SUPPLEMENTARY INFORMATION
1
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doi: 10.1038/nature08524
Supplementary Information to “Optomechanical Crystals” manuscript
Matt Eichenfield, Jasper Chan, Ryan M. Camacho, Kerry J. Vahala, and Oskar Painter
Thomas J. Watson, Sr, Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
∗
(Dated: September 21, 2009)
I. MEASURED AND SIMULATED PROPERTIES OF THE OPTOMECHANICAL CRYSTAL NANOBEAM RESONATOR
Table S1 summarizes the properties of the breathing mechanical modes. Measured values are denoted with a tilde. The
necessary RF amplitudes and linewidths are extracted from the spectra of Fig. 3c using a nonlinear least squares fit with linear
background and a sum of as many Lorentzian functions as are visible in the spectrum. Simulated values are calculated using
methods described below.
TABLE S1: Measured and Simulated properties of the breathing mechanical modes. Tildes indicate measured quantities. The experimental
effective lengths,
L
OM
, between each breathing mode and the first three optical cavity modes are calculated using the experimentally extracted
m
eff
L
2
OM
(see Fig. 3c of main text) and dividing by the
m
eff
from the model. The superscript,
n
, in
n
L
OM
, indicates coupling of that mechanical
mode to the
n
th optical mode (see Fig. 1b of main text). See §V F for discussion on modeling
Q
m
.
Mode
ν
m
(GHz)
ν
m
(GHz)
m
eff
(fg)
1
L
OM
(
μ
m)
1
L
OM
(
μ
m)
2
L
OM
(
μ
m)
2
L
OM
(
μ
m)
3
L
OM
(
μ
m)
3
L
OM
(
μ
m)
Q
m
Q
m
1
2.254
2.254
329
4.9
2.9
6.4
5.1
7.8
4.4
2050 1280
2
2.275
2.270
399
7.1
4.5
6.2
12
9.5
9.8
1180 1130
3
2.294
2.290
628
11
N/A
6.2
5.3
7.7
4.1
1290 613
4
2.322
2.326
704
110
N/A
64
49
26
N/A
387
973
5
2.369
2.361
665
38
N/A
11
25
7.1
4.7
21600 950
II. OPTICAL ACTUATION: AMPLIFICATION AND REGENERATIVE OSCILLATION
2.186
2.187
2.188
2.189
2.19
−100
−90
−80
−70
−60
−50
ν
m
(GHz)
RF PSD (dBm)
190
μ
W
150
15
2.186
2.187
2.188
2.189
2.19
−100
−95
−90
−85
−80
−75
ν
m
(GHz)
RF PSD (dBm)
μ
W
μ
W
a
b
Dropped Power
FIG. S1:
(a)
2.19 GHz breathing mode showing spectral narrowing from 800 kHz to 1.16 kHz with increasing optical input power.
(b)
Nonlinear least-squares fit (black) to redacted high power (red) curve in
a
.
Fig. S1a shows the fundamental breathing mode of the optomechanical crystal nanobeam, pumped using the fiber taper probe
coupled to the fundamental optical mode (this particular device has a scale factor 1.07 in Fig. S4a, which is nominally identical
to Device 1 of the main text but uniformly scaled by 4%). The mechanical
Q
at a dropped optical power of 15
μ
W (input power
is 74
μ
W with 20 percent of the power coupled to the device) is approximately
Q
m
=
2700. Upon increasing the dropped power
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to 190
μ
W (an 11 dB increase), the mechanical mode power rises dramatically and the linewidth narrows to below the 4.8 kHz
resolution limit of the oscilloscope (the resolution-limited effective
Q
is thus 460,000). This sort of regenerative oscillation
1,2
(sometimes called paramteric instability) arises due to the retarded part of the optical force on the mechanical mode, which, for
a blue detuned laser input, results in amplification of the mechanical motion. Even though a large part of the signal is below
the resolution bandwidth, the linewidth at 931
μ
W can still be extracted, as there is more than 20 dB of signal to noise at the
point where the lineshape becomes wider than the resolution limit. Fig. S1b shows the nonlinear least-squares fit to the redacted
dataset, which is an excellent fit to the data and gives a linewidth of 1.16 kHz (effective
Q
m
=
1
.
8
×
10
6
).
III. EXPERIMENTAL SETUP
Taper
APD
Tunable
Infrared Laser
VOA
FPC
VOA
Oscilloscope
Optomechanical Crystal Nanobeam
FIG. S2: Experimental setup used to measure optical, mechanical, and optomechanical properties of silicon optomechanical crystal nanobeam.
