Now at ]Department of Physics, University of Maryland, College Park.

# High-acoustic-index-contrast phononic circuits: numerical modeling

Wance Wang  [  Mohan Shen  Chang-Ling Zou  Department of Electrical
Engineering, Yale University, New Haven, Connecticut 06511, USA  Depratment of
Optics, University of Science and Technology of China, CAS, Hefei, Anhui
230026, China  Wei Fu  Department of Electrical Engineering, Yale University,
New Haven, Connecticut 06511, USA  Zhen Shen  Department of Electrical
Engineering, Yale University, New Haven, Connecticut 06511, USA  Depratment of
Optics, University of Science and Technology of China, CAS, Hefei, Anhui
230026, China  Hong X. Tang<EMAIL_ADDRESS>Department of Electrical
Engineering, Yale University, New Haven, Connecticut 06511, USA

###### Abstract

We numerically model key building blocks of a phononic integrated circuit that
enable phonon routing in high-acoustic-index waveguides. Our particular focus
is on Gallium Nitride-on-sapphire phononic platform which has recently
demonstrated high acoustic confinement in its top layer without the use of
suspended structures. We start with systematic simulation of various
transverse phonon modes supported in strip waveguides and ring resonators with
sub-wavelength cross-section. Mode confinement and quality factors of phonon
modes are numerically investigated with respect to geometric parameters.
Quality factor up to $10^{8}$ is predicted in optimized ring resonators. We
next study the design of the phononic directional couplers, and present key
design parameters for achieving strong evanescent couplings between modes
propagating in parallel waveguides. Last, interdigitated transducer electrodes
are included in the simulation for direct excitation of a ring resonator and
critical coupling between microwave input and phononic dissipation. Our work
provides comprehensive numerical characterization of phonon modes and
functional phononic components in high-acoustic-index phononic circuits, which
supplements previous theories and contributes to the emerging field of
phononic integrated circuits.

##  I Introduction

Surface acoustic wave (SAW) devices are widely used in electronic circuits,
finding applications such as filters and oscillators in communication devices
Campbell (1998) and a range of sensing applications Lao (1980); Friend and Yeo
(2011); Campbell (1989). Recently, SAWs are also exploited for coherent
control of various quantum systems, including the electron quantum dots Mcneil
_et al._ (2011); Hermelin  _et al._ (2011); Chen  _et al._ (2015), electron
spins in diamond Golter  _et al._ (2016a), superconducting qubits Gustafsson
_et al._ (2014); Satzinger  _et al._ (2018); Bienfait  _et al._ (2019), and
integrated photonic devices Fuhrmann  _et al._ (2011); Tadesse and Li (2014a);
Uan  _et al._ (2015); Tadesse  _et al._ (2015). Thus, SAW provides a promising
platform for hybrid quantum systems Kurizki  _et al._ (2015); Schuetz  _et
al._ (2015); Shumeiko (2016); Clerk  _et al._ (2020), where the traveling SAW
phonon can serve as a quantum bus to facilitate quantum state transfer.

The coupling strength between SAW and matter/photons is enhanced with reduced
mode volume of the SAW. The enhanced interactions such as the strong coupling
regime of cavity quantum acoustodynamics Manenti  _et al._ (2017); Moores  _et
al._ (2018) not only increases coherent coupling rates between quantum bits,
but also improve the sensitivity of SAW-based measurements Noguchi  _et al._
(2017). Also, the full potential of phononic systems can be only unraveled
when maneuverability of phonon becomes comparable to its electrical and
optical counterparts. Therefore, a strong confinement of the SAW to the
diffraction limit and the long lifetime phononic resonators are in high
demand. To that end, phonon waveguides Yen (1972); Oliner (1976, 1978) and
ring resonators Knox and Owen (1970); Sandy and Parker (1976) based on SAW
were proposed and experimentally studied in the 1970s. However, in the
following several decades, the experiments and detailed theoretical studies of
the confinement of itinerant phonons in microstructures were less active,
especially for the phonon resonators of radiating waves in nature. Only in
recent years, due to the advances of nano-fabrication technologies, ultra-low
loss phononic waveguides and resonators are achieved with with bulk acoustic
wave Goryachev  _et al._ (2014); Carvalho  _et al._ (2019) and with SAW
Biryukov  _et al._ (2007, 2009); Maznev (2009); Manenti  _et al._ (2016); Liu
_et al._ (2009); Boucher  _et al._ (2014); Fan  _et al._ (2016); Mohammadi and
Adibi (2011); Patel  _et al._ (2018); Fang  _et al._ (2016); Hatanaka  _et
al._ (2014a); Xu  _et al._ (2018), pushing the study of phononic devices,
including SAW-related waveguides and resonators back to the frontier of
researches as a strong contender for advanced phononic circuits.

Interdigital transducers (IDTs) convert the electrical RF signal into SAW or
vice versa by employing the piezoelectric materials. In most applications,
IDTs are fabricated on a uniform film, and the lateral size of IDT is much
larger than the wavelength of SAW to excite and collect the quasi-collimated
SAW. As a primary source or receiver of SAWs, the scaling of IDTs and their
integration with other SAW devices is also in critical demand in the
development of phononic circuits.

A variety of piezoelectric material platforms have been explored for SAW
devices, including ST-X quartz Manenti  _et al._ (2017); Noguchi  _et al._
(2017), zinc oxide Golter  _et al._ (2016b); Huang  _et al._ (2010); Magnusson
_et al._ (2015), lithium niobate Satzinger  _et al._ (2018); Ung  _et al._
(2017); Shao  _et al._ (2019), lithium tantalate Fu  _et al._ (2017a),
aluminum nitride Chu  _et al._ (2017); Aubert  _et al._ (2010); Tadesse and Li
(2014b); Fujii  _et al._ (2013), gallium arsenide Gustafsson  _et al._ (2014);
Moores  _et al._ (2018); Okamoto  _et al._ (2013); Metcalfe  _et al._ (2010),
and gallium nitride Xu  _et al._ (2018); Valle  _et al._ (2019); Wang  _et
al._ (2015); Fu  _et al._ (2019a). Together with substrate material such as
sapphire, diamond, and silicon, various combinations of high-acoustic-index-
contrast SAW devices can provide better confinement and thus low loss, as well
as the dispersion can be engineered. Table 1 summarizes common piezoelectric
materials and substrates. Among these piezo-materials, $\mathrm{LiNbO_{3}}$
and $\mathrm{LiTaO_{3}}$ possess much larger electromechanical coupling
coefficients ($K^{2}$) than other materials, but they have poor temperature
stabilities, whereas quartz is in a reverse case with weak coupling but good
temperature stability at room temperature Morgan and Paige (2007).
$\mathrm{ZnO}$ has wide bandgap Pang  _et al._ (2013) and good film
stoichiometry Fu  _et al._ (2017b) in addition to relatively large $K^{2}$ Fu
_et al._ (2019b). $\mathrm{ZnO/diamond}$ is also a quantum interface between
SAW and nitrogen-vacancy center in diamond Golter  _et al._ (2016b).
$\mathrm{AlN}$ has high refractive index and high phase velocity Fu  _et al._
(2019a); Deger  _et al._ (1998) which is in favor of high-frequency devices,
but in turn, the contrast of acoustic index will be low. It also finds
applications in high-temperature environments Aubert  _et al._ (2010).
Semiconductor material $\mathrm{GaN}$ and $\mathrm{GaAs}$ are compatible with
standard wafer-scale fabrication, $\mathrm{GaAs}$ is also naturally suited for
$\mathrm{InGaAs}$ quantum dots Metcalfe  _et al._ (2010) which benefit from
the telecom range. The availability of $\mathrm{GaN}$ high-performance
amplifiers enables monolithic integration with devices. For the layered system
to our knowledge, $\mathrm{ZnO/diamond}$, $\mathrm{ZnO/sapphire}$,
$\mathrm{ZnO/SiO_{2}}$ Nakahata  _et al._ (2003); Golter  _et al._ (2016b);
Huang  _et al._ (2010), $\mathrm{AlN}/\mathrm{sapphire}$,
$\mathrm{AlN}/\mathrm{SiO_{2}}$ Aubert  _et al._ (2010); Tadesse and Li
(2014b), and $\mathrm{GaN/sapphire}$ Fu  _et al._ (2019a) have been explored.
Diamond the highest known SAW phase velocity but its fabrication requires
chemical vapour deposition Nakahata  _et al._ (2003), and it suffers from
larger propagation loss. The contrast of $\mathrm{SiO_{2}}$-based platform
isn’t high. Concluding all factors, $\mathrm{GaN/sapphire}$ platform stands
out as a high-performance and ready-access choice.

Material |  $V_{l}$ ($\mathrm{m\cdot s^{-1}}$) |  $V_{t}$ ($\mathrm{m\cdot s^{-1}}$) | Piezoelectricity
---|---|---|---
Quartz Pohl  _et al._ (2002) | 5700 | 3158 | Weak
$\mathrm{ZnO}$ Azuhata  _et al._ (2003) | 5790 | 2700 | Strong
$\mathrm{LiNbO_{3}}$ Warner  _et al._ (1967) | 7316 | 4795 | Strong
$\mathrm{AlN}$ Strauch (2011) | 10,169 | 6369 | Good
$\mathrm{GaN}$ Strauch (2011) | 7350 | 4578 | Good
Diamond Flannery  _et al._ (2003) | 18,000 | 12,000 | No
Sapphire Auld (1990) | 10,658 | 5796 | No
$\mathrm{SiO_{2}}$ Hopcroft  _et al._ (2010) | 8433 | 5843 | No
Table 1: SAW waveguide and substrate materials. $V_{l},\,V_{t}$ are
longitudinal and transverse wave speed, respectively.

