††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Optomechanical crystal with bound states in the continuum

Shengyan Liu  Holonyak Micro and Nanotechnology Laboratory and Department of
Electrical and Computer Engineering, University of Illinois at Urbana-
Champaign, Urbana, IL 61801 USA  Illinois Quantum Information Science and
Technology Center, University of Illinois at Urbana-Champaign, Urbana, IL
61801 USA  Hao Tong  Holonyak Micro and Nanotechnology Laboratory and
Department of Electrical and Computer Engineering, University of Illinois at
Urbana-Champaign, Urbana, IL 61801 USA  Illinois Quantum Information Science
and Technology Center, University of Illinois at Urbana-Champaign, Urbana, IL
61801 USA  Kejie Fang<EMAIL_ADDRESS>Holonyak Micro and Nanotechnology
Laboratory and Department of Electrical and Computer Engineering, University
of Illinois at Urbana-Champaign, Urbana, IL 61801 USA  Illinois Quantum
Information Science and Technology Center, University of Illinois at Urbana-
Champaign, Urbana, IL 61801 USA

###### Abstract

Chipscale micro- and nano-optomechanical systems, hinging on the intangible
radiation-pressure force, have shown their unique strength in sensing, signal
transduction, and exploration of quantum physics with mechanical resonators.
Optomechanical crystals, as one of the leading device platforms, enable
simultaneous molding of the band structure of optical photons and microwave
phonons with strong optomechanical coupling. Here, we demonstrate a new breed
of optomechanical crystals in two-dimensional slab-on-substrate structures
empowered by mechanical bound states in the continuum (BICs) at 8 GHz. We show
symmetry-induced BIC emergence with optomechanical couplings up to
$g/2\pi\approx 2.5$ MHz per unit cell, on par with low-dimensional
optomechanical crystals. Our work paves the way towards exploration of photon-
phonon interaction beyond suspended microcavities, which might lead to new
applications of optomechanics from phonon sensing to quantum transduction.

Cavity-optomechanics has attracted extensive studies in recent years because
of the rich physics associated with the nonlinear optomechanical interaction
and a broad range of prospective applications from signal transduction to
sensing aspelmeyer2014cavity . One of the leading optomechanical device
architectures is optomechanical crystals eichenfield2009optomechanical , where
micro- and nano-scale structures give rise to strong radiation-pressure force
coupling between wavelength-similar optical photons and microwave phonons. By
band-structure engineering of suspended optomechanical crystals, both one-
dimensional and quasi-two-dimensional defect cavities
eichenfield2009optomechanical ; safavi2014two ; ren2020two  have been created
with long-lived optical and mechanical resonances maccabe2020nano . Such
optomechanical crystal microcavities have enabled groundbreaking quantum
experiments including ground-state cooling of mechanical resonators
chan2011laser , testing Bell inequality marinkovic2018optomechanical , and a
mechanical quantum memory wallucks2020quantum .

Despite the success of optomechanical microcavities, it is highly desirable to
explore two-dimensional optomechanical crystals. On one hand, two-dimensional
optomechanical crystals offer more degrees of freedom for manipulation of
photon-phonon interaction to induce collective phenomena ludwig2013quantum ;
brendel2017pseudomagnetic ; brendel2018snowflake ; ren2020topological . On the
other hand, extended optomechanical crystals, especially in unsuspended
structures sarabalis2017release ; qi2021nonsuspended ; zhang2021silicon ,
might alleviate the optical-absorption induced heating that plagues released
microcavities meenehan2014silicon . Ideally, such slab-on-substrate
optomechanical crystals should facilitate dissipation of heat phonons while
sustaining long-lived mechanical resonances in the device layer.

Recently, mechanical bound states in the continuum (BICs) are observed in two-
dimensional slab-on-substrate phononic crystals tong2020observation . Despite
having a zero Bloch wavevector and thus spectrally immersing in the sound cone
of the substrate, these mechanical BICs are confined in the slab because of
the symmetry-induced decoupling from the acoustic radiation field. There are
also proposals and demonstrations of mechanical BICs in microcavities due to,
for example, accidental radiation amplitude cancellation chen2016mechanical ;
yu2021observation . A significant step forward thus would be coupling
mechanical BICs with optical resonances in an optomechanical crystal which
will bring the effective radiation-pressure force control and associated
functionalities.

In this work, we realize two-dimensional silicon-on-insulator optomechanical
crystals with mechanical BICs coupled with optical guided resonances. In such
periodic optomechanical crystals, the radiation-pressure coupling between the
mechanical BIC and optical modes strongly depends on the mode symmetry
zhao2019mechanical , which in many cases dictates an adversely null
optomechanical coupling. Here, taking into account of both symmetry of the
optomechanical crystal and the silicon crystal lattice, we are able to achieve
optomechanical coupling up to $g/2\pi\approx 2.5$ MHz per unit cell between a
mechanical BIC at 8 GHz and an optical band-edge mode at 193 THz, which is
comparable to one-dimensional suspended optomechanical crystals
chan2012optimized . With optically-transduced mechanical spectroscopy at room
temperature, we demonstrate control of mechanical BICs and optomechanical
coupling via the interplay of symmetry of the optomechanical crystal and
crystalline material. Our work paves the way for study of photon-phonon
interaction in BIC optomechanical crystals at low temperatures, when the
benefit of slab-on-substrate device architecture is expected to arise, and
Floquet topological physics beyond the tight-binding model fang2019anomalous .

Results

Figure 1: Two-dimensional optomechanical crystal with mechanical BICs. a. A
schematic diagram of the two-dimensional silicon-on-insulator optomechanical
crystal and its unit-cell structure. The optomechanical crystal could be
rotated by an angle $\theta$ relative to the silicon crystal lattice. b.
Optical band structure near the $M$ point (top) and mechanical band structure
near the $\Gamma$ point (bottom) for $\theta=0\mathrm{\SIUnitSymbolDegree}$.
The relevant optical and mechanical bands are highlighted in color. c.
Simulated total electric field and total displacement field of the fundamental
TE-like optical mode at the $M$ point and mechanical mode 1 and 2 at the
$\Gamma$ point. d. Simulated radiative quality factor of the mechanical modes
with respect to $\theta$. e. Mechanical mode symmetry illustrated using the
$z$-direction displacement, momentum-space transverse polarization
distribution and the winding number. f. Unit-cell optomechanical coupling of
mechanical mode 1 and 2 with the fundamental TE-like optical mode.

Design of BIC optomechanical crystals. The designed optomechanical crystal in
silicon-on-oxide material system has a hexagonal “snowflake” unit cell (Fig.
1a). The reason behind this design is that six-fold symmetric structures tend
to yield mechanical BICs coupled with optical guided resonances
zhao2019mechanical  and, also, the spike feature of the “snowflake” could lead
to sizable vibrations for large optomechanical couplings. We consider optical
modes at the $M$ point below the light cone and mechanical modes at the
$\Gamma$ point. The latter is necessary because mechanical vibrations
generally induce a linear perturbation of the optical mode energy, i.e.,
$\delta U\sim\int\delta x(\bm{R})\epsilon|E(\bm{R})|^{2}d\bm{R}$, and this
integral is nonzero in a periodic structure only when the mechanical
displacement $\delta x$ has a zero Bloch wavevector (Supplementary Information
(SI)). Since the refractive index of crystalline silicon is isotropic, the
symmetry of optical modes at the $M$ point is governed by the $C_{2v}$ group.
We select the fundamental transverse-electric(TE)-like mode which is odd with
respect to the $xz-$plane. The stiffness tensor of silicon, on the other hand,
is anisotropic in the crystal plane with a $C_{4v}$ group symmetry, which is
incommensurable with the $C_{6v}$ group symmetry of the hexagonal
optomechanical crystal. As a result, the symmetry of mechanical modes at the
$\Gamma$ point will depend on the orientation angle $\theta$ between the
optomechanical crystal and the silicon crystal lattice. When
$\theta=0\mathrm{\SIUnitSymbolDegree}$, $15\mathrm{\SIUnitSymbolDegree}$,
$30\mathrm{\SIUnitSymbolDegree}$, and $45\mathrm{\SIUnitSymbolDegree}$, the
symmetry group of the mechanical mode is $C_{2v}$, while for other
orientations it will be $C_{2}$. Only $C_{2v}$ group supports mechanical BICs
which decouple from both transverse and longitudinal radiation waves
tong2020observation , whose displacement field is perpendicular and parallel
to the wavevector, respectively.

