Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity
Taofiq K. Paraïso,
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, Günther-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, Universität Erlangen-Nürnberg,
D-91058 Erlangen, Germany
(Received 27 May 2015; revised manuscript received 17 August 2015; published 12 November 2015)
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 that couple in a quadratic fashion to the motion of the beam. From independent
measurements of the anticrossing of the optical modes and of the dynamic optical spring effect, a position-
squared vacuum coupling rate as large as
~
g
0
=
2
π
¼
245
Hz is inferred between the optical supermodes and
the fundamental in-plane mechanical resonance of the structure at
ω
m
=
2
π
¼
8
.
7
MHz, which in
displacement units corresponds to a coupling coefficient of
g
0
=
2
π
¼
1
THz
=
nm
2
. For larger supermode
splittings, selective excitation of the individual optical supermodes is used to demonstrate optical trapping
of the mechanical resonator with measured
~
g
0
=
2
π
¼
46
Hz.
DOI:
10.1103/PhysRevX.5.041024
Subject Areas: Optics, Photonics, Quantum Physics
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- and
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]
, backaction 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 systems but the ultralight 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 nondemolition (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
0
≡
1
=
2
½
∂
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 on the square of the
Published by the American Physical Society under the terms of
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. Further distri-
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’
s title, journal citation, and DOI.
PHYSICAL REVIEW X
5,
041024 (2015)
2160-3308
=
15
=
5(4)
=
041024(12)
041024-1
Published by the American Physical Society
magnitude of the linear cross-coupling between the two
modes (
g
) and inversely on their frequency separation or
tunnel coupling rate (
2
J
),
g
0
¼
g
2
=
2
J
. In pioneering work
by Thompson
et al.
[12]
, a Fabry-Pérot 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-Pérot cavities
[2]
, and paddle
nanocavities
[21]
.
Despite significant technical advances made in recent
years
[21,23,25,26]
, the use of
x
2
coupling for measuring or
preparing nonclassical quantum states of a mesoscopic
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 reso-
nator,
x
2
zpf
¼
ℏ
=
2
m
ω
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 multimode 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 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
.
Because of the form of the underlying photonic band
structure, 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 anticrossing 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
0
=
2
π
¼
1
THz
=
nm
2
to the fundamental mechanical resonance of the central
beam at
ω
m
=
2
π
¼
8
.
7
MHz. Additional measurements of
g
0
through the dynamic and static optical spring effects are
also presented. In comparison to other systems, the
corresponding vacuum
x
2
-coupling rate we demonstrate
in this work (
g
0
x
2
zpf
=
2
π
¼
245
Hz) is many orders of
magnitude larger than has been obtained in conventional
Fabry-Pérot
[26]
or fiber-gap
[25]
membrane-in-the-middle
(MIM) systems. It is also orders of magnitude larger than
demonstrated in the small mode volume microdisk-
cantilever
[23]
and paddle nanocavity
[21]
devices.
Whereas the double-disk microresonators previously stud-
ied by us
[22]
reach a comparable
x
2
-coupling magnitude,
the planar photonic crystal structure of this work realizes an
order of magnitude larger vacuum coupling rate, with a
much simpler mechanical mode spectrum and a tunable
tunneling rate
J
.
