signal of 60.0 MHz. The error signal is applied directly on the AlN actuator,
such that the actuation on the microresonator stabilizes the soliton
repetition rate to the EO comb line spacing. Figure 28(b, c) compare the beat
signal and the phase noise in the cases of free-running and locked states, in
comparison with the phase noise of the 60.0 MHz reference microwave. A 600 kHz
locking bandwidth is determined from the merging point of the phase noise of
beat signal and 60 MHz reference. Note that the 600 kHz bandwidth is larger
than conventional lasers where the piezo response is typically limited to few
kilohertz, similar to integrated heaters.

Figure 29: Out-of-loop beat signal characterization using a modified setup.
(a) Modified experimental setup to characterize the _out-of-loop_ phase noise
of the beat signal between the $-1^{\text{st}}$ line of the microcomb and the
$-13^{\text{th}}$ line of the EO comb. (b) Schematic of referencing the
microcomb to the EO comb. The beatnote (in-loop) between the $+1^{\text{st}}$
line of the microcomb and $+13^{\text{th}}$ line of the EO comb is detected
and referenced to a 20.0 MHz microwave signal. The beatnote (out-of-loop)
between the $-1^{\text{st}}$ line of the microcomb and $-13^{\text{th}}$ line
of the EO comb is characterized. (c) Comparison of SSB phase noises measured
in different cases. Reprinted from Ref. [276].

The above experiment only demonstrates the stabilization of one comb line that
is involved in the feedback loop. To show the stability of the entire comb,
the out-of-loop comb needs to be characterized. To measure the _out-of-loop_
beat signal and its phase noise, a modified setup is used as shown in Fig.
29(a). The pump laser’s frequency to generate the soliton microcomb is shifted
by 77.0 MHz via a fibre-coupled acousto-optic modulator (AOM). The reason to
shift the microcomb’s pump frequency is to cancel out the drift and noise
caused by the imbalanced paths in delayed self-homodyne measurement, by
detecting the beatnote (77.0 MHz shift) between the pump lines of the
microcomb and the EO comb. By down-mixing the 77.0 MHz heterodyne beatnote
signal using the same microwave source that drives the AOM, the feedback
signal is applied to the laser current such that the pump laser’s frequency is
stabilized and the noise in the delayed self-homodyne measurement is removed.

Then, the beatnote between the $+1^{\text{st}}$ line of the microcomb and
$+13^{\text{th}}$ line of the EO comb is detected and referenced to a 20.0 MHz
microwave signal, in order to stabilize the microcomb repetition rate. The
entire schematic of referencing the microcomb to the EO comb is shown in Fig.
29(b). Figure 29(c) compares the single-sideband (SSB) phase noise for in-loop
(red) and out-of-loop (blue) beat signals when in the locked (solid) and free-
running (dashed) states. The solid green line is the phase noise between the
EO comb’s pump and the microcomb’s pump (shifted by 77 MHz), when the pump
laser’s frequency is locked, A reduction in the phase noise of the out-of-loop
beat signal is observed with the AlN actuation, which demonstrates the
stability of the repetition rate. The locking bandwidth in this case is $>300$
kHz.

Figure 30: Comparison of measured relative Allan deviation for the microcomb
and EO comb beat signals in different cases, to evaluate the long-term
stability of the locked system. Reprinted from Ref. [276].

To further evaluate the long-term stability of the locked system, frequency
counting measurements of the relative Allan deviations are performed, as shown
in Fig. 30. The 77 MHz microwave source (used to lock the pump laser’s
frequency) is referenced to the 20 MHz microwave oscillator (used to down-mix
the in-loop beat signal to derive the error signal). Similarly, the microwave
source driving the EOMs (14.6974 GHz) for EO comb generation is also
referenced to the same 20 MHz microwave oscillator. The relative Allan
deviation of the beat signals between the free-running microcomb and EO comb
do not converging, while the beat signals between the locked pump lines of the
microcomb and the EO comb re at the level of 10-2 with 1 s averaging time.
After locking the soliton repetition rate to the EO comb by actuating on AlN,
the _in-loop_ and the _out-of-loop_ beat signals show similar frequency
stability.

Figure 31:  On-chip generation of PDH error signals using the HBAR modes
induced by the AlN actuation. (a) Experimental setup. LPF: low-pass filter.
Amp.: RF power amplifier. (b) The measured $S_{21}(\omega)$ response of the
AlN actuator in the linear frequency scale. Both cases, when the laser is on-
and off-resonance, are measured. (c) The PDH error signals modulated at the
HBAR frequencies marked with stars in (b). Reprinted from Ref. [54].

In addition to the quasi-DC tuning, the HBAR modes of the AlN actuator provide
a novel functionality for error signal generation using the Pound-Drever-Hall
(PDH) technique. Figure 31(a) shows the experimental setup to generate PDH
error signals using HBAR modes. The measured $S_{21}(\omega)$ response of the
AlN actuation, up to 400 MHz, is plotted in the linear frequency scale in Fig.
31(b), showing both cases when the laser is on- and off-resonance. Different
modulation frequencies corresponding to different HBAR modes are investigated,
which are marked with stars in Fig. 31(b). The PDH error signals modulated at
these HBAR frequencies are shown in Fig. 31(c), as well as the corresponding
microresonator’s resonance. A microwave source providing $\sim$8 dBm RF power
is used to modulate the Si3N4 microresonator via AlN actuation. The decrease
in error signal contrast at higher HBAR frequency is caused by the finite
bandwidth of the optical resonance. This PDH signal can be used to stabilize
the laser frequency referenced to an on-chip high optical Q Si3N4
microresonator.

####  4.2.3 Soliton-Based Parallel LiDAR Engine

The AlN actuator can be a key component for a soliton-based, parallel,
frequency-modulated continuous-wave (FMCW) LiDAR [38]. This scheme
conventionally relies on scanning the pump laser’s frequency $\nu_{\text{L}}$
over the soliton existence range $\Delta\nu_{\text{s}}=\nu_{2}-\nu_{1}$, i.e.
$\nu_{\text{c}}-\nu_{\text{L}}\in[\nu_{1},\nu_{2}]$, with $\nu_{\text{c}}$
being the resonance frequency, $\nu_{1}$ and $\nu_{2}$ the boundary of soliton
existence detuning range. As a result, the pump frequency chirp is transferred
to all soliton comb lines, as shown in Fig. 32(a). The diffractive optics
subsequently separates the comb lines spatially, which allows high-speed and
parallel acquisition of both velocity and position in each pixel. However, the
varying soliton detuning has several limitations. First, the soliton detuning
variation leads to changes in soliton spectrum bandwidth and power, which
limits the number of usable optical channels. Second, the Raman self-frequency
shift [277, 278] causes a varied soliton repetition rate. Third, GHz frequency
excursion of the pump laser ($\Delta\nu_{\text{L}}$) requires GHz-wide soliton
existence range ($\Delta\nu_{\text{s}}$), which leads to elevated pump power
to several Watts.

Figure 32: Schematic and experimental setup of hybrid AlN-Si3N4 soliton LiDAR
engine. (a) Schematic of soliton-based parallel FMCW LiDAR. A chirped pump
translates modulation to other comb lines, with the same modulation rate and
frequency excursion. (b) Experimental setup for synchronous scan of the laser
frequency and the microresonator resonance, using the feed-forward scheme.
VCO: voltage-controlled oscillator. AFG: arbitrary waveform generator. QPSK:
quadrature phase shift keying. DSO: digital storage oscilloscope. Reprinted
from Ref. [54].

These limitations can be overcome by using the AlN actuation on the
microresonator to synchronously modulate the microresonator resonance
$\nu_{\text{c}}$ with the pump frequency $\nu_{\text{L}}$, while maintaining a
constant soliton detuning $\nu_{\text{c}}-\nu_{\text{L}}$. This scheme is
called “feed-forward scan” [54]. Figure 32(b) shows the experimental setup to
synchronously scan the microresonator and the pump laser, where a triangular
signal of frequency $f_{\text{mod}}$ is applied on both the pump laser and the
AlN actuator. A single-sideband modulator driven by a voltage-controlled
oscillator (VCO) is used to fast scan the laser frequency. The ramp signal
sent to the AlN actuator is amplified by the HVA with $\times 50$ voltage
amplification and 3-dB bandwidth of $\sim 5$ MHz. The synchronization of the
laser frequency and the microresonator resonance is performed by adjusting the
amplitude and the phase of the ramp signal applied on the VCO. A PDH lock can
further improve the synchronization by locking the resonance to the laser with
a constant detuning [279]. A reference laser is used to probe the chirp of
different comb lines (the pump line, $\pm 10^{th}$ comb lines etc). A fast
oscilloscope of 2 GHz bandwidth and 5 GSamples/s is used to capture the
heterodyne beatnote detected on the fast photodiode, for further off-line data
processing such as fast Fourier transform and fitting triangular signal.