In this paper, we numerically investigate the properties of integrated
phononic waveguides and resonators based on GaN-on-sapphire platform.
Different from $\Delta V/V$ confinement effect induced by the metal electrodes
Biryukov  _et al._ (2009, 2007); Maznev (2009); Manenti  _et al._ (2016);
Hughes (1972), we consider the pure geometric confinement of phonon on the
surface of the chip, thus the metal-induced loss is eliminated. We prove that
the phononic ring resonator can have exceptionally high quality factors ($Q$)
with the radius of tens of microns, in which the whispering gallery modes
(WGMs) can be excited either by the evanescent acoustic field through the
phononic waveguide coupling or by direct IDT excitation. These waveguide and
resonator structures will be basic elements in future phononic integrated
circuits of both fundamental and practical interests Fu  _et al._ (2019a);
Shen  _et al._ (2017).

##  II Waveguide

Figure 1: (a) Schematic illustration of an unsuspended strip phononic
waveguide. $w_{0}$ and $h_{0}$ represent the width and height of the GaN
strip, respectively. (b) The geometry setup in COMSOL simulation. The phononic
waveguide modes propagating in the uniform structure along
$\overrightarrow{x}$ with wavevector $\overrightarrow{k}$ are calculated by
applying a periodic boundary condition, and PMLs are employed to absorb
radiative acoustic waves in substrate.

Efficient confinement and guiding of phonons is essential for scalable
operation of phononic devices. In the past decades, SAW has been mostly
employed in applications that utilize its vertical confinement at the surface
of substrates. However, the lateral confinement to the SAW is seldom studied.
Alternatively, three-dimensional confinement of phonon has been mostly
achieved in suspended phononic microstructures, which imposes practical
constraints in fabrication yield and structural robustness. Here, we
numerically study a phononic architecture based on unsuspended phononic
waveugides and resonators. Shown in Fig. 1(a) is a typical strip waveguide.
The basic requirement for phonon confinement is that the speeds of both
transverse and longitudinal waves in strip material are slower than those in
the substrate Oliner (1976). Thus, we choose the material platform of GaN-on-
Sapphire satisfying these requirements, with parameters shown in Table 2. An
equivalent condition is also drawn in Ref. Tiersten (1969) where the waveguide
layer material needs to be heavier and less stiff. More detailed analyses
about the requirements for the existence of Love wave can be found in Ref. Li
_et al._ (1977); Markman  _et al._ (1977); Tiersten (1969).

Conventional Rayleigh waves have been widely studied and applied in SAW
devices, while half-space single medium also supports shear-horizontal (SH)
waves. In layered systems, in addition to layered Rayleigh waves, there are
also Love waves Morgan and Paige (2007), which is first discovered by A.E.H.
Love Love (1911). Love waves have a shear component with displacements in the
surface, which can be regarded as modified forms of the SH wave. SH waves and
Love waves are now attracting interests in spin–orbit interactions of phonons
Fu  _et al._ (2019a) and sensing applications Pang  _et al._ (2013); Fu  _et
al._ (2017a).

The basic properties of confined guiding modes are studied numerically by
three-dimensional finite-element method (COMSOL Multiphysics v5.2). The
waveguide geometrical parameters, namely its width $w_{0}$ and height $h_{0}$
marked in Fig. 1(a), determine the properties of the waveguide modes such as
frequency, confinement, loss, and energy distributions. In this paper, all the
simulations are carried out using periodic boundary conditions according to
the translational symmetry of the structure, and perfectly matched layers
(PMLs) are employed for studying the radiative loss, with the detailed
numerical model illustrated in Fig. 1(b). To reveal the basic behaviors of the
phononic waveguides, the anisotropy of GaN and sapphire are neglected.

Material | GaN Bougrov  _et al._ (2001); Strauch (2011) | Sapphire Sap
---|---|---
Density ($\mathrm{g\cdot cm^{-3}}$) | 6.15 | 3.98
Young’s modulus ($\mathrm{GPa}$) | 305 | 345
Poisson ratio | 0.183 | 0.29
Longitudinal wave speed ($\mathrm{m/s}$) | 7350.0 | 10658.0
Transverse wave speed ($\mathrm{m/s}$) | 4578.3 | 5796.4
Table 2: Elastic properties of GaN and Sapphire used in the simulation.

Figure 2: Simulated displacement field profile for low-order phonon modes
supported in an unsuspended strip waveguide. The color mappings show the
strength of displacements $|\vec{x}|$ normalized by their own maximum
$\max{(|\vec{x}|)}$, which applies to all other figures of displacement field
in this paper. Displacement amplitude is internally excited by the simulation
software, so only relative strength $|\vec{x}|/\max{(|\vec{x}|)}$ has
significance. $\max{(|\vec{x}|)}$ is extracted among the entire field of each
mode, and every following figure is normalized to their own maximum. But since
the color map setups are the same, we let them share one common color bar for
simplicity. (a) Symmetric quasi-Rayleigh mode _a_. (b) Anti-symmetric quasi-
Love mode _b_. (c) Anti-symmetric quasi-Rayleigh mode _c_. (d) Symmetric
quasi-Love mode _d_. (e)-(h) Higher-order phonon modes _e-h_. The italic
characters label different modes and will be used throughout this paper.
Wavelength $\lambda$ along guided direction $x$ are all set at $2\,\mathrm{\mu
m}$. The waveguide height is fixed at and $h_{0}=0.5\,\mathrm{\mu m}$ while
its width is set at $w_{0}=5\,\mathrm{\mu m}$ except that the $w_{0}$ of mode
h is chosen to be $6\,\mathrm{\mu m}$ because that specific mode does not
appear for $w_{0}=5\,\mathrm{\mu m}$.

Figure 3: Properties of phononic modes in a strip waveguide as the waveguide
width $w_{0}$ is varied. (a)-(c) Frequency $f$, mode area $A_{eff}$, and
energy confinement ratio $\eta$, versus $w_{0}$, respectively. (d) Evolution
of mode hybridizations when $w_{0}$ is varied with an increment of
$1\,\mathrm{\mu m}$. The color mappings show the strength of normalized
displacement $|\vec{x}|/\max{(|\vec{x}|)}$ in each mode profile. Note the
$\max{(|\vec{x}|)}$ is selected among all displacement field images for each
mode, or equivalently, each column which plots one common mode has the same
normalization. Among different columns, they use a common color mapping but
their normalizations are distinct. (e) The deformation of anti-symmetrical
flexural Lamb wave (left) and the simulation result (right) for a waveguide
with an aspect ratio $w_{0}/h_{0}=0.2$.

Shown in Figs. 2(a)-(h) are fundamental to higher-order transverse phonon
modes supported in a strip waveguide, with its width set at
$w_{0}=5\,\mathrm{\mu m}$ (for mode $h$, $w_{0}=6\,\mathrm{\mu m}$ because it
doesn’t appear at $w_{0}=5\,\mathrm{\mu m}$) and $h_{0}=0.5\,\mathrm{\mu m}$.
Wavelength along propagating direction is set at $\lambda=2\,\mathrm{\mu m}$.
The color mappings show the strength of normalized displacement
$|\vec{x}|/\max{(|\vec{x}|)}$ in each mode profile, which applies to all other
figures of displacement field in this paper. Note that we label the modes by
the corresponding characters $a,..,h$ in the following for analyzing the
evolution of mode profiles against geometry parameters. Mode _a_ (Fig. 2(a))
is dominated by out-of-plane (z-direction) displacement, which we refer as
quasi-Rayleigh mode owing to its reminiscence of Rayleigh wave Rayleigh
(1885). Mode _b_ has dominant in-plane (y-direction) motion and is a quasi-
Love mode because of its similarity to Love wave Love (1911). The modes _a_
and _b_ can be regarded as the fundamental phononic modes in the strip with
different polarizations (z- and y-polarized motion). Due to the lateral
confinement and the edges, we found that the phononic modes are distinct from
SAW modes. On one hand, the modes do not show pure flexural wave or in-plane
shear waves. Especially, at the edge, it is difficult to identify the mode
orientation because of the strong hybridization of deformation in all three
orthogonal directions. On the other hand, higher-order transverse modes appear
due to the lateral confinement (Figs. 2(c)-(h)). For example, compared to
fundamental mode _a_ and _b_ , in which vibrations on the strip edges are in-
phase, the motion of mode _c_ and _d_ on the opposite edges are out-of-phase.
In the rest of this paper, we call the symmetric (in-phase) mode _a_ as
S-Rayleigh mode and call the anti-symmetric (out-of-phase) mode _c_ as
A-Rayleigh mode, as well as A-Love mode _b_ and S-Love mode _d_. We also
observe that higher-order modes (Figs. 2(e)-(h)) exhibit more complex field
distributions.

For most applications, the concerned properties of guided phonon modes are
Oliner (1976): dispersion, confinement, radiation loss, energy distributions.
Therefore, we numerically evaluate the following properties with varying
waveguide geometrical parameters.

  1. 1.

Mode frequency $f$ for a given wavelength $\lambda$ or wavenumber
$k_{x}=2\pi/\lambda$ along guided direction.

  2. 2.

Phase velocity $v_{p}=2\pi f/k_{x}$ and group velocity $v_{g}=2\pi df/dk_{x}$.

  3. 3.

Mode area

$A_{\mathrm{eff}}=\frac{1}{L}\frac{\iiintop
W(x,y,z)dxdydz}{\mathrm{\text{$\max$}}(W(x,y,z))}$ (1)

where $L$ is the waveguide length along the propagation direction, $W(x,y,z)$
is the elastic strain energy density. The integral and maximum are calculated
in the full simulated region. The mode area is a concept borrowed from
photonics (see Ref. Hill  _et al._ (2007); Oulton  _et al._ (2008)) as a
measure of lateral confinement, where a smaller $A_{\mathrm{eff}}$ indicates
stronger confinement of phonon.

