Optimized optomechanical crystal cavity with acoustic radiation shield
Jasper Chan, Amir H. Safavi-Naeini, Jeff T. Hill, Seán Meenehan, and Oskar Painter
Citation: Appl. Phys. Lett. 101, 081115 (2012); doi: 10.1063/1.4747726
View online: http://dx.doi.org/10.1063/1.4747726
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Optimized optomechanical crystal cavity with acoustic radiation shield
Jasper Chan, Amir H. Safavi-Naeini, Jeff T. Hill, Se
an Meenehan, and Oskar Painter
a)
Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena,
California 91125, USA
(Received 12 June 2012; accepted 9 August 2012; published online 23 August 2012)
We present the design of an optomechanical crystal nanobeam cavity that combines finite-element
simulation with numerical optimization, and considers the optomechanical coupling arising
from both moving dielectric boundaries and the photo-elastic effect. Applying this methodology
results in a nanobeam with an experimentally realized intrinsic optical
Q
-factor of 1
:
2
10
6
,
a mechanical frequency of 5.1 GHz, a mechanical
Q
-factor of 6
:
8
10
5
(at
T
¼
10 K), and a
zero-point-motion optomechanical coupling rate of
g
¼
1.1 MHz.
V
C
2012 American Institute of
Physics
.[
http://dx.doi.org/10.1063/1.4747726
]
The use of radiation pressure forces to control and mea-
sure the mechanical motion of engineered micro- and nano-
mechanical objects has recently drawn significant attention
in fields as diverse as photonics,
1
precision measurement,
2
and quantum information science.
3
A milestone of sorts in
cavity circuit- and optomechanics is the recent
4
,
5
cooling of
a mechanical resonator to a phonon occupancy
h
n
i
1 using
cavity-assisted radiation pressure backaction.
6
,
7
Backaction
cooling involves the use of an electromagnetic cavity with
resonance frequency,
x
o
, sensitive to the mechanical dis-
placement,
x
, of the mechanical resonator. The canonical
system is a Fabry-Perot cavity of length
L
with one end mir-
ror fixed and with the other end mirror of mass
m
eff
mounted
on a spring with resonance frequency
x
m
. The coupling
between the electromagnetic field and mechanics is quanti-
fied by the frequency shift imparted by the zero-point motion
of the mechanical resonator, given by
g
¼
g
OM
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
=
2
m
eff
x
m
p
,
where
g
OM
¼
@
x
o
=@
x
¼
x
o
=
L
.
One of many technologies recently developed to make
use of radiation pressure effects are optomechanical crystals
(OMCs).
8
,
9
Optomechanical crystals, in their most general
form, are quasi-periodic nanostructures in which the propa-
gation and coupling of optical and acoustic waves can be
engineered. In this work, we present the comprehensive
design, fabrication, and characterization of a quasi-1D OMC
cavity formed from the silicon device layer of a silicon-on-
insulator (SOI) microchip. Our design incorporates both
moving-boundary and photo-elastic (electrostriction) radia-
tion pressure contributions, and simultaneously optimizes for
optical and acoustic parameters.
The nominal unit cell of a nanobeam OMC, geometri-
cally a silicon block with an oval hole in it, is shown sche-
matically in Fig.
1(a)
. The corresponding optical and
mechanical bandstructure diagrams are shown in Figs.
1(c)
and
1(d)
, respectively. As indicated by the gray shaded
region in the photonic bandstructure, the continuum of
unguided optical modes above the light line precludes the
existence of a complete photonic band gap (only a quasi-
bandgap exists for the guided modes of the beam). The
physical dimension of the unit cell block (see caption of
Fig.
1(a)
) is chosen to yield a photonic quasi-bandgap sur-
rounding a wavelength
k
1550 nm. The corresponding me-
chanical bandstructure has a series of acoustic bands in the
GHz frequency range (the ratio of the optical frequencies to
that of the mechanical frequencies is roughly the ratio of the
speed of light to sound in silicon). By classifying the acous-
tic bands by their vector symmetry, we can again define a
quasi-bandgap for the mechanical system in terms of modes
of common symmetry (modes of different symmetry may
couple weakly due to symmetry breaking introduced by the
fabrication process, an issue we discuss and mitigate later in
the design process).