Figure 33: Results of hybrid AlN-Si3N4 soliton LiDAR engine. (a) Comparison of
two schemes for soliton-based FMCW LiDAR. Left: The scheme employing soliton
detuning scan, with a fixed microresonator resonance $\nu_{\text{c}}$, while
the pump $\nu_{\text{L}}$ chirping within the soliton existence range (green
shaded region). The soliton spectrum varies with the soliton detuning, as
shown in the bottom panel. Right: Feed-forward scheme employing synchronous
modulation of the pump frequency and the resonance, such that the soliton
detuning $\Delta\nu$ is constant, resulting in a constant soliton spectrum.
Note that in this scheme, the pump frequency excursion can be significantly
larger than the soliton existence range. (b) Comparison of both schemes for
the soliton response on VNA. The soliton detuning scan shows a varying
C-resonance, while the resonance in the feed-forward scheme is fixed. (c)
Comparison of both schemes for the integrated soliton pulse power. The soliton
detuning scan shows a varying soliton pulse power, while the feed-forward
scheme shows a constant power. (d) Comparison of the frequency excursions of
different soliton comb lines, using different approaches. (e) Time-frequency
diagram of different comb lines chirping at 10 kHz modulation frequency, with
frequency excursion of 2.04 GHz. Reprinted from Ref. [276].

The comparison between the two schemes, i.e. (a) scanning the pump laser
within the soliton existence range, and (b) feed-forward scheme, is
experimentally studied in Figure 33. Figure 33(a-c) illustrate the differences
in soliton spectrum, soliton detuning, and integrated soliton pulse power in
both schemes, showing the advantages of the feed-forward scheme using the AlN
actuator. First, the pump frequency chirp $\Delta\nu_{\text{L}}$ is no longer
limited by the soliton existence range $\Delta\nu_{\text{s}}$, which allows
operating the soliton with tens of milliwatt power, compatible with the state-
of-art integrated laser [3]. Second, the soliton spectrum, repetition rate,
and power are maintained constant and not affected by the pump frequency
chirp. Figure 33(d) compares the frequency excursions of different soliton
comb line generated with different approaches. In the scheme of soliton
detuning scan at frequency $f_{\text{mod}}=10$ kHz (blue), the frequency
excursion depends on the comb line number, resulting from the soliton
repetition rate variation due to the change of soliton detuning. The feed-
forward scheme with modulation frequency $f_{\text{mod}}=10$ kHz and 100 kHz
(red, solid lines) allows frequency excursion of $\sim$2 GHz, which is more
than 5 times larger than the soliton existence range ($\sim$400 MHz).
Moreover, the variation of frequency excursion among comb lines is largely
suppressed. Figure 33(e) shows the time-frequency diagram of different comb
lines chirped at 10 kHz modulation frequency, with frequency excursion of 2.04
GHz.

Although the HBAR and mechanical modes of the chip are unwanted in linear and
fast tuning, they enable modulation with higher frequency and efficiency (red
dashed lines in Fig. 33(d)). By applying modulation frequency
($f_{\text{mod}}=18.308$ MHz) that coincides with the fundamental HBAR mode
and a contour mode, frequency excursion of 1.6 GHz is generated without HVA.
The voltage applied on the AlN is only 7.48 V, which is compatible with CMOS
circuits. Additionally, this modulation corresponds to an equivalent tuning
speed of 58.6 PHz/s. The corresponding resonance tuning efficiency is
$\delta\nu/\delta V=219$ MHz/V, which is 14 times larger than the DC tuning of
$\delta\nu/\delta V=15.7$ MHz/V. Applying modulation with
$f_{\text{mod}}=18.322$ MHz, which coincides with the fundamental HBAR mode
but not the contour mode, the tuning efficiency is reduced to
$\delta\nu/\delta V=133$ MHz/V. Therefore, it will be of great interest in the
future to optimize the design of the AlN actuator that is compatible with high
frequency triangular wave modulation. For instance, the higher-order HBAR
modes can be designed to match with the higher harmonics of the triangular
wave, such that the modulation signal can be linearly transduced onto the
optical microresonator.

In conclusion, integrated piezoelectric actuators that are linear, fast, non-
absorptive, bidirectional, and consume ultralow electric power endow
integrated soliton microcombs with novel capability in power-critical
applications, e.g., in space, data centers and portable atomic clocks, in
extreme environments such as cryogenic temperatures, or in emerging
applications for coherent LiDAR [38], frequency synthesizers [34] and RF
photonics [280, 281]. Therefore, this novel capability is important for tuning
and stabilization of soliton microcombs, and may allow soliton injection
locking for stabilization and microwave synthesis [274, 282]. While
polycrystalline AlN is used in current works, the operation voltage can be
reduced by more than 2 times using scandium-doped AlN [283]. In the future,
the co-integration of CMOS electronic circuitry on a close-by die paves the
way for packaged soliton microcombs with rapid electronic actuation [260].

##  5 Microwave-Frequency Acousto-Optic Modulation

While piezoelectric actuators have found wide applications in tuning of
integrated photonic devices, fast actuation at microwave frequencies will
excite acoustic waves that enable efficient acousto-optic modulation (AOM). At
the acoustic resonance, integrated AOMs have been demonstrated with efficiency
comparable to or even higher than state-of-the-art electro-optic modulators,
albeit at the expense of narrow bandwidth [15, 14, 12, 284, 285].
Nevertheless, it has been applied not only in classical applications, but also
in quantum regime working as quantum transducers between microwave and light
[189, 286]. In the following, we will review recent progress on chip-scale
AOMs that can be classified into three categories in terms of the acoustic
mode being utilized: surface acoustic wave (SAW), bulk acoustic wave (BAW),
and optomechanical crystal (OMC).

###  5.1 Surface Acoustic Wave Based AOM

The surface acoustic wave excited by interdigital transducer (IDT) has been
one of the traditional ways of launching acoustic waves on a chip [287], and
has become the key technology across a wide range of research areas, such as
microfluidics [288], biosensors [289], radio-frequency filters and delay lines
[290, 291], and quantum acoustics [292]. As nano-fabrication advances, the
ability to create sub-micron electrode fingers has extended the frequency of
the acoustic wave to beyond 10 GHz [293, 133]. By interacting with an optical
cavity, the stress and strain field of the acoustic wave can be used to
modulate the optical refractive index, achieving the chip-scale AOM.

Figure 34:  SAW AOM demonstrated with unreleased structures. (a) Schematic of
the SAW AOM with the optical waveguide made from the piezoelectric material.
Examples with the structure in (a) have been demonstrated for (b) an AlN
racetrack resonator and (c) an AlN photonic crystal (PhC) resonator. (d) and
(g) are schematics of the SAW AOMs with a hybrid optical waveguide made from a
different optical material. While the waveguide is embedded in a cladding in
(d), the waveguide is bonded on top of the piezoelectric layer in (g). (e) and
(f) are examples for the structure in (d) where a Si Mach-Zehnder
interferometer (MZI) is modulated by (e) AlN and (f) PZT SAW. (h) is the
example of (g), where a SAW excited on LiNbO3 modulates the waveguide formed
by polymer. The middle inset is the zoom-in near the IDT (yellow) and the
polymer waveguide (green). The lower inset shows the simulated fundamental and
higher order SAW Rayleigh modes. (b) reprinted from Ref. [14]. (c) reprinted
from Ref. [133]. (e) reprinted from Ref. [135]. (f) reprinted from Ref. [294].
(h) reprinted from Ref. [138].

One advantage of SAW is that the fabrication is relatively simple and easy,
where the IDT electrodes can be made by directly depositing and patterning
metal on top of the released or unreleased piezoelectric films. For the
unreleased SAW AOM demonstrated to date, there are mainly three typical
structures, as shown in Fig. 34. Since the SAW propagates in plane, the
optical waveguide needs to be brought as close as possible to the plane of
SAW. The most straightforward and widely adopted way is to etch the
piezoelectric film to form the optical waveguide, as shown in Fig. 34(a). It
has been realized on LiNbO3 [164, 139], AlN [14, 133], and AlGaAs [295]
platforms. Among them the optical resonator is mostly a racetrack with the
long straight side along the IDT fingers such that the SAW propagates
perpendicular to the waveguide [see Fig. 34(b)]. In the design shown in Fig.
34(b), only one straight segment is close to the IDT while the other side is
$\Delta L$ away. When $\Delta L$ is longer than the attenuation length of the
SAW, the acoustic energy reaching the other side of the racetrack will be
largely suppressed. One solution is to place the IDT inside the racetrack
which, however, poses challenges on the electrode routing without compromising
the optical $Q$. The other strategy is to shorten $\Delta L$ without
introducing significant bending loss of the bending part. One should also
notice that $\Delta L$ should be an integer number of the acoustic wave’s
wavelength such that modulation on both sides of the racetrack can be
constructively interfered [14]. To better confine acoustic energy inside the
racetrack, acoustic reflectors can be placed on both sides of the device to
form an acoustic cavity [139]. On the other hand, the AOM of an 1-D AlN
photonic crystal resonator has been demonstrated with the crystal along the
IDT [see Fig. 34(c)] [133]. In this case, the optical mode is confined in a
small volume and can have a good overlap with the acoustic wave.