  4. 4.

Quality factor

$Q=2\pi f\times\frac{\mathrm{Energy\>stored}}{\mathrm{Power\>loss}}.$ (2)

In this paper, we only calculate the radiation loss into substrate. A
discussion of other loss mechanisms can be found in the conclusion Sec. VI.

  5. 5.

Energy confinement ratio

$\eta=\frac{\iiintop_{\mathrm{strip}}W(x,y,z)dxdydz}{\iiintop_{\mathrm{all}}W(x,y,z)dxdydz}$
(3)

indicates the ratio between elastic strain energy stored in the phononic
waveguide and the total elastic strain energy.

Figure 4: Dispersion characteristics for phononic modes on strip waveguide,
with geometry parameters $w_{0}=5\,\mu\mathrm{m}$ and $h_{0}=0.5\,\mathrm{\mu
m}$ (aspect ratio $w_{0}/h_{0}=10$). (a) Frequency $f$ versus guide wavenumber
$k_{x}$. (b) and (c) Phase velocity and group velocity versus $k_{x}$,
respectively. (d) and (e) Mode area $A_{eff}$ and energy confinement ratio
$\eta$ versus $f$, respectively.

Figures 3(a)-(c) show the modal frequency ($f$), area ($A_{eff}$), and energy
storage ratio ($\eta$) of 8 phonon modes versus the width $w_{0}$ of strip
waveguide, with a fixed height $h_{0}=0.5\,\mathrm{\mu m}$ and wavelength
along propagation direction $\lambda=2\,\mathrm{\mu m}$. To clarify
complicated mode hybridization, we now explain the notations and colors of
plots that are adopted in this paper. By varying the waveguide geometry by
small increments, the mode properties vary smoothly, thus we can track the
frequencies or other parameters as continuous curves, which is called “branch”
and labeled by different colors. As shown in the frequency plot (Fig. 3(a)),
each branch is assigned with one specific color consistent with all other
properties plots. As explained above, due to the strong mode hybridization in
the strip waveguides, it is very difficult to identify mode orders or
polarizations by branches. Therefore, we only label the modes in the regime
that can be clearly distinguished according to the modes shown in Fig. 2.
Sections labeled with two characters indicate the modes are experiencing
hybridization.

The model frequency features versus width $w_{0}$, as shown in Fig. 3(a),
indicates a transition between single-mode and multi-mode waveguide. At large
$w_{0}$ values, all phonon mode branches asymptotically approach slab phonon
modes in GaN-on-sapphire thin films. When $w_{0}$ becomes comparable to
$\lambda$, the curves branch off and the behaviors of each branch can be
explained as follows. Since the lateral confinement imposes an approximate
quantization condition $k_{y}\sim\frac{n\pi}{w_{0}}$, an increasing $w_{0}$
results in a reduction of total phonon wavenumber as
$k=\sqrt{k_{x}^{2}+k_{y}^{2}}$ for a fixed $k_{x}$, and hence a monotonic
reduction of frequency approaching a constant. This intuitive interpretation
is valid for most modes, however, fails for the A-Love modes _b_. According to
the mode area and energy confinement ratio given in Figs. 3(b) and (c), the
confinement of A-Love mode _b_ is excellent at small $w_{0}\ll\lambda$ (rapid
reducing of mode area), and thereby almost all wave energy is confined in the
strip. It could be interpreted as the effective boundary becomes air-GaN for
the A-Love mode _b_ as opposed to sapphire-GaN for the other modes, and
consequently, it can be treated as an anti-symmetrical flexural Lamb wave of a
plate normal to $y$ direction (Fig. 3(e)) Lagasse  _et al._ (1973a). For a
plate mode, the vibration frequency $f\propto\sqrt{\frac{D}{\rho w_{0}}}$, in
which bending stiffness $D=\frac{w_{0}^{3}E}{12(1-\nu^{3})}$ Reddy (2006)
($\rho,\,E,\,\nu$ are density, Young’s modulus and Poisson ratio,
respectively.). Thus the frequency of A-Love mode _b_ scales with $w_{0}$ and
decreases for narrow waveguides.

Figure 5: (a) Schematic illustration of a strip ring resonator. (b) S-Rayleigh
(mode _a_) and (c) A-Love (mode _b_) in ring resonator. The color mappings
show the strength of normalized displacement $|\vec{x}|/\max{(|\vec{x}|)}$ in
each mode profile. (d) The evolution of mode hybridization profiles with
varying $w_{0}$. The normalization convention is the same as we used in Fig.
3(d): the $\max{(|\vec{x}|)}$ is selected among all displacement field images
for each mode, or equivalently, each column which plots one common mode has
the same normalization. (b-d) share a common color bar.

The interactions between the phonon modes lead to crossings and avoided
crossings in Figs. 3(a)-(c). For the avoided crossings, the corresponding mode
area and energy confinement ratio show crossing behavior, indicating the
coupling between modes and the mode hybridization at these device parameters.
For the crossing in frequency, the coupling between modes is negligible, so
the changes in mode area and energy confinement ratio is not noticeable.
Outside of the hybridization regimes, the confinement of quasi-Rayleigh modes
(_a_ and _c_) weakens (larger mode area) with the increase of width, while
that of quasi-Love modes (_b_ and _d_) remains relatively confined. At very
small $w_{0}$, however, the waveguide becomes too narrow to support a fully
confined mode _a_ , and its mode frequency rapidly approaches its cutoff
frequency which is the frequency of SAW in substrate for
$\lambda=2\,\mathrm{\mu m}$. The mode area of A-love mode _b_ , in contrast to
mode _a_ , drops because this mode resembles anti-symmetrical flexural Lamb
wave at small width, and its confinement is enhanced as width narrows Lagasse
_et al._ (1973a) (like a vibrating membrane whose energy is tightly
concentrated in the waveguide region thus the influence from substrate is
negligible). The quasi-Love modes (_b,d_) exhibit stronger confinement than
quasi-Rayleigh modes’ (_a,c_) due to the much larger $\eta$ of quasi-Love
modes (except the hybridization regime), as shown in Fig. 3(c).

For a strip waveguide of fixed cross-section $h_{0}=0.5\,\mathrm{\mu m}$ and
$w_{0}=5\,\mathrm{\mu m}$, the dispersion characteristics shown in Fig. 4 are
numerically calculated as a function of the mode wavenumber $k_{x}$. In Fig.
4(b), the quasi-Love modes _b_ and _d_ have faster phase velocities than their
corresponding quasi-Rayleigh modes and their behaviors approach the infinite
surface according to analytical predictions in Ref. Li  _et al._ (1977); Sinha
and Tiersten (1973); Lagasse  _et al._ (1973b). The cutoffs result in sharp
rising of group velocities in Fig. 4(c) at small wavevectors, and meanwhile,
in Fig. 4(d), mode areas rise sharply at low frequencies, indicating a
remarkable divergence of the confinement when the size of strip becomes much
smaller than the wavelength. This is confirmed in Fig. 4(e) that energy
confinement ratio decreases, i.e. phonon energy spreads into the substrate, at
small $k_{x}$. In Fig. 4(d), we also found an interesting minimum point of
mode area that occur simultaneously for branches of mode _b_ and _d_ around
$2.2$ GHz. A close comparison with results in Figs. 4(a) and (b) reveals that
these minima occur at the avoided crossing point of modes _b_ and _d_ when
they are maximally hybridized. Therefore, the mode hybridization presents an
unique approach to engineer the phononic modes and might find interesting
phononic sensing applications.

##  III Ring resonator

In this section, we study the properties of ring resonators formed by bending
and closing strip waveguide which supports phononic WGMs. The pioneering works
of the acoustic WGMs date back to the discovery by Lord Rayleigh Rayleigh
(1910, 1914), who reported the sound wave travels around a concave boundary in
in St Paul’s Cathedral due to the continuous total internal reflection. The
concept is generalized to electromagnetic waves Oraevsky (2002); Matsko and
Ilchenko (2006); Ilchenko and Matsko (2006), which holds unique
characteristics of high Q-factors, small mode volumes, and the ease of
fabrication Foreman  _et al._ (2015). These remarkable merits are also
possessed by phononic WGMs according to our investigations.

Due to the cylindrical symmetry of the ring resonator, the eigenmode profiles
of the elastic differential characteristic equation are in the form as
$u(r,z,\phi)=\psi(r,z)e^{im\phi}$, where $\left(r,z,\phi\right)$ are the
cylindrical coordinators, $\psi(r,z)$ is the field distribution at the cross-
section, $m$ is the angular momentum number. Utilizing the symmetry, we can
numerically solve the mode profiles of a sector unit in $\phi$ direction in
COMSOL, with periodic boundary condition: $u(r,z,\phi)=u(r,z,0)e^{-im\phi}$,
where $m=2\pi R/\lambda$ for an azimuthal wavelength $\lambda=2\,\mathrm{\mu
m}$ along tangent direction.

