Enhanced photon-phonon coupling via dimerization in one-dimensional
optomechanical crystals
Matthew H. Matheny
Citation:
Appl. Phys. Lett.
112
, 253104 (2018); doi: 10.1063/1.5030659
View online:
https://doi.org/10.1063/1.5030659
View Table of Contents:
http://aip.scitation.org/toc/apl/112/25
Published by the
American Institute of Physics
Enhanced photon-phonon coupling via dimerization in one-dimensional
optomechanical crystals
Matthew H.
Matheny
a)
Institute for Quantum Information and Matter and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
(Received 23 March 2018; accepted 5 June 2018; published online 19 June 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
p
¼
1.7 MHz between an optical resonance at
k
c
¼
1545 nm and a mechanical resonance at 1.14 GHz.
Published by AIP Publishing.
https://doi.org/10.1063/1.5030659
Optomechanical crystals (OMCs)
1
are periodically
structured materials in which optical and acoustic waves are
strongly coupled via radiation pressure. For typical solid-
state materials, owing to the orders of the magnitude differ-
ence between the speed of light and sound, near-infrared
photons of frequency
x
/2
p
200 THz are matched in wave-
length to acoustic waves in the GHz frequency band. Thin-
film silicon (Si) OMCs have been used to trap and localize
these disparate waves, allowing for a number of proposed
experiments in cavity optomechanics to be realized.
2
–
4
An exciting possibility is the creation of an appreciable
nonlinearity at the single photon level using patterned dielec-
tric films.
5
,
6
However, observing nonlinear photon-phonon
interactions requires a vacuum coupling rate
g
0
larger than
the intrinsic optical decay rate
j
.
7
In addition, the mechani-
cal frequencies must be larger than optical decay rates with
x
m
>
j
/2, i.e., be “sideband-resolved.” Currently, sideband-
resolved optomechanical systems are two orders of magni-
tude away from an appreciable nonlinear interaction,
5
g
0
/
j
0.01. Here, we theoretically show that single photon-
phonon strong coupling
g
0
/
j
1 is possible in optomechani-
cal crystal cavities.
The OMC design which demonstrates the strongest cou-
pling in silicon has been implemented in materials with a
lower index of refraction. However, in lower index materials,
a substantial difference in coupling is found. Chan
et al.
2
demonstrated a coupling rate of
g
0
/2
p
¼
1.1 MHz in their sil-
icon OMC cavity, while the coupling does not exceed
200 kHz in works using the same type of OMC cavity in
lower index materials.
8
–
12
The difference in coupling is
unfortunate since lower index materials may have useful
properties such as the ability to host optically addressable
spin defects,
13
–
15
trap atoms,
16
,
17
or couple to microwave
fields via piezoelectricity.
18
,
19
This difference arises from
the nature of the optomechanical interaction. Chan
et al.
found that their optomechanical interaction is primarily due
to the photoelastic effect, whose matrix element scales as the
fourth power of the index of refraction.
20
This leads to sig-
nificantly 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 bound-
ary 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 derived
by Johnson
et al.
21
Combining this with the mechanical
zero-point fluctuations,
x
zpf
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
=
2
m
eff
x
m
p
, gives the vac-
uum coupling rate
g
0,
MB
, where
m
eff
is the effective mass of
the mechanical mode with frequency
x
m
. This rate can be
written in the form
1
g
0
;
MB
¼
ffiffiffi
h
8
r
x
c
ffiffiffiffiffiffiffi
x
m
p
Ð
@
V
ð
q
ð
r
Þ
^
n
Þð
D
j
E
k
j
2
D
ð
1
Þj
D
?
j
2
Þ
d
2
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ð
V
q
j
q
ð
r
Þj
2
d
3
r
ð
r
ÞjÞ
q
Ð
V
ð
r
Þj
E
ð
r
Þj
2
d
3
r
;
(1)
where
^
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 dielec-
tric structure with volume
V
, and
q
(
r
) is the unit-normalized
mechanical displacement field.
22
A similar equation can be
expressed for the photoelastic contribution to the optome-
chanical coupling. Given this equation with fixed material
properties, the possible strategies for increasing coupling are
increasing mode overlap, decreasing mode volumes, increas-
ing optical cavity frequency, or decreasing mechanical fre-
quency. In this work, we focus on decreasing mechanical
frequencies (which boosts
x
zpf
) while leaving the other quan-
tities of the equation fixed.
