## Abstract

In this paper we study and design quasi-2D optomechanical crystals, waveguides, and
resonant cavities formed from patterned slabs. Two-dimensional periodicity allows for
in-plane pseudo-bandgaps in frequency where resonant optical and mechanical
excitations localized to the slab are forbidden. By tailoring the unit cell geometry,
we show that it is possible to have a slab crystal with simultaneous optical and
mechanical pseudo-bandgaps, and for which optical waveguiding is not compromised. We
then use these crystals to design optomechanical cavities in which strongly
interacting, co-localized photonic-phononic resonances occur. A resonant cavity
structure formed by perturbing a “linear defect” waveguide of optical and acoustic
waves in a silicon optomechanical crystal slab is shown to support an optical
resonance at wavelength *λ*
_{0} ≈ 1.5 *µ*m and a mechanical resonance of frequency
*ω _{m}*/2

*π*≈ 9.5 GHz. These resonances, due to the simultaneous pseudo-bandgap of the waveguide structure, are simulated to have optical and mechanical radiation-limited

*Q*-factors greater than 10

^{7}. The optomechanical coupling of the optical and acoustic resonances in this cavity due to radiation pressure is also studied, with a quantum conversion rate, corresponding to the scattering rate of a single cavity photon via a single cavity phonon, calculated to be

*g*/2

*π*= 292 kHz.

© 2010 Optical Society of America

## 1. Introduction

Recent efforts [1, 2] to utilize radiation pressure forces to probe and manipulate micro- and nano-scale mechanical objects has spawned a number of new types of optomechanical systems. Amongst these, guided wave nanostructures in which large gradients in the optical intensity are manifest, have been shown to possess extremely large radiation pressure effects [3–8]. The recent demonstration of strongly interacting co-localized photonic and phononic resonances in a quasi one-dimensional (1D) optomechanical crystal (OMC) [9, 10] has shown that it may be possible to coherently control phonons, photons, and their interactions on an integrated, chipscale platform. Considering the already wide applicability of quasi two-dimensional (2D) slab photonic crystals [11–16] and 2D phononic crystals [17–21], and for purposes of integration and obtaining better optical and mechanical mode localization, it is desirable to investigate the prospects of a quasi-2D OMC architecture.

Previous studies of the photonic and phononic properties of infinitely thick 2D crystal structures have shown, amongst other things, the practicality of deaf and blind structures with simultaneous photonic and phononic in-plane bandgaps [22], the experimental demonstration of a full in-plane phononic bandgap [23], and the theoretical prospects of strongly co-localizing light and sound in defect cavities [24]. More recently, studies of quasi-2D silicon slab structures have been performed in which phononic bandgap mirrors have been experimentally measured [25] and crystal structures possessing a simultaneous in-plane bandgap for guided photons and phonons have been proposed [26, 27]. Each of these past studies was performed in the linear regime, neglecting photon-phonon interactions resulting from radiation pressure. Radiation pressure effects have previously been investigated theoretically in quasi-2D photonic crystal structures [28], albeit in double-layer slab structures in which it is the flexural mechanical modes of the slabs which give rise to the optomechanical interaction. Perhaps closer to the subject of our work is the recent study of optomechanical interactions in quasi-2D photonic crystal fibers [29–32], in which measurements have shown that these crystal fiber systems can support traveling photonic and phononic modes with a strong nonlinear optical interaction.

In this article we aim to build on our previous work with quasi-1D nanobeam OMCs, and
develop quasi-2D thin-film crystal structures capable of efficient routing,
localization, and interaction of light and sound waves over the full plane. Unlike in
the infinitely thick 2D crystal structures, thin-film structures require certain limits
on the air filling fraction in order to maintain effective optical waveguiding, thus
making it more challenging to realize a phononic bandgap. We begin in Sec. 2 with the
study of two new crystal structures for this purpose, one called the “cross” substrate
and the other the “snowflake” substrate. These thin-film crystal structures possess
large in-plane phononic bandgaps by incorporating resonant elements in the substrate
[33]. After investigating the origin of these
large bandgaps in varying levels of detail, we move to studying the photonic properties
of these structures, and demonstrate in Sec. 2.4 that the “snowflake” substrate
possesses a large simultaneous optical bandgap. From here we move in Sec. 3 to creating
one-dimensional defects, i.e. waveguides, in the “snowflake” structure, and studying
their tuning properties. In Sec. 4, to understand the optomechanical coupling, we also
study the guided-mode optomechanical coupling in these waveguide structures and find
their simple relation to the expected cavity optomechanical coupling. We culminate our
analysis in Sec. 5 with the design of an optomechanical cavity on a silicon slab with an
optical *Q* > 5 × 10^{7} and an extremely large optomechanical
coupling rate.