Figures 5 (b) and (c) display the typical quasi-Rayleigh and quasi-Love WGMs
in a ring resonator. Compared to those in waveguide (Figs. 2(a)-(b)), the
displacement fields of WGMs in the ring are more concentrated at the outer
rim. Figure 5(d) shows the mode profiles of the first four branches in Fig.
6(a). In the rest of this section, except for dispersion behaviors, we study
the properties of WGMs around fixed geometry parameters
$\left\\{R_{0},\,w_{0},\,h_{0}\right\\}=\left\\{50\,\mathrm{\mu
m},\,5\,\mathrm{\mu m},\,0.5\,\mathrm{\mu m}\right\\}$. Similar to the cases
of strip waveguides, the colors of different branches are fixed in Figs. 6, 7,
and 8. Instead of the mode area, the confinement in resonator is characterized
by the mode volume Kippenberg (2004),

$V_{\mathrm{eff}}=\frac{\iiintop W(r,\theta,z)rdrd\theta
dz}{\max[W(r,\theta,z)]}.$ (4)

Figure 6: Phononic mode properties in a ring resonator of varying width
$w_{0}$. The ring radius and height are fixed respectively at
$R_{0}=50\,\mathrm{\mu m}$ and $h_{0}=0.5\,\mathrm{\mu m}$. (a)-(d) show
frequency, mode volume $V_{eff}$, $Q$ factor, and energy confinement ratio
$\eta$, respectively. The insets of (a) are the mode profiles of hybridized
modes of _$(a,\,b)$_ at $w_{0}=1.7\,\mathrm{\mu m}$.

As shown in Fig. 6(a), the dependence of mode frequencies with respect to
width $w_{0}$ follows the same trend of those of strip waveguides [Fig. 3(a)].
However, comparing to the crossing in waveguide, there is avoided crossing
between mode _a_ and _b_ , arising from the broken symmetry (hence mode
hybridization) in the bending waveguide. The mode volume and energy
confinement ratio $\eta$ are presented in Figs. 6(b) and (d), showing similar
crossing and avoid-crossing behavior as in the waveguide case.

For ring resonators, in contrast to the strip waveguides, there is always
radiation loss due to the waveguide bending, and thus the $Q$ factors of WGMs
are finite. As shown in Fig. 6(c), S-Rayleigh _a_ , A-Rayleigh _c_ and A-Love
_b_ are three modes having the highest $Q$ factors. For mode _a_ and _b_ ,
$Q$s are sensitive to the width at small $w_{0}$ value but almost constant
when $w_{0}$ exceeds $3\,\mathrm{\mu m}$, implying a saturation of loss
approaching the case of a disk resonator. We observe an abrupt increment of
$Q$ when _b_ and _a_ are hybridized at around $w_{0}=1.7\,\mathrm{\mu m}$,
suggesting that the mode _b_ and _a_ couple to a common leaky modes in
substrate and their destructive interference suppresses the phonon loss. Thus
the special width leads to a parameter-tuning type of bound state in continuum
Hsu  _et al._ (2016); Chen  _et al._ (2016).

Figure 7: Phononic mode properties in a ring resonator as functions of ring
radius $R_{0}$ with width $w_{0}=5\,\mathrm{\mu m}$, height
$h_{0}=0.5\,\mathrm{\mu m}$. (a-d) show frequency, mode volume $V_{eff}$, $Q$
factor and energy confinement ratio $\eta$, respectively.

Figure 7(a) shows the properties of phononic WGMs with varying radius. For a
fixed $\lambda$, the frequencies of the ring converge to the case of straight
waveguide at large $R_{0}$ values with their calculated modal volumes shown in
Figure 7(b). An optimal radius exists for each mode because the lateral
confinement (i.e. effective mode area $A_{\mathrm{eff}}$) becomes better for
larger $R_{0}$, while the mode volume is approximately proportional to $R_{0}$
($V_{\mathrm{eff}}\approx 2\pi R_{0}A_{\mathrm{eff}}$). The $Q$ factors in
Fig. 7(c) increase exponentially with increasing $R_{0}$, indicating radiation
loss to the substrate caused by bending decreases exponentially with the
curvature, similar to the radiation loss of optical WGM in dielectric spheres
Kippenberg (2004); Oraevsky (2002). The increment of $Q$ when _b_ and _c_ are
hybridized at around $R_{0}=54\,\mathrm{\mu m}$ can be explained with the same
reason as that of _b_ and _a_ in Fig. 6(c).

Figure 8: Dispersion characteristics for modes in a ring resonator of radius
$R_{0}=50\,\mathrm{\mu m}$, width $w_{0}=5\,\mathrm{\mu m}$, and height
$h_{0}=0.5\,\mathrm{\mu m}$. (a) Modal frequencies $f$ versus $k_{\phi}$—the
effective propagating wavenumber defined as $m/R_{0}$, with $m$ is angular
momentum number. (b) and (c) Phase velocity and group velocity versus
$k_{\phi}$. (d-f) Mode volume $V_{eff}$, $Q$ factor, and energy confinement
ratio $\eta$ versus $f$, respectively.

The dispersion characteristics of ring resonator are summarized in Fig. 8. The
$k_{\phi}=m/R_{0}$ in the figure is effective propagating wavenumber along
tangential direction. Most branches resemble their counterparts in the strip
waveguide (Fig. 4), including $f-k_{\phi}$ relation, and the decrease of phase
velocity and mode volume, and increase of ratio $\eta$ with increasing
$k_{\phi}$ or $f$. We also compute $Q$ factors which grow exponentially at
short wavelengths.

##  IV Directional Coupler

In an integrated phononic circuit, multiport devices transporting and routing
phonons between components such as the waveguide-waveguide and waveguide-
resonator coupler are indispensable circuit elements. A directional coupler
consists of two closely placed parallel waveguides. Here we study the phononic
coupling through the coupled-mode theory by analogy to its optical counterpart
widely used in the photonic community Yariv (1973); Hardy and Streifer (1985);
Haus  _et al._ (1987). The coupling between the modes in different waveguides
arises from the tunneling of the elastic wave between them. The confined
waveguide mode has a non-zero evanescent field that overlaps with the other
waveguide, leading energy transfer in between.

The spatial mode amplitude evolution along the propagation direction in
coupled waveguides for a given input signal frequency are described by the
following equations

$\displaystyle\frac{d}{dz}\alpha_{1}(z)$
$\displaystyle=-ik_{1}\alpha_{1}(z)-ig_{12}\alpha_{2}(z),$ (5)
$\displaystyle\frac{d}{dz}\alpha_{2}(z)$
$\displaystyle=-ik_{2}\alpha_{2}(z)-ig_{21}\alpha_{1}(z).$ (6)

Here, the subscript “1”, “2” represent waveguide 1 (width $w_{1}$) and 2
(width $w_{2}$), $k_{1(2)}$ is the wavenumber of the mode in waveguide $1(2)$,
and $g_{12},\,g_{21}$ are the coupling coefficients between two modes. For
lossless waveguides, coefficient $g$ must satisfy $g_{12}=g_{21}^{*}$ due to
the power conservation Haus  _et al._ (1987). Setting $g_{12}g_{21}=|g|^{2}$
and solving Eqs. (5) and (6), we obtain the wavenumber eigenvalues of

$k_{\pm}=\frac{k_{1}+k_{2}}{2}\pm\sqrt{\left(\frac{k_{1}-k_{2}}{2}\right)^{2}+|g|^{2}}$
(7)

for hybrid modes in two waveguides. When $k_{1}=k_{2}=k$, the difference
between $k_{\pm}$ reaches the minimum $2|g|$, and the eigenmodes are the
$a_{1}\pm a_{2}$, as an equal superposition of two waveguide modes. If the
phonon input at the first waveguide with $a_{1}(0)=1$ and $a_{2}(0)=0$, we
arrive at

$\displaystyle\alpha_{1}\left(z\right)$
$\displaystyle=\cos\left(|g|z\right)e^{-ikz},$ (8)
$\displaystyle\alpha_{2}\left(z\right)$
$\displaystyle=\sin\left(|g|z\right)e^{-i(kz+\frac{\pi}{2})}.$ (9)

Thus, the phonons in the first waveguide $\alpha_{1}$ could be fully
transported into the second waveguide $\alpha_{2}$ after a coupling distance
of $\pi/2|g|$, or half of the phonons can be transmitted into mode
$\alpha_{2}$ after a coupling distance $\pi/4|g|$. The larger the $|g|$, the
quicker the energy exchanges.

###  IV.1 Coupling between two identical waveguides

For two identical waveguides with $w_{1}=w_{2}$, the wavenumbers of the modes
are the same $\left(k_{1}=k_{2}\right)$, thus all supported modes can couple
between the two waveguides without phase mismatching. Here, we set the width
$w_{0}=5\,\mathrm{\mu m}$ and height $h_{0}=0.5\,\mathrm{\mu m}$ for both
waveguides, as shown in Fig. 9(a). From the analysis above, we can estimate
the coupling strength $g$ between two waveguides by the splitting of
$k_{\pm}$.

Figure 9: (a) Schematic of the directional coupler. (b) The estimated coupling
strengths $|g|$ as a function of $gap$. Set the $\lambda=2\,\mathrm{\mu m}$
along guided direction. (c) The displacement field profiles for the in-phase
and out-of-phase modes in the directional coupler, which is made by two
identical waveguides with $gap=5\,\mathrm{\mu m}$. The color mappings show the
strength of normalized displacement $|\vec{x}|/\max{(|\vec{x}|)}$ in each
profile. The in-phase and out-of-phase figures share the same normalization,
and all figures share the common color bar.

Figure 9(b) shows the coupling strength of various waveguide modes with
different gap, where the label _a-g_ corresponds to the modes introduced in
Fig. 3. We set the $\lambda=2\,\mathrm{\mu m}$. Eigenmodes _a, b_ and _c_ in
the coupled waveguides are shown in Fig. 9(c), featuring in-phase and out-of-
phase hybridized modes in the coupled waveguides as predicted by the coupled-
mode theory. Since the evanescent field exponentially decays with the distance
to the waveguide, a reduction of the $g$ with increasing gap is expected. As
shown by the numerical simulations, there is a clear exponential relation
between the coupling strength and the gap, yet the trends of high order modes
_e_ and _f_ slightly deviate from the exponential curve at small/large gaps.
This can be attributed to the weaker confinement of the high order modes, for
which the perturbative approximation in coupled-mode theory is no longer
accurate.