The drawback of using piezoelectric materials as the optical waveguide is that
it requires etching of the piezoelectric film which is still challenging so
far, and only moderate optical $Q$ can be obtained. Moreover, the waveguide
cannot be cladded, which is prone to perturbation from the environment. One
should also notice that the piezoelectric material also possesses electro-
optic effect, such that the strain can induce electric field across the
waveguide via the piezoelectricity which in turn changes its refractive index
through the electro-optic effect [15]. This may counteract the optomechanical
interaction if not designed correctly. To mitigate these problems, a hybrid
optical waveguide can be implemented. As mentioned above, the Si and Si3N4
photonic integrated circuits have evolved into a mature platform with low
optical losses and compatible with foundry processes [296]. As shown in Fig.
34(d), the waveguide is entirely embedded in the optical cladding (which is
usually SiO2), and the piezoelectric film is deposited on top. Depending on
the thickness of the top cladding, the piezoelectric layer can be either fully
separated from or partially overlapped with the optical mode. However, since
the SAW is primarily confined at the surface, it decays exponentially into the
substrate, as can be seen in Fig. 34(h). In that sense, the waveguide needs to
be brought as close as possible to the surface to have a good acousto-optic
overlap. The SAW AOM of the Si MZI has been demonstrated on both AlN [135] and
PZT [294] platforms, as shown in Fig. 34(e-f). The modulation of a hybrid
LiNbO3 on Si3N4 waveguide has also been achieved recently [297].

Another hybrid waveguide that can be designed is to place the waveguide above
the piezoelectric film, as shown in Fig. 34(g). Waveguide material may have a
refractive index either higher or lower than the piezoelectric film. While the
former case confines the light via total internal reflection [298], the latter
forms the so-called bound states in the continuum (BIC) [138]. The fabrication
of hybrid waveguides in this way is not trivial, as waveguide etching can
rough the top surface of the piezoelectric layer, reducing the SAW quality.
Despite of that, efficient AOM has been demonstrated on hybrid chalcogenide-
LiNbO3 racetrack resonator [298]. For the work shown in Fig. 34(h), the
waveguides are manufactured by patterning an electron-beam resist onto the
piezoelectric layer without etching [138]. Another solution is to use the
pick-and-place method, which will be shown later.

Figure 35:  SAW AOM demonstrated on released structures. (a) Schematic of the
AOM with the optical waveguide made from piezoelectric materials. (b) AOM of a
LiNbO3 optical racetrack resonator with the structure as in (a). The lower
inset is the false colored SEM of the region in the red dashed box. (c)
Characterization of the device in (b), showing the acousto-optic S21 response
(blue) and the microwave reflection spectrum S11 (yellow). (d) Schematic of
the hybrid optical waveguide that is on top of the piezoelectric layer. (e)
False-colored SEM of the device that modulates the Si PhC (blue) via the SAW
on the released LiNbO3 membrane. The Si waveguide is fabricated using the
pick-and-place technique. (f) Simulated Lamb wave excited by the IDT. (g)
Acousto-optic response (blue) and microwave reflection (red) of the device in
(e). (h) Schematic of the ridge waveguide that is separated from the
piezoelectric material. (i) Optical image of the device that modulates the Si
ridge waveguide using AlN SAW. (j) Modulation spectra showing a single
sideband modulation with bandwidth of 1.7 nm. Red and purple curves are the
optical power in the Stokes and anti-Stokes sidebands, respectively. (b-c)
reprinted from Ref. [15]. (e-g) reprinted from Ref. [299]. (i-j) reprinted
from Ref. [300].

Although unreleased structures are straightforward to deal with, there is a
trade-off of choosing the substrate that confines both the acoustic and
optical modes tightly. For most of the examples shown above, the device sits
on a SiO2 layer for the sake of confining the optical mode. However, SiO2 has
an acoustic velocity close to or smaller than the piezoelectric layer, which
leads to a low acoustic index contrast, such that the acoustic wave partially
leaks into the substrate [111, 166]. Therefore, to efficiently excite the SAW
and maintain a high $Q$, the substrate should not only have a low optical
refractive index, but a high acoustic velocity (equivalently a low acoustic
index). Although it is non-trivial to find a material that meets these
requirements, sapphire has attracted much attention due to its low refractive
index of 1.75 and high longitudinal and transverse acoustic velocities of
10685 m/s and 5796 m/s, respectively [301]. Tight acoustic mode confinement in
a phononic waveguide has been demonstrated with a variety of piezoelectric
materials, such as LiNbO3 [302], AlN [303], and GaN [301, 304]. With further
investigation of the optical waveguide design in these platforms, an
optomechanical integrated circuit can be realized, which confines both
mechanical and optical modes in the same waveguide [305].

An alternative way to confine the acoustic mode is by releasing the
piezoelectric layer at the cost of a complicated fabrication process. As shown
in Fig. 35, the piezoelectric thin film is suspended by isotropic etching of
the substrate through opening holes. Similar as the unreleased case, both
single and hybrid optical waveguides have been explored to date, with the
former being the most widely adopted and demonstrated on AlN [132, 293],
LiNbO3 [136, 15], and GaAs [306]. An example of the AOM of a LiNbO3 racetrack
resonator is shown in Fig. 35(b), where a ridge waveguide is formed by
partially etching the LiNbO3. Two pairs of IDT fingers are deposited on the
remaining LiNbO3 slabs. The opening slots on both sides serve as acoustic
reflectors that form an acoustic cavity. The waveguide should be placed at the
antinode of the acoustic wave in the cavity to maximize efficiency. The
modulation of a hybrid Si-on-LiNbO3 1-D photonic crystal resonator is
illustrated in Fig. 35(e) [299]. The Si PhC is fabricated on a separate chip
and attached to the piezoelectric layer using the pick-and-place technique
[307, 308]. The Lamb acoustic wave is excited by the IDT as simulated in Fig.
35(f), which is the most commonly used mode in released SAW AOMs. In contrast
to the racetrack resonator where only a small portion of the resonator is
modulated, the PhC confines the optical mode in a small volume where the whole
resonator is inside the Lamb wave. The AOM of a Si ridge waveguide has also
been demonstrated as shown in Fig. 35(h-j) [300]. The optical waveguide is
spatially separated from the piezoelectric actuator and is modulated by the
SAW propagating in the Si slab. These different structures provide us the
possibilities for exploring new materials and novel applications, aiming at
high modulation efficiency, low cost, and small size.

Table 5:  Comparison among the state-of-the-art integrated acousto-optic modulation based on surface acoustic waves. MZI: Mach-Zehnder interferometer. PhC: photonic crystal. WG: waveguide. $\gamma_{m,\text{ex}}/\gamma_{m}$ represents external coupling efficiency of the SAW. The optical loss of the optical resonator is characterized by its linewidth $\kappa_{o}$ in unit of GHz, while that for MZI (or WG) is by its propagation loss $\alpha$ in unit of dB/cm. The modulation efficiency is characterized by $V_{\pi}$ for optical resonators in unit of V, and by $V_{\pi}L$ for MZI (or WG) in unit of V$\cdot$cm. Ref. |  Year |  Structure |  Released |  Material platform |  $\Omega_{m}/2\pi$ (GHz) |  $\gamma_{m}/2\pi$ (MHz) |  $\gamma_{m,\text{ex}}/\gamma_{m}$ |  Optical loss ($\kappa_{o}/2\pi$ or $\alpha$) |  $V_{\pi}$ or $V_{\pi}L$
---|---|---|---|---|---|---|---|---|---
[15] |  2019 |  Racetrack |  Yes |  LiNbO3 |  2 |  1.28 |  0.17 |  0.095 GHz |  0.113 V
[299] |  2021 |  PhC |  Yes |  LiNbO3+Si |  1.7 |  12 |  0.015 |  6.6 GHz |  0.05 V
[298] |  2022 |  Racetrack |  No |  LiNbO3+ChG |  0.843 |  2.63 |  0.48 |  0.387 GHz |  1.74 V
[306] |  2022 |  Ring |  Yes |  GaAs |  1.94 |  6.07 |  0.013 |  9.74 GHz |  0.21 V
[295] |  2013 |  MZI |  No |  AlGaAs |  0.52 |  – |  – |  – |  0.065 V$\cdot$cm
[139] |  2019 |  MZI |  No |  LiNbO3 |  0.112 |  0.062 |  0.118 |  0.7 dB/cm |  5 V$\cdot$cm
[136] |  2021 |  PhC WG |  Yes |  LiNbO3 |  1.16 |  0.48 |  0.146 |  330 dB/cm |  0.019 V$\cdot$cm
[135] |  2021 |  MZI |  No |  AlN+Si |  3.11 |  20 |  – |  0.15 dB/cm |  1.8 V$\cdot$cm
[294] |  2022 |  MZI |  No |  PZT+Si |  2 |  – |  – |  – |  3.6 V$\cdot$cm

