Optomechanical Crystals
Matt Eichenfield,
1
Jasper Chan,
1
Ryan M. Camacho,
1
Kerry J. Vahala,
1
and Oskar Painter
1
1
Thomas J. Watson, Sr, Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
∗
Structured, periodic optical materials can be used to form photonic crystals capable of dispersing, routing, and
trapping light. A similar phenomena in periodic elastic structures can be used to manipulate mechanical vi-
brations. Here we present the design and experimental realization of strongly coupled optical and mechanical
modes in a planar, periodic nanostructure on a silicon chip. 200-Terahertz photons are co-localized with me-
chanical modes of Gigahertz frequency and 100-femtogram mass. The effective coupling length,
L
OM
, which
describes the strength of the photon-phonon interaction, is as small as 2.9
μ
m, which, together with minute oscil-
lator mass, allows all-optical actuation and transduction of nanomechanical motion with near quantum-limited
sensitivity. Optomechanical crystals have many potential applications, from RF-over-optical communication to
the study of quantum effects in mesoscopic mechanical systems.
Periodicity in materials yields interesting and useful phenomena. Applied to the propagation of light, periodicity gives rise to
photonic crystals[1], which can be precisely engineered to, among other things, transport and control the dispersion of light[2, 3],
tightly confine and trap light resonantly[4], and enhance nonlinear optical interactions[5]. Photonic crystals can also be formed
into planar lightwave circuits for the integration of optical and electrical microsystems[6]. Periodicity applied to mechani-
cal vibrations yields phononic crystals, which harness mechanical vibrations in a similar manner to optical waves in photonic
crystals[7, 8, 9, 10, 11, 12, 13]. As has been demonstrated in studies of Raman scattering in epitaxially grown vertical cavity
structures[14] and photonic crystal fibers[15], the simultaneous confinement of mechanical and optical modes in periodic struc-
tures can lead to greatly enhanced light-matter interactions. A logical next step is thus to create planar circuits that act as both
photonic
and
phononic crystals[16]: optomechanical crystals. In this spirit, we describe the design, fabrication, and charac-
terization of a planar, silicon-chip-based optomechanical crystal capable of co-localizing and strongly coupling 200 Terahertz
photons and 2 Gigahertz phonons. These planar optomechanical crystals bring the powerful techniques of optics and photonic
crystals to bear on phononic crystals, providing exquisitely sensitive (near quantum-limited), optical measurements of mechan-
FIG. 1:
Optomechanical crystal design
.
a
, Geometry of nanobeam structure.
b
, Optical and
c
, mechanical bands and defect modes calculated
via FEM for the projection of the experimentally-fabricated silicon nanobeam (
Λ
=
362 nm,
w
=
1396 nm,
h
y
=
992 nm,
h
x
=
190 nm, and
t
=
220 nm; isotropic Young’s modulus of 168
.
5 GPa;
n
=
3.493). In this particular structure, which will be referred to as “Device 1”,
N
defect
=
15 holes,
N
total
=
75 and the spacing between the holes varies quadratically from the lattice constant of the projection (362 nm) to
85% of that value (a “15% defect”) for the two holes straddling the central cross-bar (the other parameters of Device 1 are as listed above).
∗
e-mail: opainter@caltech.edu
arXiv:0906.1236v1 [physics.optics] 6 Jun 2009
2
ical vibrations, while simultaneously providing strong non-linear interactions for optics in a large and technologically-relevant
range of frequencies.
The geometry of the optomechanical crystal structure considered here is shown in Fig. 1a. The effectively one-dimensional
(1D) optomechanical crystal consists of a silicon nanobeam (thickness
t
not shown) with rectangular holes and thin cross-bars
connected on both sides by thin rails (we will refer to infinitely periodic constructs such as this as the “projection” of the
finite structure). Fig. 1b shows a finite-element-method (FEM) simulation of the optical band structure of the projection of a
nanobeam (see caption for parameters). The electric field profile for modes at the band edge (
k
x
=
π
/
Λ
, the boundary of first
Brillouin zone) are shown to the right of the band structure. The finite structure terminates at its supports on both ends, forming
a doubly-clamped beam. To form localized resonances in the center of the structure, the discrete translational symmetry of the
patterned beam is intentionally disrupted by a “defect”, consisting of a quadratic decrease in the lattice constant,
Λ
, symmetric
about the center of the beam for some odd number of holes,
N
defect
<
N
total
. The defect forms an effective potential for optical
modes at the band edges, with the spatial dependence of the effective potential closely following the spatial properties of the
defect[17] (as illustrated in the inset of the optical band diagram). Thus the optical modes of the infinitely-periodic structure are
confined by a quasi-harmonic potential. This effective potential localizes a “ladder” of modes with Hermite-Gauss envelopes,
analogous to the modes of the 1D harmonic potential of quantum mechanics. The localized optical modes of the finite structure
(hereafter referred to as Device 1) are also found by FEM simulation and shown in Fig. 1b to the right of the corresponding
mode of the projection.
