Two-dimensional phononic-photonic bandgap optomechanical crystal cavity
Amir H. Safavi-Naeini,
1, 2,
∗
Jeff T. Hill,
1, 2, †
Se
́
an Meenehan,
1, 2
Jasper Chan,
1, 2
Simon Gr
̈
oblacher,
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, CA 91125
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125
(Dated: January 8, 2014)
We present the fabrication and characterization of an artificial crystal structure formed from a thin-film of
silicon which has a full phononic bandgap for microwave
X
-band phonons and a two-dimensional pseudo-
bandgap for near-infrared photons. An engineered defect in the crystal structure is used to localize optical
and mechanical resonances in the bandgap of the planar crystal. Two-tone optical spectroscopy is used to
characterize the cavity system, showing a large coupling (
g
0
/
2
π
≈
220 kHz) between the fundamental optical
cavity resonance at
ω
o
/
2
π
=
195 THz and co-localized mechanical resonances at frequency
ω
m
/
2
π
≈
9
.
3 GHz.
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 dielec-
tric media, or photonic crystals, have led to a series of sci-
entific and technical advances in the way light can be ma-
nipulated, and has become a leading paradigm for on-chip
photonic circuits [5, 6]. Periodic mechanical structures, or
phononic crystals, have also been developed to manipulate
acoustic waves in elastic media, with myriad applications
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 wave-
length [9, 10]. Such phoxonic or optomechanical crystals can
be engineered to yield strong opto-acoustic interactions due to
the co-localization of optical and acoustic fields [11–14].
Utilizing silicon-on-insulator (SOI) wafers, similar to that
employed to form planar photonic crystal devices [6], pat-
terned silicon nanobeam structures have recently been created
in which strong driven interactions are manifest between lo-
calized photons in the
λ
=
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 opto-mechanical effects, such as the demonstra-
tion of slow light and electromagnetically induced amplifi-
cation [15], radiation-pressure cooling of mechanical motion
to its quantum ground state [16], and coherent optical wave-
length conversion [17]. Although two-dimensional (2D) pho-
tonic 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 2D crystal struc-
ture with simultaneous photonic and phononic bandgaps [21–
23] is strongly desired. In this Letter we demonstrate a 2D
OMC structure formed from a planar “snowflake” crystal [23]
which has both an in-plane pseudo-bandgap for telecom pho-
tons and a full three-dimensional bandgap for microwave
X
-
∗
Current address: ETH Z
̈
urich and Stanford University
†
Current address: Stanford University
‡
Electronic address: opainter@caltech.edu
band phonons. A photonic and phononic resonant cavity is
formed in the snowflake lattice by tailoring the properties of
a bandgap-guided waveguide for optical and acoustic waves,
and two-tone optical spectroscopy is used to characterize the
strong optomechanical coupling that exists between localized
cavity resonances.
The snowflake crystal [23], a unit cell of which is shown
in Fig. 1a, 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
=
a
−
2
r
. The bridge width
b
can be used to tune the relative frequency of the low fre-
quency acoustic-like phonon bands and the higher frequency
optical-like phonon bands of the structure. For narrow bridge
width, the acoustic-like bands are pulled down in frequency
due to a softening of the structure for long wavelength excita-
tions, whereas the internal resonances of the triangles that form
the higher frequency optical-like phonon bands are unaffected.
This gives rise to a bandgap in the crystal, exactly analogous
to phononic bandgaps in atomic crystals between their optical
and acoustic phonon branches.
As detailed in Ref. [23], for the nominal lattice parame-
ters and silicon device layer used in this work,
(
d
,
r
,
w
,
a
) =
(
220
,
210
,
75
,
500
)
nm, a full three-dimensional phononic
bandgap between 6
.
9 and 9
.