###  IV.2 Coupling between dissimilar waveguides

The coupled-mode theory is also applicable to dissimilar modes in dissimilar
waveguides, therefore opens the possibility for mode conversions with
different polarization or mode orders as long as there is a finite mode
overlap between them. From Eq. 6, the maximum ratio of energy transmittance
between the two modes is
$\mathrm{sinc}^{2}[1+\left(k_{1}-k_{2}\right)^{2}/4g^{2}]^{1/2}$. Thus, for
efficient energy transfer, we should adjust waveguide width to match the
wavenumber of the two dissimilar modes, i.e. fulfill the phase-matching
condition. In Fig. 10(a), the width of one waveguide ($w_{1}$) is changed
while the width of the other is fixed ($w_{2}=4.5\,\mathrm{\mu m}$) in order
to meet with the phase-matching condition

$k_{1}(w_{1})=k_{2}(w_{2})$ (10)

The modal frequencies of coupled waveguides are shown in Fig. 10(a) as the
$w_{1}$ is varied from $1\,\mathrm{\mu m}$ to $2.5\,\mathrm{\mu m}$. From the
plot, we observe four avoid crossings, corresponding to four regions of modal
coupling: I : $(b_{1},a_{2})$, II: $(a_{1},b_{1})$, III : $(b_{1},c_{2})$, IV
: $(a_{1},c_{2})$. Fig. 10(c) displays the displacement fields of the four
coupling regions at the minimum frequency differences. Similar to the
identical waveguides, $g$ values of different modes also exhibit an
exponential dependency on $gap$ (Fig. 10(b)). Among four cases, coupling II is
particularly interesting because it represents a special mechanism of “self
coupling” between mode _a_ and _b_ within waveguide 1, corresponding to a
double tunneling process that mode _a_ and mode _b_ in waveguide 1 both couple
to waveguide 2, which mediates the coupling of mode _a_ and _b_. The double
tunneling mechanism could also explain the much faster decaying of the
coupling strength for case II. With such a mechanism, we would expect the
realization of a single-waveguide mode converter Shen  _et al._ (2019) in the
future with the assistance of an ancillary waveguide.

Figure 10: (a) Modal frequencies in directional coupler of dissimilar
waveguides, plotted against the waveguide width $w_{1}$, with the other
waveguide width and coupling gap fixed ($w_{2}=4.5\,\mathrm{\mu m}$,
$gap=1\,\mathrm{\mu m}$). Due to the modal coupling, four avoided-crossing
regions $\left\\{\mathrm{I,\,II,\,III,\,IV}\right\\}$ are observed when the
frequencies of modes in separate waveguides approach each other. The
subscripts of waveguide mode labels $\left\\{1,2\right\\}$ denote the
waveguide 1 and 2 respectively. (b) The modal coupling strength $|g|$ as a
function of $gap$. (c) The mode profiles of the directional coupler in the
four avoided crossing regions. The color mappings show the strength of
normalized displacement $|\vec{x}|/\max{(|\vec{x}|)}$ in each profile. The in-
phase and out-of-phase figures share the same normalization, and all figures
share the common color bar.

##  V Coupling to IDT

Figure 11: (a) Schematic illustration of the IDT integrated ring resonator,
with the electrodes of IDTs connected to external microwave cable (not shown)
for signal input. (b) The influence of the IDT on the mode $Q$ factors. (c)
and (d) The equivalent impedance and the energy transmission (conversion
efficiency) from microwave to phonon as functions of the ring radius for
S-Rayleigh and A-Love modes, respectively.

Aside from the vibrational properties of phononic waveguides and ring
resonators, the excitation and detection of phonons are also of practical
importance and interests, for example, in phononic implementation of microwave
delay lines and filters. Based on the model of ring resonator presented in
Sec. III, we add IDT electrodes on the top surface of the ring resonator (Fig.
11(a)), and numerically investigate the coupling between IDT and phononic ring
resonators and its efficiency to excite phonons. In this model, only the
electrical effect of the IDT electrodes to the resonator is considered, while
the mechanical and other effects due to loading of the electrodes are ignored
(such as mass loading).

We first evaluate the quality factor of the mechanical ring resonator under
the presence of IDT electrodes. The width of each electrode is $1/4$ of the
acoustic wavelength and the electrodes cover half of the ring resonator area.
For the S-Rayleigh mode _a_ with wavelength of $2\,\mathrm{\mu m}$ in a
$5\,\mathrm{\mu m}$-wide ring resonator without IDT, the quality factor
increases exponentially with the radius of the resonator as the quality factor
is limited by the radiation loss; with the IDT electrodes, the quality factor
saturates when the radius is larger than $40\,\mathrm{\mu m}$, as shown in
Fig. 11(b). The presence of the electrodes modifies the local phonon velocity,
due to the $\delta v/v$ effect Hughes (1972). Hence, the IDTs break the
cylindrical symmetry of the perfect ring, leading to extra coupling of the
phonon mode in ring to the leaky phonon modes in the bulk substrate. The same
effect also happens to the Love mode. When the radius is small and radiation
loss is dominant, the quality factors with and without IDT electrodes are
almost the same. At larger radius in our simulation, the IDT electrodes
induced loss becomes comparable with radiation loss and the quality factor
with IDT electrodes are slightly lower.

To evaluate the external coupling to the phonon resonator, we also simulate
the equivalent impedance of the IDT coupled mechanical ring resonator in order
to achieve impedance matching to $Z_{0}=50\,\Omega$ transmission lines for
maximal acoustic launching efficiency. In the COMSOL simulation, by assigning
voltage $V$ on the electrodes and measuring the current flow $I$ on the
electrodes at mechanical resonance frequency, the equivalent impedance,
$Z=V/I$, can be extracted. The microwave reflectivity can be expressed by

$r=\frac{Z-Z_{0}}{Z+Z_{0}}$ (11)

and the corresponding microwave to mechanical energy conversion efficiency
$T=1-\left|r^{2}\right|$. The simulated results are plotted in Fig. 11(c) and
(d) for quasi-Rayleigh and quasi-Love mode, respectively. The results indicate
that the effective impedance of the IDT coupled mechanical resonator decreases
exponentially with the radius and the impedance matching condition $Z=Z_{0}$
can be satisfied with a certain radius, which gives $100\%$ microwave-to-
phonon conversion efficiency. To explain the dependence of the impedance on
the radius, we consider the Butterworth-Van Dyke circuit model of the
piezomechanical resonator which consists of a static capacitance $C_{0}$ in
parallel with a motional series RLC circuit. The admittance of the circuit is
Dahmani  _et al._ (2020)

$\displaystyle Y(\omega)=$ $\displaystyle-i\omega C_{0}+\frac{1}{1/(-i\omega
C_{m})-i\omega L_{m}+R_{m}}$ $\displaystyle=$ $\displaystyle-i\omega
C_{0}+i\omega_{s}^{2}C_{m}\frac{\omega}{\omega^{2}+i\omega_{s}\omega/Q_{i}-\omega_{s}^{2}}$
$\displaystyle=$ $\displaystyle-i\omega
C_{0}+\frac{iC_{m}\omega_{s}^{2}\omega(\omega^{2}-\omega_{s}^{2})}{(\omega^{2}-\omega_{s}^{2})^{2}+\omega_{s}^{2}\omega^{2}/Q_{i}^{2}}$
$\displaystyle+\frac{\omega_{s}^{3}\omega^{2}C_{m}/Q_{i}}{(\omega^{2}-\omega_{s}^{2})^{2}+\omega_{s}^{2}\omega^{2}/Q_{i}^{2}}$
(12)

where $C_{m}$, $L_{m}$ and $R_{m}$ are the motional capacitance, inductance
and resistance. The mechancial resonance $\omega_{s}=1/\sqrt{L_{m}C_{m}}$ and
its intrinsic quality factor $Q_{i}=1/\omega_{s}R_{m}C_{m}$. Note that
$\omega_{s}$ is the resonant frequency without the presence of the IDT
electrodes, and suppose that with IDT on it, the resonant frequency is
$\omega_{f}$. This resonant frequency shift originates from the $\delta v/v$
effect and the value $(\omega_{f}-\omega_{s})/\omega_{s}$ equals
$\frac{1}{2}\delta v/v$ (as the electrodes cover only half of the ring
resonator). In the regime where the quality factor is limited by intrinsic
radiation loss instead of the $\delta v/v$ scattering, we have
$(\omega_{f}-\omega_{s})/\omega_{s}=\delta v/v\ll 1/Q_{i}$. In this case, the
admittance can be simplified to

$Y(\omega_{f})=-i\omega_{f}C_{0}+\omega_{f}C_{m}Q_{i}$ (13)

with the real part being dominant, we get the relation between the equivalent
impedance and its quality factor

$Z(\omega_{f})=\frac{1}{\omega_{f}C_{m}Q_{i}}$ (14)

We see that the equivalent impedance is inversely proportional to its quality
factor. The motional capacitance $C_{m}$ is proportional to the electrode area
like $C_{0}$ Arnau  _et al._ (2000), so it is linearly proportional to the
radius as the surface area increases while the quality factor increases
exponentially with radius. So in Fig. 11(c) and (d), the impedance’s
dependence with radius is mostly exponential, consistent with the quality
factor simulated in Fig. 11(b). The microwave-to-phonon conversion efficiency
can also be understood from coupling condition point of view where the
external coupling between the electrodes and the resonator is weak, but by
changing the resonator’s intrinsic energy decay rate, critical coupling
condition can be satisfied and all the microwave energy can be converted into
mechanical energy.