Analogously, Fig. 1c shows a FEM simulation of the mechanical band structure of the nanobeam’s projection. Mechanical
modes at the band edge experience an effective potential analogous to the optical modes, localizing certain types of vibrations to
the defect region. The colored bands give rise to mechanical modes that, when localized by the defect, yield “ladders” of modes
with strong dispersive coupling to the localized optical modes (the frequency of the fundamental defect mode is indicated by
a horizontal bar of the same color). The red mechanical band is analogous to the acoustic (longitudinal) vibrations of a solid
or system of masses and springs, starting off at
k
x
=
0 as a pure translation of the entire structure and evolving to differential
acoustic vibrations of every pair of cross-beams at the band edge (mode at
k
x
=
π
/
Λ
shown to right of band diagram), just as
the highest-frequency acoustic vibrations of a solid have every unit cell vibrating out of phase with its nearest neighbors. The
green band also involves in-plane vibrations of the cross beams, but, in this case, the rails recoil in opposition to the cross-
beam vibration, which means that the structure can vibrate (rather than translate) even at very long-wavelengths, giving rise to
a non-zero frequency at
k
x
=
0. The blue band consists of in-plane vibrations transverse to the axis of the structure, where, at
Γ
(mode at
Γ
shown to right) the structure appears to “breathe”, as the cross-beams are stretched and compressed. We classify
these optomechanically-coupled mechanical modes, from lowest to highest frequency, as “pinch”, “accordian”, and “breathing”
modes. The localized mechanical modes of Device 1 are shown to the right of the corresponding mode of the projection.
The two kinds of waves, mechanical and optical, are on equal footing in this structure. Each mechanical mode has a frequency
ν
m
=
Ω
m
/
2
π
and displacement profile
Q
(
r
)
; each optical mode has a frequency
ν
o
=
ω
o
/
2
π
and electric field profile
E
(
r
)
.
Just as the optical mode volume,
V
o
=
R
d
V
(
√
ε
|
E
|
max
(
|
√
ε
E
|
)
)
2
, describes the electromagnetic localization of the optical mode,
the mechanical mode volume,
V
m
≡
ρ
R
d
V
(
|
Q
|
max
(
|
Q
|
)
)
2
(see App. D), describes the strain energy-averaged localization of the
mechanical mode. For both the localized optical and mechanical modes of the patterned beam cavity, the effective mode volume
is less than a cubic wavelength. The effective motional mass, being proportional to the mode volume (
m
eff
≡
ρ
V
m
), is between
50 and 1000 femtograms for the mechanical modes shown in Fig. 1c (
ρ
Si
=
2
.
33 g/cm
3
).
Drawing on recent work in the field of cavity optomechanics[18, 19], we describe the coupling between optical and mechanical
degrees of freedom (to lowest order) by an effective coupling length
L
OM
≡
(
1
ν
o
d
ν
o
d
α
)
−
1
(see App. D), where
δν
o
is the change in
the frequency of an optical resonance caused by the mechanical displacement parameterized by
α
. For this work,
α
is defined as
the maximum displacement that occurs
anywhere
for the mechanical mode. By definition then, the smaller
L
OM
, the larger the
optical response for a given mechanical displacement.
L
OM
is also the length over which a photon’s momentum is transferred
into the mechanical mode as it propagates within the structure, and thus is inversely proportional to the force per-photon applied
to the mechanical system.
To calculate
L
OM
, we employ a perturbative theory of Maxwells equations with respect to shifting material boundaries[20].