II. THEORETICAL BACKGROUND
Before we discuss the specific double-slotted photonic
crystal cavity-optomechanical system studied in this work,
we consider a more generic multimoded system consisting
of two optical modes that 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
¼
ˆ
H
0
þ
ˆ
H
OM
þ
ˆ
H
J
:
ˆ
H
0
¼
ℏ
ω
1
ˆ
a
†
1
ˆ
a
1
þ
ℏ
ω
2
ˆ
a
†
2
ˆ
a
2
þ
ℏ
ω
m
ˆ
b
†
ˆ
b;
ð
1
Þ
ˆ
H
OM
¼
ℏ
ð
g
1
ˆ
a
†
1
ˆ
a
1
þ
g
2
ˆ
a
†
2
ˆ
a
2
Þ
ˆ
x;
ð
2
Þ
ˆ
H
J
¼
ℏ
J
ð
ˆ
a
†
1
ˆ
a
2
þ
ˆ
a
†
2
ˆ
a
1
Þ
:
ð
3
Þ
Here,
ˆ
a
i
and
ω
i
are the annihilation operator and the
bare resonance frequency of the
i
th optical resonance,
ˆ
x
¼
ð
ˆ
b
†
þ
ˆ
b
Þ
x
zpf
is the quantized amplitude of motion,
x
zpf
is 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
¼
ω
2
≡
ω
0
), allowing us to rewrite the Hamiltonian in
the normal mode basis,
ˆ
a
¼ð
ˆ
a
1
ˆ
a
2
Þ
=
ffiffiffi
2
p
,as
ˆ
H
¼
ℏ
ω
þ
ð
0
Þ
ˆ
a
†
þ
ˆ
a
þ
þ
ℏ
ω
−
ð
0
Þ
ˆ
a
†
−
ˆ
a
−
þ
ℏ
ω
m
ˆ
b
†
ˆ
b
þ
ℏ
g
1
þ
g
2
2
ð
ˆ
a
†
þ
ˆ
a
þ
þ
ˆ
a
†
−
ˆ
a
−
Þ
ˆ
x
þ
ℏ
g
1
−
g
2
2
ð
ˆ
a
†
þ
ˆ
a
−
þ
ˆ
a
†
−
ˆ
a
þ
Þ
ˆ
x;
ð
4
Þ
where
ω
ð
0
Þ¼
ω
0
J
.
For
j
J
j
≫
ω
m
such that
ˆ
x
can be treated as a quasistatic
variable
[13,14]
, the Hamiltonian can be diagonalized,
resulting in eigenfrequencies
ω
ð
ˆ
x
Þ
:
ω
ð
ˆ
x
Þ
≈
ω
0
þ
ð
g
1
þ
g
2
Þ
2
ˆ
x
J
1
þ
ð
g
1
−
g
2
Þ
2
8
J
2
ˆ
x
2
:
ð
5
Þ
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
nonzero,
TAOFIQ K. PARAÏSO
et al.
PHYS. REV. X
5,
041024 (2015)
041024-2
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
¼
−
g
2
¼
g
), the
dispersion in the quasistatic normal mode basis is purely
quadratic with effective
x
2
-coupling coefficient,
g
0
¼
g
2
=
2
J;
ð
6
Þ
and quasistatic Hamiltonian,
ˆ
H
≈
ℏ
ð
ω
þ
ð
0
Þþ
g
0
ˆ
x
2
Þ
ˆ
n
þ
þ
ℏ
ð
ω
−
ð
0
Þ
−
g
0
ˆ
x
2
Þ
ˆ
n
−
þ
ℏ
ω
m
ˆ
n
b
;
ð
7
Þ
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 interpre-
tation of the
x
2
optomechanical coupling as inducing a
static optical spring,
ˆ
H
≈
ℏ
ω
þ
ð
0
Þ
ˆ
n
þ
þ
ℏ
ω
−
ð
0
Þ
ˆ
n
−
þ
ℏ
½
ω
m
ˆ
n
b
þ
g
0
ð
ˆ
n
þ
−
ˆ
n
−
Þ
ˆ
x
2
;
ð
8
Þ
where the static optical spring constant
̄
k
s
¼
2
ℏ
g
0
ð
n
þ
−
n
−
Þ
depends on the average intracavity photon number in the
even and odd optical supermodes,
n
≡
h
ˆ
n
i
.
For a sideband resolved system (
ω
m
≫
κ
), the quasistatic
Hamiltonian can be further approximated using a rotating-
wave approximation as
ˆ
H
≈
ℏ
½
ω
þ
ð
0
Þþ
2
~
g
0
ð
ˆ
n
b
þ
1
=
2
Þ
ˆ
n
þ
þ
ℏ
½
ω
−
ð
0
Þ
−
2
~
g
0
ð
ˆ
n
b
þ
1
=
2
Þ
ˆ
n
−
þ
ℏ
ω
m
ˆ
n
b
;
ð
9
Þ
where
~
g
0
≡
g
0
x
2
zpf
¼
~
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 trans-
mission through the even or odd supermode resonances one
can then perform a continuous QND measurement of the
phonon number in the mechanical resonator
[12,27
–
29]
.As
noted in Refs.