Using previously developed design intuition,
9
we choose
to focus on the optical “dielectric” band at the
X
-point and the
mechanical “breathing mode” band at the
C
-point (these bands
emphasized by thicker lines in Figs.
1(c)
and
1(e)
). Figs.
1(d)
and
1(f)
show the optical dielectric band
X
-point and the breath-
ing mode
C
-point frequency as the unit cell is transformed
smoothly from that in Fig.
1(a)
to that in Fig.
1(b)
. We see that
such a transition—simultaneousl
y reducing the lattice constant
a
and decreasing the hole aspect ratio,
r
h
h
y
=
h
x
—causes an
increase in the
X
-point optical frequency and a reduction in the
C
-point mechanical frequency, pushing both optical and me-
chanical modes into the quasi-bandgap of their respective bands
(shown as shaded regions in Figs.
1(c)
and
1(e)
.
An optomechanical cavity can be formed by transitioning
from the nominal unit cell in Fig.
1(a)
, forming the “mirror”
region, to the defect unit cell of Fig.
1(b)
. This cavity can be
parameterized by the maximum change in
a
(defined as
d
max
f
1
a
=
a
nominal
g
), the number of holes in the defect
region, the maximum curvature of
a
=
a
nominal
(plotted in
Fig.
2(a)
), and the
r
h
of the center hole. Including the unit cell
geometric parameters
a
,
w
,
h
x
,and
h
y
(
t
is fixed to 220 nm), an
entirenanobeamdesignisspecifiedbythese8values.Finite-
element-method (FEM) simulations of a complete structure are
used to determine the fundamen
tal cavity mode frequencies
(
x
o
and
x
m
), motional mass
m
eff
,
10
and radiation-limited opti-
cal
Q
-factor,
Q
o
. To quantify the coupling rate, we consider
both the frequency shift due to the moving dielectric bound-
ary
11
and the photo-elastic effect,
12
so that
g
OM
¼
g
OM
;
MB
þ
g
OM
;
PE
. The moving boundary contribution is given by
10
a)
Author to whom correspondence should be addressed. Electronic mail:
opainter@caltech.edu.
0003-6951/2012/101(8)/081115/4/$30.00
V
C
2012 American Institute of Physics
101
, 081115-1
APPLIED PHYSICS LETTERS
101
, 081115 (2012)
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g
OM
;
MB
¼
x
o
2
Þ
ð
Q
^
n
Þð
D
e
E
2
jj
D
e
1
D
2
?
Þ
dS
Ð
E
D
dV
;
(1)
where
Q
is the normalized displacement field
ð
max
fj
Q
jg
¼
1
Þ
;
^
n
is the outward facing surface normal,
E
is the elec-
tric field,
D
is the displacement field, the subscripts
jj
and
?
indicate the field components parallel and perpendicular
to the surface, respectively,
e
is the material permittivity,
D
e
e
silicon
e
air
,and
D
e
1
e
1
silicon
e
1
air
.
11
A similar
result can be derived for the photo-elastic contribution from
first-order perturbation theory,
g
OM
;
PE
¼
x
o
2
h
E
j
@
e
@
a
j
E
i
Ð
E
D
dV
;
(2)
where
a
is a generalized coordinate parameterizing the am-
plitude of
Q
. In an isotropic medium with refractive index
n
,
we have
@
e
ij
@
a
¼
e
0
n
4
p
ijkl
S
kl
, where
p
is the rank-four photo-
elastic tensor and
S
is the strain tensor.
13
For silicon, a cubic
crystal with point symmetry group
m
3
m
, with the
x
-axis and
y
-axis, respectively, aligned to the
½
100
and
½
010
crystal
direction, we can use the contracted index notation with
some simplification to write
*
E
@
e
@
a
E
+
¼
e
0
n
4
ð
h
2Re
E
x
E
y
p
44
S
xy
þ
2Re
E
x
E
z
p
44
S
xz
þ
2Re
E
y
E
z
g
p
44
S
yz
þj
E
x
j
2
ð
p
11
S
xx
þ
p
12
ð
S
yy
þ
S
zz
ÞÞ
þj
E
y
j
2
ð
p
11
S
yy
þ
p
12
ð
S
xx
þ
S
zz
ÞÞ
þj
E
z
j
2
ð
p
11
S
zz
þ
p
12
ð
S
xx
þ
S
yy
Þ
i
dV
;
(3)
where
ð
p
11
;
p
12
;
p
44
Þ¼ð
0
:
094
;
0
:
017
;
0
:
051
Þ
.