The AOM devices are generally characterized by measuring the microwave
reflection S11 and acousto-optic transduction S21 as shown in Fig. 35(c) and
(g). The S11 takes the ratio of the magnitude between the input and the
reflected microwave from the IDT, where the acoustic resonances can be found
as dips in the spectrum. The impedance of the acoustic resonator $Z_{11}$ can
be inferred from the S11 as:

$S_{11}=\frac{Z_{11}-Z_{0}}{Z_{11}+Z_{0}}$ (23)

where $Z_{0}$ is the impedance of the microwave cable, which is 50 $\Omega$ in
most cases. The S21 is measured by biasing the laser near the optical
resonance and detecting the light intensity and phase modulation through a
photodetector. S21 is then the magnitude ratio of the microwave from the
photodetector and the input microwave drive. Both S11 and S21 can be easily
measured using a commercial vector network analyzer at the same time. This
measurement can help us to obtain most of the key parameters, such as the
frequency and linewidth of the acoustic resonance, the electromechanical
coupling strength, as well as the microwave to optical transduction
efficiency.

One of the important figures of merit for traditional electro-optic modulators
is $V_{\pi}$, the voltage that induces $\pi$ phase shift of the light in the
waveguide. Similarly, an effective $V_{\pi}$ can be defined for the AOM of an
optical resonator. To do that, a modulation index $\beta$ can be defined as
[309, 284]:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{2g_{0}\sqrt{n_{\text{phon}}}}{\Omega_{m}}$ (24)
$\displaystyle a(t)$ $\displaystyle=$
$\displaystyle\alpha(t)e^{-i\beta\text{sin}(\Omega_{m}t)}$ (25)

where $a(t)$ represents the intra-cavity electric field and $n_{\text{phon}}$
is the phonon number in the acoustic mode. It can be seen $\beta$ denotes the
amplitude of the sinusoidal modulation of the light’s phase inside the cavity.
$V_{\pi}$ is thus the voltage when $\beta=\pi$ and can be measured by fitting
to the transmission spectrum of the optical mode under the driving of AOM,
where the optical resonance would split under large modulation index $\beta$
[15, 309, 284].

The $V_{\pi}$ can be also related with the key parameters of the AOM as [309]:

$V_{\pi}=\frac{\pi\gamma_{m}\Omega_{m}}{4g_{0}}\sqrt{\frac{2Z_{0}\hbar\Omega_{m}}{\gamma_{m,\text{ex}}}}$
(26)

where $\gamma_{m,\text{ex}}$ is the external coupling rate from the microwave
channel to the acoustic mode, $\gamma_{m}$ is the total loss rate, and $g_{0}$
is the single photon optomechanical coupling rate. $\gamma_{m}$ and
$\Omega_{m}$ can be found from the $S_{11}$ and $S_{21}$ measurements. There
are mainly two ways to calibrate the $g_{0}$: one is by measuring the change
in mechanical linewidth induced by the optomechanical back action [309]; the
other is by homodyne detection of the frequency fluctuation of the optical
cavity induced by mechanical thermal noise [159]. On the other hand,
$\gamma_{m,\text{ex}}$ can be measured by fitting the $S_{11}$ spectrum [15],
or measuring the injected intra-cavity phonon number by calibrating to the
thermal occupation or from the modulation index by fitting the optical
transmission spectrum (see Eq. 24) [309].

Figure 36:  BAW AOM on an unreleased structure. (a) Cross-section of the
device along the black dashed line in (b). (b) Optical image of the disk AlN
actuator with optical microring under its edge. (c) Numerical simulation of
the vertical stress distribution of one HBAR mode at 2.041 GHz. Reprinted from
Ref. [12].

The magnitude of $S_{21}$ can be related with $V_{\pi}$ as [15, 298, 217,
306]:

$|S_{21}|=\frac{\kappa_{o,\text{ex}}}{\kappa_{o}}\frac{\pi
R_{\text{PD}}I_{\text{rec}}}{V_{\pi}}$ (27)

where $\kappa_{o,\text{ex}}$ and $\kappa_{o}$ are the external and total loss
of the optical mode, $R_{\text{RD}}$ is the responsivity of the photodetector
in unit of V/W, $I_{\text{rec}}$ is the DC optical power reaching the
photodetector. From Eq. 27, $V_{\pi}$ can also be estimated from $S_{21}$ and
the smaller the $V_{\pi}$ the larger the $S_{21}$. Note that Eq. 27 is derived
by assuming a resolved sideband where the mechanical frequency is much larger
than the optical linewidth. The non-resolved sideband case is derived from
Refs. [217, 306]. Also, note that when studying the literature and in real
applications, we should pay attention to the definition of $S_{21}$ which can
be either the ratio of amplitude [298] or power [15] of the input and output
microwave signals. Also, when converting from log scale (in dB) to linear
scale, there is a factor of 2 between power and amplitude. While the above
analysis are done for AOM of optical resonator, the $V_{\pi}$ of the AOM of
MZI can be obtained from $S_{21}$ in a similar fashion [15, 136]:

$|S_{21}|=\frac{\pi R_{\text{PD}}I_{\text{rec}}}{V_{\pi}}$ (28)

The performance of the state-of-the-art SAW AOM is summarized and compared in
Table. 5. One can see that released AOMs generally has higher efficiency
(smaller $V_{\pi}$) than unreleased structures. However, the unreleased AOM
has the advantages of easier fabrication, better thermal conductivity, and
high-power handling capability. Although the photonic crystal structure
demonstrates the smallest $V_{\pi}$ due to its tight optical mode confinement,
the optical loss is still higher than in other platforms. On the material
side, the SAWs made from LiNbO3 show a higher SAW excitation efficiency due to
its high electromechanical coupling coefficient. In the future, $V_{\pi}$ can
be further reduced with the help of novel structures and new materials,
targeting at higher microwave to acoustic wave conversion efficiency, smaller
acoustic losses, and a larger optomechanical coupling rate (which benefits
from a smaller mechanical and optical mode volume, better mode overlap, and
larger photoelastic coefficient).

###  5.2 Bulk Acoustic Wave Based AOM

Apart from the surface acoustic wave, a bulk acoustic wave (BAW) that
transmits vertically in the substrate has also been utilized in modulating the
Si3N4 microring resonator [12]. As shown in Fig. 36(a, b), the device shares
the same structure as the one shown in Fig. 7, where a disk-shaped actuator is
designed to tune the optical resonance of a Si3N4 resonator. If we drive the
actuator at the microwave frequency in stead, an acoustic wave will be excited
vertically and confined in an acoustic Fabry-Pérot cavity, forming the so-
called High-overtone Bulk Acoustic wave Resonator (HBAR) mode. The mode
distribution of one typical HBAR mode is illustrated in Fig. 36(c). We can see
that the acoustic wave is confined laterally beneath the actuator and is
distributed uniformly throughout the substrate. Since the HBAR shown here is a
longitudinal wave, the stress is dominated by the vertical component. On the
other hand, transverse HBAR with large shear stress has also been demonstrated
on LiNbO3 [310, 311]. An efficient AOM can be achieved by placing the optical
waveguide on the antinode of the stress distribution.

Figure 37:  Experimental results of the unreleased HBAR AOM. (a)
Electromechanical S11 response. The inset schematic illustrates the reflection
of acoustic waves at interfaces. Optomechanical S21 spectra showing the
modulation of (b) TE and (c) TM optical modes. Zoom-in of (d) S11 and (e) S21
of TE mode in the green shaded region of (b) around 2 GHz. Periodic acoustic
resonances can be observed with spacing of 17.5 MHz. (f) Zoom-in of TM mode’s
S21 around 4 GHz (green shaded region) in (c). Reprinted from Ref. [12].