The derivative
d
ν
o
d
α
around some nominal position, where the optical fields are known, can be calculated
exactly
without actually
deforming the structure for a surface-normal displacement of the boundaries,
h
(
α
;
r
)
≡
Q
(
r
)
·
ˆ
n
=
α
q
(
r
)
·
ˆ
n
, where
q
(
r
) =
Q
(
r
)
/
α
=
d
Q
(
r
)
/
d
α
is the unitless displacement profile of the mechanical mode, and
α
parameterizes the amplitude of the
displacement. Using this perturbative formulation of Maxwell’s equations, we find
1
L
OM
=
1
2
Z
d
A
(
d
Q
d
α
·
ˆ
n
)
[
∆ε
∣
∣
E
‖
∣
∣
2
−
∆
(
ε
−
1
)
|
D
⊥
|
2
]
Z
d
V
ε
|
E
|
2
(1)
3
10
μ
m
a
1
μ
m
b
1590
1580
1570
1560
1550
1540
1530
1520
1510
1500
1490
Optical Wavelength (nm)
190
192
194
196
198
200
202
0.2
0.4
0.6
0.8
1
Normalized Transmission
ν
(THz)
Optical
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
−90
−80
−70
−60
RF PSD (dBm)
0.8
0.85
0.9
0.95
−85
−80
−75
−70
RF PSD (dBm)
1.4
1.41
1.42
−89.9
−89.8
−89.7
−89.6
2.25
2.3
2.35
−90
−85
−80
1526.60 nm
0
1
0
1
0
1
3
rd
Order
2
nd
Order
Fundamental
Q = 38000
T = 0.858
Q = 27000
T = 0.266
Q = 35000
T = 0.685
1584.89 nm
1484.03 nm
c
Pinch Modes
Accordian Modes
Breathing Modes
d
m
e
≥ 56 fg
L
e
≥ 10
μ
m
Q
m
≤ 1000
m
e
~ 1 pg
L
e
≥ 20
μ
m
Q
m
~ 1000
m
e
≥ 345 fg
L
e
≥ 2.8
μ
m
Q
m
≤ 1300
e
f
g
ν
(GHz)
Mechanical
190
μ
W
15
μ
W
−90
−70
−50
2.187
2.188
2.189
h
∆ν
=1.16 kHz
Q
m
=1.8x10
6
ν
(GHz)
Mechanical
FIG. 2:
Photonic and phoninic crystal mode spectroscopy
.
a
, and
b
, show SEM images of the fabriced silicon nanobeam optomechanical
crystal.
c
, Optical spectroscopy of Device 1 with the taper waveguide in contact.
d
, Mechanical spectroscopy of Device 1 with taper waveguide
in contact.
e
-
g
, Zoomed RF mechanical spectra of Device 1 showing pinch (red), accordian (green), and breathing (blue) modes.
h
, Funda-
mental breathing mode RF spectra (of a third device) at low (15
μ
W) and high (190
μ
w) coupled optical power . At high optical power the
breathing mode is optically amplified through dynamical back-action of the radiation pressure force, with an above threshold effective
Q
-factor
greater than 10
6
.
where
ˆ
n
is the unit normal vector on the surface of the unperturbed cavity,
D
(
r
) =
ε
(
r
)
E
(
r
)
,
∆ε
=
ε
1
−
ε
2
,
∆
(
ε
−
1
) =
ε
−
1
1
−
ε
−
1
2
,
ε
1
is the dielectric constant of the periodic structure, and
ε
2
is the dielectric constants of the surrounding medium (
ε
2
=
ε
0
in this
case). This method of calculating the coupling provides a wealth of intuition about the nature of the coupling and can be used to
engineer the structure for strong optomechanical coupling.
Figs. 2a and 2b show scanning electron microscope (SEM) images of a fabricated silicon nanobeam with the parameters of
Device 1 (see App. A). The optical modes of the nanobeam are probed with a tapered and dimpled optical fiber[21] in the near-
field of the defect cavity, simultaneously sourcing the cavity field and collecting the transmitted light in a single channel (see
App. B). Fig. 2c shows the low-pass filtered optical transmission spectrum of Device 1 at low optical input power (
∼
30
μ
W).
The optical cavity resonances are identified by comparison to FEM modeling of the optical modes of the structure. Looking
in the radio frequency (RF) spectrum provides information about the mechanical modes of the structure, as mechanical motion
gives rise to phase and amplitude modulation of the transmitted light. Figs. 2c-f show the measured photodetector RF power
spectral density (PSD) of the optical transmission through the second order cavity resonance (this mode was used due to its
deep on-resonance coupling). A series of lower frequency modes can be seen in the spectra (
∼
200 MHz and harmonics),
corresponding to compression of modes of the entire beam, followed by groups of localized phononic modes of the lattice at
850 MHz (pinch), 1
.
41 GHz (accordian), and 2
.
25 GHz (breathing). The transduced signal at low optical power corresponds to
thermally-excited motion of the mechanical modes, and is inversely proportional to
m
eff
L
2
OM
(see below). At higher optical input
power (see Fig. 2h), optical excitation of regenerative mechanical oscillation[18] of the breathing modes is possible due to the
small mass and short optomechanical coupling length of the co-localized phonon and photon modes.