Fig. 1b shows the optical band structure near the $M$ point and mechanical
band structure near the $\Gamma$ point for
$\theta=0\mathrm{\SIUnitSymbolDegree}$ and $(r,w,a,t,h)=(167,34,389,220,3000)$
nm, where $t$ and $h$ are the thickness of the silicon and oxide layers,
respectively, $a$ is the lattice constant, and $(r,w)$ are “snowflake”
dimensions. The fundamental TE-like optical mode has a frequency of 193 THz
and the relevant mechanical modes have frequencies about 7.5 and 8.2 GHz,
respectively. Their mode profiles are shown in Fig. 1c. Simulation shows that
the radiation quality factor $Q_{r}$, i.e., the ratio between the frequency
and radiation loss rate, of mechanical mode 2 (8.2 GHz) diverges and that of
mode 1 is finite but remains relatively high compared to other lossy modes. As
a result, for $\theta=0\mathrm{\SIUnitSymbolDegree}$, mode 2 and 1 are
mechanical BIC and quasi-BIC, respectively. The fact that mode 1 is a quasi-
BIC is because the two mechanical bands are degenerate at the $\Gamma$ point
when the stiffness tensor is isotropic and both modes are BICs belong to the
$E_{2}$ representation of the $C_{6v}$ group; the actual anisotropic stiffness
tensor of silicon splits the degeneracy, reducing the $E_{2}$ representation
to $A_{2}$ (BIC) and $A_{1}$ (quasi-BIC) representations of the $C_{2v}$
group. Mode 1 turns out to have the $A_{1}$ representation, leading to
coupling with the longitudinal radiation wave. When $\theta$ changes, BIC and
quasi-BIC alternate between mode 1 and 2 (while the frequencies of the two
modes remain almost unchanged) as shown in Fig. 1d. This is further
illustrated in Fig. 1e using the mode symmetry. Below, the mode symmetry under
certain symmetry operation is defined with regard to the vector parity of the
electric field of the optical mode or the displacement field of the mechanical
mode. Taking $\theta=0\mathrm{\SIUnitSymbolDegree}$ as an example again, both
mode 1 and 2 are even under the 180$\mathrm{\SIUnitSymbolDegree}$ rotation,
which leads to decoupling from the transverse radiation wave. In addition,
mode 2(1) is odd(even) with respect to the symmetry axes (dashed lines),
resulting decoupling(coupling) from/with the longitudinal radiation wave and
thus a rigorous(quasi) BIC. Same arguments apply to the other three $\theta$’s
while noticing the mirror symmetry axes rotate with $\theta$. The mechanical
BIC and quasi-BIC are also associated with transverse topological charges,
defined as the winding number of far-field transverse polarization around the
$\Gamma$ point tong2020observation . The polarization fields rotate together
with $\theta$, which is unique for the anisotropic mechanical system. For
orientations other than the four specific angles, the two modes belong to the
$A$ representation of the $C_{2}$ group and thus are quasi-BICs which only
couple to the longitudinal acoustic waves. As a result, their quality factor
over $10^{4}$ is significantly higher than unconfined modes.

The interaction between the optical and mechanical modes can be analyzed using
the mode symmetry (SI). Roughly, because the $M$-point optical mode energy
density is even with respect to the $xz-$plane, mechanical modes that are odd
with respect to the $xz-$plane, including the BIC for
$\theta=0\mathrm{\SIUnitSymbolDegree}$ and $45\mathrm{\SIUnitSymbolDegree}$,
will not interact with the optical mode. For other cases, the optomechanical
coupling could be nonzero (see Table 1 for a summary), thanks to the
incommensurable symmetry of the optomechanical crystal and silicon lattice
crystal. The bare optomechanical coupling of a unit cell, $g$, including both
moving boundary and photoelastic effects, is calculated and plotted in Fig.
1f. For example, the BIC at $\theta=15\mathrm{\SIUnitSymbolDegree}$ has
$g/2\pi=1.96$ MHz, with a contribution from the moving-boundary and
photoelastic effect of $0.61$ MHz and $1.35$ MHz, respectively. The coupling
$g$ of the “snowflake” optomechanical crystal will increase with smaller air
gaps. Here, with a practical air gap $w\approx 30$ nm, the BIC optomechanical
crystal achieves a coupling rate (per unit cell) on par with one-dimensional
suspended optomechanical crystals chan2012optimized .

Table 1: Summary of mechanical mode representation and optomechanical coupling. | $0\mathrm{\SIUnitSymbolDegree}$ | $15\mathrm{\SIUnitSymbolDegree}$ | $30\mathrm{\SIUnitSymbolDegree}$ | $45\mathrm{\SIUnitSymbolDegree}$
---|---|---|---|---
Mode 1 |  $A_{1}$, quasi-BIC |  $A_{2}$, BIC |  $A_{1}$, quasi-BIC |  $A_{2}$, BIC
$g$ | $\neq 0$ | $\neq 0$ | $\neq 0$ | 0
Mode 2 |  $A_{2}$, BIC |  $A_{1}$, quasi-BIC |  $A_{2}$, BIC |  $A_{1}$, quasi-BIC
$g$ | 0 | $\neq 0$ | $\neq 0$ | $\neq 0$
Figure 2: Scanning electron microscopy of the optomechanical crystal device.
a. Full device. b. Apodized grating coupler. c. Junction between the
optomechanical crystal and waveguide with tapered photonic crystal mirrors. d.
Boundary between the optomechanical crystal and photonic crystal mirror. The
photonic crystal mirror is displaced by $d-a$. e. Zoom-in view of the
optomechanical crystal.  Figure 3: Multimode optical spectroscopy. a.
Schematic diagram of the measurement setup. The pump light is guided into the
on-chip device using an angle-polished fiber. The reflected light is divided
by a directional coupler for optical (10%) and mechanical (90%) spectroscopy.
The optical signal is used to lock the laser-cavity detuning. VOA, variable
optical attenuator. FPC, fiber polarization controller. EDFA, erbium-doped
fiber amplifier. PD, photodetector. SA, spectrum analyzer. b. Optical spectrum
of a device with multiple standing-wave resonances, one of which has a total
quality factor of 21,000 (right). c. Mode envelop of standing-wave resonances
with given orders corresponding to the dotted resonances in b. The arrow
indicates the waveguide input. d. $\kappa_{e}/\kappa_{i}$ for different number
of nominal mirror layers at the junction between the optomechanical crystal
and waveguide. For each number of mirror layers, multiple devices with
different photonic crystal mirror displacement are measured. e. Total optical
quality factor versus the displacement of the photonic crystal mirror, i.e.,
$d-a$. e is plotted using the same set of data of d.

Photonic crystal band-edge multimodes. The optomechanical crystal device is
fabricated from a silicon-on-insulator wafer with a 220 nm thick silicon
device layer and a 3 µm buried oxide. The air gap of the “snowflake” as small
as 30 nm is achieved. The device consists of several functional components
(Fig. 2). The hexagonal BIC optomechanical crystal has $N$ unit cells along
each edge and is surrounded by photonic crystal mirrors with a triangular
lattice of cylindrical holes on five edges. The photonic crystal mirror, with
the same lattice constant as the snowflake optomechanical crystal, is designed
to have a complete TE-like bandgap with the center wavelength around 1550 nm
to suppress the lateral radiation of the optical band-edge mode. The photonic
crystal mirror is also slightly displaced to minimize the out-of-plane
radiation due to the boundary effect on the finite-size optical mode (Fig.
2d). One side of the hexagonal snowflake crystal is connected to a waveguide
terminated with an apodized grating coupler for coupling light from a single-
mode optical fiber. This configuration makes the optomechanical crystal
effectively a one-port device with (mode-dependent) external, i.e., to-
waveguide, and intrinsic loss rate of $\kappa_{e}$ and $\kappa_{i}$,
respectively. The former can be controlled by the number of photonic crystal
mirror layers between the waveguide and optomechanical crystal (Fig. 2c) to
achieve different coupling conditions including the critical coupling, i.e.,
$\kappa_{e}\approx\kappa_{i}$, which is desirable for sideband-resolved
mechanical spectroscopy.

The device is measured using the setup shown in Fig. 3a. An angle-polished
optical fiber is used to guide light via the on-chip apodized grating coupler
snyder2013packaging ; li2014silicon  to the optomechanical crystal and collect
the reflected light. The angle of the fiber ($\approx
35.5\mathrm{\SIUnitSymbolDegree}$) and the apodized grating coupler are co-
designed to realize an optimized coupling efficiency of $52\%$ for 1550 nm
light and 3-dB bandwidth of $40$ nm (SI). Because of the reflection at the
boundary of finite-size photonic/phononic crystals, band-edge standing-wave
resonances will be formed hood2016atom ; jin2019topologically ;
tong2020observation ; chua2014larger ; chen2022analytical . The mode envelope
of the standing-wave resonances can be approximated by the $(p,q)-$th order
eigenfunctions of a flat-top potential well within the boundary of the
photonic/phononic crystal xu2005confined . Fig. 3b shows the optical
reflection spectrum of a hexagonal optomechanical crystal with $N=25$, where a
series of band-edge standing-wave resonances are observed. The major order $p$
of a standing-wave resonance is identified from the group of resonances it
belongs to and the minor order $q$ is determined by the position of the
resonance in a group. Because of the hyperbolic paraboloid topology of the
band structure near the $M$ point, resonances with smaller $p$ and larger $q$
will have shorter wavelengths. In addition, only $q$-odd resonances are
observed because the excited waveguide mode is even with respect to the center
of the waveguide. Order $q=1$ modes are expected to have deepest resonance dip
given they have largest $\kappa_{e}$ close to the critical coupling. The mode
envelop of some standing-wave resonances are shown in Fig. 3c.