The HBAR modes can be found from its electromechanical S11 response as
presented in Fig. 37(a) and (d). Similarly to an optical Fabry-Pérot cavity,
there exists a series of equally spaced mechanical resonances, covering
multiple octaves in the microwave frequencies. The mode spacing of 17.5 MHz is
primarily determined by the thickness of the substrate. The resonances have a
linewidth of $\sim$3 MHz, corresponding to a mechanical $Q$ of $\sim$1000\. In
addition, the envelope of these resonances is periodically modulated. This is
due to the presence of acoustic resonance in the SiO2 layer. The coupling
between the SiO2 and Si cavities further modifies the mechanical dispersion
and perturb the mode spacing. This behavior can be captured by the Mason
model, as detailed in Ref. [12].

Each of the acoustic resonances can modulate the optical resonator as
manifested in the optomechanical S21 measurement in Fig. 37(b, c) for
transverse electric (TE) and transverse magnetic (TM) optical modes. Different
from S11 where the resonance dips are less than 1 dB, the S21 shows clear
peaks with high contrast ($>20$ dB) between resonance and anti-resonance due
to the high sensitivity and large signal to noise ratio (SNR) of the optical
readout. Maximum S21 of -65 dB can be obtained for both TE and TM modes which
leads to a $V_{\pi}$ of $\sim$200 V according to Eq.28. This is much higher
than the SAW AOM.

Figure 38:  BAW AOM on a released structure. (a) Cross-section of the released
HBAR. (b) Optical image of the fabricated device. (c) Zoom-in of the white
dashed box in (b), showing the microring to bus waveguide coupling region. (d)
Numerical simulation of the vertical stress distribution for two typical
modes. The scale bar is the stress normalized by the maximum. (e)
Electromechanical S11 spectrum. The mode order for each HBAR in the oxide
cavity is labeled. The inset shows the zoom-in of the green shaded region. (f)
Calculated electromechanical coupling efficiency $k^{2}_{t,\text{eff}}$ for
released (black square dot) and unreleased (red curve) HBAR modes. Reprinted
from Ref. [16].

The modulation efficiency can be further improved by releasing the SiO2
membrane to tightly confine the acoustic wave in smaller volume [16]. As shown
in Fig. 38(a-c), the region where the Si3N4 resides is undercut and supported
by a few cantilever beams connected to the substrate. By getting rid of the
Si, a series of HBAR modes can be excited in the SiO2 cavity, as can be found
from the S11 response in Fig. 38(e). Due to the thinner SiO2 layer, the mode
spacing is increased to around 450 MHz. The mode order is labeled for each
resonance, which corresponds to the number of wavelengths in one round-trip
inside the cavity. Two examples of HBAR modes on the order of 4 and 9 are
simulated in Fig. 38(d). The electromechanical response around 4 GHz (green
shaded region) is enhanced by the fundamental acoustic resonance in the AlN
cavity, where the acoustic wavelength is twice of the AlN thickness. Besides
the deep resonances, there exist small resonances with a much smaller FSR of
17 MHz (see the inset in Fig. 38(e)). This is due to the HBAR resonances from
the Si substrate since part of the contact pads and traces are not released.
The electromechanical coupling efficiency $k^{2}_{t,\text{eff}}$ is calculated
based on the Mason model [217] for released and unreleased structures, as
shown in Fig. 38(f). The efficiency is maximized around the AlN resonance at 4
GHz and improved by more than one order of magnitude upon releasing.

The AOM is characterized by the optomechanical S21 response in Fig. 39(a).
Each of the SiO2 HBAR modes manifests itself as a peak in S21. The Si HBAR
modes do not show up since they mainly reside at the outer periphery that is
far away from the inner Si3N4 microring. Modulation up to 9.2 GHz can be
achieved that enters the microwave X-band, as shown in Fig. 39(c). A maximum
of -45 dB of S21 is reached at 1.5 GHz and $V_{\pi}$ is estimated to be 18 V,
which is much smaller than the unreleased HBAR AOM. The improvement mainly
comes from the enhancement of the electromechanical and optomechanical
interactions as a result of the better confinement of the HBAR mode.

Figure 39:  Experimental results of the released HBAR AOM. Optomechanical S21
for TE (top) and TM (bottom) modes. Zoom-in of (b) S11 and (c) S21 of TM mode
for higher order modes around 9 GHz. Reprinted from Ref. [16].  Figure 40:
Piezo-optomechanical crystal based on single piezoelectric material. (a) GaAs
OMC mechanically excited by the Lamb wave from IDT. Reprinted from Ref. [312].
(b) GaP OMC excited by a pair of electrodes. Reprinted from Ref. [162]. The
lower inset shows the zoom-in of the OMC above the electrodes. (c) LiNbO3 OMC
coupled through a mechanical waveguide to an IDT. The lower left inset shows
the IDT, while the lower right inset is the OMC. Reprinted from Ref. [284].
(d) AlN OMC excited by a focusing IDT. Reprinted from Ref. [313].

Although the BAW AOM demonstrated so far has lower efficiency than the SAW
AOM, it possesses advantages that are preferred in specific applications.
First, since the HBAR mode is distributed uniformly in the vertical direction
of the substrate, the optical waveguide can be buried deep inside the cladding
to maintain a high optical $Q$ and immunity to the environment perturbation.
Also, the out-of-plane transmission of the acoustic wave makes it possible for
independent optimization of the in-plane optical circuits. Secondly, the
acoustic wave is confined laterally by the actuator, which guarantees a
compact size and a low cross-talk between adjacent actuators [314]. Lastly,
since the acoustic resonance is mainly dictated by the thickness of each
layer, the lateral degree of freedom is released such that larger critical
dimensions of the actuator can be designed, in contrast to the sub-micron
electrodes of SAW IDT. This not only eases the fabrication, but also increases
the power handling capability. On the other hand, since the thickness of each
layer can be well controlled over the whole wafer, HBAR actuators with the
same frequency can be massively produced, which would lower the cost. In the
future, the AOM efficiency can be improved by engineering the mechanical and
optical mode, such as coupling HBAR with a photonic crystal. In addition,
materials with larger piezoelectric coefficient, such as LiNbO3 and BaTiO3,
will be of great interest for excitation of HBAR modes [310].

Figure 41:  Hybrid piezo-optomechanical system with Piezo-on-Si structure. (a)
Si OMC excited by a focusing AlN IDT. The lower insets are the optical (left)
and mechanical (right) modes. Reprinted from Ref. [315]. Si OMC coupled with
(b) an AlN piezoelectric resonator, (c) a half-wavelength AlN piezo-acoustic
cavity (p-a cavity), and (d) a LiNbO3 (LN) piezoelectric resonator (purple)
through a phonon waveguide. The insets in (b) and (d) show the optical mode
and the hybridized mechanical mode. (b) Reprinted from Ref. [285]. (c)
Reprinted from Ref. [316]. (d) Reprinted from Ref. [317].

###  5.3 Piezo-Optomechanical Crystal

Since its first demonstration by Eichenfield et al. [318] in 2009, the
optomechanical crystal (OMC) has provided a way to tightly confine mechanical
and optical modes together in a small volume by co-design of the photonic and
phononic band structure. This has enabled us to achieve the highest
optomechanical coupling rate on the order of 1 MHz. OMC has been realized in
most common optical materials including Si [146], Si3N4 [319], diamond [320]
and SiC [321], as well as piezoelectric materials, such as AlN [322], LiNbO3
[323], GaP [154, 161], and GaAs [153]. By depositing electrodes, the OMC made
from piezoelectric material can be directly coupled to the external microwave
channel for optical modulation, as shown in Fig. 40. Either a single pair of
electrodes or an IDT can be designed to excite the mechanical vibration.
However, to maintain a high optical $Q$, the electrodes need to be placed a
few microns away from the OMC. A phononic crystal waveguide is required to
deliver the acoustic energy to the OMC (see Fig. 40(a)), which must be
tailored to support a large acoustic wavenumber but in the same time serve as
an optical mirror for the optical mode [309]. On the other hand, due to the
small size of the OMC mechanical mode, there is a large mismatch with the
acoustic mode of the IDT, making the coupling efficiency low. One solution is
to confine the IDT mode in a wavelength-scale mechanical waveguide, as
illustrated in Fig. 40(c) [284, 324]. Another way is to design an IDT that
focuses the acoustic wave onto the phonon waveguide, as demonstrated in Fig.
40(d) [313, 325, 326].