Fig. 3a, top panel, shows how the frequency of the fundamental mechanical breathing mode scales with a uniform geometric
scaling in the plane. A series of 12 devices have been fabricated, identical except that the entire geometry in the plane is
scaled incrementally by 1% per device. For each device, one of the first two optical modes is selected and used to measure the
mechanical frequency of the fundamental breathing mode (cyan dots, top panel). The frequencies plotted in magenta are the
normalized frequencies, i.e. the bare frequencies (cyan) times the scale factor for the device. Because the vibrations are entirely
two-dimensional (in the plane), the frequency of the mechanical mode scales perfectly with the two-dimensional scale factor.
This is in contrast to the optical modes (Fig. 3a, bottom panel), which clearly do not scale with the planar geometry, a result
of the coupling of in-plane and vertical optical mode confinement (scaling in all three dimensions is thus required). Since the
planar scaling for the lattice-localized mechanical modes is trivial, this method could be used with a larger span of devices to
measure the frequency dependence of the Young’s (or bulk) modulus of the material.
Significant shifts in the frequency of the lattice-localized mechanical modes can be obtained through a non-uniform planar
scaling. Fig. 3B shows the RF PSD for Device 1 and a second device, Decice 2, which has an essentially identical lattice
constant,
Λ
=
365 nm, and total length,
L
, as compared to Device 1, but a considerably smaller width (
w
=
864 nm,
h
y
=
575
4
FIG. 3:
Phononic mode tuning and transduction
.
a
, Geometric scaling (planar) of the fundamental breathing mode. Device 1 is the
device with scale factor 1.03. The best linear least-squares fit lines in the top panel correspond to the mechanical frequency changing by
−
0
.
9%
±
0
.
2% per device; the normalized frequency changes by
−
0
.
01%
±
0
.
2% per device. The optical frequency of the mode used to make
the mechanical measurement is filled (the other optical mode is open).
b
, Engineering of pinch mode frequencies, showing two devices with
pinch mode frequencies of 850 MHz and 1.75 GHz. The mechanical band diagrams of each structure are shown to the right of the measured
RF spectrum, with the pinch mode band highlighted in red.
c
, Transduction of breathing mode motion.
d
, Ideal and actual (“primed”) modes
of the silicon nanobeam optomechanical crystal due to the ideal (dashed) and actual defect (solid, red).
nm,
h
x
=
183 nm). Simulations show that the pinch modes are the lowest-frequency group of localized and optomechanically-
coupled mechanical modes in both structures (see right panels of Fig. 3B). Experimentally, the ratio of the localized pinch mode
frequencies (highlighted in red) in these two devices is 1.749 GHz/805 MHz = 2.17. The ratio of the frequency of the localized
pinch-mode manifold, after accounting for the defect, is theoretically 1.826 GHz/846 MHz = 2.16. It is interesting to note that
the mechanical modes of the entire doubly-clamped beam (as opposed to the lattice-localized modes) depend very weakly on the
structural differences between Device 1 and Device 2. For instance, the second-order acoustic vibration mode of the nanobeam
(highlighted in yellow in Fig. 3B) has a frequnecy which should be
3
π
2
L
√
E
〈
ρ
〉
, where
E
is Young’s modulus and
〈
ρ
〉
is the average
linear density. The frequency of this mode is measured to be 234 MHz/195 MHz = 1.20 times higher in Device 2 than for Device
1, which is in good agreement with the ratio
√
〈
ρ
1
〉
/
〈
ρ
2
〉
=
1
.
23. The difference between the change in the frequencies of
the lattice-localized versus beam modes illustratres the independence of these two “systems”; once the wavelength of the global
beam modes approach the scale of the lattice periodicity, the vibrations become localized and behave independently of the global
beam structure (such as the end clamps).
Fig. 3c shows the RF optical transmission spectrum due to Brownian motion of the breathing modes of Device 1 (i.e., at low
optical input power), for the three optical modes shown in Figs. 1c and 2c. Because the various optical modes have different
spatial profiles, each mechanical mode has a
different L
OM
for each optical mode
. The root-mean-square (rms) mechanical
amplitude of a mode due to Brownian motion is
〈
α
2
〉
=
k
B
T
/
(
m
eff
Ω
2
)
. It can be shown analytically that the factor 1
/
(
m
eff
L
2
OM
)
uniquely determines the transduction of the Brownian motion for these sideband-resolved optomechanical oscillations (see App.
E). To the right of each measured spectrum is the experimentally-extracted mechanical frequency and value of 1
/
(
m
eff
L
2
OM
)
,
together with the values of these quantities obtained from the FEM model (using Eq. D4 to determine
L
OM
). Good corre-
spondence, in both frequency and transduced signal amplitude, is found across all optical and mechanical mode pairs. In order
5
to achieve this level of correspondence, imperfections in the fabricated structure are taken into account by extracting the ge-
ometry from high-resolution SEM images of the device and calculating the modified optical and mechanical modes (Fig. 3d).