Figure 4: Room-temperature mechanical spectroscopy. a. Measured mechanical
noise power spectrum for one device each for
$\theta=0\mathrm{\SIUnitSymbolDegree}$, $15\mathrm{\SIUnitSymbolDegree}$,
$30\mathrm{\SIUnitSymbolDegree}$ and $45\mathrm{\SIUnitSymbolDegree}$. b.
Mechanical frequency distribution of a group of five devices for each
$\theta$. Error bar is the standard deviation. Dashed lines are the simulated
mechanical frequencies. c. Unit-cell optomechanical coupling. The dashed lines
correspond to simulated coupling for $w=28$ nm (top) and $40$ nm (bottom). d.
Total quality factor of the mechanical modes. The dashed lines are the
estimated bound and mean from the simulation of a $1\times 4$ super-cell with
disorders.

The number of photonic crystal mirror layers at the waveguide-optomechanical
crystal junction is varied to tune $\kappa_{e}/\kappa_{i}$, which is extracted
by fitting the optical resonance spectrum using the normalized reflection
coefficient of the one-port waveguide-coupled resonator,
$R[\omega]=\left|1-\frac{\kappa_{e}}{i(\omega-\omega_{o})+(\kappa_{e}+\kappa_{i})/2}\right|^{2}$.
Fig. 3d shows $\kappa_{e}/\kappa_{i}$ of the $(1,1)$ standing-wave resonance
of a group of devices, where a decreasing trend over the number of junction
mirror layers is observed as a result of the reduced coupling between the
waveguide mode and standing-wave resonances. The displacement of the photonic
crystal mirror around the optomechanical crystal is also varied to optimize
the total quality factor of the standing-wave resonance. As shown in Fig. 3e,
a displacement of $-60$ nm from the nominal lattice constant is optimal, which
is consistent with the simulation (SI). We note the variation of photonic
crystal mirrors not only changes $\kappa_{e}$ but also $\kappa_{i}$ via the
perturbation of the evanescent field, leading to fluctuations of
$\kappa_{e}/\kappa_{i}$. An optimized $(2,1)$ optical resonance is shown in
Fig. 3b with a total quality factor of $21,000$ while being close to the
critical coupling. The quality factor of the optical resonance and its
variation is largely due to the disorder induced scattering to the modes above
the light cone jin2019topologically .

Room-temperature mechanical spectroscopy. We performed the mechanical
spectroscopy at room temperature using a blue-detuned laser with a frequency
$\omega_{l}=\omega_{o}+\omega_{m}$, where $\omega_{o}$ and $\omega_{m}$ are
the frequencies of the optical and mechanical band-edge modes, respectively.
We stabilized the laser-cavity-detuning by locking the device-reflected power
with feedback control of the laser frequency (Fig. 3a). For each device, an
optical standing-wave resonance with $Q_{t}$ about $2\times 10^{4}$ and
$\kappa_{e}/\kappa_{i}$ close to 1 is chosen, which means the device is
operated near the sideband-resolved regime, i.e.,
$\omega_{m}\approx\omega_{o}/Q_{t}$. The reflected pump light with sidebands
due to the modulation of mechanical modes is sent to a high-speed
photodetector. The beating between the pump and sidebands thus yields the
mechanical spectrum which is observed by a spectrum analyzer.

Devices with different orientations between the optomechanical crystal and the
silicon crystal lattice are fabricated to reveal the mechanical (quasi-)BICs
and the impact of symmetry on the optomechanical interaction. Fig. 4a shows
the mechanical spectroscopy of a group of devices with
$\theta=0\mathrm{\SIUnitSymbolDegree}$, $15\mathrm{\SIUnitSymbolDegree}$,
$30\mathrm{\SIUnitSymbolDegree}$, and $45\mathrm{\SIUnitSymbolDegree}$. The
spectrum is normalized with respect to the total background noise. At
$\theta=0\mathrm{\SIUnitSymbolDegree}$ and $45\mathrm{\SIUnitSymbolDegree}$,
only the quasi-BIC mode is observed while the BIC mode is invisible because of
the symmetry-inhibited optomechanical coupling. At
$\theta=15\mathrm{\SIUnitSymbolDegree}$ and $30\mathrm{\SIUnitSymbolDegree}$,
both BIC and quasi-BIC modes are detected. These observed mechanical modes are
the (1,1) standing-wave resonance, while higher order resonances are obscure
because of both lower quality factor and optomechanical coupling. Fig. 4b
shows the frequency distribution of a group of devices. The occurrence of the
mechanical BIC and quasi-BIC modes and their frequencies are consistent with
the simulation.

Under a blue-detuned pump, the optomechanical interaction between the
mechanical and optical resonances is described by a linearized two-mode
squeezing Hamiltonian,
$\hat{H}=G(\hat{a}^{\dagger}\hat{b}^{\dagger}+\hat{a}\hat{b})$, where
$\hat{a}^{\dagger}$($\hat{b}^{\dagger}$) and $\hat{a}$($\hat{b}$) are the
creation and annihilation operators of the optical(mechanical) resonance. The
parametrically-enhanced optomechanical mode coupling $G$ of standing-wave
resonances is given by

$G=\zeta g\sqrt{n_{c}},$ (1)

where $n_{c}$ is the average number of photons in one unit cell and $\zeta$ is
a resonance-dependent $O(1)$ parameter due to finite-mode-size correction
(SI). Given the nature of weak radiation-pressure force, the important
coupling in generic optomechanical systems is the parametric mode coupling,
which is related to the cooperativity given by $C=4G^{2}/\kappa\gamma$. It is
seen from Eq. 1 that the parametric mode coupling of a finite optomechanical
crystal is primarily determined by the unit-cell coupling and the per-unit-
cell photon number, and is independent of the size of the crystal. In other
words, despite that the bare mode coupling of optomechanical crystals roughly
scales as $1/N$, it can be compensated by the large number of photons
available which scales with $N^{2}$. The per-unit-cell photon number is
largely constrained by the thermo-optic effect and the heat capacity of the
unit cell, which the slab-on-substrate structure can ameliorate, especially
when the thermal conductivity of the substrate is comparable to the slab. In
our experiments, the device is typically operated under a pump power $P$ of a
few mW and
$n_{c}=\kappa_{e}P/((\omega_{m})^{2}+(\kappa/2)^{2})/\frac{3\sqrt{3}}{2}N^{2}$
is on the order of $10-20$. The unit-cell optomechanical coupling $g$ is
extracted from the noise power spectrum (see Methods and SI) and plotted in
Fig. 4c. Large optomechanical coupling $g/2\pi\approx 2.5$ MHz is observed for
the mechanical BIC. The deviation from the simulated value could be due to the
variation of the actual snowflake gap size and disorders in the optomechanical
crystal. Despite being unreleased and two-dimensional, the optomechanical
interaction (per unit cell) in the BIC optomechanical crystal is on par with
the suspended low-dimensional optomechanical crystal devices, such as
nanobeams.

The quality factor of the observed mechanical modes is shown in Fig. 4d. There
are two main factors limiting the mechanical quality factor of the current
device. First, the size of the optomechanical crystal is $N=25$, corresponding
to a finite wavevector of $k\approx 2\pi/(3Na)$ for the fundamental standing-
wave resonance. The radiation quality factor of mechanical standing-wave
resonances with finite wavevectors decreases drastically as they deviate from
the BIC at the $\Gamma$ point tong2020observation . Second, the fabricated
optomechanical crystal has critical dimensions as small as 30 nm, which cause
random variations among unit cells. These disorders induce scattering between
different orders of band-edge resonances, which effectively introduces more
radiation channels to a given resonance and degrades its quality factor
regan2016direct ; ni2017analytical ; jin2019topologically . The lateral
radiation for this size of crystal is not a dominant loss according to the
simulation and can be optimized with a phononic crystal mirror. In addition,
at room temperature, the quality factor of GHz-frequency mechanical modes is
ultimately limited by material absorption to around 1000-2000
eichenfield2009optomechanical . Because of these, there is no significant
statistical difference of the quality factor of BIC and quasi-BIC modes. The
observed quality factor is verified with numerical simulations of finite
super-cells with realistic disorders (SI). However, the mechanical radiation
quality factor will increase with the size of optomechanical crystals
tong2020observation , which will be critical to low-temperature measurements
when the material absorption is suppressed.

Discussion
In summary, we have realized the first two-dimensional slab-on-substrate
optomechanical crystals with mechanical BICs. Guided by a symmetry-based
design approach, this architecture offers optomechanical interaction (per unit
cell) on par with suspended optomechanical crystal devices. The two-
dimensional optomechanical crystal with tunable symmetry provides an arena for
exploration of rich multimode physics nielsen2017multimode ; renninger2018bulk
. In addition, the cavity-less optomechanical crystal might realize Floquet
topological physics beyond the tight-binding model fang2019anomalous . The
benefit of the slab-on-substrate device architecture in terms of heat
dissipation is obscured at room temperature, especially for optomechanical
crystals with a large area-to-perimeter ratio, because of the low thermal
conductivity of the oxide substrate two orders smaller than silicon. However,
such benefit will become evident at low temperatures ($<1$ K), relevant for
quantum experiments involving GHz mechanical modes, or in a different material
systems, when/where the thermal conductivity of the substrate and slab becomes
comparable zeller1971thermal ; thompson1961thermal . Besides the suppressed
material-absorption loss at low temperatures, the mechanical quality factor
could be enhanced using the merging BIC mechanism jin2019topologically  and
implementing appropriate phononic crystal mirrors. As a consequence, slab-on-
substrate BIC optomechanical crystals with improved optical and mechanical
losses could be unique at low temperatures for exploration of modalities
including phonon sensing and macroscopic mechanical oscillators in the quantum
regime chu2017quantum ; kotler2021direct .