5 GHz is formed. A corresponding
pseudo-bandgap also exists for the fundamental even-parity
optical guided-wave modes of the slab, extending from optical
frequencies of 185 THz to 235 THz, or a wavelength range of
λ
=
1620 nm to 1275 nm. Plots of the photonic and phononic
bandstructures are shown in Figs. 1b and c, respectively. A
significant benefit of the planar snowflake crystal is that the
optical guided-wave bandgap lies substantially 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 de-
fect is created, which acts as a linear waveguide for the prop-
agation of optical and acoustic waves at frequencies within
their respective bandgaps (see Figs. 1d-f). Second, the prop-
arXiv:1401.1493v1 [physics.optics] 7 Jan 2014
2
Γ
M
K
Γ
0
2
4
6
8
10
0
100
200
300
400
500
12
Γ
M
K
Γ
6.5
10
7.0
7.5
8.0
8.5
9.0
9.5
Γ
X
Γ
X
150
170
190
210
230
250
ν
o
(THz)
ν
o
(THz)
ν
m
(
GHz)
ν
m
(GHz)
b
r
a
w
d
∆
y
y
x
a
b
c
d
e
f
500 nm
g
h
FIG. 1:
a,
Snowflake crystal unit cell.
b,
Photonic and
c,
phononic bandstructure of a silicon planar snowflake crystal with
(
d
,
r
,
w
,
a
) =
(
220
,
210
,
75
,
500
)
nm. Photonic bandstructures are computed with the MPB [24] mode solver and phononic bandstructures are computed with
the COMSOL [25] finite-element method (FEM) solver. In the photonic bandstructure only the fundamental even-parity optical modes (solid
blue curves) of the silicon slab are shown and the grey shaded area indicates the region above the light line of the vacuum cladding. The
dashed grey 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
∆
y
=
√
3
a
−
2
W
.
e,
Photonic and
f,
phononic bandstructure of the linear waveguide with
(
d
,
r
,
w
,
a
,
W
) = (
220
,
210
,
75
,
500
,
200
)
nm. Solid blue curves are
waveguide bands of interest; shaded light blue regions are bandgaps of interest; green tick mark indicates the cavity mode frequencies.
g,
FEM simulated mode profile of the fundamental optical resonance at
ω
o
/
2
π
=
195 THz (
λ
o
=
1530 nm).
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
/
2
π
=
9
.
35 GHz and
g
0
/
2
π
=
250 kHz. Here the magnitude of the displacement is represented by color (large
displacement in red, zero displacement in blue).
erties of the waveguide are modulated along its length, locally
shifting the bands to frequencies that cannot propagate within
the waveguide. 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
/
2
π
=
195 THz) and strongly coupled
X
-band
acoustic resonance (
ω
m
/
2
π
=
9
.
35 GHz) of such a snowflake
crystal cavity are shown in Figs. 1g and h, respectively. Note
that here we have slightly rounded features in the simulation to
better approximate the properties of the crystal that is actually
fabricated. The localized acoustic mode has a theoretical op-
tomechanical coupling of
g
0
/
2
π
=
250 kHz to the co-localized
optical resonance, an effective motional mass of 4 femtograms,
and a zero-point-motion amplitude of
x
zpf
=
1
.
5 femtometers.
Fabrication of the snowflake OMC cavity design consists 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 an SOI chip,
and a HF wet etch to remove the underlying SiO
2
layer to re-
lease the patterned structure. A zoom-in of the cavity region
of a fabricated device is shown in the scanning electron mi-
croscope (SEM) image of Fig. 2a. 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
devices with high efficiency (see Fig. 2b). The schematic of
the full optical test set-up used to characterize the snowflake
cavities is shown in Fig. 2c and described in the figure cap-
tion. The optical properties of the localized resonances of the
snowflake cavity are determined by scanning the frequency
of a narrowband tunable laser across the
λ
=
1520-1570 nm
wavelength band, and measuring the transmitted optical power
on a photodetector. From the normalized transmission spec-
trum, 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
=
1529
.
9 nm,
κ
/
2
π
=
2
.
1 GHz,
and
κ
e
/
2
π
=
1
.
0 GHz, respectively, corresponding to a loaded
(intrinsic) optical
Q
-factor of 9
.
3
×
10
4
(1
.
8
×
10
5
).
