for nanometer-scale-sensitive positioning
to achieve appreciable mode overlap. Nano-positioning is difficult and expensive to implement
in cryogenic setups due to footprint and imaging requirements, whereas the coupler presented
here can be installed directly into any system with a fiber port. While the reported coupling
method is particularly useful for attaining single-sided coupling to photonic crystal cavities, it
could also be adapted to traveling wave resonators by employing a side-coupled geometry in
an on-chip analogue to fiber-taper probing.
We now describe the optimization of the optical efficiency. To determine the optimal width
of the Si
3
N
4
waveguide, we compute the guided transverse modes of both the waveguide and
the optical fiber at the target wavelength of
λ
=
1550 nm using a finite-element-method (FEM)
solver [16]. The coupling efficiency at the fiber-waveguide junction is calculated from the mode
profiles using a mode overlap integral [17]. For a 400 nm thick Si
3
N
4
membrane, the optimal
waveguide width is
w
=
230 nm with a transmission efficiency of
η
overlap
=
95% as depicted
in Fig. 1(c). Then
w
increases gradually to the nanobeam width of 850 nm. To obtain high
transmission efficiency in this tapered waveguide section, the rate of change of
w
along the
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| Vol.
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| OPTICS
EXPRESS
11230
propagation direction
z
must be small enough to satisfy the adiabatic condition [17]
dw
/
dz
Δ
n
eff
at every point along the taper, where
Δ
n
eff
is the difference in effective index between the
fundamental waveguide mode and any other guided or radiation mode. The actual transmission
efficiency is calculated using a finite-difference-time-domain (FDTD) simulation [18]. For a
400
μ
m long taper with a cubic shape between the junctions shown in Figs. 1(c,d), we obtain a
transmission efficiency of
η
taper
=
98%.
The high-stress Si
3
N
4
film used here, desirable for its excellent optical and mechanical qual-
ity, deforms out of plane significantly when suspended over lengths greater than 10 - 20
μ
m. For
this reason a 70 nm wide tether is placed near the fiber-waveguide junction (Fig. 1(c)) to fix the
waveguide taper in position for optimal fiber alignment [19]. The scattering loss of the tether is
computed using FDTD and the transmission efficiency is calculated to be
η
tether
=
95%. A sec-
ond set of 150 nm wide tethers (Fig. 1(d)) is placed just before the cavity, in order to isolate the
optomechanical crystal from the low-frequency vibrational modes of the tapered waveguide.
The waveguide is temporarily widened to 1
.
5
μ
m at this point, rendering the scattering loss due
to the tethers negligible.
Finally, the uniform dielectric waveguide adiabatically transitions into a one-dimensional
photonic crystal mirror by linearly increasing the radius of the holes while keeping the lattice
constant fixed. An 8 hole photonic crystal taper (Fig. 1(e)) is sufficient to achieve an efficiency
of
η
mirror
=
95%. The coupling rate to the cavity is controlled by varying the number of mir-
ror periods after the taper. The photonic crystal cavity mode (Fig. 1(f)) is shared between the
waveguide beam and a near-field, flexibly supported test beam, with the optomechanical cou-
pling arising from the sensitivity of the resonance frequency to the beam separation. In this
manner the waveguide structure serves as an optical readout of the test beam motion with an
optimal round-trip efficiency of
η
rt
=(
η
overlap
η
taper
η
tether
)
2
η
mirror
=
74%. For the purposes of
optomechanical transduction, the relevant figure is the single-pass transmission efficiency be-
tween the cavity output and the fiber, which ideally is
η
cpl
=
√
η
rt
=
86%.
3. Optical characterization
A broadband scan of the reflection from the fiber-coupled device is shown in Fig. 2(a). For
probe wavelengths detuned from cavity resonances, the photonic crystal depicted in Figs. 1(d,e)
functions as a near-unity reflectivity mirror. Thus a low-finesse Fabry-Perot cavity is formed
in the waveguide between the photonic crystal and the cleaved fiber facet (with reflectivity
R
=
3
.