## 2. Crystal Design

In this section we study two types of crystal structures, one with a square lattice and
the other with a hexagonal lattice. Most photonic and phononic crystals to date have
utilized circular holes. Here we investigate the degree of freedom associated with hole
shape. Circular holes are the simplest holes to fabricate and the most symmetric.
Considering that it is desirable for the hole to have at least as much (point-group)
symmetry as the underlying lattice, circular holes seem like the obvious choice.
Unfortunately, circular holes fail to provide large mechanical bandgaps at desirable
frequencies, and fail completely at providing a bandgap for a variety of slab
thicknesses. In cases where full simultaneous phononic-photonic bandgaps are achievable,
e.g. the square lattice proposed by Mohammadi, et al., in Refs. [26, 27], the large hole sizes
make the crystal unsuitable for fabrication of ultrahigh-*Q* optical
cavities. Thankfully, by exploring the shape degree of freedom of the hole it is
possible to do better when it comes to phononic bandgap materials.

#### 2.1. Origin of the Gap: From Effective Medium to Tight-Binding

We start with the toy model consisting of alternating small and large masses coupled
by springs typically studied in solid-state physics [34]. As shown in Fig. 1(a), this
system results in a large bandgap in the frequency spectrum of the mechanical
vibrations of this linear chain. The bandgap arises from the existence of modes of
oscillation with widely different frequencies. Traveling phonon modes where the large
masses are predominantly excited (low frequency “acoustic”-like phonons) are split in
frequency from those where the small masses are predominantly excited (high frequency
“optical”-like phonons). The frequency bandgap between low and high frequency
vibrations becomes more pronounced with increasing difference in particle masses. A
quasi-1D nanomechanical realization of the linear chain of alternating small and
large masses coupled by springs is shown in Fig.
1(b). This structure consists of a linear array of square “drumheads”
joined together by narrower “connector” pieces. Here, and throughout the rest of this
article, the material of the mechanical structure is assumed to be silicon (Si), with
Young’s modulus of 170 GPa and mass density of 2329 kg/m^{3}. These and the
following mechanical simulations were done using COMSOL Multiphysics [35], a finite-element solver. We see that large
phononic bandgaps exist in this structure just as in the elementary periodic array of
small and large masses coupled via springs.

To more accurately understand the origin of the bandgap in the quasi-1D
nanomechanical structure of Fig. 1(b), we
consider the low and high frequency bands separately. In the lowfrequency, small
wavelength (*λ* ≪ *a*) regime, an effective medium
theory [36, 37] can be used to calculate the exact values of wave velocities and the
dispersion at the origin. For our purposes, we wish to push these low frequency bands
to as low a frequency as possible. This can be achieved by reducing the width of the
connector piece, *a*−2*h*. The reasoning heuristically
is as follows. The stiffness of a beam is in a sense dependent on the stiffness of
its weakest link. In this case, the connectors, acting as contacts between the larger
square drumhead sections, make the beam much floppier than an unperturbed beam of
uniform cross-section. At higher frequencies, as *ν* approaches the
resonances, *ν _{j}*, of each square drumhead, a tight-binding model may be used to model the
dispersion. This sort of model is valid as long as the interaction between each
square is made small, which we have achieved by making

*a*−2

*h*small. To get a bandgap then, the effective medium bands are squeezed down to low frequencies by reducing

*a*−2

*h*, which does not affect the tight-binding bands since their frequency is set by the internal resonances of the larger square drumhead section. Simultaneously, the interaction strength between the coupled drumhead sections is reduced, and therefore so is the slope of the tight-binding bands making them more flat. These two effects conspire together to produce a large phononic bandgap.