Methods
Device fabrication. Devices are fabricated in silicon-on-insulator microchips
(220 nm silicon device layer and 3 $\upmu$m buried oxide layer) using electron
beam lithography with ZEP520A mask, followed by inductively coupled plasma
reactive ion etch of silicon using SF_6 and CHF_3.
Mechanical noise power spectrum. The total noise power spectral density
measured by the photodetector is given by safavi2013laser ;
meenehan2014silicon

$\displaystyle S[\omega]$
$\displaystyle={S_{e}}+\frac{G_{e}^{2}}{R_{I}}\left({S_{\mathrm{EDFA}}}+G^{2}_{\mathrm{EDFA}}{S^{2}_{\mathrm{SN}}}{\Big{(}1+\eta\frac{{{\kappa_{e}}}}{\kappa}\frac{{8{G^{2}}}}{\kappa}{{S}_{m}}[\omega]\Big{)}}\right)$
(2)

where $S_{e}$ is the electronic noise of the detector, $S_{\mathrm{EDFA}}$ is
the noise of EDFA, $S_{\mathrm{SN}}=\sqrt{2P_{\mathrm{out}}\hbar\omega_{l}}$
is the optical shot noise, $S_{m}[\omega]$ is the mechanical noise spectrum,
$\kappa=\kappa_{i}+\kappa_{e}$ is the total loss rate of the optical
resonance, $G$ is the parametrically-enhanced optomechanical coupling, $\eta$
is the total detection efficiency, $G_{\mathrm{EDFA}}$ is the EDFA gain,
$G_{e}$ is the detector gain factor from optical power to voltage, and $R_{I}$
is the input impedance of the spectrum analyzer. The optically-transduced
mechanical noise spectrum is given by

${S_{m}}[\omega]=\frac{1}{2}\left(\frac{{\gamma(\mathrm{bar}n_{m}+1)}}{{{{\left({{\omega_{m}}-\omega}\right)}^{2}}+{{\left({\gamma/2}\right)}^{2}}}}+\frac{{\gamma\mathrm{bar}n_{m}}}{{{{\left({{\omega_{m}}+\omega}\right)}^{2}}+{{\left({\gamma/2}\right)}^{2}}}}\right)$
(3)

with $\mathrm{bar}n_{m}=\frac{k_{B}T}{\hbar\omega_{m}}$ the thermal occupation
of the mechanical mode. The total detection efficiency is
$\eta=\eta_{\mathrm{cpl}}\eta_{\mathrm{t}}\eta_{\mathrm{det}}$, where
$\eta_{\mathrm{cpl}}$ is the coupling efficiency of the grating coupler at the
pump wavelength, $\eta_{\mathrm{t}}$ is the total transmission efficiency in
the optical fiber path from the chip to the detector, and
$\eta_{\mathrm{det}}$ is the quantum efficiency of the photodetector. We
measured $\eta_{\mathrm{t}}=0.80$ and $\eta_{\mathrm{cpl}}\approx 0.5$
depending on the pump wavelength, while $\eta_{\mathrm{det}}=0.68$ is given by
the photodetector, which results in $\eta\approx 0.27$. ${S_{e}}$ is
determined by blocking the light and
$\frac{G_{e}^{2}}{R_{I}}G^{2}_{\mathrm{EDFA}}{S^{2}_{\mathrm{SN}}}$ is
determined by removing EDFA while keeping the optical power incident onto the
photodetector the same. Then the measured noise power spectrum is fitted using
Eqs. 2 and 3, with $\omega_{m}$, $\gamma$, and $G$ the only fitting
parameters. The unit-cell optomechanical coupling is calculated from
$g=G/(\zeta\sqrt{n_{c}})$, using
$n_{c}=\kappa_{e}P/((\omega_{m})^{2}+(\kappa/2)^{2})/\frac{3\sqrt{3}}{2}N^{2}$
and $\zeta$ depending on the optical standing-wave resonance that is used for
the mechanical spectroscopy (SI).

Acknowledgements
This work is supported by US National Science Foundation (Grant No. 1944728
and 2016136) and Office of Naval Research (Grant No. N00014-21-1-2136).

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##  Appendix A Symmetry analysis of mechanical BICs and optomechanical
coupling

Figure 5: Schematic illustration of the symmetry of the silicon crystal plane
and optomechanical crystal.

The symmetry group of the stiffness tensor of silicon in the crystal plane and
the hexagonal optomechanical crystal is $C_{4v}$ and $C_{6v}$, respectively.
For both $C_{4v}$ and $C_{6v}$ groups, there are two sets of equivalent mirror
planes $\sigma_{v}$ and $\sigma_{d}$ as shown in Fig. 5. When one mirror plane
of the silicon crystal lattice aligns with one mirror plane of the hexagonal
optomechanical crystal, i.e., $\theta=0\mathrm{\SIUnitSymbolDegree}$,
$15\mathrm{\SIUnitSymbolDegree}$, $30\mathrm{\SIUnitSymbolDegree}$, and
$45\mathrm{\SIUnitSymbolDegree}$, the mechanical system restores mirror
symmetry and is described by $C_{2v}$ group. Otherwise, the symmetry group for
the mechanical system is $C_{2}$.

The far-field acoustic plane wave propagating along the $z-$direction can be
written as

$\bm{Q}_{0}=\bm{Q}_{T,0}+\bm{Q}_{L,0}=\left(u\bm{e}_{x}+v\bm{e}_{y}\right)e^{ik_{T,z}^{0}z}+w\bm{e}_{z}e^{ik_{L,z}^{0}z}.$
(4)

According to Tables 2 and 3, $\bm{e}_{x}$ and $\bm{e}_{y}$ belong to $B_{1}$
and $B_{2}$ representations of $C_{2v}$ and $B$ representation of $C_{2}$,
respectively, and $\bm{e}_{z}$ belongs to $A_{1}$ representation of $C_{2v}$
and $A$ representation of $C_{2}$. Therefore, a mechanical mode at the
$\Gamma$ point can be a BIC, i.e., decouples from both transverse and
longitudinal radiation waves, only if it belongs to the $A_{2}$ representation
of $C_{2v}$ group when $\theta=0\mathrm{\SIUnitSymbolDegree}$,
$15\mathrm{\SIUnitSymbolDegree}$, $30\mathrm{\SIUnitSymbolDegree}$, or
$45\mathrm{\SIUnitSymbolDegree}$.

Table 2: Character table of the $C_{2v}$ point group $C_{2v}$ | $E$ | $C_{2}$ | $\sigma_{v}$ | $\sigma_{d}$
---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1
$A_{2}$ | 1 | 1 | -1 | -1
$B_{1}$ | 1 | -1 | 1 | -1
$B_{2}$ | 1 | -1 | -1 | 1
Table 3: Character table of the $C_{2}$ point group $C_{2}$ | $E$ | $C_{2}$
---|---|---
$A$ | 1 | 1
$B$ | 1 | -1

Next we analyze the optomechanical coupling of a unit cell based on the
symmetry. The unit-cell optomechanical coupling is given by

$g={g_{{\mathrm{MB}}}}+{g_{{\mathrm{PE}}}}\equiv\sqrt{\frac{\hbar}{{2{m_{{\mathrm{eff}}}}{\omega_{m}}}}}\left({{g_{{\mathrm{OM,MB}}}}+{g_{{\mathrm{OM,PE}}}}}\right)$
(5)

where $m_{\mathrm{eff}}=\int_{\mathrm{u.c.}}\rho|\bm{Q}|^{2}\mathrm{~{}d}V$ is
the unit-cell effective mass, $\omega_{m}$ is the mechanical frequency. The
moving-boundary and photoelastic components are calculated as

$g_{\mathrm{OM},\mathrm{MB}}=-\frac{\omega_{o}}{2}\frac{\int_{\mathrm{u.c.}}(\bm{Q}\cdot\hat{\bm{n}})\left(\Delta\epsilon\left|\bm{E}_{\|}\right|^{2}-\Delta\epsilon^{-1}\left|\bm{D}_{\perp}\right|^{2}\right)\mathrm{~{}d}S}{\int_{\mathrm{u.c.}}\bm{E}^{*}\cdot\bm{D}\mathrm{~{}d}V},$
(6)

and

$g_{\mathrm{OM},\mathrm{PE}}=\frac{\omega_{o}}{2}\frac{\int_{\mathrm{u.c.}}\epsilon_{0}n^{4}E_{i}^{*}E_{j}p_{ijkl}S_{kl}\mathrm{~{}d}V}{\int_{\mathrm{u.c.}}\bm{E}^{*}\cdot\bm{D}\mathrm{~{}d}V},$
(7)

where $\omega_{o}$ is the optical frequency, $\hat{\bm{n}}$ is the normal
vector of the interfaces, $\bm{Q}$ is displacement, $S_{kl}$ is strain tensor,
the subscripts $\|$ and $\perp$ indicate the field components parallel and
perpendicular to the surface and
$\Delta\epsilon=\epsilon_{\mathrm{int}}-\epsilon_{\mathrm{ext}}$ and
$\Delta\epsilon^{-1}=\epsilon_{\mathrm{int}}^{-1}-\epsilon_{\mathrm{ext}}^{-1}$
($\epsilon_{\mathrm{ext}}$ is the permittivity of the media which
$\hat{\bm{n}}$ points to and $\epsilon_{\mathrm{int}}$ is the permittivity of
the media on the other side).