12
,
13
In order to optimize the nanobeam OMC design, we have
chosen to assign a fitness value
F
g
min
f
Q
o
;
Q
cutoff
g
=
Q
cutoff
to the simulations. The additional
Q
cutoff
term (set to
3
10
6
) is used to avoid unrealizably high simulated
radiation-limited
Q
o
values from unfairly weighting the fit-
ness. With this choice, the nanobeam design has been reduced
to an 8 parameter optimization problem amenable to a variety
of numerical minimization techniques. For a computationally
expensive fitness function with a large parameter space, a
good choice of optimization algorithm is the Nelder-Mead
method;
14
as a downhill simplex method (compared to a gra-
dient descent method) the search technique is resistant to sim-
ulation noise and discontinuities. To mitigate the problem of
converging on local minima, the optimization procedure is
applied to many randomly generated initial conditions. The
resulting optimized nanobeam OMC design is shown in
FIG. 2. (a) Plot of the unit cell parameters,
a
(blue
),
h
x
(green dashed
(
),
and
h
y
(red
(
), along the length of the nanobeam, in units of
a
nominal
. (b)
The normalized optical
E
y
field and (c) the normalized mechanical displace-
ment field
j
Q
j
of the localized optical and mechanical modes, respectively.
(d) The normalized surface density of the integrand in Eq.
(1)
, showing the
contributions to
g
OM
;
MB
. (e) The normalized volumetric density of the inte-
grand in Eq.
(3)
, showing the contributions to
g
OM
;
PE
. (f) Scanning electron
microscope (SEM) image of the experimentally realized cavity.
ГX
0
50
100
150
200
250
ГX
0
2
4
6
8
frequency (GHz)
(c)
(d)
(e)
(f )
frequency (THz)
(b)
(a)
a
t
w
h
x
h
y
z
x
y
FIG. 1. The (a) nominal unit cell with
ð
a
;
t
;
w
;
h
x
;
h
y
Þ¼ð
436
;
220
;
529
;
165
;
366
Þ
nm and (b) defect unit cell with
ð
a
;
t
;
w
;
h
x
;
h
y
Þ¼ð
327
;
220
;
529
;
199
;
170
Þ
nm of the OMC nanobeam cavity. The (c) optical and (e) mechani-
cal band structure for propagation along the
x
-axis in the nominal unit cell,
with quasi-bandgaps (red regions) and cavity mode frequencies (black
dashed) indicated. In (c), the light line (green curve) divides the diagram
into two regions: the gray shaded region above representing a continuum of
radiation and leaky modes, and the white region below containing guided
modes with
y
-symmetric (red bands) and
y
-antisymmetric (blue bands) vec-
tor symmetries. In (e), modes that are
y
- and
z
-symmetric (red bands),
and modes of other vector symmetries (blue bands) are indicated. These me-
chanical simulations use the full anisotropic elasticity matrix, where
ð
C
11
;
C
12
;
C
44
Þ¼ð
166
;
64
;
80
Þ
GPa, and assume a
½
100
crystallographic ori-
entation for the
x
-axis (
½
110
orientation simulations exhibit a small,
<
10%
frequency shift in the bands below 8 GHz). The bands from which the local-
ized cavity modes are formed are shown as thicker curves. Tuning of the (d)
X
-point optical and (f)
C
-point mechanical modes of interest as the unit cell
is smoothly transformed from the nominal to the defect unit cell.
081115-2 Chan
etal.
Appl. Phys. Lett.
101
, 081115 (2012)
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Fig.
2
, with a simulated
x
o
=
2
p
of 194 THz, radiation-limited
Q
o
of 2
:
2
10
7
;
x
m
=
2
p
of 5.1 GHz, and
m
eff
of 136 fg. The
total optomechanical coupling rate,
g
=
2
p
, 860 kHz, composed
of a
90 kHz contribution from the moving boundary and a
950 kHz contribution from the photo-elastic effect.