#### 2.2. Quasi-1D Phononic Tight-Binding Bands: Symmetry and Dispersion

As will be evident below in the design of phononic waveguides and cavities, it is
useful to study the properties of the tight-banding bands in a little more detail.
Considering the simple linear lattice example introduced above [refer to Fig. 1(b)], the group of the wavevector at the Γ
and *X* points of the Brillouin zone possesses the full point group
symmetry of the crystal itself. As such, the Bloch modes (**Q**) at these
high symmetry points can be characterized according to their vector symmetry with
respect to reflection *σ _{x}*(

*x,y,z*) = (−

*x,y,z*) about the

*x*-axis in a plane intersecting the middle of the unit cell of the linear nanomechanical structure. We have for

*x*-symmetric (

**Q**

^{(x+)}) and

*x*-antisymmetric (

**Q**

^{(x−)}) Bloch modes, for which

*σ*

_{x}**Q**

^{(x±)}(

*σ*

_{x}**r**) = ±

**Q**

^{(x±)}(

**r**), the following relation for the displacement vector (

**Q**) at the unit-cell boundaries (

**r**

*):*

_{b}On the other hand, we have the usual phase shift acquired by the different Bloch
modes, which for the *X*-point and Γ-point modes yields,

These constraints together imply that *x*-symmetric modes at the Γ-
and *X*-points must obey respectively the conditions

In many cases the excitation is of a dominant polarization. For example, for the
(*σ _{y},σ_{z}*) = (+,+) vector symmetry vibrational modes, the dominant polarization is
found to be

*Q*(not suprising as for this symmetry the displacement in

_{x}*y*or

*z*would have to be of higher order, involving a stretching/compression of the membrane along these directions). For this symmetry of modes then, the boundary condition given by Eq. (5) is approximately equivalent to a fixed boundary condition, whereas the boundary condition of Eq. (6) can be approximated as a free boundary condition. Since intuitively, one expects the frequency of a free resonator to increase by fixing part of it, the above argument implies that for the given polarization mode symmetry, (

*σ*) = (+,+),

_{y},σ_{z}*x*-symmetric modes bend up while

*x*-antisymmetric modes slope downwards as the in-plane wavevector

*k*varies from Γ −

*X*. This correspondence is elucidated in Fig. 2, where the tight-binding bands are shown for the aforementioned quasi-1D system. In all cases, the frequencies calculated with fixed or free boundary conditions on the square drumhead correspond to their respective high symmetry Bloch function frequencies to within a factor 10

^{−2}. In what follows, we use these features of the tight-binding bands to help identify vibrational bands that will strongly couple to optical waves.

#### 2.3. Quasi-2D “Cross” Crystal

In going to a quasi-2D system, we begin with the simplest extension of the quasi-1D
structure of Fig. 1(b), that being the same
elements of square drumheads and thin connectors, but now arrayed in two dimensions
as shown in Fig. 3(c). We call this the
“cross” substrate, since it results from a square array of crosses cut into a slab.
Each cross is characterized by a height *h* and a width
*w*, which along with the lattice spacing *a* and
slab thickness *d*, serve to fully parametrize geometrically the
system. For reference, the reciprocal space representation of the lattice is shown in
Fig. 3(d), in which the common notation of
the high symmetry points of the first Brillouin zone in a square lattice are used.
The phononic bandstructure, including all symmetries of vibrational modes, of the
cross substrate is shown in Fig. 3(b) for the
same set of parameters as used in the quasi-1D structure of Fig. 1. Clearly this Si structure has excellent mechanical
properties, with a large bandgap opening up between 5.3 and 6.8 GHz. The phononic gap
maps for a variety of relevant parameters of the cross substrate are shown in Figure 4, indicating the robustness of the
phonon bandgap to each parameter.

