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Position-squared coupling in a tunable photonic crystal optomechanical cavity
Taofiq K. Para ̈ıso,
1, 2
Mahmoud Kalaee,
2, 3
Leyun Zang,
1
Hannes Pfeifer,
1
Florian Marquardt,
1, 4
and Oskar Painter
2, 3
1
Max Planck Institute for the Science of Light, G ̈unther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany
2
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
3
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA
4
Institute for Theoretical Physics, Department of Physics, Universit ̈at Erlangen-N ̈urnberg, 91058 Erlangen
We present the design, fabrication, and characterization of a planar silicon photonic crystal
cavity in which large position-squared optomechanical coupling is realized. The device consists
of a double-slotted photonic crystal structure in which motion of a central beam mode couples
to two high-
Q
optical modes localized around each slot. Electrostatic tuning of the structure is
used to controllably hybridize the optical modes into supermodes which couple in a quadratic
fashion to the motion of the beam. From independent measurements of the anti-crossing of
the optical modes and of the optical spring effect, the position-squared vacuum coupling rate is
measured to be as large as ̃
g
/
2
π
= 245 Hz to the fundamental in-plane mechanical resonance of the
structure at
ω
m
/
2
π
= 8
.
7 MHz, which in displacement units corresponds to a coupling coefficient
of
g
/
2
π
= 1 THz/nm
2
. This level of position-squared coupling is approximately five orders of
magnitude larger than in conventional Fabry-Perot cavity systems.
PACS numbers: 42.50.Wk, 42.65.-k, 62.25.-g
I. INTRODUCTION
In a cavity-optomechanical system the electromagnetic
field of a resonant optical cavity or electrical circuit is
coupled to the macroscopic motional degrees of freedom
of a mechanical structure through radiation pressure [1].
Cavity-optomechanical systems come in a multitude
of different sizes and geometries, from cold atomic
gases [2] and nanoscale photonic structures [3], to the
kilogram/kilometer scale interferometers developed for
gravitational wave detection [4]. Recent technological
advancements in the field have led to the demonstration
of optomechanically induced transparency [5, 6],
back-action cooling of a mechanical mode to its quantum
ground state [7–9], and ponderomotive squeezing of the
light field [10, 11].
The interaction between light and mechanics in a
cavity-optomechanical system is termed dispersive when
it couples the frequency of the cavity to the position or
amplitude of mechanical motion. To lowest order this
coupling is linear in mechanical displacement, however,
the overall radiation pressure interaction is inherently
nonlinear due to the dependence on optical intensity. To
date, this nonlinear interaction has been too weak to
observe at the quantum level in all but the ultra-light
cold atomic gases [2], and typically a large optical drive
is used to parametrically enhance the optomechanical
interaction.
Qualitatively novel quantum effects are
expected when one takes a step beyond the standard
linear coupling and exploits higher order dispersive
optomechanical coupling. In particular, “
x
2
-coupling”
where the cavity frequency is coupled to the square
of the mechanical displacement has been proposed as
a means for realizing quantum non-demolition (QND)
measurements of phonon number [12–14], measurement
of phonon shot noise [15], and the cooling and squeezing
of mechanical motion [16–18]. In addition to dispersive
coupling, an effective
x
2
-coupling via optical homodyne
measurement has also been proposed, with the capability
of generating and detecting non-Gaussian motional
states [19].
The dispersive
x
2
-coupling between optical and
mechanical resonator modes in a cavity-optomechanical
system is described by the coefficient
g
2
ω
c
/∂x
2
,
where
ω
c
is the frequency of the optical resonance of
interest and
x
is the generalized amplitude coordinate
of the displacement field of the mechanical resonance.
One can show via second-order perturbation theory [20,
21] that
x
2
-coupling arises due to linear cross-coupling
between the optical mode of interest and other modes
of the cavity.
In the case of two nearby resonant
modes, the magnitude of the
x
2
-coupling coefficient
depends upon the square of the magnitude of the linear
cross-coupling between the two modes (
g
) and inversely
on their frequency separation or tunnel coupling rate
(2
J
),
g
=
g
2
/
2
J
. In pioneering work by Thompson,
et al. [12], a Fabry-Perot cavity with an optically-thin
Si
3
N
4
membrane positioned in between the two end
mirrors was used to realize
x
2
-coupling via hybridization
of the degenerate modes of optical cavities formed on
either side of the partially reflecting membrane. More
recently, a number of cavity-optomechanical systems
displaying
x
2
-coupling have been explored, including
double microdisk resonators [22], microdisk-cantilever
systems [23],
microsphere-nanostring systems [24],
atomic gases trapped in Fabry-Perot cavities [2], and
paddle nano-cavities [21].