Therefore, the IDT integrated mechanical resonator can enhance microwave-
phonon conversion efficiency, boost the performances of applications that
require high excitation and collection efficiency and provide coherent
interface between phononic and superconducting circuits.

##  VI Conclusion

In conclusion, we have quantitatively studied phononic mode properties in
unsuspended strip waveguides and ring resonators. Our numerical results
demonstrate efficient confinement of phonon in the practical GaN-on-sapphire
microstructures, which provide a scalable platform of phononic integrated
circuits for acoustic signal processing and enhanced phonon-matter or phonon-
light interactions. Basic components for a phononic circuit are thoroughly
investigated, including the directional coupler made by two waveguides, quasi-
Rayleigh mode to quasi-Love mode conversion in coupled non-identical
waveguides, ring resonator, as well as microwave photon-to-phonon conversion
for efficient input and output coupling. It is worth noting that the studies
in this paper can be generalized to other frequencies by rescaling the
geometry parameters. For example, $10\,$GHz phononic waveguide modes could be
realized with a thickness of $~{}125\,$nm and a width of $500\,$nm, whose
geometry is compatible with the photonic integrated circuits and promises the
applications in optical Brillouin scattering.

Although not addressed in this paper, it is worth noting that there are
various other loss mechanisms Imboden and Mohanty (2014). Some losses don’t
exist or can be neglected in our strip waveguide SAW devices, for example, our
system doesn’t use any suspended structures so it isn’t subject to clamping or
support losses. For the circuit damping caused by drive-and-detection circuit,
we evaluate the equivalent impedance and $Q$ factor in Sec. V. Scattering
losses from crystal defects or thermal phonons are ignored, though
thermoelastic damping can be one primary limits of the $Q$ factors at room
temperature. Recently, people has done phononic band structure engineering to
reduce scattering of SAW into bulk modes Xu  _et al._ (2018); Shao  _et al._
(2019). However, scattering losses from surface roughness due to fabrication
disorder, can be critical for miniaturizing phononic systems and thus for
high-frequency application Safavi-Naeini  _et al._ (2019), since the
scattering loss coefficient has a quadratic dependency with repect to the
guided wave frequency Payne and Lacey (1994); Hughes  _et al._ (2005); Melati
_et al._ (2014), it could be the main limitation to efficient photo-phonon
interaction Safavi-Naeini  _et al._ (2019); Liu  _et al._ (2019). In addition,
scaling to smaller geometry increases surface-to-volume ratios which may give
rise to more surface loss, this loss mechanism has been found in other
micromechanical systems Yasumura  _et al._ (2000). Besides, we do not consider
material losses due to any intrinsic microscopic processes. The damping from
surrounding medium contributes to additional loss if the devices are operating
under atmosphere or viscous medium.

We believe the phononic waveguides and ring resonators studied in this paper
can be applied in future studies on phononic circuits Olsson III and El-Kady
(2009); Maldovan (2013); Hatanaka  _et al._ (2014b), gyroscopic sensors Lao
(1980); Zou  _et al._ (2016); Fu  _et al._ (2019a), integrated acoustic-optics
modulators Fuhrmann  _et al._ (2011); Tadesse and Li (2014a); Uan  _et al._
(2015); Tadesse  _et al._ (2015), circulators Fu  _et al._ (2015), integrated
delay line and data bus that communicates the quantum bits Gustafsson  _et
al._ (2014).