5%) of Fig. 1(b). By fitting the visibility of the fringes to
V
=
η
cpl
(
1
−
R
)
/
(
√
R
(
1
−
η
2
cpl
))
,
the curve in Fig. 2(a) provides a convenient calibration of optical efficiency, which is plotted in
Fig. 2(b) versus wavelength and reveals
η
cpl
=
74
.
6% for resonant measurements of the primary
mode at
λ
=
1538 nm. The coupling depth of the mode is determined by fitting the resonance
dip to a coupled-cavity model incorporating the photonic crystal and Fabry-Perot interference
effects, yielding
κ
e
/
κ
=
0
.
7. To verify this model, we study a series of devices with varying
numbers of mirror-type holes between the photonic crystal taper and cavity. In Fig. 2(c), the
total optical quality factor (
Q
t
) is plotted in green alongside intrinsic (
Q
i
) and extrinsic (
Q
e
)
components as determined by the coupled cavity fit (blue and red points respectively). As the
mirror holes are removed,
Q
e
decreases in good agreement with simulation (dotted line), and
the limiting component of
Q
t
transitions from
Q
i
to
Q
e
. By converting quality factor into loss
rate and calculating the coupling depth of the device series, a clear trend from undercoupling
to overcoupling is evident in Fig. 2(d). The device under test in this work features
κ
e
/
κ
=
0
.
7,
bringing the fiber collection efficiency figure of merit to
η
CE
=
52%. For comparison, systems
in which ground-state occupancy of a single mechanical mode has been achieved have featured
η
CE
=
37 % [2] and
η
CE
=
9 % [3].
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Received 8 Feb 2013; revised 22 Apr 2013; accepted 22 Apr 2013; published 1 May 2013
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OSA
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| Vol.
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No.
9 | DOI:10.1364/OE.21.011227
| OPTICS
EXPRESS
11231
10
4
10
5
10
6
Quality Factor
a
b
c
0.2
0.4
0.6
0.8
1.0
543210
Coupling Mirror Periods
Critical Coupling,
Perfect Overcoupling,
d
1560
Wavelength (nm)
0.90
5
5
5
1520
1510
1500
1530
1540
1550
0.85
0.80
0.75
0.70
0.65
3.5
Reflection (a.u.)
2.5
1.5
0.5
h
cp
l
Fig. 2. Optical response of the system. (a) A wide range reflection scan reveals Fabry-
Perot interference fringes off-resonant from the photonic crystal cavity, and sharp dips on
resonance. The visibility of the fringes reveals that
η
cpl
has the wavelength dependence
shown in solid green in (b), with
η
cpl
=
74
.
6 % at the cavity resonance wavelength of
λ
≈
1538 nm. The dashed green line denotes the simulated ideal efficiency of the coupler. (c)
By tuning the number of mirror-type-hole periods between the coupling section and cavity
section of the photonic crystal, the total quality factor
Q
t
(green points) of the resonance
transitions from being limited by intrinsic loss
Q
i
(blue points) to extrinsic loss
Q
e
(red
points) in good agreement with simulation (dashed red curve). (d) The ratio of extrinsic
loss rate
κ
e
to total loss rate
κ
progresses from the strongly undercoupled regime to the
strongly overcoupled regime with the mirror variation.
Laser
L
L
La
La
La
La
La
a
se
se
se
se
r
r
r
r
Laser
VOA
FPC
0.1%
99.9%
VBS1
FS
50%
50%
VBS2
PID
DUT
BPD
Vacuum
LO
Signal
ESA
Diplexer
Fig. 3. Experimental setup. FPC: fiber polarization controller. VOA: variable optical at-
tenuator. VBS: variable beam splitter. LO: local oscillator. DUT: device under test. FS:
fiber stretcher. PID: proportional-integral-derivative controller. BPD: balanced photodiode.
ESA: electronic spectrum analyzer.
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Received 8 Feb 2013; revised 22 Apr 2013; accepted 22 Apr 2013; published 1 May 2013
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OSA
6 May
2013
| Vol.
21,
No.
9 | DOI:10.1364/OE.21.011227
| OPTICS
EXPRESS
11232