Table 6:  Comparison among the state-of-the-art AOM based on piezo-optomechanical crystals. WG: waveguide. WL: wavelength. $V_{\pi}$ is calculated using Eq. 26 based on the reported experimental results.  Ref. |  Year |  Piezo coupling scheme |  Material platform |  $\Omega_{m}/2\pi$ (GHz) |  $\gamma_{m}/2\pi$ (MHz) |  $\gamma_{m,\text{ex}}/\gamma_{m}$ |  Optical linewidth (GHz) |  $g_{0}/2\pi$ (kHz) |  $V_{\pi}$ (V)
---|---|---|---|---|---|---|---|---|---
[313] |  2016 |  Focusing SAW IDT |  AlN |  3.78 |  5 |  $2\times 10^{-4}$ |  15 |  115 |  0.162
[312] |  2016 |  SAW IDT |  GaAs |  2.42 |  1.6 |  $3\times 10^{-10}$ |  5.2 |  1100 |  0.625
[163] |  2020 |  SAW IDT |  GaAs |  2.7 |  0.2 |  $3.6\times 10^{-10}$ |  5.8 |  1300 |  1.29
[323] |  2019 |  Electrodes |  LiNbO3 |  2 |  0.5 |  $1.8\times 10^{-8}$ |  0.78 |  120 |  1.99
[284] |  2020 |  WL-scale IDT |  LiNbO3 |  1.85 |  1.9 |  0.001 |  1.2 |  80 |  0.024
[162] |  2022 |  Electrodes |  GaP |  3.31 |  2.8 |  $1.36\times 10^{-9}$ |  3 |  280 |  15.6
[315] |  2022 |  Focusing SAW IDT |  AlN on Si |  4.74 |  6.2 |  $\sim$0.002 |  10.3 |  734 |  0.013

Apart from piezo-optomechanical systems made from single material, a hybrid
Piezo-on-Si platform has been widely studied, showing the advantages of high
mechanical and optical $Q$ and large optomechanical coupling strength. This
primarily benefits from the low acoustic losses, large refractive index and
photoelastic coefficients of Si [327]. AlN and LiNbO3 have been the two main
piezoelectric materials demonstrated so far, as shown in Fig. 41. Different
acoustic modes can be excited by the IDT, such as Lamb wave (Fig. 41(a)) and
wavelength-scale acoustic cavity (Fig. 41(b)). The acoustic wave is then
guided by the phononic crystal waveguide to the OMC. The coupling between the
OMC and piezoelectric modes leads to a hybridized mechanical mode when they
are designed to have the same frequency. This hybridized mode is partially
coupled with the optical field and the electric field of IDT for transduction
between optical and microwave photons. The acoustic energy participated in the
piezoelectric cavity does not contribute to the optomechanical interaction,
which could lower the optomechanical coupling rate $g_{\text{om}}$. Therefore,
it is preferable to have a piezo-acoustic cavity with a small mode volume.
Recently, a half-wavelength scale acoustic resonator has been designed [327]
and demonstrated [316], as shown in Fig. 41(c). A Si-like hybrid mode is
formed, which supports a large $g_{0}$ and a strong coupling to a high
impedance microwave resonator.

The performances of the state-of-the-art AOMs achieved in piezo-optomechanical
crystal have been summarized in Table 6. Despite of the large $g_{0}$, the
efficiency is largely limited by the external coupling rate from microwave to
OMC, especially for the excitation schemes using SAW IDT and simple
electrodes. In that sense, the wavelength-scale IDT has been more popular due
to its better mode matching with OMC, and a $V_{\pi}$ much smaller than the
SAW AOM has been demonstrated [284]. On the other hand, GaAs exhibits the
largest $g_{0}$ such that a much improved $V_{\pi}$ can be foreseen with
optimized IDT design in the future.

In summary, this chapter reviews the integrated AOMs realized using various
acoustic modes, which have their own advantages in different aspects. While
the SAW has a high microwave-to-acoustic-wave conversion efficiency
($\gamma_{m,\text{ex}}/\gamma_{m}$), the HBAR supports high optical $Q$ by
cladding the optical waveguide, and the OMC is superior in optomechanical
coupling $g_{0}$ through sub-wavelength mode confinement. Therefore, depending
on the requirement of the application, the appropriate scheme should be
chosen, taking into account the material compatibility, efficiency, and
fabrication cost. A few of the most widely explored applications will be
outlined in the next chapter, and the comparison between different acoustic
modes will be made.

##  6 Applications of On-chip Acousto-Optic Modulator

While the on-chip AOM features a narrow modulation bandwidth that is primarily
limited by the mechanical resonance, there are many advanced applications that
only require single frequency or narrow band microwave drive. In this chapter,
we will review three of the most widely explored applications that will
benefit from the high modulation efficiency of the AOM. Despite of being made
from AOMs with similar structures, they realize unique functions, ranging from
frequency converter, optical isolator, to frequency shifter. This manifests
the universality of the AOM and the possibility of performing multiple
functions on the same chip.

###  6.1 Microwave to Optical Converter

Superconducting (SC) circuits and qubits have experienced a rapid progress in
recent decades, featuring a full set of microwave quantum gates with high
fidelity [328]. Although the computation supremacy over the classical computer
has been demonstrated [329], the requirement of working at cryogenic
temperature limits its scalability, due to the finite cooling power of current
dilution refrigerators and the limited space imposed by the demand coaxial
cables for the driving and readout of SC qubits. These challenges prevent the
realization of fault-tolerant quantum computing in a foreseeable future, which
requires millions of qubits [330]. A potential solution is to interconnect
multiple quantum processors hosted in separate dilution refrigerators.
Although the connection via coaxial cables has been demonstrated [331], they
must be kept at cryogenic temperatures to maintain a low thermal noise, which
is expensive and not scalable. However, optical photons possess frequencies
four orders of magnitude higher than the microwave, leading to negligible
thermal noise at room temperature. Indeed, the realization of the optical
quantum key distribution (QKD) [332] has demonstrated its capability to
maintain quantum superposition and entanglement at room temperature. Moreover,
optical fiber telecommunication systems have become one of the major
infrastructures which can be readily upgraded for building large-scale quantum
processor and quantum network. Therefore, conversion between microwave and
optical photons with high quantum efficiency has become an active research
area recently, and promising progress has been made.

For an ideal microwave to optical converter in quantum transduction, it should
have high quantum efficiency with low added noise and wide transduction
bandwidth. Although classical optical modulator based on electro-optic effect
has been well studied, due to the large frequency and energy difference
between microwave and optical photons, it is nontrivial to convert directly
with high quantum efficiency through the weak nonlinearity of most optical
materials. Nevertheless, electro-optic converter has been explored in various
platforms, such as integrated AlN [187, 333] and LiNbO3 [167, 311, 334]
photonic circuits, as well as bulk LiNbO3 disk [335, 336, 337]. Quantum
transduction via an intermediary mode has attracted much attention, such as
mechanical mode [338, 339], magnon mode [340, 341], and atomic transition
[342]. Among them, the mechanical mode has been more popular due to its high
efficiency, ease of fabrication, and compatibility with SC qubits.

There are mainly two ways of coupling mechanics with electrical signals,
including electrostatic and piezoelectric. The highest conversion efficiency
has been achieved in electrostatically coupled mechanical membrane, from
10$\%$ [338] to 47$\%$ [343], reaching the 50$\%$ efficiency required for
having finite quantum capacity in quantum communication [344]. However, since
the membrane operates at MHz frequencies, it only supports a limited
bandwidth, and the added noise is high even at cryogenic temperature.
Progresses have been made recently in electrostatically exciting Si
optomechanical crystal (OMC) which supports GHz mechanical modes [345, 309].
Despite of being applicable to most materials, the electrostatic coupling is
still weaker than the piezoelectric force. In that sense, piezo-optomechanical
transducer has drawn much attention in coherent conversion between microwave
and optics. The variety of mechanical modes as presented in the previous
chapter for AOM have also been explored in quantum transducer, from SAW [15,
306], BAW [346, 347, 348], to OMC [323, 284, 285].

