Two-Dimensional Phononic-Photonic Band Gap Optomechanical Crystal Cavity
Amir H. Safavi-Naeini,
1,2,*
Jeff T. Hill,
1,2,†
Seán Meenehan,
1,2
Jasper Chan,
1,2
Simon Gröblacher,
1,2
and Oskar Painter
1,2,‡
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology,
Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
(Received 7 January 2014; published 14 April 2014)
We present the fabrication and characterization of an artificial crystal structure formed from a thin film of
silicon that has a full phononic band gap for microwave X-band phonons and a two-dimensional pseudo-
band gap for near-infrared photons. An engineered defect in the crystal structure is used to localize optical
and mechanical resonances in the band gap of the planar crystal. Two-tone optical spectroscopy is used
to characterize the cavity system, showing a large coupling (g
0
=2π ≈ 220 kHz) between the funda mental
optical cavity resonance at ω
o
=2π ¼ 195 THz and colocalized mechanical resonances at frequency
ω
m
=2π ≈ 9.3 GHz.
DOI: 10.1103/PhysRevLett.112.153603 PACS numbers: 42.50.Wk, 42.65.-k, 62.25.-g
The control of optical [1,2] and mechanical waves [3,4]
by periodic patterning of materials has been a focus of
research for more than two decades. Periodically patterned
dielectric media, or photonic crystals, have led to a series
of scientific and technical advances in the way light can be
manipulated and have become a leading paradigm for on-
chip photonic circuits [5,6]. Periodic mechanical structures,
or phononic crystals, have also been developed to manipu-
late acoustic waves in elastic media, with myriad applica-
tions from radio-frequency filters [7] to the control of heat
flow in nanofabricated systems [8]. It has also been realized
that the same periodic patterning can simultaneously be
used to modify the propagation of light and acoustic waves
of similar wavelength [9,10]. Such phoxonic or optome-
chanical crystals can be engineered to yield strong opto-
acoustic interactions due to the colocalization of optical and
acoustic fields [11–14].
Utilizing silicon-on-insulator (SOI) wafers, similar to
that employed to form planar photonic crystal devices [6],
patterned silicon nanobeam structures have recently been
created in which strong driven interactions are manifest
between localized photons in the λ ¼ 1500 nm telecom
band and GHz-frequency acoustic modes [13,15,16].
These quasi-one-dimensional (1D) optomechanical crystal
(OMC) devices have led to new optomechanical effects,
such as the demonstration of slow light and electromag-
netically induced amplification [15], radiation pressure
cooling of mechanical motion to its quantum ground state
[16], and coherent optical wavelength conversion [17].
Although two-dimensional (2D) photonic crystals have
been used to study localized phonons and photons
[18–20], in order to create circuit-level functionality for
both optical and acoustic waves, a planar 2 D crystal
structure with simultaneous photonic and phononic band
gaps [21–23] is strongly desired. In this Letter, we
demonstrate a 2D OMC structure formed from a planar
“snowflake” crystal [23] that has both an in-plane pseudo-
band gap for telecom photons and a full three-dimensional
band gap for microwave X-band phonons. A photonic and
phononic resonant cavity is formed in the snowflake lattice
by tailoring the properties and inducing a defect in a band-
gap-guided waveguide for optical and acoustic waves, and
two-tone optical spectroscopy is used to characterize the
strong optomechanical coupling that exists between local-
ized cavity resonances.
The snowflake crystal [23], a unit cell of which is
shown in Fig. 1(a), is composed of a triangular lattice of
holes shaped as snowflakes. The dimensional parameters
of the snowflake lattice are the radius r, lattice constant a,
snowflake width w, and silicon slab thickness d.
Alternatively, the structure can be thought of as an array
of triangles connected to each other by thin bridges of
width b ¼ a − 2r. The bridge width b can be used to tune
the relative frequency of the low-frequency acousticlike
phonon bands and the higher-frequency optical-like pho-
non bands of the structure. For narrow bridge widths, the
acousticlike bands are pulled down in frequency due to a
softening of the structure for long wavelength excitations,
whereas the internal resonances of the triangles that form
the higher-frequency optical-like phonon bands are unaf-
fected. This gives rise to a band gap in the crystal, exactly
analogous to phononic band gaps in atomic crystals
between their optical and acoustic phonon branches.
As detailed in Ref. [23], for the nominal lattice parameters
and silicon device layer used in this work, ðd; r; w; aÞ¼
ð220; 210; 75; 500Þ nm, a full three-dimensional phononic
band gap between 6.9 and 9.5 GHz is formed. A corres-
ponding pseudo-band gap also exists for the fundamental
even-parity optical guided-wave modes of the slab, extending
from optical frequencies of 185–235 THz or a wavelength
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range of λ¼1620−1275 nm. Plots of the photonic and
phononic band structures are shown in Figs. 1(b) and 1(c),
respectively. A significant benefit of the planar snowflake
crystal is that the optical guided-wave band gap lies sub-
stantially below the light line at the zone boundary
(ν
ll
≳ 350 THz), enabling low-loss guiding and trapping
of light within the 2D plane.