Despite the encouraging mechanical properties of the cross substrate, as we will see
it is difficult to realize this substrate as an OMC due to its unfavorable optical
properties. In general, design of photonic bandgaps in materials with refractive
index of order *n* ~ 3 (semiconductors) is more suited to a plane-wave
expansion approach as opposed to the tight-binding picture discussed above for
nanomechanical vibrations. The square lattice, with its low symmetry, behaves
differently for plane waves propagating in different directions, such as at the high
symmetry *X* and *M* points of the first Brillouin zone
boundary. This results in a much smaller in-plane photonic bandgap for the square
lattice in comparison to a higher symmetry lattice such as the hexagonal lattice.
Note also that for quasi-2D photonic structures we usually talk about an
*inplane* bandgap only, as light (unlike sound) can also propagate
vertically into the surrounding vacuum cladding. The photonic bandstructure of the Si
cross substrate, assuming a refractive index for Si of 3.4, is shown in Fig. 3(a) for the even vertically symmetric
optical modes of the Si slab (these modes include the fundamental TE-like bands).
These and the following photonic simulations were done using the software package MPB
[38]. The photonic bandstructure consists
of two distinct regions: (i) the guided mode region below the light line (shown as a
black line) in which there are a discrete number of mode bands, all of which are
evanescent into the surrounding cladding, and (ii) the region above the light (shown
in grey) in which *leaky* guided mode resonances and a continuum of
radiation modes exist. We have shown the extension of the lowest lying guided mode
band above the light line as a red line, where it behaves as a
*leaky*, but highly localized resonance of the Si slab.

Due to the presence of the continuum of radiation modes above the light line, it is
clear that in the photonic case one can only talk about a pseudo in-plane bandgap in
the case of a quasi-2D slab structure. Much more problematic is the presence of
*leaky* guided mode resonances above the light line, which can
strongly couple to the guided modes of the structure in the presence of perturbations
of the lattice. These perturbations can both be unintentional, such as in fabrication
imperfections, or intentianal such as in the formation of resonant cavities and
linear waveguides. Here we will adopt a practical definition of a pseudo in-plane
photonic bandgap as one where the bandgap extends across *both* guided
mode and leaky resonances. As can be seen in the photonic bandstructure of Fig. 3(d), the cross structure lacks even a
pseudo in-plane photonic bandgap for the TE-like modes of the slab structure (the odd
symmetry modes of the slab, which include the fundamental TM-like modes, do not even
possess a guided mode bandgap). Due then, to leaky resonances with large local
density of states in the slab, the cross substrate is ruled out as a suitable
structure from which to form important photonic elements such as
ultrahigh-*Q* optical cavities. This is the same problem faced by
the “honey-comb” crystal proposed in Ref. [26].

#### 2.4. Quasi-2D “Snowflake” Crystal

The hexagonal lattice counterpart of the cross substrate is shown in Fig. 5, which we term the “snowflake” substrate.
Each snowflake pattern is characterized by radius *r*, and a width
*w*; which along with the lattice spacing *a* and
slab thickness *d*, serve to fully parametrize geometrically the
system. The phononic gap maps for a variety of relevant parameters of the snowflake
substrate are shown in Figs. 6(a) and 6(b). The snowflake substrate, unlike the cross
substrate, does possess favorable optical properties. Gap maps for the photonic
properties are shown in Figs. 6(c) and 6(d), where again we focus on the even vertical
symmetry optical modes of the slab (which includes the fundamental TE-like bands of
primary interest). Due to the fact that the phononics is more sensitive to connector
width, *a* − 2*r*, while the photonics is more
sensitive to the air-slot width, *w*, this crystal provides us with
two different tunable parameters for control over the photonic and phononic
properties of the system. This property, which is apparent from the respective gap
maps, is highly valuable as it pertains to designing optomechanical devices which
simultaneous must manipulate sound and light. For this reason, and for its superior
optical bandgap properties, in what follows we focus on the snowflake crystal. We
begin with a study of the light and sound waveguiding properties of this
structure.