[13,14]
, however, the quasistatic picture
described by the dispersion of Eq.
(5)
fails to capture
residual effects resulting from the nonresonant scattering
between the
a
þ
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
j
2
J
j
∼
ω
m
is also very interesting, and is
explored in depth in Refs.
[14,30]
. Transforming to a
reference frame that 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
≈
ℏ
ω
þ
ð
0
Þ
ˆ
n
þ
þ
ℏ
ω
−
ð
0
Þ
ˆ
n
−
þ
ℏ
ω
m
ˆ
n
b
þ
ℏ
~
g
2
2
1
2
J
−
ω
m
þ
1
2
J
þ
ω
m
×
ð
ˆ
a
†
þ
ˆ
a
þ
−
ˆ
a
†
−
ˆ
a
−
Þð
ˆ
b
þ
ˆ
b
†
Þ
2
þ
ℏ
~
g
2
2
1
2
J
−
ω
m
−
1
2
J
þ
ω
m
×
ð
ˆ
a
†
þ
ˆ
a
−
þ
ˆ
a
†
−
ˆ
a
þ
Þ
2
;
ð
10
Þ
where we assume
j
~
g=
δ
j
≪
1
for
δ
≡
j
2
J
j
−
ω
m
, and terms
of order
~
g
3
=
ð
2
J
ω
m
Þ
2
and higher are neglected. In the
limit
j
J
j
≫
ω
m
, we recover the quasistatic result of Eq.
(7)
,
whereas in the near-resonant limit of
j
δ
j
≪
j
J
j
,
ω
m
,we
arrive at
ˆ
H
eff
≈
ℏ
ω
þ
ð
0
Þ
ˆ
n
þ
þ
ℏ
ω
−
ð
0
Þ
ˆ
n
−
þ
ℏ
ω
m
ˆ
n
b
þ
ℏ
~
g
2
2
δ
½
2
sgn
ð
J
Þð
ˆ
n
þ
−
ˆ
n
−
Þð
ˆ
n
b
þ
1
Þ
þ
2
ˆ
n
þ
ˆ
n
−
þ
ˆ
n
þ
þ
ˆ
n
−
:
ð
11
Þ
Here, we neglect highly oscillatory terms such as
ð
ˆ
a
†
þ
ˆ
a
−
Þ
2
and
ˆ
b
2
, a good approximation in the sideband-resolved
regime (
κ
≪
ω
m
,
j
J
j
). From Eq.
(11)
, we find that the
frequency shift per phonon of the optical resonances is
much larger than in the quasistatic case (
~
g
2
=
2
j
δ
j
≫
~
g
2
=
2
j
J
j
). Although a QND measurement of phonon num-
ber 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 intracavity photon
number stored in one optical supermode can be performed
by monitoring the transmission of light through the other
supermode
[14,31]
.
III. DOUBLE-SLOTTED PHOTONIC CRYSTAL
OPTOMECHANICAL CAVITY
A sketch of the double-slotted photonic crystal cavity
structure is shown in Fig.
1(a)
. As we detail below, the
optical cavity structure can be thought of as being formed
from two coupled photonic crystal waveguides, one around
each of the nanoscale slots, and each with propagation
direction along the
x
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 reso-
nances of interest in this work are the fundamental in-plane
POSITION-SQUARED COUPLING IN A TUNABLE
...
PHYS. REV. X
5,
041024 (2015)
041024-3
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
x
axis of the
central nanobeam, the linear dispersive coupling coeffi-
cients 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
¼
ω
2
[cf. Eq.
(5)
]. Figure
1(b)
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 super-
mode branches (
a
þ
,
a
−
) in the presence of tunnel coupling
J
. The mechanical modes of the outer slabs (
b
1
,
b
2
) provide
degrees of freedom for postfabrication tuning of the slotted
waveguide optical modes, i.e., to symmetrize the structure
such that
ω
1
¼
ω
2
. This is achieved in practice by integrat-
ing metallic electrodes which form capacitors at the outer
edge of the two slabs of the structure as schematically
shown in Fig.