Unaddressed so far in our design is mechanical losses, a
source of which is coupling (made unavoidable by inevitable
symmetry breaking during the fabrication process) to uncon-
fined modes of alternate symmetries present in the quasi-
bandgap of the patterned nanobeam (Fig.
1(e)
). These acoustic
radiation losses can be significantly suppressed by surround-
ing the entire nanobeam inside a second acoustic shield con-
sisting of a patterning with a complete phononic band gap.
The 2D cross phononic crystal design
10
has previously dem-
onstrated this to great effect,
15
and the wide tuneability of the
acoustic gap essentially allows the mechanical loss engineer-
ing to be decoupled from the design of the co-localized cavity
modes of the nanobeam. By adjusting
c
a
;
c
h
,and
c
t
in the 2D
unit cell (Fig.
3(a)
), it can be seen from Fig.
3(b)
that the com-
plete bandgap can be tailored to be centered on
x
m
.
The optimized nanobeam design was fabricated from
the [001] silicon device layer of a silicon-on-insulator wafer
from SOITEC (resistivity 4
20
X
cm, device layer thick-
ness 220 nm, buried-oxide layer thickness 2
3
m
m). The cav-
ity geometry was defined by electron beam lithography
followed by inductively coupled-plasma reactive-ion etching
to transfer the pattern through the 220 nm silicon device
layer. The cavities were then undercut using a 1:1 HF
:
H
2
O
solution to remove the buried oxide layer, and cleaned using
a piranha/HF cycle. The silicon surface was hydrogen-
terminated with a weak 1:20 HF
:
H
2
O solution for chemical
passivation.
16
Characterization of the OMC nanobeam devices was
performed in a continuous-flow helium cryostat, under vac-
uum and at a sample mount temperature of
T
s
6K. A
tapered optical fiber,
17
positioned in the optical near-field
(
100 nm) using a set of low-temperature-compatible piezo-
electric stages, is used to evanescently couple laser light into
and out of the devices. A tunable external cavity diode
laser (New Focus, model 6728) is used to scan across the
wavelength band from
k
¼
1520 to 1570 nm. The resulting
normalized optical transmission scan of an OMC nanobeam
cavity with resonance wavelength at
k
o
¼
1544
:
8nm
ð
x
o
=
2
p
¼
194 THz
Þ
is shown in Fig.
4(a)
. A Lorentzian fit
to the measured optical linewidth
ð
d
k
¼
1
:
7pm
Þ
yields a
fiber-taper-loaded optical
Q
-factor of
Q
o
¼
9
:
06
10
5
,which
for the measured on-resonance transmission of 55% corre-
sponds to an intrinsic
Q
-factor of
Q
o
;
i
¼
1
:
22
10
6
. The corre-
sponding total cavity (energy) decay rate, (bi-directional) fiber
taper waveguide coupling rate, and intrinsic decay rate are
j
=
2
p
¼
214 MHz
;
j
e
=
2
p
¼
55 MHz, and
j
i
=
2
p
¼
159 MHz,
respectively.
Spectroscopy of the mechanical mode is performed by
tuning the frequency of the input laser (
x
l
) to either a me-
chanical frequency red- or blue-detuned from the optical
cavity resonance (
D
ð
x
o
x
l
Þ¼
6
x
m
). The optome-
chanical backaction under such conditions results in an opti-
cally induced damping of
c
OM
¼
6
4
G
2
=
j
for
D
¼
6
x
m
,
18
where
G
¼
ffiffiffiffiffiffiffiffi
n
cav
p
g
is the parametrically enhanced optome-
chanical coupling. The intracavity photon number can be
related to the laser input power (
P
i
) through the relation,
n
cav
¼
P
i
ð
j
e
=
2
h
x
l
ðð
j
=
2
Þ
2
þ
D
2
ÞÞ
. The thermal Brownian
motion of the mechanical mode is
imprinted on the transmitted
laser light as a sideband of the input laser resonant with the op-
tical cavity resonance. As shown in Fig.