Mechanical properties of the cavity device are measured
using a variant of the optical two-tone spectroscopy used to
demonstrate slow-light and electromagnetically induced trans-
parency (EIT) in optomechanical cavities [15, 26]. In this mea-
surement 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,
∆
≡
ω
o
−
ω
l
≈
3
500 nm
VOA
calibration and EIT
spectroscopy
FPC
FPC
PD2
tap
er
λ
-meter
λ
=
1520
nm
- 1570 nm
VNA
EOM
PD1
SW1
snowake OMC
a
b
c
FIG. 2:
a,
SEM image of fabricated snowflake crystal structure.
b,
Schematic showing the fiber-taper-coupling method used to optically
excite and probe the snowflake cavity.
c
, Experimental setup for opti-
cal and mechanical spectroscopy of the snowflake cavity. PD
≡
pho-
todetector, VOA
≡
variable optical attenuator, FPC
≡
fiber polariza-
tion controller,
λ
-meter
≡
optical wavemeter, EOM
≡
electro-optic
modulator, and VNA
≡
vector network analyzer.
ω
m
. Optical sideband tones are generated on the input laser
beam by using an electro-optic intensity modulator driven by
a microwave vector network analyzer (VNA). This modulated
laser light is then sent into the cavity, and the optical cavity
transmission is detected by a high speed photodetector, the
output of which is connected to the input of the VNA. A sweep
of the VNA modulation frequency (
δ
) scans the upper modu-
lated laser sideband across the optical cavity resonance, from
which the
s
12
(
δ
)
scattering parameter of the VNA yields the
optomechanical response function.
The normalized phase response (angle[
s
12
(
δ
)
]) of the
snowflake cavity is shown in Figs. 3a and 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). A
zoom-in of the
s
12
spectra near cavity resonance are shown
in the insets to Figs. 3a and b, where two sharp dips are evi-
dent, corresponding to coupling to mechanical resonances of
the snowflake cavity. The frequency of the mechanical modes
are in the
X
-band as expected, with
ω
m
,
1
/
2
π
=
9
.
309 GHz and
ω
m
,
2
/
2
π
=
9
.
316 GHz.
In the sideband resolved, weak-coupling limit, the optome-
chanical coupling is given by
γ
OM
≡
4
g
2
0
n
c
/
κ
, where
g
0
is the
vacuum coupling rate and
γ
OM
is an optically-induced damp-
ing of the mechanical resonance. The depth of each resonance
is given by the cooperativity
C
=
γ
OM
/
γ
i
, whereas the reso-
nance width is given by
γ
=
γ
OM
+
γ
i
, where
γ
i
is the intrinsic
mechanical damping. From the visibility and width of the me-
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
] (r
adians)
c
0
500
1000
1500
2000
2500
0
0.5x10
4
mechanical linewidth (kH
z)
intracavity photon number,
n
c
modulation fr
equenc
y,
δ
(
GH
z)
2
4
6
8
10
12
14
-0.1
0
0.1
0.2
0.3
0.4
angle[
s
12
] (r
adians)
1.0x10
4
1.5x10
4
2.0x10
4
FIG. 3:
a,
Low (
n
c
=
550) and
b,
high (
n
c
=
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 interaction between the optics and mechan-
ics. The fits shown in the insets are used to extract the optomechanical
coupling (
γ
OM
) and intrinsic mechanical loss rate (
γ
i
) for every opti-
cal 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.
chanical resonance dips,
γ
OM
and
γ
i
are extracted and plotted
versus
n
c
in Fig. 3c. The linear slope of
γ
OM
versus
n
c
yields
a vacuum coupling rate for the higher (lower) frequency me-
chanical mode of
g
0
/
2
π
=
220 kHz (180 kHz). The intrinsic
mechanical damping rate is also seen to slowly rise with opti-
cal 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 me-
chanical 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. 1f) 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 distri-
bution with 2% standard deviation. Roughly 10% of the sim-
ulated disordered structures yielded localized mechanical res-
onances with frequency-splitting less than 20 MHz and large