## 3. Waveguide Design

A line-defect in a photonic bandgap material will act as a waveguide for light [39, 40]. In a
quasi-2D crystal structure with simultaneous in-plane photonic and phononic bandgaps,
such a defect should be able to direct both light and sound around in the plane of the
crystal. In this section, we study the guided modes in these types of linear-defect
waveguides of the snowflake crystal slab. It is of interest to understand the properties
of these modes, since it has been shown for photonic crystal slabs that good waveguides
also yield ultrahigh-*Q* optical cavities [41, 42]. In Sec. 5 we follow
this design technique to demonstrate ultrahigh-*Q* optomechanical
cavities in which both light and sound are effectively localized in the same volume with
very little radiative loss. Although we are primarily interested here in
*cavity* optomechanics, the coupling between guided photons and
phonons in periodic structures is also an interesting subject of study which has been
recently explored in photonic crystal fiber systems [32]. As will be discussed further below, studying
*guided-mode* optomechanical properties also allows one to simplify
the design and optimization of cavity optomechanical devices.

When designing optical and mechanical cavities and waveguides, it is desirable to have
control over where the waveguide bands are placed within the frequency bandgap [39]. For example, previous demonstrations of
photonic crystal cavities have often involved a line defect waveguide in which a
localized cavity resonance was formed by locally tuning the line defect guided mode out
of the bandwidth of the waveguide band and into the bandgap. In previous work, this
tuning has been achieved by changing locally the longitudinal lattice constant along the
guiding direction [41], the width of the
line-defect forming the waveguide [42], or the
radii of the holes adjacent to the line-defect region [43]. Figure 7(a) shows an example of
a linear defect waveguide formed in the snowflake crystal slab. This waveguide consists
of a row of snowflake holes which have been removed (a *W1*-like
waveguide), and a transverse variation of the slowflake hole size has been applied (see
Fig. 7 caption for details). Figure 7(b) shows a corresponding resonant cavity
structure formed from the linear defect waveguide.

The photonic and phononic bandstructures of a linear defect waveguide with
*W* = 200 nm is shown in Fig.
8. In this diagram only the vertically (*z*) symmetric optical
bands are shown. The waveguide dielectric structure also has a transverse mirror
symmetry, *σ _{y}*, about the

*y*-axis in the middle of the waveguide. The transverse symmetry of each of the mechanical bands is indicated in Fig. 8 by the color of the band, whereas for the optical waveguide bands we use the labels

*E*(even) and

*O*(odd) to indicate the

*σ*parity of the fields. A similar

_{y}*E*and

*O*labelling scheme is used for the mechanical waveguide modes at the Γ-point, although in this case the parity relates to the

*σ*symmetry of the mechanical displacement field within each unit cell along the

_{x}*x*-direction of the waveguide. Also, in the photonic band diagram we have indicated the light cone with a dark grey shade. The regions above and below the guided mode bandgap of the unperturbed snowflake crystal are shaded a light grey in both the photonic and phononic band diagrams, with the leaky regions of the photonic waveguide bands colored in red. For this waveguide width, a significant pseudo-bandgap can be seen in the photonic bandstructure (shaded in light blue). At the same time, several phononic bandgaps can be seen in the bands of the mechanical band diagram of the waveguide. Our primary interest when forming a resonant cavity in the next section will be the bandgap highlighted in light blue between the highest frequency phononic waveguide band and the upper frequency band-edge of the unperturbed snowflake crystal. This mechanical waveguide band has the desirable property of very flat dispersion which allows for highly localized phonon cavity states in the presence of waveguide perturbations.