Despite significant technical advances made in recent
years [21, 23, 25, 26], the use of
x
2
-coupling for measuring
or preparing non-classical quantum states of a mesoscopic
arXiv:1505.07291v1 [physics.optics] 27 May 2015
2
mechanical resonator remains an elusive goal. This is a
direct result of the small coupling rate to motion at the
quantum level, which for
x
2
-coupling scales as the square
of the zero-point motion amplitude of the mechanical
resonator,
x
2
zpf
=
~
/
2
m
, where
m
is the motional mass
of the resonator and
ω
m
is the resonant frequency. As
described in Ref. [14], one method to greatly enhance
the
x
2
-coupling in a multi-mode cavity-optomechanical
system is to fine tune the mode splitting 2
J
to that of
the mechanical resonance frequency.
In this work we utilize a quasi two-dimensional
(2D) photonic crystal structure to create an optical
cavity supporting a pair of high-
Q
optical resonances
in the 1500 nm wavelength band exhibiting large linear
optomechanical coupling. The double-slotted structure
is split into two outer slabs and a central nanobeam,
all three of which are free to move, and electrostatic
actuators are integrated into the outer slabs to allow for
both the trimming of the optical modes into resonance
and tuning of the tunnel coupling rate
J
.
Due to
the form of the underlying photonic bandstructure the
spectral ordering of the cavity supermodes in this
structure may be reversed, enabling arbitrarily small
values of
J
to be realized. Measurement of the optical
resonance anti-crossing curve, along with calibration of
the linear optomechanical coupling through measurement
of the dynamic optical spring effect, yields an estimated
x
2
-coupling coefficient as large as
g
/
2
π
= 1 THz/nm
2
( ̃
g
/
2
π
= 245 Hz) to the fundamental mechanical
resonance of the central beam at
ω
m
/
2
π
= 8
.
7 MHz.
Additional measurements of
g
through the dynamic
and static optical spring effects are also presented.
This level of
x
2
-coupling is approximately five orders
of magnitude larger than in conventional Fabry-Perot
MIM systems demonstrated to date [26], and two orders
of magnitude larger than in the smaller mode volume
fiber-gap cavities [25].
II. THEORETICAL BACKGROUND
Before we discuss the specific double-slotted photonic
crystal cavity-optomechanical system studied in this
work, we consider a more generic multi-moded system
consisting of two optical modes which are dispersively
coupled to the same mechanical mode, and in which the
dispersion of each mode is linear with the amplitude
coordinate
x
of the mechanical mode. If we further
assume a purely optical coupling between the two
optical modes, the Hamiltonian for such a three-mode
optomechanical system in the absence of drive and
dissipation is given by
ˆ
H
=
ˆ
H
0
+
ˆ
H
OM
+
ˆ
H
J
:
ˆ
H
0
=
~
ω
1
ˆ
a
1
ˆ
a
1
+
~
ω
2
ˆ
a
2
ˆ
a
2
+
~
ω
m
ˆ
b
ˆ
b,
(1)
ˆ
H
OM
=
~
(
g
1
ˆ
a
1
ˆ
a
1
+
g
2
ˆ
a
2
ˆ
a
2
x,
(2)
ˆ
H
J
=
~
J
a
1
ˆ
a
2
+ ˆ
a
2
ˆ
a
1
)
.
(3)
b
10
μ
m
1
μ
m
500
nm
c
d
e
y
x
x
3
ω
ω
+
ω
{
ω
1
ω
2
2
J
a
1
a
2
V
2
V
1
b
3
b
1
b
2
a
Figure 1. (a) Double-slotted photonic crystal cavity with
optical cavity resonances (
a
1
,a
2
) centered around the two
slots, and three fundamental in-plane mechanical resonances
corresponding to motion of the outer slabs (
b
1
,b
2
) and the
central nanobeam (
b
3
). Tuning the equilibrium position of the
outer slabs
b
1
and
b
2
, and consequently the slot size on either
side of the central nanobeam, is achieved by pulling on the
slabs (red arrows) through an electrostatic force proportional
to the square of the voltage applied to capacitors on the outer
edge of each slab. (b) Dispersion of the optical modes as
a function of
x
3
, the in-plane displacement of the central
nanobeam from its symmetric equilibrium position. Due to
tunnel coupling at a rate
J
the slot modes
a
1
and
a
2
hybridize
into the even and odd supermodes
a
+
and
a
, which have
a parabolic dispersion near the central anti-crossing point
(
ω
1
=
ω
2
). (c) SEM image of a fabricated double-slotted
photonic crystal device in the SOI material system.
(d)
Zoom-in SEM image showing the capacitor gap (
100 nm)
for the capacitor of one of the outer slabs. (e) Zoom-in SEM
image showing some of the suspending tethers of the outer
slabs which are of length 2
.
5
μ
m and width 155 nm. The
central beam, which is much wider, is also shown in this
image.