The experimental setup used to measure the optical, mechanical, and optomechanical properties of the silicon optomechanical
crystal nanobeam is shown in Fig. S2. The setup consists of a bank of fiber-coupled tunable infrared lasers spanning approx-
imately 200 nm, centered around 1520 nm. After a variable optical attenuator (VOA) and fiber polarization controller (FPC),
light enters the tapered and dimpled optical fiber, the position of which can be controlled with nanometer-scale precision (al-
though vibrations and static electric forces limit the minimum stable spacing between the fiber and device to about 50 nm). The
transmission from the fiber is (optionally) passed through another VOA and finally reaches an avalanche photodiode (APD) with
a transimpedance gain of 11,000 and a bandwidth (3 dB rolloff point) of 1.2 GHz. The APD has an internal bias tee, and the RF
voltage is connected to the 50 Ohm input impedance of the oscilloscope. The oscilloscope can perform a Fourier transform (FT)
to yield the RF power spectral density (RF PSD). The RF PSD is calibrated using a frequency generator that outputs a variable
frequency sinusoid with known power.
IV. FABRICATION
The optomechanical crystal nanobeam is formed in the 220 nm thick silicon device layer of a [100] Silicon-On-Insulator (SOI)
wafer. The pattern is defined in electron beam resist by electron beam lithography. The resist pattern is transfered to the device
layer by an inductively-coupled plasma reactive ion etch with a C
4
F
8
/SF
6
gas chemistry. The nanobeam is then undercut and
released from the silica BOX layer by wet undercutting with hydrofluoric acid.
V. NUMERICAL MODELING
Modeling of both the optical and mechanical modes is done via finite element method (FEM), using COMSOL Multiphysics
3
.
Mechanical band structures are done in COMSOL. Photonic bands are done with MIT Photonic Bands
4
. The following subsec-
tions provide the description of the method used to define the FEM model of the optomechanical crystal system.
A. Extracting the geometry in the plane
To model the optomechanical crystal system, the geometry of the as-fabricated structure must be measured. As the features are
smaller than an optical wavelength, the measurements must be done by scanning electron microscope (SEM). FIG. S3a shows
an “eagle’s-eye” high-resolution SEM micrograph of a portion of device 1, with the defect centered in the image.
Digital line-scans of the micrograph are used to detect the edges of the geometry. From the extracted edge positions, the
geometry is approximated as a series of rectangular holes with two filleted ends inside of a rectangle (the beam), giving an
approximate planar geometric representation of the structure shown as an overlay in Fig. S3b. This geometric representation
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a
b
FIG. S3:
(a)
Scanning electron micrograph of fabricated silicon optomechanical crystal.
(b)
Approximated geometry shown as blue overlay
on SEM micrograph from
(a)
.
takes into account the size, position, and any curvature of each hole, giving an accurate approximation of the geometry. In the
defect region, each hole is given by its measured value. Outisde the defect region, a series of holes is used to get the average
hole shape, which is used in the model.
The SEM has been calibrated, and the dimesnions as measured by the SEM are too large by 5%. Thus, the entire planar
geometry is uniformly scaled down by 5%. Since the lattice constant,
Λ
, is a center-to-center distance between features, it is not
affected by erosion during processing, which makes it the most reliable measure of distance on the sample. After applying the
SEM calibration factor, the average lattice constant outside of the defect as measured by the SEM agrees with the value written
by the electron beam lithography tool to better than 1%. Since the SEM and lithography tool are independent, this is yet another
confirmation that the geometry has been measured correctly (the fine spectral features of the simulation are the other way to
check the geometry measurements, after comparing to measured mechanical and optical spectra).
The SOI wafer thickness is specified as 220 nm by the manufacturer. We will assume that the planar geometry extends
uniformly into the vertical direction for the entire 220 nm, since the “eagle’s-eye” view used to measure the planar geometry
does not capture any asymmetries in the vertical dimension. These vertical asymmetries are much more difficult to extract
without sacrificing the device (by focused ion beam, cleaving, etc.).
B. Young’s modulus and index of refraction
The nanobeam structures are fabricated such that the long axis (ˆ
x
) is parallel to the SOI wafer flat, which is oriented along
[
110
]
(
±
0.5
◦
). We decompose the displacement field in FEM simulations along the crystal axes and find the majority of the
strain energy is primarily stored in deformations along the family of equivalent directions specified by
110
. Because the strain
for the modes of interest are primarily along
110
, an isotropic elasticity tensor (two independent elements; see §V D) derived
from a single Young’s modulus and Poisson’s ratio is appropriate for the current level of detail. The index of refraction will also
be treated as an isotropic scalar.