In the design of an optomechanical device, one in which light and sound must be
simultaneously manipulated, the independent control of the two types of wave excitations
is desired. The tuning of the optical and mechanical waveguide bands of a
*W*1-like line-defect waveguide is shown in Fig. 9 for two different types of waveguide geometry
perturbations. For simplicity we have only shown the tuning of the waveguide modes at
the zone boundary (*X*-point for the optical and Γ-point for the
mechanical waveguide modes). From these plots, it is evident that radius modulations of
the snowflake hole tend to tune the optical and mechanical modes in differing
directions, whereas for width modulations (through *W*) of the waveguide
the optical and mechanical frequencies tend to tune in a similar direction (this is not
true of the odd symmetry mechanical mode in this narrow waveguide). A heuristic argument
for this behavior goes as follows. For an optical mode, regions of high refractive
index, such as the silicon, tend to reduce the optical frequency for a given curvature
(wavevector) of the optical wave. Quite the opposite is true for mechanical excitations
in which the material adds to the stiffness of the structure, thereby generally raising
the frequency of acoustic waves. This suggests that by using a perturbation where the
hole sizes are slightly reduced, since we are increasing the amount of
high-index(stiffness) material the photon (phonon) sees while keeping the wavelength
constant, the frequency of the mode will decrease (increase). On the other hand, when
the waveguide width is increased, one is in some sense increasing the “transverse”
lattice constant of the crystal. Since the lattice constant sets the wavelength for both
the optical and mechanical modes, increasing it will cause the frequencies of both waves
to drop. By using these perturbations simultaneously, within a certain small range,
photonic and phononic bands may be raised and lowered independently. This is a powerful
consequence of using differing tuning mechanisms, and allows us to design independently
the longitudinal (cavity) confining potentials for phonons and photons.

## 4. Optomechanical Coupling Relations

In the design methodology followed in this paper, a resonant cavity is formed by locally
modulating the properties of a linear defect, or waveguide, in a planar crystal
structure. This methodology has been used previously in the design of
high-*Q* photonic crystal cavities, and due to the similarities in the
kinematic properties of the wave equations of phonons and photons, we expect it to also
produce phononic crystal cavities in the snowflake crystal structure. For our purposes,
however, having the photonic and phononic resonances simply co-localized doesn’t
suffice, as their interaction must also be tailored, and maximized. To better understand
the origin of the optomechanical coupling, it is useful to study the interaction at the
level of the waveguide modes, from which the localized cavity resonances are formed.

The coupling between guided optical and mechanical waves has been studied previously in
both theoretical and experimental settings [32,
44] for photonic crystal fiber structures with
*continuous* longitudinal symmetry. These analyses have generally
expanded on calculations of acousto-optical scattering in bulk materials [45]. For this work, we are interested in the case of
*discrete* longitudinal symmetry, and a calculation of the coupling
per unit cell. Instead of extending the aforementioned analysis to the case of discrete
longitudinal symmetry, our approach will be to start with the known
*cavity* optomechanical coupling relations, and then to work backwards
to a relavant per unit cell guided-mode coupling. This has the benefit of providing a
direct relation between the guided-mode and cavity-mode optomechanical couplings.
Specifically, using Johnson’s formulation of perturbation theory for moving dielectric
boundaries [46], which has been previously
applied successfully to the calculation of cavity optomechanical properties [9, 10], we
find the cavity optomechanical coupling in terms of the localized mechanical vibration
field and an effective optical energy density. Then, using the Wannier function
formalism [47–50], we relate (approximately) the
cavity and waveguide modes to one another, providing a relation for the guided-mode
optomechanical coupling from the cavity-mode optomechanical coupling.

The formula for the (lowest-order) optomechanical coupling rate in a deformable cavity has been shown to be most generally given by [9]:

This is a pure rate, and is found by multiplying the dispersion of the optical cavity
resonance with mechanical oscillator displacement (*g*
_{OM} ≡ ∂*ω _{c}*/∂

*x*) by the zero-point fluctuation amplitude of the mechanical oscillator $({x}_{\mathrm{ZPF}}=\sqrt{\frac{\overline{h}}{{2m}_{\mathrm{eff}}\Omega}})$ . To relate the cavity optomechanical coupling to the properties of the waveguides, we assume that our acoustic and optical cavity fields can both be written in terms of a waveguide Bloch function multiplied by a smoothly varying envelope function. In general these cavity fields can be represented as superpositions of terms of the type ${\mathbf{E}}_{\pm}\left(\mathbf{r}\right)={\mathbf{E}}_{e}\left(\mathbf{r}\right){e}^{\pm i{\mathbf{k}}_{e}\xb7\mathbf{r}}{f}_{e}\left(x\right)({\mathbf{Q}}_{\pm}\left(\mathbf{r}\right)={\mathbf{Q}}_{m}\left(\mathbf{r}\right){e}^{\pm i{\mathbf{k}}_{m}\xb7\mathbf{r}}{f}_{m}\left(x\right))$ , where

