Enhanced photon-phonon coupling via dimerization in one-dimensional
optomechanical crystals
Matthew H. Matheny
∗
Institute for Quantum Information and Matter and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
(Dated: March 28, 2018)
We show that dimerization of an optomechanical crystal lattice, which leads to folding of the band
diagram, can couple flexural mechanical modes to optical fields within the unit cell via radiation
pressure. When compared to currently realized crystals, a substantial improvement in the coupling
between photons and phonons is found. For experimental verification, we implement a dimerized
lattice in a silicon optomechanical nanobeam cavity and measure a vacuum coupling rate of
g
0
/
2
π
=
1
.
7 MHz between an optical resonance at
λ
c
= 1545 nm and a mechanical resonance at 1
.
14 GHz.
Optomechanical crystals (OMCs) [1] are periodically
structured materials in which optical and acoustic waves
are strongly coupled via radiation pressure. For typi-
cal solid-state materials, owing to the orders of magni-
tude difference between the speed of light and sound,
near-infrared photons of frequency
ω/
2
π
∼
200 THz are
matched in wavelength to acoustic waves in the GHz fre-
quency band. Thin-film silicon (Si) OMCs have been
used to trap and localize these disparate waves, allow-
ing for a number of proposed experiments in cavity-
optomechanics to be realized [2–4].
An exciting possibility is the creation of an apprecia-
ble nonlinearity at the single photon level using patterned
dielectric films [5]. However, observing nonlinear photon-
phonon interactions requires a vacuum coupling rate
g
0
larger than the intrinsic optical decay rate
κ
[6]. In ad-
dition, the mechanical frequencies must be larger than
optical decay rates with
ω
m
> κ/
2, i.e. be ’sideband-
resolved’. Currently, sideband-resolved optomechanical
systems are two orders of magnitude away from an ap-
preciable nonlinear interaction [5],
g
0
/κ
≈
0
.
01. Here, we
theoretically show single photon-phonon strong coupling
g
0
/κ
≈
1 is possible in optomechanical crystals cavities.
The OMC design which demonstrates the strongest
coupling in silicon [2] has been implemented in materi-
als with a lower index of refraction, such as silicon nitride
(Si
3
N
4
) [7], aluminum nitride (AlN) [8], and Diamond [9].
In those works the coupling does not exceed 200 kHz,
while Chan, et al. [10] show a coupling
g
0
= 1
.
1 MHz.
This difference arises from the nature of the optomechan-
ical interaction. Chan, et al. find their optomechani-
cal interaction is primarily due to the photoelastic effect,
whose matrix element scales as the fourth power of the
index of refraction [10]. This leads to significantly smaller
coupling in materials with a lower index. Here we show
that the coupling can be significantly improved in a lower
index material using the moving boundary interaction,
whose matrix element scales as the square of the index.
The optical frequency shift per unit displacement for
a moving boundary in a dielectric optical cavity was de-
∗
matheny@caltech.edu
rived by Johnson, et. al [11]. Combining this with the
mechanical zero-point fluctuations,
x
zpf
=
√
~
/
2
m
eff
ω
m
,
gives the vacuum coupling rate
g
0
,MB
, where
m
eff
is the
effective mass of the mechanical mode with frequency
ω
m
.
This rate can be written in the form [1],
g
0
,MB
=
−
√
~
8
∗
ω
c
√
ω
m
∗
́
∂V
(
q
(
r
)
·
ˆ
n
)(∆
̄
̄
|
E
||
|
2
−
∆(
−
1
)
|
D
⊥
|
2
)
d
2
r
√
́
V
ρ
|
q
(
r
)
|
2
d
3
r
(
r
)
|
)
́
V
̄
̄
(
r
)
|
E
(
r
)
|
2
d
3
r
(1)
where ˆ
n
is the outward vector normal to the surface
of the dielectric boundary,
̄
̄
is the dielectric tensor,
E
(
D
) is the electric (displacement) field,
∂V
is the surface
of the dielectric structure with volume
V
, and
q
(
r
) is
the unit-normalized mechanical displacement field [12].
A similar equation can be expressed for the photoelas-
tic contribution to the optomechanical coupling. Given
this equation with fixed material properties, the possible
strategies for increasing coupling are: increasing mode
overlap, decreasing mode volumes, increasing optical cav-
ity frequency, or decreasing mechanical frequency. In
this work, we focus on decreasing mechanical frequencies
(which boosts
x
zpf
) while leaving the other quantities of
the equation fixed.
Since the optical intensity profile of the unit cell com-
prising photonic crystal cavities is typically mirror sym-
metric about the midpoint, previous OMC designs fo-
cused on using fully symmetric extensional-type mechani-
cal modes within the unit cell [1, 10]. However, the lowest
frequency eigenmodes usually involve mechanical torsion
or flexure. Thus, we outline a method for using flexural
modes within the unit cell.
Engineering coupling between flexural mechanical
modes and optical modes in resonant OMC cavities is
problematic. Flexural modes are usually not symmet-
ric at the Γ-point of the band diagram. Thus, they do
not couple into symmetric optical modes at the
X
-point.