For the moving boundary term, we write the numerator using a shorthanded
notation $\int_{\mathrm{u.c.}}(\bm{Q}\cdot\hat{\bm{n}})f\mathrm{~{}d}S$. When
there is mirror symmetry, this integration can be calculated as

$\displaystyle\int_{\mathrm{u.c.}}{\bm{Q}\cdot\hat{\bm{n}}f\mathrm{~{}d}S}$
$\displaystyle=\left({\int_{\mathrm{u.c.}}{\bm{Q}(\bm{r})\cdot\hat{\bm{n}}(\bm{r})f(\bm{r})\mathrm{~{}d}S}+\int_{\mathrm{u.c.}}{\bm{Q}(\sigma\bm{r})\cdot\hat{\bm{n}}(\sigma\bm{r})f(\sigma\bm{r})\mathrm{~{}d}S}}\right)/2$
(8)
$\displaystyle=\left({\int_{\mathrm{u.c.}}{\bm{Q}(\bm{r})\cdot\hat{\bm{n}}(\bm{r})f(\bm{r})\mathrm{~{}d}S}+\int_{\mathrm{u.c.}}{{\sigma^{-1}}\bm{Q}(\bm{r})\cdot\hat{\bm{n}}(\bm{r}){\sigma^{-1}}f(\bm{r})\mathrm{~{}d}S}}\right)/2$
$\displaystyle=(1+{\chi_{\bm{Q}}}(\sigma))\int_{\mathrm{u.c.}}{\bm{Q}(\bm{r})\cdot\hat{\bm{n}}(\bm{r})f(\bm{r})\mathrm{~{}d}S}/2,$

where $\chi_{\bm{Q}}(\sigma)$ stands for the character of mirror symmetry
operation $\sigma$. Thus, the integral vanishes when the mechanical mode is
odd with respect to the mirror plane. We have already used the fact that
$f(\bm{r})$ is even under the mirror operation as it is a quadratic function
of $\bm{E}$ and $\bm{D}$. This derivation is valid only when both mechanical
and optical modes share the same mirror plane, which is true for
$\theta=0\mathrm{\SIUnitSymbolDegree}$ and
$\theta=45\mathrm{\SIUnitSymbolDegree}$. For
$\theta=15\mathrm{\SIUnitSymbolDegree}$ and
$\theta=30\mathrm{\SIUnitSymbolDegree}$, $g_{\mathrm{OM},\mathrm{MB}}$ in
general will be nonzero.

To calculate the photoelastic term, we note the photoelastic tensor for cubic
crystal, such as silicon, in Voigt notation is given by

$p_{\mathrm{C}}=\left(\begin{array}[]{cccccc}p_{11}&p_{12}&p_{12}&&&\\\
&p_{11}&p_{12}&&&\\\ &&p_{11}&&&\\\ &&&p_{44}&&\\\
&\mathrm{symm.}&&&p_{44}&\\\ &&&&&p_{44}\end{array}\right).$ (9)

We perform the computation in the frame of optomechanical crystal by rotating
the silicon crystal lattice. Under a rotating $R(\theta)$, the photoelastic
tensor transforms as
$p_{ijkl}^{\prime}(\theta)=R_{ip}(\theta)R_{jq}(\theta)R_{kr}(\theta)R_{ls}(\theta)p_{pqrs}$,
or in the Voigt notation,

$\begin{array}[]{*{20}{c}}{{{p}_{11}^{\prime}}={{p}_{22}^{\prime}}=\frac{1}{4}\left[{{p_{11}}(\cos
4\theta+3)+\left({{p_{12}}+2{p_{44}}}\right)(1-\cos 4\theta)}\right]}\\\
\begin{array}[]{c}{{p}_{33}^{\prime}}={p_{33}}\\\
{{p}_{12}^{\prime}}={{p}_{21}^{\prime}}=\frac{1}{4}\left[{{p_{12}}(\cos
4\theta+3)+\left({{p_{11}}-2{p_{44}}}\right)(1-\cos
4\theta)}\right]\end{array}\\\
\begin{array}[]{c}{{p}_{13}^{\prime}}={{p}_{23}^{\prime}}={{p}_{31}^{\prime}}={{p}_{32}^{\prime}}={p_{12}}\\\
\end{array}\\\ {{{p}_{44}^{\prime}}={{p}_{55}^{\prime}}={p_{44}}}\\\
{{{p}_{66}^{\prime}}=\frac{1}{4}\left[{2{p_{44}}(\cos
4\theta+1)+\left({{p_{11}}-{p_{12}}}\right)(1-\cos 4\theta)}\right]}\\\
\begin{array}[]{c}{{p}_{16}^{\prime}}={{p}_{61}^{\prime}}=\frac{1}{4}\sin
4\theta\left({{p_{11}}-{p_{12}}-2{p_{44}}}\right)\\\
{{p}_{26}^{\prime}}={{p}_{62}^{\prime}}=-{{p}_{16}^{\prime}}.\end{array}\end{array}$
(10)

The integral in the photoelastic coupling term is expressed as

$\int{{\epsilon_{0}}{n^{4}}E_{i}^{*}{E_{j}}{p_{ijkl}}{S_{kl}}{\mathrm{~{}d}}V}=\int{{\epsilon_{0}}{n^{4}}(p_{11}^{\prime}{f_{11}}+p_{12}^{\prime}{f_{12}}+p_{13}^{\prime}{f_{13}}+p_{31}^{\prime}{f_{31}}+p_{33}^{\prime}{f_{33}}+p_{44}^{\prime}{f_{44}}+p_{66}^{\prime}{f_{66}}+p_{16}^{\prime}{f_{16}})}\mathrm{~{}d}V,$
(11)

where

$\begin{array}[]{c}{f_{11}=\left|E_{x}\right|^{2}S_{xx}+\left|E_{y}\right|^{2}S_{yy}}\\\
{f_{12}=\left|E_{x}\right|^{2}S_{yy}+\left|E_{y}\right|^{2}S_{xx}}\\\
{f_{13}=\left|E_{x}\right|^{2}S_{zz}+\left|E_{y}\right|^{2}S_{zz}}\\\
{f_{31}=\left|E_{z}\right|^{2}S_{xx}+\left|E_{z}\right|^{2}S_{yy}}\\\
{f_{33}=\left|E_{z}\right|^{2}S_{zz}}\\\
{f_{44}=2\mathrm{Re}\left\\{E_{y}^{*}E_{z}\right\\}\cdot
2S_{yz}+2\mathrm{Re}\left\\{E_{x}^{*}E_{z}\right\\}\cdot 2S_{xz}}\\\
{f_{16}=\left(\left|E_{x}\right|^{2}-\left|E_{y}\right|^{2}\right)\cdot
2S_{xy}+2\mathrm{Re}\left\\{E_{x}^{*}E_{y}\right\\}\cdot\left(S_{xx}-S_{yy}\right)}.\end{array}$
(12)

When $\theta=0\mathrm{\SIUnitSymbolDegree}$ and
$45\mathrm{\SIUnitSymbolDegree}$, there are mirror planes $\sigma_{x}$ and
$\sigma_{y}$. We list the symmetry of $E_{i}^{*}E_{j}$, $S_{kl}$ and $f_{ij}$
in Tables 4 and 5, where ‘e’ stands for even and ‘o’ stands for odd.

|  | $\left|E_{x}\right|^{2}$ | $\left|E_{y}\right|^{2}$ | $\left|E_{z}\right|^{2}$ | ${\mathop{\rm Re}\nolimits}\\{E_{x}^{*}{E_{y}}\\}$ | ${\mathop{\rm Re}\nolimits}\\{E_{x}^{*}{E_{z}}\\}$ |  ${\mathop{\rm Re}\nolimits}\\{E_{y}^{*}{E_{z}}\\}$ [b]
---|---|---|---|---|---|---|---
$\chi({\sigma_{x}})=\pm 1$ | under $\sigma_{x}$ | e | e | e | o | o | e
$\chi({\sigma_{y}})=\pm 1$ | under $\sigma_{y}$ | e | e | e | o | e | o
|  | $S_{xx}$ | $S_{yy}$ | $S_{zz}$ | $S_{xy}$ | $S_{xz}$ |  $S_{yz}$
$\chi({\sigma_{x}})=1$ | under $\sigma_{x}$ | e | e | e | o | o | e
$\chi({\sigma_{x}})=-1$ | under $\sigma_{x}$ | o | o | o | e | e | o
$\chi({\sigma_{y}})=1$ | under $\sigma_{y}$ | e | e | e | o | e | o
$\chi({\sigma_{y}})=-1$ | under $\sigma_{y}$ | o | o | o | e | o | e
Table 4: The behavior of $E_{i}^{*}E_{j}$ and $S_{ij}$ under $\sigma_{x}$ and $\sigma_{y}$. |  | $f_{11}$ | $f_{12}$ | $f_{13}$ | $f_{31}$ | $f_{33}$ | $f_{44}$ | $f_{66}$ |  $f_{16}$
---|---|---|---|---|---|---|---|---|---
$\chi({\sigma_{x}})=1$ | under $\sigma_{x}$ | e | e | e | e | e | e | e | o
$\chi({\sigma_{x}})=-1$ | under $\sigma_{x}$ | o | o | o | o | o | o | o | e
$\chi({\sigma_{y}})=1$ | under $\sigma_{y}$ | e | e | e | e | e | e | e | o
$\chi({\sigma_{y}})=-1$ | under $\sigma_{y}$ | o | o | o | o | o | o | o | e
Table 5: The behavior of $f_{ij}$ under $\sigma_{x}$ and $\sigma_{y}$.