Since the optical intensity profile of the unit cell com-
prising photonic crystal cavities is typically mirror symmet-
ric about the midpoint, previous OMC designs focused on
using fully symmetric extensional-type mechanical modes
within the unit cell.
1
,
20
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.
a)
Electronic mail: matheny@caltech.edu
0003-6951/2018/112(25)/253104/5/$30.00
Published by AIP Publishing.
112
, 253104-1
APPLIED PHYSICS LETTERS
112
, 253104 (2018)
Flexural modes are usually not symmetric at the
C
-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
C
-point via dimerization of
the lattice. Thus, we can preserve the mode volumes of the
unit cell while decreasing mechanical mode frequencies. In
addition, the major contribution to the coupling is from the
moving boundary of the dielectric due to mechanical flexure.
Lattice dimerization was first discussed by Peierls
23
who
predicted an energy gap in the electronic band structure of
atomic systems. By doubling the size of the unit cell and
breaking the degeneracy (via different hole sizes), we engi-
neer a dimer unit cell with the
k
-vector and the sum of the
two constituent
k
-vectors. If we choose a constituent
k
-vector
of alternating
6
p
/
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 displace-
ment 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 lat-
tice and the
z
-axis is out of page. In Fig.
1
, we show a simu-
lation of the band structure for the “ladder” OMC in silicon,
before (dashed lines) and after (solid lines) dimerization. It
was previously found that the largest coupling occurred
between the X-point optical “dielectric” mode [Fig.
1(a)
,
right-hand side, green-dashed] and the
C
-point “breathing”
mode of the mechanics [Fig.
1(b)
, left-hand side, red-
dashed]. These modes exhibit strong overlap and are both
fully symmetric; 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 “ladder”
OMC are the first X-point mode [Fig.
1(b)
, right-hand side,
green-dashed] and the second
C
-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 Eq.
(1)
due to the
anti-symmetry of the displacement and strain fields in the
x
-
axis. We can conceptually understand the lack of a moving
boundary interaction by visualizing the mechanical displace-
ment of the 1st “flexural” mode moving through the optical
field of the “dielectric” mode (green-dashed boxes of Fig.
1
).
From zero displacement, the left
yz
-surface moves to a
region of higher optical intensity, while the right
yz
-surface
moves to a region of lower optical intensity; however, at
zero displacement, both of these surfaces start at the same
value of optical intensity. Thus, the contributions to the
change in the radiation pressure from the two surfaces cancel
each other for infinitesimal displacements.
Dimerization leads to the folding of both band struc-
tures, sending X-point modes to the
C
-point of the new lat-
tice, and doubling the number of bands. This is shown in
Fig.
1
, where the band structure is folded at
k
x
¼
p
/2
a
. The
new mechanical “flexural” band is now fully symmetric at
the
C
-point [green-solid, Fig.
1(b)
], and a finite coupling
between this mechanical mode and the optical modes is
generated. Note that this dimerized mechanical mode at the
C
-point is basically the X-point mode of the “ladder” com-
posed with its mirror (inverse
k
-vector). Visualizing the
dimerized “flexural” mode displacement within the dimer-
ized optical “dielectric” mode helps to understand this finite
coupling (green-solid boxes, Fig.
1
). Essentially, since all of
the
yz
-surfaces of the dimerized mode displace towards
regions of lower intensity (for the pictures shown), there is a
finite change to the radiation pressure on the dielectric
boundaries from the optical field.
Additionally, the lowest “dielectric” mode will now be
split into optical modes whose electric field intensity is the
strongest at different pairs of interior dielectric boundaries.
This creates a strong overlap between the electric field inten-
sity at the surface and the displacement field. The band split-
ting of these two new optical “dielectric” modes at the
X-point is determined by degeneracy of the two constituents
of the dimer. In the limit of the
x
-axis size of one of the holes
going to zero (say
h
x
1
!