**E**

*(*

_{e}**Q**

*) is a periodic Bloch function,*

_{m}*k*(

_{e}*k*) the reduced wavevector, and

_{m}*f*(

_{e}*x*) (

*f*(

_{m}*x*)) the envelope of the electric (mechanical displacement) cavity field. Note that both the co- and counter-propagating terms (±

*k*) are necessary to describe the localized standing-wave resonances of a linear cavity.

_{e,m}While a general analysis is possible, we limit ourselves here to the case where the
optical cavity mode is formed from the *X*-point of the waveguide band
diagram [see Fig. 8(a)], with **k**
* _{e}* =

**k**

*, and the mechanical cavity mode is a Γ-point mode with*

_{X}**k**

*= 0. The condition on the optical mode is necessary in a quasi-2D slab structure to achieve a high-*

_{m}*Q*optical cavity, as small

*k*-vector components in the plane of slab can radiate into the light cone of the low-index cladding surrounding the slab. The mechanical mode condition is a phase matching requirement for the coupling of the two counter-propagating optical waves of a standing-wave cavity resonance,

**k**

*+(−*

_{X}**k**

*) =*

_{X}**k**

*= 0. Hence, starting with*

_{m}and assuming that the envelope functions vary slowly over a lattice spacing, and that they have no zero-crossings, we separate the integrals into a product of two integrals, one over a single waveguide unit-cell and the other across multiple unit-cells. For example,

From here, we arrive at the following expression for *g*:

where *g*
_{Δ} is the guided-mode optomechanical coupling given by

Equation (11) shows that the
optomechanical coupling achievable is the product of a term *g*
_{Δ} depending only on the linear waveguide properties and a second term which
is a function of the envelope functions *f _{e}*(

*x*) and

*f*(

_{m}*x*) describing the localization of the cavity resonances along the length of the waveguide. In many relevant systems, the optical and mechanical modes may be approximated as Gaussians with standard deviation in intensity profile of

*L*and

_{m}*L*. In this case the envelope-dependent component of the cavity optomechanical coupling is,

_{e}The largest value of the envelope dependent part of the optomechanical coupling from
Eq. (13) is achieved by making
*L _{m}* =

*L*/√2. For this ratio of mechanical and optical cavity Gaussian profiles one arrives at a maximum optomechanical coupling rate of

_{e}Clearly, the more localized the optical and mechanical resonances the larger the optomechanical coupling, all other things being equal.

## 5. Optomechanical Cavity Design

As mentioned above, higher-*Q* optical cavity resonances will be formed
from optical modes near the *X*-point of the linear-defect waveguide as
they lie underneath the out-of-plane light cone. Phase-matching then requires the
mechanical resonance to be formed primarily from the Γ-point in order to have
significant optomechanical interaction of the two localized resonances. Given the even
symmetry along *x* within each unit cell of the
*intensity* of the optical field for waveguide modes at the
*X*-point (again, all *X*-point modes can be classified
by their *σ _{x}* parity, and thus their intensity must be symmetric), and considering the form
of Eq. (12) for the per unit cell
optomechanical coupling, we see that only the even symmetry (

*E*) mechanical modes at the Γ-point yield a non-zero

*g*

_{Δ}. As such, we choose to form the localized phononic cavity resonance from the uppermost phononic waveguide band in Fig. 8(b) (the lowermost phononic waveguide band, also of even parity at the Γ-point, was found to have a smaller

*g*

_{Δ}). To avoid coupling to mechanical waveguide bands below this upper band, we choose to form a cavity defect perturbation which

*increases*the frequency of the upper mechanical waveguide band, thus localizing the phononic resonance in the highlighted blue pseudo-bandgap of Fig. 8(b).