In this work, we design a fully symmetric flexural mode
at the Γ-point via dimerization of the lattice. Thus, we
can preserve the mode volumes of the unit cell while de-
creasing mechanical mode frequencies. In addition, the
arXiv:1803.09324v1 [cond-mat.mes-hall] 25 Mar 2018
2
major contribution to the coupling is from the moving
boundary of the dielectric due to mechanical flexure.
Lattice dimerization was first discussed by Peierls [13]
who predicted an energy gap in the electronic band struc-
ture of atomic systems. By doubling the size of the unit
cell and breaking the degeneracy (via different hole sizes),
we engineer a dimer unit cell with
k
-vector the sum of the
two constituent
k
-vectors. If we choose a constituent
k
-
vector of alternating
±
π/a
where
a
is the lattice constant,
we can create a null
k
-vector for the dimer. In essence,
we imbue the OMC lattice with a two-’atom’ basis of
flexing beams. We illustrate how this corresponds to a
symmetrized displacement vector in the 1-D lattice.
We begin the discussion by dimerizing the first OMC
design [1], which is based on a simple ”ladder” structure.
Throughout this Letter, the
x
-axis is in the direction of
the lattice and the
z
-axis is out of page. In Fig. 1, we
show a simulation of the band structure for the ”ladder”
OMC in silicon, before (dashed lines) and after (solid
lines) dimerization. It was found that the largest cou-
pling occured between the
X
-point optical ”dielectric”
mode (Fig. 1(a) right hand side, green-dashed) and the
Γ-point ”breathing” mode of the mechanics (Fig. 1(b)
left hand side, red-dashed). These modes exhibit strong
overlap; this gives a large photoelastic contribution to
the coupling in high index materials. Since the electric
field is not designed to be maximum at the boundaries,
this type of design does not emphasize optomechanical
coupling due to a moving boundary.
The flexural mechanical modes of the simple ”lad-
der” OMC are the first
X
-point mode (Fig. 1(b), right
hand side, green-dashed) and the second Γ-point mode
(Fig. 1(b), left hand side, blue-dashed). These modes do
not couple to any of the
X
-point optical modes according
to Eqn. (1) due to antisymmetry of the displacement and
strain fields in the
x
-axis.
However, dimerization can give a symmetric flexural
mode (Fig. 1(b) left hand side, green solid line). Also,
the lowest ”dielectric” mode will now be split into optical
modes whose electric field intensity is strongest at differ-
ent pairs of interior dielectric boundaries. This creates
a strong overlap between the electric field intensity and
the displacement field. Essentially, dimerization leads to
folding of both band structures, sending
X
-point modes
to the Γ-point of the new lattice, and doubling the num-
ber of bands. This is shown in Fig. 1, where the band
structure is folded at
k
x
=
π/
2
a
. The overlap between
the new optical modes at the new
X
-point (Fig. 1(a),
green solid line) and folded bottom mechanical mode
(Fig. 1(b), left hand side, green solid line) now gives a
finite coupling.
Next we study the differences in optomechanical cou-
pling between the ”breathing” OMC and the dimerized
”flexural” OMC as we scale the unit cell along
x
and
y
. This analysis emphasizes the benefits of a dimerized
design when using lower index materials. Here, we ana-
lyze a recently reported OMC [8] unit cell with the ma-
terial properties of AlN. In Fig. 2(a,b,c) we show simu-
Fold Line
0
2.2
(THz)
Undimerized
Dimerized
(GHz)
0
2
Fold Line
Undimerized
Dimerized
a)
b)
Γ
X
Γ
X
FIG. 1.
For both optical and mechanical band
structures the dotted lines are bands for the unit
cell before dimerization, the solid lines after dimeriza-
tion.
The unit cell has dimensions
{
a,w
y
,h
x
,h
y
,t
}
=
{
250
nm,
1500
nm,
125
nm,
1300
nm,
220
nm
}
, where
a
is the
lattice constant,
w
y
is the extent of the body in the y-
dimension, and
h
x
,h
y
are the hole dimensions in the x
and y directions, respectively. In the dimerized unit cell
{
h
x
1
,h
x
2
}
=
{
100
nm,
150
nm
}
.
a)
Simulated optical band
structure of a ”ladder” OMC in silicon. Here we show only
the modes with electric field (vector) symmetry in the
y
and
z
axes. Color in the unit cell plots indicates the value of the
electric field in y (
E
y
(
r
)). Simulations are performed with the
MIT Photonic Bands (MPB) package.
b)
Mechanical band
structure of the same OMC as (a). We show the acoustic
modes with vector displacement symmetry in the
y
and
z
axes. Color in the unit cell plots indicates total displacement
(
|
Q
(
r
)
|
). Simulations are performed in COMSOL.
lations of the optomechanical coupling via the photoe-
lastic effect between the 1st optical dielectric mode and
the ”breathing” mechanical mode, similar to previous de-
signs [2, 7, 8]. In Fig. 2(d,e,f) we show simulations for
the dimerized lattice, where the unit cell degeneracy has
been strongly broken to generate a large optical band gap
useful for making high quality cavities. The simulation
parameters are initially set so that the optical wavelength
of the Γ-point eigenmode of the OMC unit cell is 1550nm.
The bare optomechanical couplings are scaled by the op-
tical frequency found in simulation, which removes the
contribution due to
ω
c
from Eqn. 1. Thus, the figure