Creation of localized defect states for phonons and
photons in the quasi-2D crystal is a two-step procedure.
First, a line defect is created, which acts as a linear
waveguide for the propagation of optical and acoustic
waves at frequencies within their respective band gaps [see
Figs. 1(d)–1(f)]. Second, the properties of the waveguide
are modulated along its length, locally shifting the bands
to frequencies that cannot propagate within the waveguide.
Thus, localized resonances are created from the band edge
of the guided modes. For the snowflake cavity studied
here, a small (3%) quadratic variation in the radius of the
snowflake holes is used to localize both the optical and
acoustic waveguide modes [23]. Simulated field profiles of
the fundamental optical resonance (ω
o
=2π ¼ 195 THz)
and strongly coupled X-band acoustic resonance (ω
m
=2π ¼
9.35 GHz) of such a snowflake crystal cavity are shown in
Figs. 1(g) and 1(h), respectively. Note that here we have
slightly rounded the features in the simulation to better
approximate the properties of the crystal that is actually
fabricated. The localized acoustic mode has a theoretical
vacuum optomechanical coupling rate of g
0
=2π ¼ 250 kHz
to the colocalized optical resonance, an effective motional
mass of 4 fg, and a zero-point-motion amplitude of
x
zpf
¼ 15 fm. The coupling rate g
0
denotes the frequency
shifts imparted on the optical cavity resonance by the
zero-point motion of the mechanical resonator.
Fabrication of the snowflake OMC cavity design con-
sists of electron beam lithography to define the snowflake
pattern, a C
4
F
8
∶SF
6
inductively coupled plasma dry etch
to transfer the pattern into the 220 nm silicon device
layer of a SOI chip, and a HF wet etch to remove the
underlying SiO
2
layer to release the patterned structure.
A zoom in of the cavity region of a fabricated device is
shown in the scanning electron microscope (SEM) image
of Fig. 2(a). Testing of the fabricated devices is performed
at cryogenic temperatures (T
b
∼ 20 K) and high vacuum
(P ∼ 10
−6
Torr) in a helium continuous-flow cryostat.
An optical taper with a localized dimple region is used
to evanescently couple light into and out of individual
Γ
MK
Γ
0
2
4
6
8
10
0
100
200
300
400
500
12
Γ
MK
Γ
6.5
10
7.0
7.5
8.0
8.5
9.0
9.5
Γ XΓ X
150
170
190
210
230
250
ν
o
(THz)ν
o
(THz)
ν
m
(GHz)
ν
m
(GHz)
b
r
w
d
∆
y
y
x
y
y
(a) (b) (c)
(d) (e) (f)
500 nm
(g)
(h)
FIG. 1 (color online). (a) Snowflake crystal unit cell. (b) Photonic and (c) phononic band structure of a silicon planar snowflake crystal
with ðd; r; w; aÞ¼ð220; 200; 75; 500Þ nm. Photonic band structures are computed with the MPB [24] mode solver, and phononic band
structures are computed with the
COMSOL
[25] finite-element method (FEM) solver. In the photonic band structure, only the fundam ental
even-parity optical modes (solid blue curves) of the silicon slab are shown and the gray shaded area indicates the region above the light
line of the vacuum cladding. The dashed gray curves are leaky resonances above the light line. (d) Unit cell schematic of a linear
waveguide formed in the snowflake crystal, in which a row of snowflake holes are removed and the surrounding holes are moved
inwards by W, yielding a waveguide width Δ
y
¼
ffiffiffi
3
p
a − 2W. Guided modes of the waveguide propagate along x. (e) Photonic and (f)
phononic band structure of the linear waveguide with ðd; r; w; a; WÞ¼ð220; 210; 75; 500; 200Þ nm. The solid blue curves are
waveguide bands of interest; dashed lines are the other guided modes; shaded light blue regions are band gaps of interest; green tick
mark indicates the cavity mode frequencies; gray regions denote the continua of propagating modes outside of the snowflake crystal
band gap. (g) FEM simulated mode profile of the fundamental optical resonance at ω
o
=2π ¼ 195 THz (λ
o
¼ 1530 nm). The E
y
component of the electric field is plotted here, with red (blue) corresponding to positive (negative) field amplitude. (h) FEM simulated
mechanical resonance displacement profile for mode with ω
m
=2π ¼ 9.35 GHz and g
0
=2π ¼ 250 kHz. Here, the magnitude of the
displacement is represented by color (large displacement in red, zero displacement in blue).