The same mode coupling scheme as presented in Fig. 6 for AOM also holds for
the quantum transduction, where the mechanical and optical modes are coupled
through the optomechanical coupling rate $g_{\text{om}}$. The Hamiltonian that
describes the interaction is [186]:

$H_{\text{int}}=\hbar
g_{0}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})$ (29)

where $\hat{a}$ and $\hat{b}$ are the quantum operators for optical and
mechanical modes, $g_{0}$ is the single photon optomechanical rate. The above
equation is usually linearized by substituting $\hat{a}$ by
$\bar{\alpha}+\delta\hat{a}$, where $\bar{\alpha}$ is the average amplitude of
the pump optical field, and $\delta\hat{a}$ is the fluctuation. Under
rotating-wave approximation, the interaction Hamiltonian can be deduced as:

$\displaystyle H_{\text{int}}=\hbar
g_{\text{om}}(\delta\hat{a}^{\dagger}\hat{b}+\delta\hat{a}\hat{b}^{\dagger})+\hbar
g_{\text{om}}(\delta\hat{a}^{\dagger}\hat{b}^{\dagger}+\delta\hat{a}\hat{b})$
(30)

where $g_{\text{om}}=g_{0}\sqrt{n_{\text{cav}}}$ with $n_{\text{cav}}$ the
intra-cavity pump photon number. The first and second terms in the above
equation correspond to the beam-splitter and two-mode squeezing interaction,
respectively. Depending on the detuning of the pump laser (assuming resolved
sideband $\omega_{m}\gg\kappa_{o}$), one interaction will dominate the other,
with red detuning for beam-splitter and blue detuning for two-mode squeezing
[186]. The quantum efficiency is defined as the ratio of the output optical
photon to input microwave photon in up-conversion, and vice versa in down-
conversion. Practically, the total efficiency consists of the coupling
efficiency from the off-chip (microwave or optical) sources / detectors to the
on-chip waveguides and the on-chip conversion efficiency $\eta_{\text{oc}}$.
The $\eta_{\text{oc}}$ can serve as a figure of merit for comparing different
works and can be derived from the Heisenberg-Langevin equations of motion (see
Eqs. 20-21) as [348]:

$\eta_{\text{oc}}=\eta_{\text{ext}}\eta_{\text{int}}=\frac{\kappa_{o,\text{ex}}}{\kappa_{o}}\frac{\gamma_{m,\text{ex}}}{\gamma_{m}}\frac{4C_{\text{om}}}{(1+C_{\text{om}})^{2}}$
(31)

where
$\eta_{\text{ext}}=\kappa_{o,\text{ex}}\gamma_{m,\text{ex}}/\kappa_{o}\gamma_{m}$
is the external coupling efficiency from the waveguides to the optical and
mechanical modes. The internal efficiency $\eta_{\text{int}}$ is the
efficiency between mechanical and optical modes and is determined by the
optomechanical cooperativity
$C_{\text{om}}=4g_{\text{om}}^{2}/\kappa_{o}\gamma_{m}$, which leads to its
maximum of 1 when $C_{\text{om}}$ is equal to 1. The external efficiency will
become larger when $\kappa_{o,\text{ex}}$ and $\gamma_{m,\text{ex}}$ increase.
However, this would lower the $C$ and the internal efficiency. Therefore, an
optimal efficiency can be achieved when both of the mechanical and optical
modes are critically coupled to their waveguides [348].

Figure 42:  Mode coupling scheme of the microwave to optical transduction by
including a microwave mode to boost the mechanical excitation efficiency and a
second optical mode to increase the intra-cavity photon number.

As discussed in the previous chapter, the direct coupling rate from microwave
transmission line to the mechanical mode is generally small due to impedance
mismatch, especially for the high impedance, sub-micron OMC mode. To further
improve the excitation efficiency, a new scheme has been proposed that
involves a microwave resonator, as illustrated in Fig. 42. It helps by
recycling the microwave reflected by the piezoelectric transducer. The
microwave and mechanical modes are coupled through $g_{\text{em}}$, which can
be calculated as [185]:

$\displaystyle g_{\text{em}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\sqrt{\Omega_{\mu}\Omega_{m}}\sqrt{\frac{C_{m}}{C_{\mu}+C_{m}+C_{0}}}$
(32) $\displaystyle\approx$
$\displaystyle\frac{1}{2}\sqrt{\frac{\gamma_{m,\text{ex}}}{Z_{0}}}\sqrt{\frac{1}{C_{\mu}+C_{m}+C_{0}}}$
(33)

where $\Omega_{\mu}$ and $C_{\mu}$ are the frequency and capacitance of the
microwave resonator, respectively. When $\Omega_{\mu}$ equals $\Omega_{m}$,
$g_{\text{em}}$ can be related with the external coupling rate of the
mechanical mode to the transmission line $\gamma_{m,\text{ex}}$ by Eq. 33. It
can be seen for large $g_{\text{em}}$, the microwave resonator is desirable to
have small capacitance $C_{\mu}$ which corresponds to a high impedance
$Z_{\mu}$ ($=\sqrt{L_{\mu}/C_{\mu}}$). The development of high-impedance and
compact superconducting resonator has benefited from the study of thin film
superconductors with high kinetic inductance and has been applied in microwave
to optical transduction, such as TiN [309], NbTiN [317], NbN [316], and MoRe
[349]. Moreover, the design of these resonators with ladder geometry allows
tuning via external magnetic flux [350], as shown in Fig. 43.

Figure 43:  High-impedance microwave resonator with ladder geometry. (a)
Optical images of an NbTiN microwave resonator fabricated on a chip separate
from the transducer chip. The resonator is formed in a coplanar waveguide made
from nanowires patterned in a ladder shape. The scale bar is 200 $\mu$m. (b)
Assembly of the transducer chip (top) with the microwave chip (bottom) via
wirebonds. (c) False-colored SEM of a MoRe microwave resonator made on the
same chip with the transducer. (d) Finite element simulation of the current
density of the microwave resonator. (a-b) Reprinted from Ref. [317]. (c-d)
Reprinted from Ref. [349].

The Hamiltonian that describes the interaction between the microwave and
mechanical resonators is $\hbar
g_{\text{em}}(\hat{c}^{\dagger}\hat{b}+\hat{c}\hat{b}^{\dagger})$. The
conversion efficiency by including the microwave resonator can be obtained
similarly from the equations of motion as [309, 284]:

$\eta_{\text{oc}}=\eta_{\text{ext}}\eta_{\text{int}}=\frac{\kappa_{o,\text{ex}}}{\kappa_{o}}\frac{\Gamma_{\text{ex}}}{\Gamma}\frac{4C_{\text{om}}C_{\text{em}}}{(1+C_{\text{om}}+C_{\text{em}})^{2}}$
(34)

where $\Gamma$ and $\Gamma_{\text{ex}}$ are the total and external coupling
rate of the microwave resonator, and
$C_{\text{em}}=4g_{\text{em}}^{2}/\Gamma\gamma_{m}$ is the electromechanical
cooperativity. The internal efficiency $\eta_{\text{int}}$ reaches maximum
when $C_{\text{om}}=C_{\text{em}}$, and approaches 1 as $C_{\text{om}}$ and
$C_{\text{em}}$ increase. Interestingly, the previous two-mode case with only
optical and mechanical modes can be considered as a special case of Eq. 34 by
treating the mechanical mode coupled to an overcoupled broad microwave
resonator such that $C_{\text{em}}\ll 1$ and
$C_{\text{em}}=\gamma_{m,\text{ex}}/{\gamma_{m}}$ [181, 284, 309].

Figure 44:  Effects of optical absorption induced hot bath. (a) Schematic of
the model describing different baths the mechanical mode is coupled to.
Reprinted from Ref. [351]. (b) SEM of a two-dimensional optomechanical crystal
with better thermal conductance. (c) Experimental measurement of $n_{p}$
versus intra-cavity photon number $n_{c}$ for 1D OMC (red dashed line) and 2D
OMC (red dots). Solid red line is the fitting leads to power-law scaling of
$n_{c}^{0.3}$. (d) Measurement of mechanical linewidth versus $n_{c}$ for 2D
OMC (blue dots) with theory fitting (blue solid line). Red solid line is the
fitting for $\gamma_{p}$. Dashed red line is $\gamma_{p}$ for 1D OMC. (b-d)
Reprinted from Ref. [352].

Since the conversion efficiency is proportional to the intracavity pump photon
number, and thus the on-chip optical pump power, high efficiency can be
achieved by increasing the pump power. However, the coherent quantum
transduction requires the added noise to be smaller than one. Therefore, the
maximum power is ultimately limited by the added noise resulting from optical
absorption, which has two main negative effects on the transducer. One is the
generation of quasi-particles on the superconductor, and the other is the
increased thermal occupancy of the mechanical mode. The quasi-particle is
generated from broken copper pairs which can come either directly from the
light absorption or through thermal heating [349, 285]. It has been
experimentally examined that the quasi-particle could lower the lifetime of
superconducting qubits [285] and shift the frequency and increase the
intrinsic loss of the superconducting resonator [349]. The quasi-particle can
be effectively mitigated by reducing the time of optical illumination through
optical pulses [349]. Also, the use of superconductors with shorter quasi-
particle lifetime could accelerate the recovery time of the superconductor,
such as Nb and NbTiN which have quasi-particle lifetime on the order of
nanoseconds [353].