We find that if the mechanical mode is odd under either $\sigma_{x}$ or
$\sigma_{y}$, all the $f_{ij}$s except $f_{16}$ are odd and their integrals
are zero. For $f_{16}$, because $p_{16}^{\prime}=0$ when
$\theta=0\mathrm{\SIUnitSymbolDegree}$ or $45\mathrm{\SIUnitSymbolDegree}$,
the term $\int{p_{16}^{\prime}{f_{16}}\mathrm{~{}d}V}$ also vanishes. For
$\theta=15\mathrm{\SIUnitSymbolDegree}$ and $30\mathrm{\SIUnitSymbolDegree}$,
$g_{\mathrm{OM},\mathrm{PE}}$ is in general nonzero.

In summary, $A_{2}$ modes for $\theta=0\mathrm{\SIUnitSymbolDegree}$ and
$45\mathrm{\SIUnitSymbolDegree}$, which are BICs, have a zero optomechanical
coupling. For other cases, optomechanical coupling is in general nonzero. This
is also summarized in Table I of the main text.

##  Appendix B Optomechanical coupling of finite optomechanical crystals

The optomechanical coupling for a sufficiently large optomechanical crystal
can be calculated with the result of the unit-cell coupling. Performing the
integrals of Eqs. 6 and 7 in the whole optomechanical crystal and apply the
Bloch theorem, i.e.,
$\bm{Q}(\bm{r}+\bm{R})=e^{i{\bm{k}_{m}}\cdot\bm{R}}\bm{Q}(\bm{r})$ and
$\bm{E}(\bm{r}+\bm{R})=e^{i{\bm{k}_{o}}\cdot\bm{R}}\bm{E}(\bm{r})$, where
$\bm{k}_{m}$ and $\bm{k}_{o}$ are the Bloch wavevector of the mechanical and
optical modes, respectively, we have

${\mathrm{bar}g_{\mathrm{OM,MB}}}=-\frac{{{\omega_{o}}}}{2}\frac{{\sum\limits_{\bm{R}}{{e^{i{\bm{k}_{m}}\cdot\bm{R}}}}\int_{\mathrm{u.c.}}{(\bm{Q}\cdot\hat{\bm{n}})(\Delta\epsilon{{\left|{{\bm{E}_{\parallel}}}\right|}^{2}}-\Delta{\epsilon^{-1}}{{\left|{{\bm{D}_{\bot}}}\right|}^{2}})\mathrm{~{}d}S}}}{{\sum\limits_{\bm{R}}{\int_{\mathrm{u.c.}}{{\bm{E}^{*}}\cdot\bm{D}\mathrm{~{}d}V}}}}=\frac{{\sum\limits_{\bm{R}}{{e^{i{\bm{k}_{m}}\cdot\bm{R}}}}}}{N_{c}}{g_{\mathrm{OM,MB}}}$
(13)

where $N_{c}$ is the number of unit cells and $\sum\limits_{\bm{R}}$ is the
summation for all the lattice vectors. Same expression holds for
$g_{\mathrm{OM,PE}}$. Because
$\sum\limits_{\bm{R}}{{e^{i{\bm{k}_{m}}\cdot\bm{R}}}}$ is nonzero only when
$\bm{k}_{m}=0$, optomechanical coupling exists only for mechanical modes at
the $\Gamma$ point. Because the total mass
${\mathrm{bar}m_{\mathrm{eff}}}=N_{c}{m_{\mathrm{eff}}}$, the optomechanical
coupling of a sufficiently large optomechanical crystal is related to its
unit-cell coupling as

$\mathrm{bar}g=\sqrt{\frac{\hbar}{{2N_{c}{m_{\mathrm{eff}}}{\omega_{m}}}}}({g_{\mathrm{OM,MB}}}+{g_{\mathrm{OM,PE}}})=\frac{{{g}}}{{\sqrt{N}_{c}}}.$
(14)

For finite optomechanical crystals, we have to consider the field envelop of
the standing-wave resonances. The fields across unit cells now are related by
$\bm{Q}(\bm{r}+\bm{R})=e^{i{\bm{k}_{m}}\cdot\bm{R}}{\phi_{({p_{m}},{q_{m}})}}(\bm{R})\bm{Q}(\bm{r})$
and
$\bm{E}(\bm{r}+\bm{R})=e^{i{\bm{k}_{o}}\cdot\bm{R}}{\phi_{({p_{o}},{q_{o}})}}(\bm{R})\bm{E}(\bm{r})$,
where ${\phi_{({p},{q})}}(\bm{R})$ is the envelop function of the $(p,q)$-th
order standing-wave resonance. The envelope function can be approximately
solved as the eigenfunction of a flat-top potential well within the boundary
of the finite optomechanical crystal. Then we have

$\displaystyle{\mathrm{bar}g_{\mathrm{OM,MB}}}$
$\displaystyle=-\frac{{{\omega_{o}}}}{2}\frac{{\sum\limits_{\bm{R}}e^{i{\bm{k}_{m}}\cdot\bm{R}}{{\phi_{({p_{m}},{q_{m}})}}(\bm{R}){{\left|{{\phi_{({p_{o}},{q_{o}})}}(\bm{R})}\right|}^{2}}}\int_{\mathrm{u.c.}}{(\bm{Q}\cdot\hat{\bm{n}})(\Delta\epsilon{{\left|{{\bm{E}_{\parallel}}}\right|}^{2}}-\Delta{\epsilon^{-1}}{{\left|{{\bm{D}_{\bot}}}\right|}^{2}})\mathrm{~{}d}S}}}{{\sum\limits_{\bm{R}}{{{\left|{{\phi_{(p,q)}}(\bm{R})}\right|}^{2}}\int_{\mathrm{u.c.}}{{\bm{E}^{*}}\cdot\bm{D}\mathrm{~{}d}V}}}}$
(15)
$\displaystyle=\frac{{\sum\limits_{\bm{R}}e^{i{\bm{k}_{m}}\cdot\bm{R}}{{\phi_{({p_{m}},{q_{m}})}}(\bm{R}){{\left|{{\phi_{({p_{o}},{q_{o}})}}(\bm{R})}\right|}^{2}}}}}{{\sum\limits_{\bm{R}}{{{\left|{{\phi_{(p_{o},q_{o})}}(\bm{R})}\right|}^{2}}}}}{g_{\mathrm{OM,MB}}}$
$\displaystyle\approx\frac{{\int
e^{i{\bm{k}_{m}}\cdot\bm{R}}{{\phi_{(p_{m},q_{m})}}(\bm{R}){{\left|{{\phi_{(p_{o},q_{o})}}(\bm{R})}\right|}^{2}}\mathrm{~{}d}S}}}{{\int{{{\left|{{\phi_{(p_{o},q_{o})}}(\bm{R})}\right|}^{2}}\mathrm{~{}d}S}}}{g_{\mathrm{OM,MB}}}.$

where the last integrals are performed in the whole optomechanical crystal.
Similar expression applies to the photoelastic term. For the total effective
mass, we have

$\displaystyle{\mathrm{bar}m_{{\mathrm{eff}}}}=$
$\displaystyle\sum\limits_{\bm{R}}{\int_{\mathrm{u.c.}}{\rho{{\left|\bm{Q}\right|}^{2}}\mathrm{~{}d}V}}=\sum\limits_{\bm{R}}{{{\left|{{\phi_{(p_{m},q_{m})}}(\bm{R})}\right|}^{2}}\int_{\mathrm{u.c.}}{\rho{{\left|\bm{Q}\right|}^{2}}\mathrm{~{}d}V}}=\sum\limits_{\bm{R}}{{{\left|{{\phi_{(p_{m},q_{m})}}(\bm{R})}\right|}^{2}}}{m_{\mathrm{eff}}}\approx
N_{c}\frac{{\int{{{\left|{{\phi_{(p_{m},q_{m})}}(\bm{R})}\right|}^{2}}\mathrm{~{}d}S}}}{{\int{\mathrm{d}S}}}m_{\mathrm{eff}}.$
(16)