###### Acknowledgements.

This work was supported by DARPA/MTO’s PRIGM: AIMS program through a grant
from SPAWAR (N66001-16-1-4026). H.X.T. acknowledges funding support from Army
Research Office (W911NF-18-1-0020) and Packard Fellowship in Science and
Engineering. C.L.Z. thanks Liang Jiang for helpful discussions.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * Campbell (1998) C. Campbell,  _Surface acoustic wave devices for mobile and wireless communications_ (Academic press, 1998).
  * Lao (1980) B. Y. Lao, “Gyroscopic Effect in Surface Acoustic Waves,” 1980 Ultrason. Symp. , 687 (1980).
  * Friend and Yeo (2011) J. Friend and L. Y. Yeo, “Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics,” Rev. Mod. Phys. 83, 647 (2011).
  * Campbell (1989) C. K. Campbell, “Applications of surface acoustic and shallow bulk acoustic wave devices,” Proc. IEEE 77, 1453 (1989).
  * Mcneil  _et al._ (2011) R. P. G. Mcneil, M. Kataoka, C. J. B. Ford, C. H. W. Barnes, D. Anderson, G. A. C. Jones, I. Farrer, and D. A. Ritchie, “On-demand single-electron transfer between distant quantum dots,” Nature 477, 439 (2011).
  * Hermelin  _et al._ (2011) S. Hermelin, S. Takada, M. Yamamoto, S. Tarucha, A. D. Wieck, L. Saminadayar, C. Bäuerle, and T. Meunier, “Electrons surfing on a sound wave as a platform for quantum optics with flying electrons,” Nature 477, 435 (2011).
  * Chen  _et al._ (2015) J. C. H. Chen, Y. Sato, R. Kosaka, M. Hashisaka, K. Muraki, and T. Fujisawa, “Enhanced electron-phonon coupling for a semiconductor charge qubit in a surface phonon cavity,” Sci. Rep. 5, 15176 (2015).
  * Golter  _et al._ (2016a) D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Phys. Rev. Lett. 116, 143602 (2016a).
  * Gustafsson  _et al._ (2014) M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekstrom, G. Johansson, and P. Delsing, “Propagating phonons coupled to an artificial atom,” Science 346, 207 (2014).
  * Satzinger  _et al._ (2018) K. J. Satzinger, Y. Zhong, H.-S. Chang, G. A. Peairs, A. Bienfait, M.-H. Chou, A. Cleland, C. R. Conner, É. Dumur, J. Grebel, _et al._ , “Quantum control of surface acoustic-wave phonons,” Nature 563, 661 (2018).
  * Bienfait  _et al._ (2019) A. Bienfait, K. J. Satzinger, Y. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, É. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, _et al._ , “Phonon-mediated quantum state transfer and remote qubit entanglement,” Science 364, 368 (2019).
  * Fuhrmann  _et al._ (2011) D. A. Fuhrmann, S. M. Thon, H. Kim, D. Bouwmeester, P. M. Petroff, A. Wixforth, and H. J. Krenner, “Dynamic modulation of photonic crystal nanocavities using gigahertz acoustic phonons,” Nat. Photonics 5, 605 (2011).
  * Tadesse and Li (2014a) S. A. Tadesse and M. Li, “Sub-optical wavelength acoustic wave modulation of integrated photonic resonators at microwave frequencies.” Nat. Commun. 5, 5402 (2014a).
  * Uan  _et al._ (2015) H. L. I. Uan, S. E. A. T. Adesse, Q. I. Y. U. L. Iu, H. Li, S. A. Tadesse, Q. Liu, and M. Li, “Nanophotonic cavity optomechanics with propagating acoustic waves at frequencies up to 12 GHz,” Optica 2, 826 (2015).
  * Tadesse  _et al._ (2015) S. A. Tadesse, H. Li, Q. Liu, and M. Li, “Acousto-optic modulation of a photonic crystal nanocavity with Lamb waves in microwave K band,” Appl. Phys. Lett. 107 (2015), 10.1063/1.4935981.
  * Kurizki  _et al._ (2015) G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. 112, 3866 (2015).
  * Schuetz  _et al._ (2015) M. J. A. Schuetz, E. M. Kessler, G. Giedke, L. M. K. Vandersypen, M. D. Lukin, and J. I. Cirac, “Universal Quantum Transducers Based on Surface Acoustic Waves,” Phys. Rev. X 5, 031031 (2015).
  * Shumeiko (2016) V. S. Shumeiko, “Quantum acousto-optic transducer for superconducting qubits,” Phys. Rev. A 93, 023838 (2016).
  * Clerk  _et al._ (2020) A. Clerk, K. Lehnert, P. Bertet, J. Petta, and Y. Nakamura, “Hybrid quantum systems with circuit quantum electrodynamics,” Nature Physics , 1 (2020).
  * Manenti  _et al._ (2017) R. Manenti, A. F. Kockum, A. Patterson, T. Behrle, J. Rahamim, G. Tancredi, F. Nori, and P. J. Leek, “Circuit quantum acoustodynamics with surface acoustic waves,” Nature communications 8, 1 (2017).
  * Moores  _et al._ (2018) B. A. Moores, L. R. Sletten, J. J. Viennot, and K. Lehnert, “Cavity quantum acoustic device in the multimode strong coupling regime,” Physical review letters 120, 227701 (2018).
  * Noguchi  _et al._ (2017) A. Noguchi, R. Yamazaki, Y. Tabuchi, and Y. Nakamura, “Qubit-assisted transduction for a detection of surface acoustic waves near the quantum limit,” Physical review letters 119, 180505 (2017).
  * Yen (1972) K. H. Yen, “Broadband Efficient Excitation of the Thin-Ribbon Waveguide for Surface Acoustic Waves,” Appl. Phys. Lett. 20, 284 (1972).
  * Oliner (1976) A. Oliner, “Waveguides for acoustic surface waves: A review,” Proc. IEEE 64, 615 (1976).
  * Oliner (1978) A. A. Oliner, “Waveguides for surface waves,” in  _Acoust. Surf. Waves_ (Springer Berlin Heidelberg, 1978) pp. 187–223.
  * Knox and Owen (1970) R. Knox and D. Owen, “Distributed Components in Microwave Elastic Surface Wave Circuits,” in  _G-MTT 1970 Int. Microw. Symp._, Vol. 1 (IEEE, 1970) pp. 370–374.
  * Sandy and Parker (1976) F. Sandy and T. Parker, “Surface Acoustic Wave Ring Filter,” in  _1976 Ultrason. Symp._, Vol. 79 (IEEE, 1976) pp. 391–396.
  * Goryachev  _et al._ (2014) M. Goryachev, D. Creedon, E. Ivanov, S. Galliou, R. Bourquin, and M. Tobar, “Extremely high q-factor mechanical modes in quartz bulk acoustic wave resonators at millikelvin temperature,” in  _AIP Conference Proceedings_, Vol. 1633 (American Institute of Physics, 2014) pp. 90–92.
  * Carvalho  _et al._ (2019) N. Carvalho, J. Bourhill, M. Goryachev, S. Galliou, and M. Tobar, “Piezo-optomechanical coupling of a 3d microwave resonator to a bulk acoustic wave crystalline resonator,” Applied Physics Letters 115, 211102 (2019).
  * Biryukov  _et al._ (2007) S. V. Biryukov, G. Martin, and M. Weihnacht, “Ring waveguide resonator on surface acoustic waves,” Appl. Phys. Lett. 90, 173503 (2007).
  * Biryukov  _et al._ (2009) S. V. Biryukov, H. Schmidt, A. V. Sotnikov, M. Weihnacht, T. Y. Chemekova, and Y. N. Makarov, “Ring waveguide resonator on surface acoustic waves: First experiments,” J. Appl. Phys. 106, 130 (2009).
  * Maznev (2009) A. Maznev, “Laser-generated surface acoustic waves in a ring-shaped waveguide resonator,” Ultrasonics 49, 1 (2009).
  * Manenti  _et al._ (2016) R. Manenti, M. J. Peterer, A. Nersisyan, E. B. Magnusson, A. Patterson, and P. J. Leek, “Surface acoustic wave resonators in the quantum regime,” Phys. Rev. B 93, 041411 (2016).
  * Liu  _et al._ (2009) F. Liu, S. Peng, H. Jia, M. Ke, and Z. Liu, “Strongly localized acoustic surface waves propagating along a V-groove,” Appl. Phys. Lett. 94, 2007 (2009).
  * Boucher  _et al._ (2014) P. Boucher, S. Rauwerdink, A. Tahraoui, C. Wenger, Y. Yamamoto, and P. V. Santos, “Ring waveguides for gigahertz acoustic waves on silicon,” Appl. Phys. Lett. 105, 161904 (2014).
  * Fan  _et al._ (2016) L. Fan, C.-L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Noise-free quantum optical frequency shifting driven by mechanics,” arXiv preprint arXiv:1607.01823 (2016).
  * Mohammadi and Adibi (2011) S. Mohammadi and A. Adibi, “On chip complex signal processing devices using coupled phononic crystal slab resonators and waveguides,” AIP Advances 1, 041903 (2011).
  * Patel  _et al._ (2018) R. N. Patel, Z. Wang, W. Jiang, C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Single-mode phononic wire,” Phys. Rev. Lett. 121, 040501 (2018).
  * Fang  _et al._ (2016) K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photon. 10, 489 (2016).
  * Hatanaka  _et al._ (2014a) D. Hatanaka, I. Mahboob, K. Onomitsu, and H. Yamaguchi, “Phonon waveguides for electromechanical circuits,” Nat. Nanotech. 9, 520 (2014a).
  * Xu  _et al._ (2018) Y. Xu, W. Fu, C.-l. Zou, Z. Shen, and H. X. Tang, “High quality factor surface fabry-perot cavity of acoustic waves,” Applied Physics Letters 112, 073505 (2018).
  * Golter  _et al._ (2016b) D. A. Golter, T. Oo, M. Amezcua, K. A. Stewart, and H. Wang, “Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond,” Physical Review Letters 116, 143602 (2016b).
  * Huang  _et al._ (2010) C.-Y. Huang, J.-H. Sun, and T.-T. Wu, “A two-port ZnO/silicon Lamb wave resonator using phononic crystals,” Applied Physics Letters 97, 031913 (2010).
  * Magnusson  _et al._ (2015) E. B. Magnusson, B. H. Williams, R. Manenti, M.-S. Nam, A. Nersisyan, M. J. Peterer, A. Ardavan, and P. J. Leek, “Surface acoustic wave devices on bulk ZnO crystals at low temperature,” Applied Physics Letters 106, 063509 (2015).
  * Ung  _et al._ (2017) W. L. Ung, K. Mutafopulos, P. Spink, R. W. Rambach, T. Franke, and D. A. Weitz, “Enhanced surface acoustic wave cell sorting by 3D microfluidic-chip design,” Lab on a Chip 17, 4059 (2017).
  * Shao  _et al._ (2019) L. Shao, S. Maity, L. Zheng, L. Wu, A. Shams-Ansari, Y.-I. Sohn, E. Puma, M. Gadalla, M. Zhang, C. Wang, E. Hu, K. Lai, and M. Loncar, “Phononic Band Structure Engineering for High-Q Gigahertz Surface Acoustic Wave Resonators on Lithium Niobate,” Physical Review Applied 12, 014022 (2019).
  * Fu  _et al._ (2017a) C. Fu, A. J. Quan, J. T. Luo, H. F. Pang, Y. J. Guo, Q. Wu, W. P. Ng, X. T. Zu, and Y. Q. Fu, “Vertical jetting induced by shear horizontal leaky surface acoustic wave on 36Â° Y-X LiTaO3,” Applied Physics Letters 110, 173501 (2017a).
  * Chu  _et al._ (2017) Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart, L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, “Quantum acoustics with superconducting qubits,” Science 358, 199 (2017).
  * Aubert  _et al._ (2010) T. Aubert, O. Elmazria, B. Assouar, L. Bouvot, and M. Oudich, “Surface acoustic wave devices based on AlN/sapphire structure for high temperature applications,” Applied Physics Letters 96, 203503 (2010).
  * Tadesse and Li (2014b) S. A. Tadesse and M. Li, “Sub-optical wavelength acoustic wave modulation of integrated photonic resonators at microwave frequencies,” Nature Communications 5, 5402 (2014b).
  * Fujii  _et al._ (2013) S. Fujii, T. Odawara, H. Yamada, T. Omori, K.-y. Hashimoto, H. Torii, H. Umezawa, and S. Shikata, “Low propagation loss in a one-port SAW resonator fabricated on single-crystal diamond for super-high-frequency applications,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 60, 986 (2013).
  * Okamoto  _et al._ (2013) H. Okamoto, A. Gourgout, C.-Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nature Physics 9, 480 (2013).
  * Metcalfe  _et al._ (2010) M. Metcalfe, S. M. Carr, A. Muller, G. S. Solomon, and J. Lawall, “Resolved Sideband Emission of $\mathrm{InAs}/\mathrm{GaAs}$ Quantum Dots Strained by Surface Acoustic Waves,” Physical Review Letters 105, 037401 (2010).