Here, ˆ
a
i
and
ω
i
are the annihilation operator and the
bare resonance frequency of the
i
th optical resonance,
ˆ
x
= (
ˆ
b
+
ˆ
b
)
x
zpf
is the quantized amplitude of motion,
x
zpf
the zero point amplitude of the mechanical resonance,
ω
m
is the bare mechanical resonance frequency, and
g
i
is the
linear optomechanical coupling constant of the
i
th optical
mode to the mechanical resonance.
Without loss of
generality, we take the bare optical resonance frequencies
to be equal (
ω
1
=
ω
2
ω
0
), allowing us to rewrite the
Hamiltonian in the normal mode basis ˆ
a
±
= (ˆ
a
1
±
ˆ
a
2
)
/
2
as,
3
ˆ
H
=
~
ω
+
(0)ˆ
a
+
ˆ
a
+
+
~
ω
(0)ˆ
a
ˆ
a
+
~
ω
m
ˆ
b
ˆ
b
+
~
(
g
1
+
g
2
2
)
(
ˆ
a
+
ˆ
a
+
+ ˆ
a
ˆ
a
)
ˆ
x
+
~
(
g
1
g
2
2
)
(
ˆ
a
+
ˆ
a
+ ˆ
a
ˆ
a
+
)
ˆ
x,
(4)
where
ω
±
(0) =
ω
0
±
J
.
For
|
J
| 
ω
m
such that ˆ
x
can be treated as a
quasi-static variable [13, 14], the Hamiltonian can be
diagonalized resulting in eigenfrequencies
ω
±
x
),
ω
±
x
)
ω
0
+
(
g
1
+
g
2
)
2
ˆ
x
±
J
(
1 +
(
g
1
g
2
)
2
8
J
2
ˆ
x
2
)
.
(5)
As shown below, in the case of the fundamental in-plane
motion of the outer slabs of the double-slotted photonic
crystal cavity we have only one of
g
1
or
g
2
non-zero,
whereas in the case of the fundamental in-plane motion
of the central nanobeam we have
g
1
≈−
g
2
.
For a system in which the mechanical mode couples
to the
a
1
and
a
2
optical modes with linear dispersive
coupling of equal magnitude but opposite sign (
g
1
=
g
2
=
g
), the dispersion in the quasi-static normal
mode basis is purely quadratic with effective
x
2
-coupling
coefficient,
g
=
g
2
/
2
J,
(6)
and quasi-static Hamiltonian,
ˆ
H
~
(
ω
+
(0) +
g
ˆ
x
2
)
ˆ
n
+
+
~
(
ω
(0)
g
ˆ
x
2
)
ˆ
n
+
~
ω
m
ˆ
n
b
,
(7)
where ˆ
n
±
are the number operators for the
a
±
supermodes and ˆ
n
b
is the number operator for the
mechanical mode. Rearranging this equation slightly
highlights the interpretation of the
x
2
optomechanical
coupling as inducing a static optical spring,
ˆ
H
~
ω
+
(0)ˆ
n
+
+
~
ω
(0)ˆ
n
+
~
[
ω
m
ˆ
n
b
+
g
n
+
ˆ
n
) ˆ
x
2
]
,
(8)
where the static optical spring constant
̄
k
s
=
2
~
g
(
n
+
n
) depends upon the average intra-cavity
photon number in the even and odd optical supermodes,
n
±
≡〈
ˆ
n
±
.
For a sideband resolved system (
ω
m

κ
), the
quasi-static Hamiltonian can be further approximated
using a rotating-wave approximation as,
ˆ
H
~
[
ω
+
(0) + 2 ̃
g
n
b
+ 1
/
2)]ˆ
n
+
+
~
[
ω
(0)
2 ̃
g
n
b
+ 1
/
2)]ˆ
n
+
~
ω
m
ˆ
n
b
,
(9)
where ̃
g
g
x
2
zpf
= ̃
g
2
/
2
J
and ̃
g
gx
zpf
are the
x
2
and linear vacuum coupling rates, respectively. It is
tempting to assume from eq. (9) that by monitoring the
optical transmission through the even or odd supermode
resonances, that one can then perform a continuous
quantum non-demolition (QND) measurement of the
phonon number in the mechanical resonator [12, 27–29].
As noted in Refs. [13, 14], however, the quasi-static
picture described by the dispersion of eq. (5) fails to
capture residual effects resulting from the non-resonant
scattering between the
a
+
and
a
supermodes which
depends linearly on ˆ
x
(last term of eq. (4)). Only in the
vacuum strong coupling limit ( ̃
g/κ
&
1) can one realize
a QND measurement of phonon number [13, 14].