4(b)
, the resulting in-
tensity modulation of the photodetected signal gives rise to a
Lorentzian response in the photoc
urrent electronic power spec-
trum around the 5.1 GHz resonance frequency of the breathing
mechanical mode. The intrinsic mechanical damping of the
breathing mode is found by averaging the measured mechanical
linewidths under red and blue detuning,
c
i
¼ð
c
þ
þ
c
Þ
=
2
¼
7
:
5kHz, where
c
6
¼
c
i
6
c
OM
. The corresponding
t
c
a
c
t
c
h
ГXM
Г
0
1
2
3
4
5
6
7
8
frequency (GHz)
(a)
(b)
(c)
FIG. 3. (a) The unit cell of the phononic shield with parameters
ð
c
a
;
c
h
;
c
t
;
t
Þ
¼ð
534
;
454
;
134
;
220
Þ
nm. (b) Full in-plane mechanical band diagram with
complete phononic bandgap (red region) and frequency of the acoustic
mode of the cavity indicated (black dashed line). (c) SEM image showing
the phononic shield (green) around the nanobeam.
PSD (dBm/Hz)
−20
−15
−10
−5
0
5
10
15
20
λ
–
λ
o
(pm)
transmission (%)
(b)
0
20
40
60
80
100
120
cooperativity,
C
0
10
20
30
(c)
−50 −40 −30 −20 −10
0
10
20
30
40
−132.2
−132.0
−131.8
−131.6
−131.4
50
(
ω
–
ω
m
)/2
π
(kHz)
n
cav
80
100
60
40
1.7 pm
(a)
FIG. 4. (a) Normalized optical transmission spectrum, centered at
1544.8 nm, showing the fundamental optical cavity mode of the nanobeam,
with a measured intrinsic
Q
o
;
i
¼
1
:
22
10
6
. (b) Optically transduced ther-
mal noise power spectral density centered at the mechanical frequency,
x
m
=
2
p
¼
5
:
1 GHz, of the breathing mode, taken at
T
b
¼
10 K with the input
laser red-detuned (
D
¼
x
m
; red curve) and blue-detuned (
D
¼
x
m
; blue
curve) from the cavity. (c) Measured cooperativity,
C
, as function of intra-
cavity photon number,
n
cav
, for red detuning
D
¼
x
m
.
081115-3 Chan
etal.
Appl. Phys. Lett.
101
, 081115 (2012)
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intrinsic mechanical
Q
-factor is
Q
m
;
i
¼
6
:
8
10
5
. Experi-
ments with different surface preparations (with and without
HF-dip prior to testing) indicate that
Q
m
;
i
is extremely sensi-
tive to the surface quality, and is likely not limited by bulk
material damping at these temperatures.
15
A plot of the coop-
erativity,
C
¼
c
OM
=
c
i
¼
c
þ
=
c
i
1, is shown in Fig.
4(b)
ver-
sus
n
c
, yielding the zero-point optomechanical coupling rate
of
g
¼
1.1 MHz for the breathing mode, very close to the
simulated value (discrepancy here is attributed to the uncer-
tainty in the silicon photo-elastic coefficients, which were
extrapolated from data at wavelengths of 3
:
4
l
m and
1
:
15
l
m (Refs.
12
and
19
)).
These measured device parameters place the system in
the deeply resolved sideband regime (
x
m
=
j
30), suitable
for efficient laser cooling of the mechanical resonator into its
quantum ground state. Calibration of the thermal Brownian
motion
5
at low incident laser power (
n
cav
1) indicates that
the local bath temperature of the 5.1 GHz breathing mode
is
T
b
10 K, corresponding to a thermal bath occupancy of
only
n
b
¼
43, and a thermal decoherence time of
s
th
¼
hQ
m
=
k
B
T
b
0
:
5
l
s(
>
10
4
cycles of the mechanical resonator).
With a measured (simulated) granularity parameter of
g
=
j
i
0
:
0069 (0.12), such devices with further improve-
ment in the optical
Q
-factor represent a promising platform
for realizing quantum nonlinear optics and mechanics.
20
This work was supported by the DARPA/MTO ORCHID
program through a grant from AFOSR, and the Kavli Nano-
science Institute at Caltech. J.C
. and A.S.N. gratefully acknowl-
edges support from NSERC.
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