In the case of the optical field, we can choose to form the localized cavity resonance
from either the upper or the lower frequency waveguide bands that define the photonic
pseudo-bandgap of the waveguide [see Fig. 8(a)].
We choose here to use the upper frequency waveguide band due to its more central
location in the pseudo-bandgap of the unperturbed snowflake crystal. The curvature of
the upper frequency waveguide band near the *X*-point is positive, so we
need to have the cavity perturbation cause a local *decrease* in the
band-edge frequency; the opposite frequency shift required for that of the mechanical
waveguide band. From Fig. 9, it is evident that
the radius modulation satisfies this requirement, i.e. it tunes the optical and
mechanical band-edge modes in opposing directions. Here we use a combination of
transverse and longitudinal quadractic modulations with parameters
(*r _{d},N_{d},N*

^{WG}

*) = (0.03,14,5) to form the optomechanical cavity. The resulting cavity geometry is depicted in Fig. 7(b).*

_{d}As hoped, the resulting optomechanical cavity supports a localized fundamental
mechanical mode of frequency *ν _{m}* = 9.5 GHz and a fundamental optical mode of frequency

*ν*= 205.6 THz, the latter corresponding to a free-space wavelength of

_{o}*λ*

_{0}= 1.459

*µ*m. Plots of the FEM-simulated electric and mechanical displacement fields of both localized cavity resonances are shown in Fig. 10. Owing to the extremely flat dispersion of the mechanical waveguide band, the cavity phonon resonance is localized almost entirely to the central unit cell of the cavity. The resulting effective motional mass of this localized phonon is only

*m*

_{eff}= 3.85 fg (where the maximum displacement of the mechanical vibration is used as the normal coordinate defining the mode [10]). The optomechanical coupling of photons and phonons in this cavity is calculated to be

*g*/2

*π*= 292 kHz. This value represents an enormous radiation pressure coupling of the optical and mechanical fields, being only a factor of two or so weaker than that found in the strongly-coupled zipper cavity [4] and only slightly below the approximate upper-bound calculated using Eq. (14) of

*g*

_{optimal}/2

*π*= 382 kHz. We should also note that an estimate of the elasto-optic contribution to g indicates that this effect in the Si snowflake cavity is several orders of magnitude smaller than the optomechanical coupling computed using Eq. (7).

The radiation-limited optical *Q*-factor was also computed for this
cavity using perfectly-matched-layer radiation boundary conditions, and found to be
limited to *Q* ≈ 5.1 × 10^{7} due to out-of-plane radiation. The
corresponding mechanical *Q*-factor of the localized phonon resonances is
*Q _{m}* ≫ 10

^{7}for 40 unit cells surrounding the cavity region. Unlike in the optical case, this mechanical

*Q*can be made arbitrarily large by increasing the number of unit cells surrounding the cavity due to the lack of an out-of-plane loss mechanism and a pseudo-bandgap in-plane. The important practical limiting factor in both the mechanical and optical

*Q*will of course be the presence of perturbations in a real fabricated structure. From Fig. 8 we see that in the absence of perturbations breaking the

*z*-mirror symmetry, the mechanical radiation loss can be made effectively zero due to the complete lack of states to which the phononic resonance can couple. Perturbations breaking the

*z*-mirror symmetry will however induce loss by coupling to odd vertical symmetry mechanical waveguide modes, colored green in Fig. 8. Terminating the waveguide, i.e. transitioning back into the bulk snowflake crystal after some number of unit cells, eliminates this component of mechanical radiative loss as well.

## 6. Conclusions

By introducing a new crystal structure, the “snowflake” structure, with photonic and phononic properties amenable to the formation of low-loss optical and acoustic resonances, we have shown that the on-chip control and interaction of photons and phonons may be possible in a realistic setting. This crystal structure, along with the waveguides and cavities proposed in this paper, allow for the unification of the quasi-2D photonic and phononic crystal systems. In the future, systems utilizing this unification may not only have wide applicability in classical optics [51], but could also provide new ways to do quantum information science by allowing for a new class of hybrid quantum systems [51–53].

## Acknowledgements

This work was supported by the DARPA/MTO ORCHID program through a grant from AFOSR. ASN gratefully acknowledges support from NSERC.

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