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devices with high efficiency [see Fig. 2(b)]. The schematic
of the full optical test setup used to characterize the
snowflake cavities is shown in Fig. 2(c) and described in
the figure caption. The optical properties of the localized
resonances of the snowflake cavity are determined by
scanning the frequency of a narrow-band tunable laser
across the λ ¼ 1520–1570 nm wavelength band and meas-
uring the transmitted optical power on a photodetector.
From the normalized transmission spectrum, the resonance
wavelength, the total optical cavity decay rate, and the
external coupling rate to the fiber taper waveguide of the
fundamental optical resonance for the device studied here
are determined to be λ
o
¼ 1529.9 nm, κ=2π ¼ 2.1 GHz,
and κ
e
=2π ¼ 1.0 GHz, respectively, corresponding to a
loaded (intrinsic) optical Q factor of 9.3 × 10
4
(1.8 × 10
5
).
The mechanical properties of the cavity device are
measured using a variant of the optical two-tone spectros-
copy used to demonstrate slow light and electromag-
netically induced transparency (EIT) in optomechanical
cavities [15,26]. In this measurement scheme, the input
laser frequency (ω
l
) is locked off resonance from the cavity
resonance (ω
o
) at a red detuning close to the mechanical
frequency of interest Δ ≡ ω
o
− ω
l
≈ ω
m
. Optical side-
bands of the input laser are generated using an electro-
optic intensity modulator (EOM) driven by a port of a
microwave vector network analyzer (VNA). This modu-
lated laser light is then sent into the cavity. A sweep of the
VNA modulation frequency (δ) scans the modulated laser
sidebands, causing the higher-frequency sideband to cross
over the optical cavity resonance for Δ > 0. The carrier and
sidebands transmitted through the system are then mixed
at a photodetector, with the resulting microwave signal at
frequency δ sent into the second port of the VNA. Thus, a
reading of the s
12
ðδÞ scattering parameter detected by
the VNA yields the optical response of the structure to
the higher-frequency modulated sideband of the laser.
In addition to a broad (∼GHz) feature due to the optical
cavity response, a series of narrow features (∼MHz) can be
observed in the phase of s
12
ðδÞ, corresponding to coherent
coupling between the sideband photons and mecha-
nical resonances of the structure [15,26], as shown in
Figs. 3(a) and 3(b).
The normalized phase response (angle[s
12
ðδÞ]) of the
snowflake cavity is shown in Figs. 3(a) and 3(b) for low
and high optical input power, respectively. Here, laser
power is indicated by estimated intracavity photon number
n
c
, and the measured s
12
parameter is normalized by the
response of the system with the laser detuned far from
the cavity resonance (> 20 GHz). The measured laser
power P
l
along with κ, κ
e
, and Δ are used to obtain
n
c
¼ðP
l
=ℏω
o
Þfðκ
e
=2Þ=½Δ
2
þðκ=2Þ
2
g. A zoom in of the
λ 1520 nm
- 1570 nm
VOA
calibration and EIT
spectroscopy
FPC
FPC
PD2
taper
λ-meter
=
VNA
EOM
PD1
SW1
(a) (b)
(c)
500 nm
FIG. 2 (color online). (a) SEM image of the fabricated snow-
flake crystal structure. (b) Schematic showing the fiber-taper-
coupling method used to optically excite and probe the snowflake
cavity. (c) Experimental setup for optical and mechanical
spectroscopy of the snowflake cavity; PD ≡ photodetector,
VOA ≡ variable optical attenuator, FPC ≡ fiber polarization
controller, λ meter ≡ optical wave meter, EOM ≡ electro-optic
modulator, and VNA ≡ vector network analyzer.
n
c
=22,000
9.3 9.31 9.32
0.1
0.3
9.3 9.31 9.32
0.1
0.3
n
c
=550
a
b
(b)
(a)
-0.1
0
0.1
0.2
0.3
0.4
angle[
s
12
] (radians)
(c)
0
500
1000
1500
2000
2500
0
0.5x10
4
mechanical linewidth (kHz)
intracavity photon number, n
c
modulation frequency, δ (GHz)
2468101214
-0.1
0
0.1
0.2
0.3
0.4
angle[
s
12
] (radians)
1.0x10
4
1.5x10
4
2.0x10
4
FIG. 3 (color online). (a) Low (n
c
¼ 550) and (b) high
(n
c
¼ 2.2 × 10
4
) power EIT spectra of the snowflake cavity with
nominal parameters described in the text. The insets show a zoom
in of the interference resulting from the optomechanical inter-
action between the optics and mechanics. The fits shown in the
insets are used to extract the optomechanical coupling (γ
OM
)
and intrinsic mechanical loss rate (γ
i
) for every optical power.
(c) Plot of the resulting fit mechanical damping rates versus n
c
. γ
i
data are shown as squares, and γ
OM
are shown as circles, with the
low (high) frequency mode shown in green (purple). Dashed lines
correspond to linear fits to the γ
OM
data.