The regime of
|
2
J
| ∼
ω
m
is also very interesting,
and explored in depth in Refs. [14, 30]. Transforming
to a reference frame which removes in eq. (4) the
radiation pressure interaction between the even and
odd supermodes to first order in
g
, yields an effective
Hamiltonian given by [14, 31],
ˆ
H
eff
~
ω
+
(0)ˆ
n
+
+
~
ω
(0)ˆ
n
+
~
ω
m
ˆ
n
b
+
~
̃
g
2
2
[
1
2
J
ω
m
+
1
2
J
+
ω
m
]
(
ˆ
a
+
ˆ
a
+
ˆ
a
ˆ
a
)(
ˆ
b
+
ˆ
b
)
2
+
~
̃
g
2
2
[
1
2
J
ω
m
1
2
J
+
ω
m
]
(
ˆ
a
+
ˆ
a
+ ˆ
a
ˆ
a
+
)
2
,
(10)
where we assume
|
̃
g/δ
|
1 for
δ
≡|
2
J
|−
ω
m
, and terms
of order ̃
g
3
/
(2
J
±
ω
m
)
2
and higher have been neglected.
In the limit
|
J
| 
ω
m
we recover the quasi-static result
of eq. (7), whereas in the near-resonant limit of
|
δ
| 
|
J
|
m
we arrive at,
ˆ
H
eff
~
ω
+
(0)ˆ
n
+
+
~
ω
(0)ˆ
n
+
~
ω
m
ˆ
n
b
+
~
̃
g
2
2
δ
[
2 sgn(
J
) (ˆ
n
+
ˆ
n
) (ˆ
n
b
+ 1)
+ 2ˆ
n
+
ˆ
n
+ ˆ
n
+
+ ˆ
n
]
.
(11)
Here we have neglected highly oscillatory terms such as
a
+
ˆ
a
)
2
and
ˆ
b
2
, a good approximation in the sideband
resolved regime (
κ

ω
m
,
|
J
|
).
From eq. (11) we
find that the frequency shift per phonon of the optical
resonances is much larger than in the quasi-static case
( ̃
g
2
/
2
|
δ
| 
̃
g
2
/
2
|
J
|
). Although a QND measurement of
phonon number still requires the vacuum strong coupling
limit, this enhanced read-out sensitivity is attainable
even for ̃
g/κ

1. Equation (11) also indicates that,
much like the QND measurement of phonon number, in
the near-resonant limit a measurement of the intra-cavity
photon number stored in one optical supermode can
be performed by monitoring the transmission of light
through the other supermode [14, 31].
4
III. DOUBLE-SLOTTED PHOTONIC CRYSTAL
OPTOMECHANICAL CAVITY
A sketch of the double-slotted photonic crystal cavity
structure is shown in Fig. 1a.
As detailed below
and elsewhere [32], the optical cavity structure can be
thought of as formed from two coupled photonic crystal
waveguides, one around each of the nanoscale slots, and
each with propagation direction along the
z
-axis. A
small adjustment (
5%) in the lattice constant is used
to produce a local shift in the waveguide band-edge
frequency, resulting in trapping of optical resonance to
this “defect” region. Optical tunneling across the central
photonic crystal beam, which in this case contains only
a single row of holes, couples the cavity mode of slot 1
(
a
1
) to the cavity mode of slot 2 (
a
2
).
The two outer photonic crystal slabs and the central
nanobeam are all mechanically compliant, behaving as
independent mechanical resonators.
The mechanical
resonances of interest in this work are the fundamental
in-plane flexural modes of the top slab, the bottom slab,
and the central nanobeam, denoted by
b
1
,
b
2
and
b
3
,
respectively. For a perfectly symmetric structure about
the
z
-axis of the central nanobeam, the linear dispersive
coupling coefficients of the
b
3
mode of the central
nanobeam to the two slot modes
a
1
and
a
2
are equal in
magnitude but opposite in sign, resulting in a vanishing
linear coupling at the resonant point where
ω
1
=
ω
2
(c.f.,
eq. (5)). Figure 1b shows a plot of the dispersion of
the optical resonances as a function of the nanobeam’s
in-plane displacement (
x
3
), illustrating how the linear
dispersion of the slot modes (
a
1
,a
2
) transforms into
quadratic dispersion for the upper and lower supermode
branches (
a
+
,a
) in the presence of tunnel coupling
J
.
The mechanical modes of the outer slabs (
b
1
,b
2
) provide
degrees of freedom for post-fabrication tuning of the
slotted waveguide optical modes, i.e., to symmetrize the
structure such that
ω
1
=
ω
2
. This is achieved in practice
by integrating metallic electrodes which form capacitors
at the outer edge of the two slabs of the structure as
schematically shown in Fig. 1a.