  * Valle  _et al._ (2019) S. Valle, M. Singh, M. J. Cryan, M. Kuball, and K. C. Balram, “High frequency guided mode resonances in mass-loaded, thin film gallium nitride surface acoustic wave devices,” Applied Physics Letters 115, 212104 (2019).
  * Wang  _et al._ (2015) S. Wang, L. C. Popa, and D. Weinstein, “Tapered Phononic Crystal sawresonator in GaN,” in  _2015 28th IEEE International Conference on Micro Electro Mechanical Systems (MEMS)_ (2015) pp. 1028–1031.
  * Fu  _et al._ (2019a) W. Fu, Z. Shen, Y. Xu, C.-L. Zou, R. Cheng, X. Han, and H. X. Tang, “Phononic integrated circuitry and spin–orbit interaction of phonons,” Nature communications 10, 1 (2019a).
  * Morgan and Paige (2007) D. Morgan and E. Paige, “2 - acoustic waves in elastic solids,” in  _Surface Acoustic Wave Filters (Second edition)_, Studies in Electrical and Electronic Engineering, edited by D. Morgan and E. Paige (Academic Press, Oxford, 2007) second edition ed., pp. 38 – 67.
  * Pang  _et al._ (2013) H.-F. Pang, Y.-Q. Fu, Z.-J. Li, Y. Li, J.-Y. Ma, F. Placido, A. J. Walton, and X.-T. Zu, “Love mode surface acoustic wave ultraviolet sensor using ZnO films deposited on 36Â° Y-cut LiTaO3,” Sensors and Actuators A: Physical 193, 87 (2013).
  * Fu  _et al._ (2017b) Y. Q. Fu, J. K. Luo, N. T. Nguyen, A. J. Walton, A. J. Flewitt, X. T. Zu, Y. Li, G. McHale, A. Matthews, E. Iborra, H. Du, and W. I. Milne, “Advances in piezoelectric thin films for acoustic biosensors, acoustofluidics and lab-on-chip applications,” Progress in Materials Science 89, 31 (2017b).
  * Fu  _et al._ (2019b) S. Fu, W. Wang, Q. Li, Z. Lu, Z. Chen, J. Luo, J. Shen, R. Wang, C. Song, F. Zeng, and F. Pan, “High-frequency V-doped ZnO/SiC surface acoustic wave devices with enhanced electromechanical coupling coefficient,” Applied Physics Letters 114, 113504 (2019b).
  * Deger  _et al._ (1998) C. Deger, E. Born, H. Angerer, O. Ambacher, M. Stutzmann, J. Hornsteiner, E. Riha, and G. Fischerauer, “Sound velocity of AlxGa1xN thin films obtained by surface acoustic-wave measurements,” Applied Physics Letters 72, 2400 (1998).
  * Nakahata  _et al._ (2003) H. Nakahata, S. Fujii, K. Higaki, A. Hachigo, H. Kitabayashi, S. Shikata, and N. Fujimori, “Diamond-based surface acoustic wave devices,” Semiconductor Science and Technology 18, S96 (2003).
  * Pohl  _et al._ (2002) R. O. Pohl, X. Liu, and E. Thompson, “Low-temperature thermal conductivity and acoustic attenuation in amorphous solids,” Reviews of Modern Physics 74, 991 (2002).
  * Azuhata  _et al._ (2003) T. Azuhata, M. Takesada, T. Yagi, A. Shikanai, S. Chichibu, K. Torii, A. Nakamura, T. Sota, G. Cantwell, D. B. Eason, and C. W. Litton, “Brillouin scattering study of ZnO,” Journal of Applied Physics 94, 968 (2003).
  * Warner  _et al._ (1967) A. W. Warner, M. Onoe, and G. A. Coquin, “Determination of Elastic and Piezoelectric Constants for Crystals in Class (3m),” The Journal of the Acoustical Society of America 42, 1223 (1967).
  * Strauch (2011) D. Strauch, “Gan: Poisson ratio, young’s modulus, bulk modulus,” in  _New Data and Updates for IV-IV, III-V, II-VI and I-VII Compounds, their Mixed Crystals and Diluted Magnetic Semiconductors_, edited by U. Rössler (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 409–412.
  * Flannery  _et al._ (2003) C. M. Flannery, M. D. Whitfield, and R. B. Jackman, “Acoustic wave properties of CVD diamond,” Semiconductor Science and Technology 18, S86 (2003).
  * Auld (1990) B. A. Auld,  _Acoustic fields and waves in solids_ , Vol. 2 (Krieger, 1990).
  * Hopcroft  _et al._ (2010) M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?” Journal of Microelectromechanical Systems 19, 229 (2010).
  * Hughes (1972) A. J. Hughes, “Elastic Surface Wave Guidance by ($\Delta v/v$) Effect Guidance Structures,” J. Appl. Phys. 43, 2569 (1972).
  * Shen  _et al._ (2017) Z. Shen, X. Han, C.-L. Zou, and H. X. Tang, “Phase sensitive imaging of 10 ghz vibrations in an aln microdisk resonator,” Review of Scientific Instruments 88, 123709 (2017).
  * Tiersten (1969) H. Tiersten, “Elastic surface waves guided by thin films,” Journal of Applied Physics 40, 770 (1969).
  * Li  _et al._ (1977) R. Li, A. Oliner, and H. Bertoni, “Microwave network analyses of surface acoustic waveguides. i-flat overlay guides,” IEEE Transactions on Sonics Ultrasonics 24, 66 (1977).
  * Markman  _et al._ (1977) S. Markman, R. C. Li, A. Oliner, and H. Bertoni, “Microwave network analyses of surface acoustic waveguides. ii-rectangular ridge guides,” IEEE Transactions on Sonics Ultrasonics 24, 79 (1977).
  * Love (1911) A. Love, “Some problems of geodynamics,” Some Problems of Geodynamics Publisher: Cambridge University Press, Cambridge (1911).
  * Bougrov  _et al._ (2001) V. Bougrov, M. Levinshtein, S. Rumyantsev, and A. Zubrilov,  _Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe_ (John Wiley & Sons, Inc., New York, 2001) pp. 1–30.
  * (77) “MolTech GmbH website sapphire al2o3,” http://www.mt-berlin.com/frames_cryst/descriptions/sapphire.htm, accessed: 2016-07-17.
  * Rayleigh (1885) L. Rayleigh, “On waves propagated along the plane surface of an elastic solid,” Proceedings of the London Mathematical Society 1, 4 (1885).
  * Hill  _et al._ (2007) M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. De Vries, P. J. Van Veldhoven, F. W. Van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. De Waardt, _et al._ , “Lasing in metallic-coated nanocavities,” Nature Photonics 1, 589 (2007).
  * Oulton  _et al._ (2008) R. Oulton, G. Bartal, D. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New Journal of Physics 10, 105018 (2008).
  * Lagasse  _et al._ (1973a) P. E. Lagasse, I. M. Mason, and E. A. Ash, “Acoustic surface waveguides—analysis and assessment,” IEEE Transactions on Microwave Theory and Techniques 21, 225 (1973a).
  * Reddy (2006) J. N. Reddy,  _Theory and analysis of elastic plates and shells_ (CRC press, 2006).
  * Sinha and Tiersten (1973) B. Sinha and H. Tiersten, “Elastic and piezoelectric surface waves guided by thin films,” Journal of Applied Physics 44, 4831 (1973).
  * Lagasse  _et al._ (1973b) P. E. Lagasse, I. M. Mason, and E. A. Ash, “Acoustic surface waveguides—analysis and assessment,” IEEE Transactions on Microwave Theory and Techniques 21, 225 (1973b).
  * Rayleigh (1910) L. Rayleigh, “Cxii. the problem of the whispering gallery,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 20, 1001 (1910).
  * Rayleigh (1914) L. Rayleigh, “Ix. further applications of bessel’s functions of high order to the whispering gallery and allied problems,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 27, 100 (1914).
  * Oraevsky (2002) A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electronics 32, 377 (2002).
  * Matsko and Ilchenko (2006) A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering gallery modes i: basics,” IEEE J. Sel. Top. Quantum Electron 12, 3 (2006).
  * Ilchenko and Matsko (2006) V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes-part ii: applications,” IEEE Journal of selected topics in quantum electronics 12, 15 (2006).
  * Foreman  _et al._ (2015) M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode sensors,” Advances in optics and photonics 7, 168 (2015).
  * Kippenberg (2004) T. J. A. Kippenberg,  _Nonlinear optics in ultra-high-Q whispering-gallery optical microcavities_ , Ph.D. thesis, California Institute of Technology (2004).
  * Hsu  _et al._ (2016) C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nature Reviews Materials 1, 1 (2016).
  * Chen  _et al._ (2016) Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New Journal of Physics 18, 063031 (2016).
  * Yariv (1973) A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE Journal of Quantum Electronics 9, 919 (1973).
  * Hardy and Streifer (1985) A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” Journal of lightwave technology 3, 1135 (1985).
  * Haus  _et al._ (1987) H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” Journal of Lightwave Technology 5, 16 (1987).
  * Shen  _et al._ (2019) Z. Shen, W. Fu, R. Cheng, H. Townley, C.-L. Zou, and H. X. Tang, “Polarization mode hybridization and conversion in phononic wire waveguides,” Applied Physics Letters 115, 201901 (2019).
  * Dahmani  _et al._ (2020) Y. D. Dahmani, C. J. Sarabalis, W. Jiang, F. M. Mayor, and A. H. Safavi-Naeini, “Piezoelectric transduction of a wavelength-scale mechanical waveguide,” Physical Review Applied 13, 024069 (2020).
  * Arnau  _et al._ (2000) A. Arnau, T. Sogorb, and Y. Jiménez, “A continuous motional series resonant frequency monitoring circuit and a new method of determining butterworth–van dyke parameters of a quartz crystal microbalance in fluid media,” Review of scientific instruments 71, 2563 (2000).
  * Imboden and Mohanty (2014) M. Imboden and P. Mohanty, “Dissipation in nanoelectromechanical systems,” Physics Reports Dissipation in nano-electromechanical systems, 534, 89 (2014).
  * Safavi-Naeini  _et al._ (2019) A. H. Safavi-Naeini, D. V. Thourhout, R. Baets, and R. V. Laer, “Controlling phonons and photons at the wavelength scale: integrated photonics meets integrated phononics,” Optica 6, 213 (2019).
  * Payne and Lacey (1994) F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Optical and Quantum Electronics 26, 977 (1994).
  * Hughes  _et al._ (2005) S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity,” Physical Review Letters 94, 033903 (2005).
  * Melati  _et al._ (2014) D. Melati, A. Melloni, and F. Morichetti, “Real photonic waveguides: guiding light through imperfections,” Advances in Optics and Photonics 6, 156 (2014).
  * Liu  _et al._ (2019) Q. Liu, H. Li, and M. Li, “Electromechanical Brillouin scattering in integrated optomechanical waveguides,” Optica 6, 778 (2019).
  * Yasumura  _et al._ (2000) K. Yasumura, T. Stowe, E. Chow, T. Pfafman, T. Kenny, B. Stipe, and D. Rugar, “Quality factors in micron- and submicron-thick cantilevers,” Journal of Microelectromechanical Systems 9, 117 (2000).
  * Olsson III and El-Kady (2009) R. H. Olsson III and I. El-Kady, “Microfabricated phononic crystal devices and applications,” Meas. Sci. Technol. 20, 012002 (2009).
  * Maldovan (2013) M. Maldovan, “Sound and heat revolutions in phononics,” Nature 503, 209 (2013).
  * Hatanaka  _et al._ (2014b) D. Hatanaka, I. Mahboob, K. Onomitsu, and H. Yamaguchi, “Phonon waveguides for electromechanical circuits,” Nat. Nanotechnol. 9, 520 (2014b).
  * Zou  _et al._ (2016) C.-L. Zou, W. Fu, and H. X. Tang, “Surface acoustic gyroscope,” In Preparation (2016).
  * Fu  _et al._ (2015) W. Fu, F.-J. Shu, Y.-L. Zhang, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Integrated optical circulator by stimulated Brillouin scattering induced non-reciprocal phase shift,” Opt. Express 23, 025118 (2015).