As the Young’s modulus,
E
, and index of refraction,
n
, determine the phase velocity of the waves (and thus the frequency),
they can be “tuned” to make a single simulated frequency (mechanical for
E
and optical for
n
) come out exactly as measured.
The free spectral range of the modes and the relative frequencies of different types of modes are determined by the details of
the geometry, in conjunction with
E
and
n
; so although a single frequency can always be made to match experiment exactly
by scaling
E
or
n
, the wider details of the spectrum are a more accurate reflection of whether the model is a good match to the
experimental values.
After accounting for the planar geometry and scale factor, the Young’s modulus and index are tuned until the fundamental
optical mode and the fundamental breathing mechanical mode each come out exactly as measured, which occurs for
E
=
168
.
5
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GPa and
n
=
3
.
493 (the Poissons’s ratio,
ν
, is 0.28 in this work). These parameters, along with the measured geometry (as
discussed above) yields a model that produces the values in Table S1 and Fig. 3c in the main text.
C. Optics: mode maps and modeling
0.95
1
1.05
1.1
190
195
200
205
210
ν
(THz)
Λ
/
Λ
nominal
10
5
10
4
10
3
10
2
10
1
140
180
120
160
200
220
ν
(Ghz)
Q
1
2
3
4
5
2
1
3
5
4
a
b
FIG. S4:
(a)
Optical modes measured in a 200 nm laser wavelength span for a series of 20 devices.
(b)
Simulation of Device 1, which is the
device with scaling factor 1.03 (dashed line). Filled blue circles correspond to modes of the fundamental (valence) optical band; the region
shaded blue corresponds to frequencies that are no longer within the defect potential barrier hight (i.e. propagating modes). Open blue circles
correspond to transverse valence band modes; the pink shaded region shows the edge of the effective optical potential for the transverse valence
band modes. Black circles correspond to conduction band modes.
Fig. S4a shows all the optical modes measured for a series of 20 devices, which are identical up to a uniform planar scaling
that changes by 1% per device (Device 1 is the device with scale factor 1.03; this figure is just an expanded version of the bottom
panel of Fig. 3a in the main text). Because of the limited laser range, only a limited number of modes can be measured on any
given device. By measuring this series of uniformly scaled devices, a large number of modes can be seen as they are “scanned”
through the laser range.
The devices, taken together, display a lot of information about the optical spectrum, which contains a number of conspicuous
features that match well with an FEM model of the optical properties. First, the devices display a series of five relatively high
Q
modes (analogous modes of different devices are connected with red lines across different device measurements). Second, the
smallest devices show a number of low-
Q
modes at frequencies below the fifth mode. Finally, at high frequencies, the devices
display another set of low-
Q
modes, which are higher-
Q
than waveguide-like modes but not as high-
Q
as the other five modes.
These features are all consistent with the FEM optical model of Device 1. Fig. S4b shows the simulated modes of Device 1,
plotted as a function of their optical
Q
. The simulation shows that the defect confines 5 modes, with a precipitous drop in
Q
as
the modal frequencies exit the defect potential (go below the negative energy barrier height); the region of frequency space below
the negative barrier height is shaded in light blue in Figs. S4a and b. The simulation also explains the series of modes higher
in frequency, which are
not
the conduction band modes. These modes, indicated with open circles in both Fig. S4a and S4b (as
opposed to filled) circles, are the Hermite-Gauss ladder of modes with a single node transverse to the direction of propagation
(
y
direction). These modes have a lower effective index, which reduces their radiation-limited
Q
relative to modes without
transverse nodes. The simulated effective optical potential for the transverse optical modes is shown in pink. Conduction band
modes (which are not measured due to their very low optical
Q
) are shown as filled black circles.
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While the optical information provided by any single device would be difficult to unravel, the measurements of the series of
devices coupled with simulation allow us to unambiguously identify the optical spectra of every device in the series.
D. Mechanical Band Diagram of the Nanobeam’s Projection
Λ
z=0
y=0
Λ
0
1
Normalized |Q|
FIG. S5: Mechanical band diagram and corresponding normalized displacement profiles of the unit cell at the
Γ
(
k
x
=
0) and
X
(
k
x
=
π
/
Λ
)
points. In the band diagram, the mirror symmetry
σ
z
, (across the plane defined by
z
=
0) is indicated by color: red corresponds to even vector
parity (
p
z
=
1) and blue to odd vector parity (
p
z
=
−
1). Mirror symmetry
σ
y
(across the plane defined by
y
=
0) plane is indicated by the
line shape: solid corresponds to even vector parity (
p
y
=
1) and dashed to odd vector parity (
p
y
=
−
1). The mechanical mode profiles are all
viewed from a direction normal to the
z
=
0 plane unless labeled “yz”, in which case the viewing angle is normal to the
x
=
0 plane. The pinch,
accordion, and breathing mode bands are b, i, and j, respectively. As torsional modes can be difficult to interpret without isometric views, it is
noted for the reader that the mechanical modes for band e at
X
, band f at
Γ
, and band h at
X
are all torsional mechanical modes.