The double-slotted photonic crystal cavity of this
work is realized in the silicon-on-insulator (SOI) material
system, with a top silicon device layer thickness of
220 nm and an underlying buried oxide (BOX) layer
of 3
μ
m. Fabrication begins with the patterning of the
metal electrodes of the capacitors, and involves electron
beam (ebeam) lithography followed by evaporation and
lift-off of a bi-layer consisting of a 5 nm sticking layer of
chromium and a 150 nm layer of gold. After lift-off we
deposit uniformly a
4 nm protective layer of silicon
dioxide. A second electron beam lithography step is
performed, aligned to the first, to form the pattern of the
photonic crystal and the nanoscale slots which separate
the central nanobeam from the outer slabs. At this
step, we also pattern the support tethers of the outer
slabs and the cut lines which define and isolate the outer
capacitors. A fluorine based (C
4
F
8
and SF
6
) inductively
coupled reactive-ion etch (ICP-RIE) is used to transfer
the ebeam lithography pattern through the silicon device
layer.
The remaining ebeam resist is stripped using
trichloroethylene, and then the sample is cleaned in a
heated Piranha (H
2
SO
4
:H
2
O
2
) solution. The devices
are then released using a hydrofluoric (HF)acid etch to
remove the sacrificial BOX layer (this also removes the
deposited protective silicon dioxide layer), followed by a
water rinse and critical point drying.
A scanning electron microscope (SEM) image showing
the overall fabricated device structure is shown in Fig. 1c.
Zoom-ins of the capacitor region of one of the outer
slabs and the tether region at the end of the nanobeam
are shown in Figs. 1d and e, respectively. Note that
the geometry of the capacitors and the stiffness of the
support tethers determine how tunable the structure is
under application of voltages to the capacitor electrodes.
The outermost electrode of each slab is connected to
an independent low-noise DC voltage source, while the
innermost electrodes are connected to a common ground,
thereby allowing one to independently pull on each outer
slab with voltages
V
1
and
V
2
. In this configuration, we
are limited to increasing the slots defining the optical
modes around the central nanobeam.
A. Photonic bandstructure
To further understand the optical properties of the
double-slotted photonic crystal cavity, we display in
Fig. 2a the photonic bandstructure of the periodic
waveguide structure. The parameters of the waveguide
are given in the caption of Fig. 2a.
Here we only
show photonic bands that are composed of waveguide
modes with even vector symmetry around the “vertical”
mirror plane (
σ
z
), where the vertical mirror plane is
defined by the
z
-axis normal and lies in the middle of
the thin-film silicon slab. The fundamental (lowest lying)
optical waveguide bands are of predominantly transverse
(in-plane) electric field polarization, and are thus called
TE-like. In the case of a perfectly symmetric structure,
we can further classify the waveguide bands by their
odd or even symmetry about the “horizontal” mirror
plane (
σ
y
) defined by the
y
-axis normal and cutting
through the middle of the central nanobeam. The two
waveguide bands of interest that lie within the quasi-2D
photonic bandgap of the outer photonic crystal slabs,
shown as bold red and black curves, are labeled “even”
and “odd” depending on the spatial symmetry with
respect to
σ
y
of their mode shape for the dominant
electric field polarization in the
y
-direction,
E
y
(note that
this labeling is opposite to their vector symmetry). The
E
y
spatial mode profiles at the
X
-point for the odd and
even waveguide supermodes are shown in Figs. 2b and c,
respectively.
An optical cavity is defined by decreasing the lattice
constant 4
.
5% below the nominal value of
a
0
= 480 nm
for the middle five periods of the waveguide (see Fig. 2d).
5
frequenc
y
( x10
2
TH
z)
2.2
2.1
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
X
Γ
ev
en
odd
odd
even
1μm
ev
en
odd
a
0
y
x
b
c
d
e
a
wavevector
a
0
0.955
a
0
Figure 2. (a) Bandstructure diagram of the periodic (along
x
) double-slotted photonic crystal waveguide structure. Here
we only show photonic bands that are composed of modes
with even vector symmetry around the “vertical” (
σ
z
) mirror
plane. The two waveguide bands of interest lie within the
quasi-2D photonic bandgap of the outer photonic crystal
slabs and are shown as bold red and black curves. These
waveguide bands are labeled “even” (bold black curve) and
“odd” (bold red curve) due to the spatial symmetry of their
mode shape for the dominant electric field polarization in
the
y
-direction,
E
y
. The simulated structure is defined by
the lattice constant between nearest neighbor holes in the
hexagonal lattice (
a
0
= 480 nm), the thickness of the silicon
slab (
d
= 220 nm), the width of the two slots (
s
= 100 nm),
and the refractive index of the silicon layer (
n
Si
= 3
.
42).
The hole radius in the outer slabs and the central nanobeam
is
r
= 144 nm.
The grey shaded region represents a
continuum of radiation modes which lie above the light cone
for the air cladding which surrounds the undercut silicon slab
structure. (b) Normalized
E
y
field profile at the
X
-point of
the odd waveguide supermode, shown for several unit cells
along the
x
guiding axis. (c)
E
y
field profile of the even
waveguide supermode. Waveguide simulations of (a-c) were
performed using the plane-wave mode solver MPB [33, 34].