0), we recover the simple “ladder”
FIG. 1. For both optical and mechanical band structures, the dotted lines are
bands for the unit cell before dimerization and the solid lines after dimeriza-
tion. The unit cell has dimensions as follows:
f
a
;
w
y
;
h
x
;
h
y
;
t
g¼f
250 nm
;
1500 nm
;
125 nm
;
1300 nm
;
220 nm
g
, where
a
is the lattice constant,
w
y
is
the extent of the body in the y-dimension, and
h
x
and
h
y
are the hole dimen-
sions in the x and y directions, respectively. In the dimerized unit cell,
f
h
x
1
;
h
x
2
g¼f
100 nm
;
150 nm
g
. (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. The 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. The color in the unit cell plots indicates the total
displacement (
j
Q
ð
r
Þj
). Simulations are performed using COMSOL.
253104-2 Matthew H. Matheny
Appl. Phys. Lett.
112
, 253104 (2018)
lattice with band splitting similar to that shown by the
dashed lines in Fig.
1(a)
. Thus, the degeneracy tunes the
bandgap.
The band diagram from Fig.
1
is meant as an illustrative
example, not as an optimized design. In creating a smooth,
tapered phonon mode in an extended crystal defect, mono-
tonic dispersion of the phonon band is desired.
24
However,
the dimerized mechanical “flexural” band in Fig.
1(b)
(green-solid line) is relatively flat. In order to increase the
dispersion of the dimerized “flexural” mode of the solid-
green band, degeneracy of the
y
-axis parameters should also
be broken (which connects adjacent unit cells at points of
larger displacement). Since we now focus on OMC cavities,
designs using the dimerized lattice in the rest of the letter
also break the
y
-axis degeneracy.
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 analyze a recently
reported OMC unit cell with the material properties of alu-
minum nitride (AlN).
10
In Figs.
2(a)–2(c)
, we show simula-
tions of the optomechanical coupling via the photoelastic
effect between the 1st optical dielectric mode and the
“breathing” mechanical mode, similar to previous
designs.
2
,
8
,
10
In Figs.
2(d)–2(f)
, we show simulations for the
dimerized lattice, where the unit cell degeneracy between
the constituents has been strongly broken in the
x
and
y
dimensions. The simulation parameters are initially set so
that the optical wavelength of the
C
-point eigenmode of the
OMC unit cell is 1550 nm. The bare optomechanical cou-
plings are scaled by the optical frequency found in simula-
tion, which removes the contribution due to
x
c
from Eq.
(1)
.
Thus, the figure highlights the contributions from field over-
lap and
x
zpf
.
Figure
2(a)
shows the structure of the “breathing” unit
cell along with the optical mode’s electric field energy and
mechanical mode’s
y
-strain. Figure
2(b)
shows the photoe-
lastic coupling as the
x
and
z
parameters are changed in one
axis of the surface plot, with the
y
parameters changed along
the other axis of the surface plot. Note that the moving
boundary coupling in the “breathing” mode can either add or
subtract from the overall coupling in this type of unit cell
and is thus not included. This does not detract from the point
of the analysis, which is primarily concerned with the effects
of
x
zpf
. The value found for the photoelastic coupling within
the unit cell is consistent with previous work.
10
,
11
In Fig.
2(c)
, we show the individual contributions from
g
0
/
x
opt
and
x
zpf
along two different diagonals of the surface plot.
Figure
2(d)
shows the “flexural” OMC structure with
electric field energy and total displacement. The overall cou-
pling (photoelastic and moving boundary contributions) is
plotted in Fig.
2(e)
, where the photoelastic part only adds to,
but is less than 1% of, the coupling due to the moving
FIG. 2. Simulated scaling of aluminum nitride unit cell coupling. The coupling is scaled by the optical mode frequency in order to highlight the effect
of both
the optical/elastic field overlap and enhanced
x
zpf
of the new design. In the simulation, since the
z
-axis thickness is chosen by the wavelength and the
x
-axis
parameters strongly determine the wavelength, the parameters related to
x
and
z
are swept simultaneously. The y-parameters are stepped separately. The initial
AlN thickness is set to 330 nm. (a) and (d) Unit cell, optical field intensity [left (a) and (d)] of the first dielectric band and mechanical y-axis strain [
right (a)] of
the “breathing” mode, and displacement [right (d)] of the “flexural” mode, respectively. (b) and (e) Optomechanical coupling for the unit cell as diff
erent sets
of dimensions are scaled. (c) and (f) Contributions along different diagonals of (b) and (e) to
g
0
. Simulations are performed using COMSOL.
253104-3 Matthew H. Matheny
Appl. Phys. Lett.
112
, 253104 (2018)