The equation of motion for the displacement field,
Q
(
r
)
, of a non-piezoelectric body is given by
5
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∇
·
�
c
:
∇
s
Q
(
r
)
=
ρ
∂
2
Q
(
r
)
∂
t
2
,
(S1)
where
∇
s
≡
�
∇
+
∇
T
/
2 is the symmetric gradient operator,
ρ
is the mass density, the colon denotes the double scalar (a.k.a.
double dot) product of a fourth rank and a second rank tensor, and
c
is the (fourth rank) elasticity tensor. As we are treating the
material as isotropic, the elasticity tensor reduces in Voigt notation to
5
c
−
1
=
1
E
1
−
ν
−
ν
000
−
ν
1
−
ν
000
−
ν
−
ν
10 0 0
0002
(
1
+
ν
)
00
000 0 2
(
1
+
ν
)
0
000 0 0 2
(
1
+
ν
)
,
(S2)
where
E
is Young’s modulus and
ν
is Poisson’s ratio.
The projection of the nanobeam optomechanical crystal (the infinite extension of the structure without the defect) has discrete
periodicity, satisfying
Q
(
r
)=
Q
(
r
+
Λ
ˆ
x
)
, where
Λ
is the periodicity of the lattice. Thus, by Bloch’s theorem, the solutions can
be classified according to a wave vector,
k
x
, in the first Brillouin zone,
k
x
∈
[
−
π
/
Λ
,
π
/
Λ
]
(time-reversal symmetry guarantees
that positive and negative wave vectors yield identical solutions, and the solutions can further be restricted to the first half of the
first Brillouin zone,
k
x
∈
[
0
,
π
/
Λ
]
) . As the Bloch solution is effectively restricted to a finite region of space (the unit cell), the
solutions for a given
k
x
have a discrete spectrum of eigenfrequencies, which can be labeled by a band index,
n
. Furthermore, the
structure is mirror symmetric about the
y
=
0 and
z
=
0 planes (see Fig. S5), and it can be shown that the mirror operators,
σ
y
=
100
0
−
10
001
σ
z
=
10 0
01 0
00
−
1
,
(S3)
commute with the differential operator
Ξ
≡
∇
·
c
:
∇
s
. The solutions to the wave equation can thus be further classified with
respect to their vector parity about these planes, each solution having an eigenvalue of the mirror operator such that
σ
j
Q
(
σ
−
1
r
)=
p
j
Q
(
r
)
, where
j
can be
y
or
z
, and
p
j
=
±
1. We accordingly classify the solutions to the wave equation by the wave vector
k
x
∈
[
0
,
π
/
Λ
]
,
p
y
, and
p
z
. Figure S5 shows the mechanical band diagram of the nanobeam optomechanical crystal’s projection, with
the first ten band indices,
n
, labeled a to j,
p
z
indicated by color, and
p
y
indicated by line shape. The mechanical displacement
profiles of the unit cell are shown for each band at
Γ
and
X
.
E. Mechanical Modes of the Defect
The defect in the structure breaks the discrete periodicity in the
x
direction, and the solutions to the wave equation (Eq. S1) for
the structure can no longer be classified by wave vectors and band indices. The structure still retains its
σ
y
and
σ
z
mirror planes.
In addition, the structure now has a third mirror plane (the plane
x
=
0), which divides the structure in half in the
x
direction. As
with the solutions of the projection (see discussion above), the wave equation commutes with the mirror operator
σ
x
, where
σ
x
=
−
100
0 10
0 01
.
(S4)
Each solution of the wave equation is thus an eigenvector of
σ
j
, with corresponding vector parity
p
j
=
±
1,
j
∈
[
x
,
y
,
z
]
.
The solutions to the wave equation in the defect can be viewed as being drawn from the band edges of the projection. Localized
modes are formed whenever the modes of the defect exist at a frequency for which the density of states in the projection is small
or zero. Thus, many localized mechanical modes are formed at the various band edges, as shown in Figs. 3c and 3d of the
main text, where at least 5 mechanical modes are predicted and measured for the breathing modes alone. Manifolds of localized
modes, such as the breathing modes, have identical parities with respect to
σ
y
and
σ
z
, and the parity with respect to
σ
x
alternates
as one climbs the ladder of states in the manifold.