Normalized
E
y
field profile of the corresponding localized
cavity supermodes of (d) odd and (e) even spatial symmetry
about the horizontal mirror plane. The lattice constant
a
0
is decreased by 4
.
5% for the central five lattice constants
between the dashed lines to localize the waveguide modes.
Simulations of the full cavity modes were performed using the
COMSOL finite-element method mode solver package [35].
This has the effect of locally pushing the bands toward
higher frequencies [36, 37], which creates an effective
potential that localizes the optical waveguide modes
along the
x
-axis of the waveguide.
The resulting
odd and even TE-like cavity supermodes are shown
in Figs. 2d and e, respectively. These optical modes
correspond to the normal modes
a
+
and
a
in Section II,
which are symmetric and anti-symmetric superpositions,
respectively, of the cavity modes localized around each
slot (
a
1
and
a
2
). Due to the non-monotonic decrease in
the even waveguide supermode as one moves away from
the
X
-bandedge (c.f., Fig. 2a), we find that the simulated
optical
Q
-factor of the even
a
+
cavity supermode is
significantly lower than that of the odd
a
cavity
90
92
94
96
98
100
85
87
89
91
93
95
1553
1555
1557
1558
1560
90
92
94
96
98
1544
1546
1548
1550
1552
100
wavelength (nm)
wavelength (nm)
)
m
n
(
h
t
d
i
w
t
o
l
s
)
m
n
(
h
t
d
i
w
t
o
l
s
a
b
c
d
odd
even
-100
-50
0
50
100
wavelength (nm)
1540
1545
1550
1555
1560
2
J
/
2
π
(GHz)
Figure 3.
Symmetric tuning of the slot widths of the
double-slotted photonic crystal cavity showing (a) the mean
wavelength shift and (b) the splitting 2
J
=
ω
+
ω
of the
even and odd cavity supermodes versus slot width
s
=
s
1
=
s
2
. (c-d) Avoided crossing of the cavity supermodes obtained
by tuning
s
1
while keeping
s
2
fixed at (c)
s
2
= 90 nm and (d)
s
2
= 95 nm. For all simulations in (a-d) the parameters of the
cavity structure are the same as in Fig. 2, except for the slot
widths. The simulations were performed using the COMSOL
FEM mode solver [35].
supermode. This will be a key distinguishing feature
found in the measured devices as well.
B. Optical tuning simulations
The slot width in the simulated waveguide and cavity
structures of Fig. 2 is set at
s
= 100 nm. For this
slot width we find a lower frequency for the even (
a
+
)
supermode than the odd (
a
) supermode at the
X
-point
photonic bandedge of the periodic waveguide and in the
case of the localized cavity modes. Figure 3 presents
finite-element method (FEM) simulations of the optical
cavity for slot sizes swept from 90 nm to 100 nm in steps
of 1 nm, all other parameters the same as in Fig. 2. For
the slot widths tuned symmetrically (
s
1
=
s
2
=
s
), the
mean wavelength of the even and odd cavity supermodes
and their frequency splitting 2
J
=
ω
+
ω
are plotted
in Fig. 3a and Fig. 3b, respectively.
As expected
the mean wavelength drops for increasing slot width.
The frequency splitting, however, also monotonically
decreases with slot width, going from a positive value
for
s
= 90 nm to a negative for
s
= 100 nm slots and
crossing zero for a slot width of
s
= 95 nm. In Figs. 3c
and d the symmetry is broken by keeping
s
2
fixed and
scanning
s
1
; the cavity supermodes are driven through
an anti-crossing with a splitting determined by the fixed
slot width
s
2
.
The spectral inversion of the even
a
+
and odd
a
cavity supermodes predicted in Fig. 3b originates
in the unequal overlap of each mode with the air
slots separating the two outer slabs from the central
nanobeam. The odd supermode tends to be pushed
6
further from the middle of the central nanobeam, having
slightly larger overlap with the air slots. An increase
in the air region for increased slot size leads to a blue
shift of both cavity supermodes. The odd mode having
a larger electric field energy density in the air slots
than the even mode is more affected by a change in
the slot widths. Therefore, upon equal increase of the
slot widths, the odd mode experiences larger frequency
shifts than the even mode, which results in a tuning
of the frequency splitting. For particular geometrical
parameters of the central nanobeam [32], a change in the
slot widths is sufficient to invert the spectral ordering
of the supermodes. This means that arbitrarily small
splittings can potentially be realized, which is important
for applications in
x
2
detection where the splitting enters
inversely in the coupling (for the quasi-static case).