The resulting measured value for optomechanical coupling between the fundamental breathing and optical mode (assumming
a FEM-calculated motional mass of
m
eff
=
330 fg) is
L
OM
=
2
.
9
μ
m, approaching the limit of the wavelength of light. The
sensitivity of the mechanical transduction of the fundamental breathing mode can be appreciated by comparing the mode’s rms
thermal amplitude at
T
=
300 K,
α
th
=
245 fm, to its quantum zero-point motion of
α
zp
=
3
.
2 fm. The sensitivity limit, as given
by the background level in the middle panel of Fig. 3c, is thus a factor of
∼
7
.
5 times that of the standard quantum limit.
The loss of mechanical energy from confined mechanical modes of a phononic crystal can, in principal, be made arbitrarily
low (and thus the mechanical
Q
arbitrarily high) by including a large number of unit cells outside the localizing potential region.
Of course, other forms of mechanical damping, such as thermo-elastic damping, phonon-phonon scattering, or surface damping
effects, would eventually become dominant[22]. This makes optomechanical crystals ideal structures for studying these loss
mechanisms. The fundamental breathing mode of the 1D phononic crystal structure studied here, at 2
.
254 GHz, has a room
temperature mechanical
Q
of 1300 in air, and in contact with the taper waveguide (power-dependent measurements confirm that
this mechanical Q is not enhanced by dynamical back-action). This corresponds to a frequency-
Q
product of 3
×
10
12
Hz, a value
close to largest demonstrated to date[23]. Although further tests (as a function of temperature and lattice periods) are required
to determine the contribution of various mechanical loss mechanisms, numerical simulations show that mode coupling between
localized and leaky phonon modes exist in these 1D cavity structures and can significantly limit the
Q
-factor (see App. C). This
obstacle can be overcome in two-dimensional periodic slab structures, which have been shown to possess complete gaps for both
optical and mechanical modes simultaneously[24].
The experimental demonstration of optomechanical coupling between 200 Terahertz photons and 2 Gigahertz phonons in a
planar optomechanical crystal paves the way for new methods of probing, manipulating, and stimulating linear and non-linear
mechanical and optical interactions in a chip-scale platform. As the study of quantum mesoscale mechanical oscillators has
nearly become a reality[25, 26, 27, 28, 29], high frequency mechanics will provide a distinct experimental advantage due to
the lower thermal phonon occupancy. In addition, optomechanical crystals with full phononic bandgaps provide a platform to
decouple the direct decoherence (phonon leakage) of mechanical modes from their supports. This could allow the preparation
of mechanical vibrations with ultra-long lifetimes, the study of the intrinsic mechanical material losses, and narrow-linewidth
Gigahertz frequency sources. Optomechanical crystals could also be used as high-spatial resolution mass sensors; with
m
eff
=
62 fg and
ν
m
=
850 MHz, the mass of a single Hemoglobin A protein (
∼
10
−
19
g) would change the frequency of the pinch
mode by 700 Hz, allowing sensitivity paralleling NEMS zeptogram mass sensors[30].
Acknowledgments
Funding for this work was provided by a DARPA seedling effort managed by Prof. Henryk Temkin, and through an EMT
grant from the National Science Foundation.
APPENDIX A: 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.
APPENDIX B: EXPERIMENTAL SETUP
Taper
APD
Tunable
Infrared Laser
VOA
FPC
VOA
Oscilloscope
Optomechanical Crystal Nanobeam
FIG. 4: Experimental setup used to measure optical, mechanical, and optomechanical properties of silicon optomechanical crystal nanobeam.
6
The experimental setup used to measure the optical, mechanical, and optomechanical properties of the silicon optomechanical
crystal nanobeam is shown in Fig. 4. The setup consists of a bank of fiber-coupled tunable infrared lasers spanning approximately
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 (although 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.
APPENDIX C: SIMULATION PARAMETERS
Modeling of both the optical and mechanical modes is done via finite element method (FEM), using COMSOL Multiphysics.
The following subsections provide the description of the method used to define the FEM model of the optomechanical crystal
system.
1. 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). Figure 5a
shows an “eagle’s-eye” high-resolution SEM micrograph of a portion of device 1, with the defect centered in the image.
FIG. 5: (a) Scanning electron micrograph of fabricated silicon optomechanical crystal. (b) Approximated geometry shown as blue overlay on
SEM micrograph from (a).
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. 5b. This geometric representation 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