1(a)
.
The double-slotted photonic crystal cavity of this work is
realized in the silicon-on-insulator material system, with a
top silicon device layer thickness of 220 nm and an
underlying buried oxide layer of
3
μ
m. Fabrication begins
with the patterning of the metal electrodes of the capacitors
and involves electron-beam (
e
-beam) lithography followed
by evaporation and lift-off of a bilayer 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 that 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 that define and isolate the outer capacitors. A fluorine-
based (C
4
F
8
and SF
6
) inductively coupled reactive-ion etch
is used to transfer the
e
-beam lithography pattern through
the silicon device layer. The remaining
e
-beam 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 acid etch to
remove the sacrificial buried oxide layer (this also removes
the deposited protective silicon dioxide layer), followed by
a water rinse and critical point drying.
Ascanningelectronmicroscope(SEM)imageshowingthe
overall fabricated device structure is shown in Fig.
1(c)
.
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.
1(d)
and
1(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 band structure
To further understand the optical properties of the
double-slotted photonic crystal cavity, we display in
Fig.
2(a)
the photonic band structure of the periodic
waveguide structure. The parameters of the waveguide
are given in the caption of Fig.
2(a)
. Here, we show only
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
(a)
(c)(d)
(e)
(b)
FIG. 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. Because of tunnel coupling at a rate
J
, the
slot modes
a
1
and
a
2
hybridize into the even and odd supermodes
a
þ
and
a
−
, which have a parabolic dispersion near the central
anticrossing point (
ω
1
¼
ω
2
). (c) SEM image of a fabricated
double-slotted photonic crystal device in the silicon-on-insulator
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.
TAOFIQ K. PARAÏSO
et al.
PHYS. REV. X
5,
041024 (2015)
041024-4
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 band gap 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.
2(b)
and
2(c)
,
respectively.
An optical cavity is defined by decreasing the lattice
constant 4.5% below the nominal value of
a
0
¼
480
nm for
the middle five periods of the waveguide [see Fig.
2(d)
].
This has the effect of locally pushing the bands toward
higher frequencies
[35,36]
, 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.
2(d)
and
2(e)
,
respectively. These optical modes correspond to the normal
modes
a
þ
and
a
−
in Sec.
II
, which are symmetric and
antisymmetric superpositions, respectively, of the cavity
modes localized around each slot (
a
1
and
a
2
). Because of the
nonmonotonic decrease in the even waveguide supermode
as one moves away from the
X
band edge [cf. Fig.
2(a)
], we
find that the simulated optical
Q
factor of the even
a
þ
cavity
supermode is significantly lower than that of the odd
a
−
cavity 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
¼
100
nm. For this slot width
we find a lower frequency for the even (
a
þ
) supermode
than for the odd (
a
−
) supermode at the
X
-point photonic
band edge 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 to 100 nm in steps of 1 nm, all other
parameters are the same as in Fig.
2
. For the slot widths
(a)
(b)
(c)
(d)
FIG. 3. Tuning of the slot widths of the double-slotted photonic
crystal cavity showing (a) the mean wavelength shift and (b) the
splitting
2
J
¼
ω
þ
−
ω
−
of the even and odd cavity supermodes
versus slot width
s
¼
s
1
¼
s
2
. (c),(d) Avoided crossing of the
cavity supermodes obtained by tuning
s
1
while keeping
s
2
fixed
at (c)
s
2
¼
90
nm and (d)
s
2
¼
95
nm. The red and black data
points correspond to the supermode branch with even and odd
symmetry at the center of the anticrossing, respectively. Note that
the upper and lower supermode branch switch symmetry between
small slots (
s
2
¼
90
nm) and large slots (
s
2
¼
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
are performed using the
COMSOL
FEM mode solver
[34]
.
(a)(b)
(c)
(d)
(e)
FIG. 2. (a) Band structure diagram of the periodic (along
x
)
double-slotted photonic crystal waveguide structure. Here, we
show only 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
band gap 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
¼
480
nm), the thickness of the silicon
slab (
d
¼
220
nm), the width of the two slots (
s
¼
100
nm), and
the refractive index of the silicon layer (
n
Si
¼
3
.