IV. EXPERIMENTAL MEASUREMENTS
Optical testing of the fabricated devices is performed
in a nitrogen-purged enclosure at room temperature and
pressure. A dimpled optical fiber taper is used to locally
excite and collect light from the photonic crystal cavity,
details of which can be found in Ref. [38]. The light from
a tunable, narrow-bandwidth laser source in the telecom
1550 nm wavelength band (New Focus, Velocity series) is
evanescently coupled from the fiber taper into the device
with the fiber taper guiding axis parallel with that of
the photonic crystal waveguide axis, and the fiber taper
positioned laterally at the center of the nanobeam and
vertically a few hundreds of nanometers above the surface
of the silicon chip. Relative positioning of the fiber taper
to the chip is accomplished using a multi-axis set of
encoded DC-motor stages with 50 nm step resolution.
The polarization of the light in the fiber is polarized
parallel with the surface chip in order to optimize the
coupling to the in-plane polarization of the cavity modes.
With the taper placed suitably close to a photonic
crystal cavity (
200 nm), the transmission spectrum
of the laser probe through the device features resonance
dips at the supermode resonance frequencies, as
shown in the intensity plots of Figs. 4a-c.
The
resonance frequencies of the cavity modes are tuned
via displacement of the top and bottom photonic
crystal slabs, which can be actuated independently using
their respective capacitor voltages
V
1
and
V
2
.
The
capacitive force is proportional to the applied voltage
squared [37], and thus increasing the voltage
V
i
on
a given capacitor widens the waveguide slot
s
i
and
(predominantly) increases the slot mode frequency
a
i
(note the other optical slot mode frequency also increases
slightly). For the devices studied in this work, the slab
tuning coefficient with applied voltage (
α
cap
) is estimated
from SEM analysis of the resulting structure dimensions
and FEM electromechanical simulations to be
α
cap
=
25 pm/V
2
.
We fabricated devices with slot widths targeted
for a range of 75-85 nm, chosen smaller than the
expected zero-splitting slot width of
s
= 95 nm so
that the capacitors could be used to tune through
the zero-splitting point. While splittings larger than
150 GHz were observed in the nominal 85 nm slot width
devices, splittings as small as 10 GHz could be resolved
in the smaller 75 nm slot devices. As such, in what
follows we focus on the results from a single device with
as-fabricated slot size of
s
75 nm.
A. Anti-crossing measurements
Figure 4 shows intensity plots of the normalized
optical transmission through the optical fiber taper when
evanescently coupled to the photonic crystal cavity of
a device with nominal slot width
s
= 75 nm. Here a
series of optical transmission spectrum are measured by
sweeping the probe laser frequency and the voltage
V
1
,
with
V
2
fixed at three different values. The estimated
anti-crossing splitting from the measured dispersion of
the cavity supermodes is 2
J/
2
π
= 50 GHz, 12 GHz, and
25 GHz for
V
2
= 1 V, 15 V, and 18 V, respectively.
In order to distinguish between the odd and even cavity
supermodes at the anti-crossing point, we use the fact
that both the coupling rate to the fiber taper
κ
e
and
the intrinsic linewidth
κ
i
depend upon the symmetry
of the cavity mode. First, the odd supermode branch
becomes dark at the anti-crossing because it cannot
couple to the symmetric fiber taper mode. Second, in
the vicinity of the anti-crossing point the linewidth of
the odd supermode branch narrows while the linewidth
of the even supermode branch broadens [32]. Far from
the anti-crossing region, the branches are asymptotic to
individual slot modes and their linewidths and couplings
to the fiber taper are similar.
These features are clearly evident in the optical
transmission spectra of Figs. 4a-c, as well as in the
measured linewidth of the optical supermode resonances
shown in Figs. 4g-h. Figure 4a was taken with a small
voltage
V
2
= 1 V, corresponding to a small slot width
at the anti-crossing point, and thus consistent with the
even mode frequency being higher than the odd mode
frequency for small slot widths (c.f., Fig. 3b). The exact
opposite identification is made in Fig. 4c where
V
2
= 18 V
is much larger, corresponding to a larger slot width at the
anti-crossing point. Fig. 4b with
V
2
= 15 V is close to the
zero-splitting condition. For comparison, a simulation of
the expected anti-crossing curves are shown in Figs. 4d,
e, and f for
s
2
= 93, 95, and 97 nm, respectively. Here we
have taken the even superposition of the slot modes to
have a lower
Q
-factor than the odd superposition of the
slot modes, and the coupling of the fiber taper to be much
stronger to the even mode than the odd mode, consistent
with results from numerical FEM simulations. Good
qualitative correspondence is found with the measured
transmission curves of Figs. 4a-c.