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s
12
spectra near cavity resonance is shown in the insets
to Figs. 3(a) and 3(b), where two sharp dips are evident,
corresponding to coupling to mechanical resonances of the
snowflake cavity. The frequencies of the mechanical modes
are in the X-band as expected, with ω
m;1
=2π ¼ 9.309 GHz
and ω
m;2
=2π ¼ 9.316 GHz.
Radiation pressure forces couple the mechanical motion
to the optical field. This can lead to damping of the
mechanical resonance and an optically induced increase
in its linewidth [16]. In the sideband resolved, weak-
coupling limit, the optomechanical coupling is given by
γ
OM
≡ 4g
2
0
n
c
=κ, where g
0
is the vacuum coupling rate and
γ
OM
is an optically induced damping of the mechanical
resonance. The depth of each resonance is given by the
cooperativity C ¼ γ
OM
=γ
i
, whereas the resonance width is
given by γ ¼ γ
OM
þ γ
i
, where γ
i
is the intrinsic mechanical
damping [15]. From the visibility and width of the
mechanical resonance dips, γ
OM
and γ
i
are extracted and
plotted versus n
c
in Fig. 3(c). The linear slope of γ
OM
versus n
c
yields a vacuum coupling rate for the higher
(lower) frequency mechanical mode of g
0
=2π ¼ 220 kHz
(180 kHz). The intrinsic mechanical damping rate is also
seen to slowly rise with optical input power, a result of
parasitic optical absorption in the patterned silicon cavity
structure [16].
In order to explain the presence of two strongly coupled
mechanical resonances in the measured s
12
spectrum, we
note that the flat dispersion of the acoustic waveguide mode
from which the cavity is formed [see Fig. 1(f)] causes the
spectrum of localized mechanical cavity modes to be highly
sensitive to unavoidable fabrication disorder. The localized
optical and mechanical modes for 50 different disordered
structures were calculated numerically, the results of which
are summarized in Fig 4. Disorder was introduced into the
structures by varying the width and radius of the snowflake
holes in a normal distribution with 2% standard deviation.
Roughly 10% of the simulated disordered structures
yielded localized mechanical resonances with frequency
splitting less than 20 MHz and large optomechanical
coupling, similar to that of the measured device.
The snowflake 2D-OMC structure presented here
provides the foundation for developing planar circuits
for interacting optical and acoustic waves. Such circuits
allow for the realization of coupled arrays of devices for
advanced photonic or phononic signal processing, such as
the dynamic trapping and storage of optical pulses [27] or
the tunable filtering and routing of microwave-over-optical
signals. In the realm of quantum optomechanics, planar
2D-OMC structures should enable operation at much
lower millikelvin temperatures, due to their improved
connectivity and thermal conductance, where thermal noise
is absent and quantum states of mechanical motion may be
prepared and measured via quantum optical techniques.
2D-OMCs have also been theoretically proposed as the
basis for quantum phononic networks [28] and for the
exploration of quantum many-body physics in optome-
chanical metamaterials [29].
The authors would like to thank T. P. M. Alegre for
contributions. This work was supported by the DARPA
ORCHID and MESO programs, the Institute for Quantum
Information and Matter, a NSF Physics Frontiers Center
with support of the Gordon and Betty Moore Foundation,
and the Kavli Nanoscience Institute at Caltech. A. H. S.-N.
and J. C. gratefully acknowledge support from NSERC. S.
G. was supported by a Marie Curie International Outgoing
Fellowship within the 7th European Community
Framework Programme.
*
Present address: Department of Physics, ETH Zürich,
CH-8093 Zürich, Switzerland.
†
Present address: Edward L. Ginzton Laboratory, Stanford
University, Stanford CA 94305, USA.
‡
opainter@caltech.edu
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![FIG. 1 (color online). (a) Snowflake crystal unit cell. (b) Photonic and (c) phononic band structure of a silicon planar snowflake crystal with ðd; r; w; aÞ ¼ ð220; 200; 75; 500Þ nm. Photonic band structures are computed with the MPB [24] mode solver, and phononic band structures are computed with the COMSOL [25] finite-element method (FEM) solver. In the photonic band structure, only the fundamental even-parity optical modes (solid blue curves) of the silicon slab are shown and the gray shaded area indicates the region above the light line of the vacuum cladding. The dashed gray curves are leaky resonances above the light line. (d) Unit cell schematic of a linear waveguide formed in the snowflake crystal, in which a row of snowflake holes are removed and the surrounding holes are moved inwards by W, yielding a waveguide width Δy ¼ ffiffiffi 3](/figures/fig-1-color-online-a-snowflake-crystal-unit-cell-b-photonic-vjmfx8je.webp)