42
). The hole
radius in the outer slabs and the central nanobeam is
r
¼
144
nm.
The gray 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) are performed using the plane-wave mode solver MPB
[32,33]
. Normalized
E
y
field profile of the corresponding
localized cavity supermodes of (d) odd and (e) even spatial
symmetry about the
y
axis 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 are performed using the
COMSOL
finite-
element method mode solver package
[34]
.
POSITION-SQUARED COUPLING IN A TUNABLE
...
PHYS. REV. X
5,
041024 (2015)
041024-5
tuned symmetrically (
s
1
¼
s
2
¼
s
), the mean wavelength
of the even and odd cavity supermodes and their frequency
splitting
2
J
¼
ω
þ
−
ω
−
are plotted in Figs.
3(a)
and
3(b)
,
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
¼
90
nm to a negative value for
s
¼
100
nm slots and crossing zero for a slot width of
s
¼
95
nm. In Figs.
3(c)
and
3(d)
, the symmetry is broken
by keeping
s
2
fixed and scanning
s
1
; the cavity supermodes
are driven through an anticrossing with a splitting deter-
mined by the fixed slot width
s
2
.
The spectral inversion of the even
a
þ
and odd
a
−
cavity
supermodes predicted in Fig.
3(b)
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 farther 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
blueshift of both cavity supermodes. The odd mode having a
larger electricfieldenergydensityintheair slotsthantheeven
mode is more affected by a change in the slot widths.
Therefore, upon equal increase of the slot widths, the odd
modeexperienceslarger frequency shifts than the even mode,
which results in a tuning of the frequency splitting. For
particular geometrical parameters of the central nanobeam, 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 quasistatic 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.
[37]
. 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 multiaxis set of encoded dc-motor stages with 50-nm
step resolution. The light in the fiber is polarized parallel
with the surface of the 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.
4(a)
–
4(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
[36]
, and thus increasing the voltage
V
i
on a
given capacitor widens the waveguide slot
s
i
and (predomi-
nantly) 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
¼
25
pm
=
V
2
.
We fabricate devices with slot widths targeted for a range
of 75
–
85 nm, chosen smaller than the expected zero-
splitting slot width of
s
¼
95
nm so that the capacitors
could be used to tune through the zero-splitting point.
While splittings larger than 150 GHz are 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 the following we focus on the results
from a single device with an as-fabricated slot size
of
s
≈
75
nm.
A. Anticrossing measurements
Figure
4
shows intensity plots of the normalized optical
transmission through the optical fiber taper when evan-
escently coupled to the photonic crystal cavity of a device
with nominal slot width
s
¼
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 anticrossing split-
ting from the measured dispersion of the cavity supermodes
is
2
J=
2
π
¼
50
, 12, and
−
25
GHz for
V
2
¼
1
, 15, and 18 V,
respectively. In order to distinguish between the odd and
even cavity supermodes at the anticrossing point, we use
the fact that both the coupling rate to the fiber taper
κ
e
and
the intrinsic linewidth
κ
i
depend on the symmetry of the
cavity mode. First, the odd supermode branch becomes
dark at the anticrossing because it cannot couple to the
symmetric fiber taper mode. Second, from numerical FEM
simulation we find that in the vicinity of the anticrossing
point the linewidth of the odd supermode branch narrows
while the linewidth of the even supermode branch broad-
ens. Far from the anticrossing 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.
4(a)
–
4(c)
, as well as in the
measured linewidth of the optical supermode resonances
shown in Figs.
4(g)
and
4(h)
. Figure
4(a)
was taken with a
small voltage
V
2
¼
1
V, corresponding to a small slot
width at the anticrossing point, and is thus consistent with
the even mode frequency being higher than the odd mode
frequency for small slot widths [cf. Fig.
3(b)
]. The exact
opposite identification is made in Fig.
4(c)
, where
TAOFIQ K. PARAÏSO
et al.
PHYS. REV. X
5,
041024 (2015)
041024-6