Diamond optomechanical crystals: supplementary
material
M
ICHAEL
J.
B
UREK
,
1
J
USTIN
D.
C
OHEN
,
2
S
EÁN
M.
M
EENEHAN
,
2
N
AYERA
E
L
-
S
AWAH
,
1,3
C
LEAVEN
C
HIA
,
1
T
HIBAUD
R
UELLE
,
1,4
S
RUJAN
M
EESALA
,
1
J
AKE
R
OCHMAN
,
1,3
H
AIG
A.
A
TIKIAN
,
1
M
ATTHEW
M
ARKHAM
,
5
D
ANIEL
J.
T
WITCHEN
,
5
M
IKHAIL
D.
L
UKIN
,
6
O
SKAR
P
AINTER
,
2
M
ARKO
L
ON
Č
AR
,
1,*
1
John A. Paulson School of Engineering and Applied Sciences,
Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA
2
Kavli Nanoscience Institute, Institute for
Quantum Information and Matter and Thomas J. Watson, Sr., Laboratory of Applied Phys
ics,
California Institute of Techno
logy, Pasadena, CA 91125, USA
3
University of Waterloo, 200 University Aven
ue West, Waterloo, ON, N2L 3G1, Canada
4
École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
5
Element Six Innovation, Fermi Avenue, Harwel
l Oxford, Didcot, Oxfordshire OX110QR, UK
6
Department of Physics, Harvard University
, 17 Oxford Street, Cambridge, MA 02138, USA
*Corresponding author:
loncar@seas.harvard.edu
Published
18
November
2016
This
document
provides
supplementary
information to "
Diamond
optomechanical
crystals,”
http://dx.doi.org/10.1364/optica.
3
.0
01404.
©
2016 Optical Society
of America
http://dx.doi.org/10.1364/optica.
3
.0
01404.s00
1
1. GUIDED ACOUSTIC PHONON MODES IN DIAMOND
OPTOMECHANICAL CRYST
ALS
To supplement our discussion of the guided acoustic phonon
modes supported by diamond optomechanical crystals (OMCs),
we present normalized displacement profiles of the nominal unit
cell at the Γ (
k
x
= 0) and X (
k
x
= π/a) points of its mechanical
bandstructure (originally displayed in Figure 1(c) of the main
text).
Figures S1 and S2 reveal the guided acoustic modes categorized
by
even (solid black lines) and odd (dashed blue lines) vector
symmetries about the
y
-axis, respectively, with displacement
profiles originating from the indicated band edges shown as ins
ets
(three dimensional, top down and cross-section views included).
Note, the unit cell lattice constant in the displacement profil
es is
displayed between the (
h
x
,n
,
h
y,n
) and (
h
x,n+1
,
h
y,n+1
) center points, in
order to clearly reveal displacement components within the air
holes. Mechanical simulations included here and throughout the
main text use the full anisotropic elasticity matrix of diamond
[1],
where (C
11
, C
12
, C
44
) = (1076, 125, 578) GPa. However, due to
considerations expanded upon in section 5 of this supplementary
material, devices characterized in this work were ultimately
fabricated with their
x
-axis oriented with the in-plane [110]
crystallographic direction. Thus, a rotated version of the
anisotropic elasticity matrix ensured proper device orientation
in
our simulations, with guided
mode propagation along the
x
-axis
aligned with the [110] crystallographic direction, with the z-a
xis
aligned with [001]. Only a small (< 10 %) change in the guided
mode frequencies was observed between simulations with unit
cell x-axis alignment to the [100] and [110] in plane crystal
directions.
While the mechanical bandstructures reveal a rich library of
guided acoustic modes in the few to 16 GHz frequency range, onl
y
guided modes originating from
y
-symmetric bands ultimately
couple to the optical cavity [2]. Additionally, modes originating
from the Γ-point ensure large optomechanical coupling rates in
the
final design [3]. With this in mind, two modes from the Γ-point
of y-
symmetric bands enable design of diamond OMCs with large
single-photon optomechanical coupling rates,
g
o
. Specifically, the Γ-
point modes from the 4
th
and 7
th
y
-symmetric bands, referred to as
the “flapping” and “swelling” modes, respectively, were both
investigated.
2.
OPTIMIZED DIAMOND OPTOMECHANICAL
CRYSTAL DESIGN
As discussed in the main text, the final diamond OMC design rel
ies
on transitioning from a “mirror” region formed by the base unit
cell in Figure 1(a) to a “defect” cell, which localizes the tar
get