An estimate of the
x
2
-coupling coefficient
g
b
3
can found
7
1550
1551
1552
1553
1554
90
91
92
93
94
95
96
150
1557
1558
1559
1560
1561
140
130
120
110
100
90
80
V
1
2
(V
2
)
slot width (nm)
wavelength (nm)
a
d
1546
1547
1548
1549
1550
1551
98
97
96
95
94
93
92
310
1550
1551
1552
1553
1554
1555
300
290
280
270
260
250
240
b
e
wavelength (nm)
1543
1544
1545
1546
1547
100
99
98
97
96
95
94
370
1548
1549
1550
1551
1552
360
350
340
330
320
310
300
c
f
wavelength (nm)
V
1
2
(V
2
)
g
h
linewidth (GHz)
linewidth (GHz)
50
100
150
0
5
10
15
20
25
30
35
300
320
340
360
380
0
10
20
30
40
50
Figure 4. (a-c) Optical transmission measurements versus the wavelength of the probe laser showing the cavity mode
anti-crossing and tuning of the photon tunneling rate. In these measurements the probe laser wavelength (horizontal axis)
is scanned across the optical cavity resonances as the voltage across the first capacitor
V
1
is swept from low to high (vertical
axis shows
V
2
1
in V
2
, proportional to slab displacement). The second capacitor is held fixed at (a)
V
2
= 1V, (b)
V
2
= 15V and
(c)
V
2
= 18V. The colorscale indicates the fractional change in the optical transmission level, ∆
T
, with blue corresponding
to ∆
T
= 0 and red corresponding to ∆
T
0
.
25. From the three anti-crossing curves we measure a splitting 2
J
equal to (a)
50 GHz, (b) 12 GHz, and (c)
25 GHz. (d-f) Corresponding simulations of the normalized optical transmission spectra for the
slot width
s
1
varied and the second slot width held fixed at (d)
s
2
= 93 nm, (e)
s
2
= 95 nm and (f)
s
2
= 97 nm. The dispersion
and tunneling rate of the slot modes are taken from simulations similar to that found in Fig. 3. (g) and (h) show the measured
linewidths of the high frequency upper (black) and low frequency lower (red) optical resonance branches as a function of
V
2
1
,
extracted from (a) and (c), respectively. The narrowing (broadening) is a characteristic of the odd (even) nature of the cavity
supermode, indicating the inversion of the even and odd supermodes for the two voltage conditions
V
2
= 1 V and
V
2
= 18 V.
The lines are guides for the eye.
from the simulated value of
α
cap
and a fit to the measured
tuning curves of Fig. 4 away from the anti-crossing point.
Consider the anti-crossing curve of Fig. 4b with the
smallest splitting. Far from the anti-crossing point the
tuning of the
a
1
and
a
2
slot modes are measured to be
linear with the square of
V
1
:
g
a
1
,V
2
1
/
2
π
= 3
.
9 GHz/V
2
and
g
a
2
,V
2
1
/
2
π
= 0
.
5 GHz/V
2
. For the simulated value of
α
cap
= 0
.
025 nm/V
2
the corresponding linear dispersive
coefficients versus the first slot width are
g
a
1
,δs
1
/
2
π
=
156 GHz/nm and
g
a
2
,δs
1
/
2
π
= 20 GHz/nm. Noting
that a displacement amplitude
x
3
for the fundamental
in-plane mechanical mode of the central nanobeam is
approximately equivalent to a reduction in the width
of one slot by -
x
3
and an increase in the other slot
by +
x
3
, the linear optomechanical coupling coefficient
between optical slot mode
a
1
and mechanical mode
b
3
is estimated to be
g
a
1
,b
3
(
g
a
1
,δs
1
+
g
a
1
,
δs
2
) =
(
g
a
1
,δs
1
g
a
2
,δs
1
) = 2
π
[136 GHz/nm], where by symmetry
g
a
1
,
δs
2
=
g
a
2
,δs
1
. Along with a measured splitting of
2
J/
2
π
= 12 GHz, this yields through eq. (6) an estimate
for the
x
2
-coupling coefficient of
g
b
3
/
2
π
1
.
54 THz/nm
2
.
B. Transduction of mechanical motion
Figure.
5
shows
the
evolution
of
the
optically-transduced mechanical noise power spectral
density (PSD) near the anti-crossing region of Fig. 4a.
In this plot
s
2
is fixed and
s
1
is varied over an estimated
range of
δs
1
=
±
0
.
3 nm around the anti-crossing. Optical
motion is imprinted as intensity modulations of the
probe laser which is tuned to the blue side of the upper
frequency supermode.
Here we choose the detuning
point corresponding to ∆
L
ω
L
ω
+
κ/
2
3,
where
ω
L
is the probe laser frequency and
κ
is the
full-width at half-maximum linewidth of the optical
resonance.
This detuning choice ensures (maximal)
linear transduction of small fluctuations in the
frequency of the cavity supermode, which allows us
to relate nonlinear transduction of motion with true
nonlinear optomechanical coupling [21, 23]. A probe
power of
P
in
= 10
μ
W is used in order to avoid
any nonlinear effects due to optical absorption, and
the transmitted light is first amplified through an
erbium-doped fiber amplifier before being detected
on a high gain photoreceiver (transimpedance gain