Finally, the optomechanical coupling of a finite optomechanical crystal is
related to its unit-cell coupling by

$\mathrm{bar}g=\frac{{\zeta{g}}}{{\sqrt{N}_{c}}},$ (17)

where the finite-size correction factor $\zeta$ is given by

$\zeta=\frac{{\int
e^{i{\bm{k}_{m}}\cdot\bm{R}}{{\phi_{(p_{m},q_{m})}}(\bm{R}){{\left|{{\phi_{(p_{o},q_{o})}}(\bm{R})}\right|}^{2}}\mathrm{~{}d}S}}}{{\int{{{\left|{{\phi_{(p_{o},q_{o})}}(\bm{R})}\right|}^{2}}\mathrm{~{}d}S\sqrt{\int{{{\left|{{\phi_{(p_{m},q_{m})}}(\bm{R})}\right|}^{2}}\mathrm{~{}d}S}/\int{\mathrm{d}S}}}}}.$
(18)

For the mechanical Bloch wavevector $\bm{k}_{m}=0$, the numerically-calculated
correction factor for a few standing-wave mechanical and optical resonances of
a hexagonal crystal is listed in Table 6. The field envelops of some orders
are shown in Fig. 3c. The order of the hexagonal crystal mode can be hard to
identify. We determined it by adiabatically deforming the hexagonal crystal to
a rectangular crystal and tracing the corresponding mode, where $(p,q)$-th
resonance has $p-1$ and $q-1$ nodes along the $x$ and $y$ direction,
respectively.

Table 6: Correction factor $\zeta$ for some standing-wave mechanical and optical resonances of a hexagonal crystal. | Optical | (1,1) | (1,2) | (2,1) | (2,2) |  (3,1) [t]
---|---|---|---|---|---|---
Mechanical |  |  [b]
(1,1) | 1.4075 | 1.1427 | 1.1415 | 0.9737 | 0.9740 [t]
(3,1) | 0.0185 | 0.8250 | 0.8322 | 0.0183 | 0.0805
(1,3) | 0.3987 | 0.4442 | 0.3692 | 0.7307 | 0.7398 [b]

The interaction Hamiltonian of the finite optomechanical crystal involving a
pair of mechanical and optical standing-wave resonances is given by
$\hat{H}=\mathrm{bar}g\hat{a}^{\dagger}\hat{a}(\hat{b}^{\dagger}+\hat{b})$,
where $\hat{a}^{\dagger}$($\hat{a}$) and $\hat{b}^{\dagger}$($\hat{b}$) are
the creation(annihilation) operators of the optical and mechanical resonances,
respectively. When the optomechanical crystal is driven by a pump detuned from
the optical resonance by the mechanical frequency, the interaction Hamiltonian
could be linearized to be
$\hat{H}=G(\hat{a}^{\dagger}\hat{b}^{\dagger}+\hat{a}\hat{b})$ for blue-
detuned pump or $\hat{H}=G(\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}^{\dagger})$
for red-detuned pump, with

$G=\mathrm{bar}g\sqrt{n}=\zeta g\sqrt{n_{c}}$ (19)

the parametrically-enhanced optomechanical coupling, where $n$ and
$n_{c}=n/N_{c}$ are the total photon number in the optical resonance and
average photon number per unit cell, respectively.

##  Appendix C Mechanical noise power spectrum

The total noise power spectral density $S$ measured by the photodetector is
given by meenehan2014silicon ; safavi2013laser

$\displaystyle S$
$\displaystyle={S_{e}}+\frac{G_{e}^{2}}{R_{I}}\left({S_{\mathrm{EDFA}}}+G^{2}_{\mathrm{EDFA}}{S^{2}_{\mathrm{SN}}}{\Big{(}1+\eta\frac{{{\kappa_{e}}}}{\kappa}\frac{{8{G^{2}}}}{\kappa}{{S}_{m}}[\omega]\Big{)}}\right),$
(20)

where $S_{e}$ is the electronic noise of the detector, $S_{\mathrm{EDFA}}$ is
the noise of EDFA, $S_{\mathrm{SN}}=\sqrt{P_{\mathrm{out}}\hbar\omega_{l}}$ is
the optical shot noise,, $S_{m}[\omega]$ is the mechanical noise spectrum,
$\kappa=\kappa_{i}+\kappa_{e}$ is the total optical dissipation, $G$ is the
parametrically-enhanced optomechanical coupling, $\eta$ is the total detection
efficiency, $G_{e}$ is the detector gain factor from optical power to voltage,
$G_{\mathrm{EDFA}}$ is the EDFA gain, and $R_{I}$ is the input impedance of
the spectrum analyzer. The optically-transduced mechanical noise spectrum is
given by

${S_{m}}[\omega]=\frac{1}{2}\left(\frac{{\gamma(\mathrm{bar}n_{m}+1)}}{{{{\left({{\omega_{m}}-\omega}\right)}^{2}}+{{\left({\gamma/2}\right)}^{2}}}}+\frac{{\gamma\mathrm{bar}n_{m}}}{{{{\left({{\omega_{m}}+\omega}\right)}^{2}}+{{\left({\gamma/2}\right)}^{2}}}}\right)$
(21)

where $\mathrm{bar}n_{m}=\frac{k_{B}T}{\hbar\omega_{m}}$ the thermal
occupation of the mechanical mode. The total detection efficiency is
$\eta=\eta_{\mathrm{cpl}}\eta_{\mathrm{t}}\eta_{\mathrm{det}}$, where
$\eta_{\mathrm{cpl}}$ is the coupling efficiency of the grating coupler at the
pump wavelength, $\eta_{\mathrm{t}}$ is the total transmission efficiency in
the optical fiber path, and $\eta_{\mathrm{det}}$ is the quantum efficiency of
the photodetector. We measured $\eta_{\mathrm{t}}=0.80$ and
$\eta_{\mathrm{cpl}}\approx 0.5$ depending on the pump wavelength, while
$\eta_{\mathrm{det}}=0.68$ is given by the detector, which gives $\eta\approx
0.27$.

We normalize the noise power spectrum with respect to the total noise floor
which yields

$\displaystyle\overline{S}$
$\displaystyle=1+\frac{{S^{\prime}_{\mathrm{SN}}}}{{S_{e}}+{S^{\prime}_{\mathrm{EDFA}}}+{S^{\prime}_{\mathrm{SN}}}}\eta\frac{{{\kappa_{e}}}}{\kappa}\frac{{8{G^{2}}}}{\kappa}{{S}_{m}}[\omega],$
(22)

where we have used
${S^{\prime}_{\mathrm{EDFA}}}\equiv\frac{G_{e}^{2}}{R_{I}}{S_{\mathrm{EDFA}}}$
and
$S^{\prime}_{\mathrm{SN}}\equiv\frac{G_{e}^{2}G_{\mathrm{EDFA}}}{R_{I}}{S_{\mathrm{SN}}}$
to denote the photodetector-measured EDFA noise and laser shot noise. The
electronic noise $S_{e}$ is measured by blocking all the light.
$S^{\prime}_{\mathrm{SN}}+S_{e}$ can be determined by removing the EDFA while
keeping the laser power incident to the detector unchanged. Thus we obtain
$S^{\prime}_{\mathrm{SN}}$ and the ratio of
$\frac{{S^{\prime}_{\mathrm{SN}}}}{{S_{e}}+{S^{\prime}_{\mathrm{EDFA}}}+{S^{\prime}_{\mathrm{SN}}}}$.
Finally, the measured normalized noise power spectrum is fitted using Eqs. 22
and 21, with $\omega_{m}$, $\gamma$, and $G$ the only fitting parameters. The
unit-cell optomechanical coupling is calculated from
$g=G/(\zeta\sqrt{n_{c}})$, using
$n_{c}=\kappa_{e}P/((\omega_{m})^{2}+(\kappa/2)^{2})/\frac{3\sqrt{3}}{2}N^{2}$
and $\zeta$ depending on the optical standing-wave resonance that is deployed
(see Table 6). For the four devices shown in Fig. 4, we used optical
resonances of the following order, $0\mathrm{\SIUnitSymbolDegree}$ (1,1),
$15\mathrm{\SIUnitSymbolDegree}$ (1,1), $30\mathrm{\SIUnitSymbolDegree}$
(1,1), $45\mathrm{\SIUnitSymbolDegree}$ (2,1), while the observed mechanical
resonances are (1,1).

##  Appendix D Fabrication results

The silicon-on-insulator microchip (220 nm silicon device layer and 3 $\upmu$m
buried oxide layer) is first rinsed with Acetone, isopropyl alcohol and
deionized water to clean the surface. The pattern is then defined using
electron beam lithography (Elionix ELS-G150) with 300 nm thick ZEP520A mask,
followed by inductively coupled plasma reactive ion etch (Oxford PlasmaPro 100
Cobra) of silicon using SF6 and CHF3 . The mask residual is removed by
n-methyl-2-pyrrolidone hot bath and oxygen plasma cleaning (Diener Descum).

The SEM image of a typical unit cell is shown in Fig.6a. The disorder of the
snowflake patterns are measured both inside a same optomechanical crystal and
across different optomechanical crystals (Table 7). The standard deviation of
$r$ and $w$ is about 2.7% and 7.22% for the former and 2.6% and 6.7% for the
latter. For comparison of different optomechanical crystals, we examined unit
cells from the same location. Roughly speaking, the variation across different
devices accounts for the the resonance frequency fluctuation and disorder
within an optomechanical crystal induces scattering loss and decrease the
resonance quality factor (see Section E).

Figure 6: a. An SEM image of a typical unit cell. The dashed line shows the fitted snowflake structure. b. An SEM image of a few unit cells within an optomechanical crystal. The white bar refers to 200 nm for both images. Table 7: Mean and standard deviation of $r$ and $w$. | Across different devices | Inside one device
---|---|---
$\mathrm{bar}{r}$ (nm) | 172.09 | 167.35
$r_{\mathrm{max}}$ (nm) | 177.60 | 174.65
$r_{\mathrm{min}}$ (nm) | 161.00 | 154.65
$\Delta r$ (nm) | 4.49 | 4.57
$\mathrm{bar}{w}$ (nm) | 37.76 | 37.26
$w_{\mathrm{max}}$ (nm) | 44.00 | 43.10
$w_{\mathrm{min}}$ (nm) | 31.70 | 32.70
$\Delta w$ (nm) | 2.53 | 2.67

##  Appendix E Simulation of Scattering Loss

In multimode photonic/phononic crystals, structural disorders could cause
scattering among different modes and thus introduce more loss channels to a
given mode jin2019topologically . To estimate the quality factor of standing-
wave mechanical resonances in disordered crystals, we simulated a $1\times 4$
super-cell structure (Fig. 7a), where variations of both length ($\Delta r$)
and width ($\Delta w$) of each snowflake hole are introduced. Each $\Delta r$
and $\Delta w$ are independently sampled and the standard deviation of $\Delta
r$ and $\Delta w$ are controlled to be 4 nm and 2 nm, respectively, in
accordance with the fabricated devices. We run 50 sets of simulations each for
$\theta=0\mathrm{\SIUnitSymbolDegree}$, $15\mathrm{\SIUnitSymbolDegree}$,
$30\mathrm{\SIUnitSymbolDegree}$, and $45\mathrm{\SIUnitSymbolDegree}$. We set
the Bloch wavevector to be $k=2\pi/(3aN)$ with $N$ = 25, approximately
corresponding to the $(1,1)$ standing-wave resonance in size $N=25$ hexagonal
optomechanical crystals. The simulated radiative quality factors $Q_{r}$
combined with room-temperature material absorption $Q_{a}$, which we assumed
to be 3000, gives the total mechanical quality factor:
$1/Q_{m}=1/Q_{r}+1/Q_{a}$. The result is plotted in Fig. 7b. Due to the small
size of the optomechanical crystal, there are no significant variations of
$Q_{m}$ among four different $\theta$’s. We use the data set of all four
angles here to calculate the upper and lower bounds of $Q_{m}$ as shown in
Fig. 4.

Figure 7: a. A schematic diagram of a $1\times 4$ super-cell structure with
disorders. b. The simulated $Q_{m}$ for different $\theta$. The lower and
upper edges of the boxes represent the 16% and 84% bounds of the data,
respectively, and the middle lines is the median value.

##  Appendix F Design of grating coupler

Figure 8: a. Schematic plot of the angle-polished fiber and grating coupler.
b. Parameters $l_{i}$ and $g_{i}$ for the optimized grating coupler. c.
Measured and simulated efficiency of the optimized grating coupler.

Fig. 8a shows the fiber-optic coupler for the measurement of the
optomechanical crystal. Light is guided by an angle-polished single-mode
fiber, reflects at the polished surface, and is incident onto a grating
coupler. The angle $\alpha$ of the fiber is chosen to ensure total internal
reflection. Because of the inevitable air gap between the fiber and grating
coupler, the incident angle onto the grating coupler is given by
$\beta=\arcsin(n_{\mathrm{fiber}}\cdot\sin(90\mathrm{\SIUnitSymbolDegree}-2\alpha))$.
In general, the grating coupler is designed to convert the incident light to a
guided wave by satisfying the phase-matching condition
$\frac{{2\pi{n_{\mathrm{eff}}}}}{\lambda}=\frac{{2\pi}}{\lambda}\sin\beta+\frac{{2\pi}}{\Lambda}$,
where $n_{\mathrm{eff}}$ is the effective refractive index of the grating and
$\Lambda$ is the lattice constant of the grating. It turns out an apodized
grating coupler could yield higher coupling efficiency than a uniform grating
coupler because of the better mode-matching with the single-mode optical fiber
zhao2020design . Thus, we adopt the apodized grating coupler scheme. The
apodized grating coupler consists of 20 unit-cells with 40 individual tunable
parameters $l_{i}$ and $g_{i}$. The parameters are optimized using a standard
inverse design approach molesky2018inverse  to maximize the coupling
efficiency for different fiber angles. The optimized fiber angle is found to
be 35.5° and the corresponding $l_{i}$ and $g_{i}$ are shown in Fig. 8b. Fig.
8c shows the simulated transmission and measured device spectrum (inferred
from the reflection spectrum). The peak efficiency is observed to be $52$%
with a 3-dB bandwidth of $40\,$nm.

##  Appendix G Design of photonic crystal mirror

Figure 9: a and b Mechanical and optical bandstructure of the nominal
optomechanical crystal. c. Band structure of the photonic crystal mirror. d.
Super-cell used in the simulation. Inset: Top view of the junction between the
optomechanical crystal and waveguide. Dashed circles are the original location
of the photonic crystal mirror holes before displacement. e. Simulated total
quality factor $Q_{t}$ versus the displacement $d-a$. f. Simulated
$\kappa_{e}/\kappa_{i}$ versus the number of nominal mirror layers $N_{m}$ for
$Q_{i}=10^{6}$ and $Q_{i}=2\times 10^{4}$.

Fig. 9a and b show the full mechanical and optical bandstructure of the
nominal optomechanical crystal. The photonic crystal mirror consists of a
triangular lattice of cylindrical holes with $r=78\,$nm and the same lattice
constant $a=389\,$nm as the snowflake optomechanical crystal. The simulated
band structure with the two lowest TE-like bands (Fig. 9c) shows a complete
bandgap from $187\,$THz ($1606\,$nm) to $201\,$THz ($1493\,$nm). The photonic
crystal mirror effectively suppresses the side-leakage but also perturbs the
tail of the standing-wave resonance, inducing wavevector components that fall
inside the continuum and causing out-of-plane radiation. We could alleviate
such out-of-plane radiation by tuning the separation $d$ between the photonic
crystal mirror and optomechanical crystal akahane2005fine ;
chen2021observation. To simulate this, we take a strip of optomechanical
crystal of 25 unit cells as one super-cell and add periodic boundary
conditions on the transverse direction, as shown in Fig. 9d. A sufficient
number of photonic crystal mirror unit-cells are included and so the total
quality factor $Q_{t}$ of the resonance is dominated by the out-of-plane
radiation. Fig. 9e shows
$Q_{t}\equiv\frac{\mathrm{Re}[\omega_{o}]}{\mathrm{Im}[\omega_{o}]}$ of the
fundamental standing-wave resonance for different $d$. An optimal $d=319$ nm
is observed, corresponding to a displacement of the photonic crystal from the
nominal lattice constant by $d-a=-70$ nm.

The fabricated optomechanical crystal connects to a waveguide and the external
coupling rate could be controlled by the number of phtonic crystal mirror
layers between the waveguide and optomechanical crystal. In the simulation, we
change the number of phtonic crystal mirror layers $N_{m}$ on one side (the
waveguide side) of the optomechanical crystal super-cell while keeping the
other side with a sufficient amount of photonic crystal mirror layers. We have
also included four additional tapered mirror layers on the waveguide side. The
external and internal quality factors are inferred from the energy flux as

${Q_{e}}=\frac{{{P_{e}}+{P_{i,u}}+{P_{i,d}}}}{{{P_{e}}}}{Q_{t}},\quad{Q_{i}}=\frac{{{P_{e}}+{P_{i,u}}+{P_{i,d}}}}{{{P_{i,u}}+{P_{i,d}}}}{Q_{t}},$
(23)

where $P_{e}$ is the energy flux towards the waveguide and $P_{i,u}$ and
$P_{i,d}$ are out-of-plane energy flux upward and downward, respectively. Fig.
9f shows $\kappa_{e}/\kappa_{i}\equiv Q_{i}/Q_{e}$. The simulation is done for
two cases. One case is with a large number of photonic crystal mirrors on the
other side of the super-cell such that $Q_{i}\approx 10^{6}$. In the other
case, we purposely reduced the number of photonic crystal mirrors on the other
side of the super-cell such that $Q_{i}\approx 2\times 10^{4}$ to mimic the
fabricated optomechanical crystal. For this case. We find the critical
coupling $\kappa_{e}/\kappa_{i}\approx 1$ is achieved around $N_{m}=4$.