The optical-absorption-induced hot bath becomes more prominent at sub-Kelvin
temperature, which has been experimentally studied in Ref. [351]. The effect
of the hot bath on the mechanical resonator can be described by the model in
Fig. 44(a), where the mechanical mode couples not only to the refrigerator
bath $n_{f}$ at a rate of $\gamma_{0}$, but also to the hot bath $n_{p}$ from
the optical absorption with rate $\gamma_{p}$. Both $n_{p}$ and $\gamma_{p}$
depend on the temperature of the hot bath $T_{p}$ which is a function of the
intra-cavity photon number $n_{c}$. The thermal occupancy of the mechanical
mode can be described as:

$\left<n\right>(n_{c})=\frac{\gamma_{0}n_{f}+\gamma_{p}(n_{c})n_{p}(n_{c})}{\gamma_{0}+\gamma_{p}(n_{c})+\gamma_{\text{om}}(n_{c})}$
(35)

The dependence of $n_{p}$ and $\gamma_{p}$ on $n_{c}$ for an 1D OMC is
measured as shown in Fig. 44(c-d). It can be seen the optical absorption not
only increase the thermal occupancy but also the dissipation of the mechanical
resonator.

A few strategies are under investigation to mitigate the optical absorption.
The most widely adopted method is the use of optical pulses, where high
optical peak power can be achieved while the average power remains low [354,
355]. However, the repetition rate of the pulses is ultimately limited by the
power handling capability of the transducer, which limits the transduction
bandwidth. By increasing the thermal conductance of the optical cavity to the
surrounding cold environment, the heating from the optical absorption can be
effectively dissipated. For instance, a recent study has shown that the two-
dimensional optomechanical crystal has thermal conductance 40 times larger
than the 1D OMC, and the corresponding occupancy of the hot bath is 7 times
smaller (see Fig. 44(b-c)) [352]. The thermal conductance can be further
increased by utilizing unreleased OMC structure which has been demonstrated in
both 1D [356] and 2D OMC [357]. Bulk acoustic wave has also demonstrated good
optical power handling at cryogenic temperature [358].

For most of the optomechanical system in quantum transduction, it works in the
resolved sideband regime where the mechanical frequency is much larger than
the optical linewidth. This inevitably leads to low pump efficiency when the
laser is red or blue detuned by $\Omega_{m}$ where most of the pump light is
reflected by the optical cavity. Moreover, it is shown that the heating due to
the pump light in the coupling waveguide is comparable to that in the cavity
[352]. To circumvent this problem, it has been proposed to include a second
optical mode that aligns with the pump frequency, as shown in Fig. 42. In this
way, the same intra-cavity photon number can be achieved with much smaller
pump power. This approach is initially proposed and demonstrated in electro-
optic converters [359, 360, 167, 311, 334, 333]. Recently, it has been
implemented in HBAR [348] and OMC [361] schemes, where a second optical cavity
is evanescently coupled to the cavity with optomechanical interaction. The two
cavities form hybrid symmetric and asymmetric modes with spacing equal to the
mechanical frequency. Depending on the application, either the lower or higher
frequency mode can be pumped to drive the quantum transduction.

Table 7:  Comparison of piezo-optomechanical microwave to optical converters based on different mechanical modes. The last two columns show the comparison with other schemes using electro-optic (EO) and electromechanical (EM) coupling.  Scheme |  SAW |  BAW |  OMC |  OMC |  OMC |  OMC |  OMC |  EO |  EM
---|---|---|---|---|---|---|---|---|---
Ref. |  [15] |  [348] |  [284] |  [128] |  [316] |  [317] |  [349] |  [167] |  [309]
Year |  2019 |  2023 |  2020 |  2022 |  2023 |  2023 |  2023 |  2020 |  2023
Material platform |  LiNbO3 |  AlN on Si3N4 |  LiNbO3 |  GaP |  AlN on Si |  LiNbO3 on Si |  LiNbO3 on Si |  LiNbO3 |  Si
$\Omega_{m}/2\pi$ (GHz) |  2 |  3.48 |  1.85 |  2.8 |  5 |  3.596 |  5.043 |  6.8 |  4.93
$\gamma_{m}/2\pi$ (MHz) |  1.28 |  13 |  1.9 |  0.067 |  0.15 |  1.1 |  2.63 |  - |  3.3
$\gamma_{m,\text{ex}}/\gamma_{m}$ |  0.17 |  0.11 |  0.001 |  $3.6\times 10^{-6}$ |  - |  - |  - |  - |  $2.58\times 10^{-7}$
$\kappa_{o}/2\pi$ (GHz) |  0.095 |  0.17 |  1.2 |  4.17 |  1.3 |  1.12 |  4.99 |  0.6 |  1.39
$g_{0}/2\pi$ (kHz) |  1.1 |  0.042 |  80 |  700 |  270 |  410 |  530 |  1.2 |  577
$\Gamma/2\pi$ (MHz) |  - |  - |  - |  - |  1.75 |  3.06 |  3.44 |  21.6 |  -
$\Gamma_{\text{ex}}/\Gamma$ |  - |  - |  - |  - |  0.69 |  0.993 |  0.64 |  0.2 |  -
$g_{\text{em}}/2\pi$ (kHz) |  - |  - |  - |  $\sim$68 |  800 |  420 |  7400 |  - |  -
Pump mode |  CW |  CW & Pulse |  CW |  CW & Pulse |  Pulse |  CW & Pulse |  CW & Pulse |  CW |  CW
Optical power |  1 mW |  126 mW |  3.3 $\mu$W |  0.5 $\mu$W |  6.2 |  10.2 $\mu$W |  1 $\mu$W |  1.2 mW |  -
$C_{\text{om}}$ |  $1.7\times 10^{-4}$ |  $5\times 10^{-4}$ |  $6.5\times 10^{-3}$ |  1.04 |  $1.2\times 10^{-3}$ |  0.3 |  0.13 |  $3.7\times 10^{-5}$ |  0.268
$C_{\text{em}}$ |  - |  - |  - |  - |  9.75 |  0.2 |  24.2 |  - |  -
$\eta_{\text{int}}$ |  $6.7\times 10^{-4}$ |  $2\times 10^{-3}$ |  $2.6\times 10^{-2}$ |  0.032 |  - |  0.11 |  0.019 |  $1.48\times 10^{-4}$ |  0.67
$\eta_{\text{oc}}$ |  $1.7\times 10^{-5}$ |  $7.9\times 10^{-5}$ |  $1.1\times 10^{-5}$ |  $6.8\times 10^{-8}$ |  - |  0.049 |  0.009 |  $6.6\times 10^{-6}$ |  $1.72\times 10^{-7}$

The comparison of microwave to optical converters realized in different
schemes is summarized in Table 7. It can be seen the OMC has an optomechanical
coupling strength $g_{0}$ orders of magnitude larger than SAW and BAW due to
the tight mode confinement, which leads to large cooperativity $C_{\text{om}}$
under modest optical power. However, the sub-micron mechanical mode makes it
hard to be driven directly through microwave, showing a small
$\gamma_{m,\text{ex}}$. The development of high-impedance microwave resonator
in recent works [316, 317, 349] helps boost the drive efficiency. On the other
hand, the advantages of SAW and BAW include lower optical losses and better
optical power handling capability. Nevertheless, OMC is the most promising
platform so far that would meet most of the requirements for quantum
transduction. The highest efficiency of 5$\%$ has been achieved on the
platform of LiNbO3 on Si OMC with only 10 $\mu$W optical pump power.

Figure 45:  Potential applications of microwave-to-optical converter in near
future. (a) Schematic of photonic links, where the converters are used to
coherently drive and readout SC qubits, with the potential of replacing
coaxial cables. (b) Schematic of the optically heralded remote entanglement
generation between distant SC qubits. The inset shows the generation of
microwave-optical photon pair by a high frequency pump light. SPD: single
photon detector.

Although the quantum transduction efficiency of current converters is still
far below the requirement for building the quantum internet [362, 344], the
superconducting quantum processor can already benefit from the development of
this technology in the near furture. One potential application is the
classical optical control and readout of superconducting (SC) qubit.
Conventionally, each qubit is connected via the coaxial cable to the control
electronics at room temperature. However, this poses challenges on the scaling
of the number of qubits which is limited to a thousand qubits based on current
technology [363]. Apart from the space constraint, the heat load from coaxial
cables would ultimately overwhelm the cooling power of the dilution
refrigerator [364]. Optical fibers made from silica, on the other hand, show
negligible thermal conductivity at cryogenic temperatures. Also, the ultra-
wide optical bandwidth allows signal multiplexing in a single fiber. In that
sense, replacing the coaxial cable with the photonic link could become an
attracting solution to large-scale quantum computer in the future. As shown in
Fig. 45(a), the converter down-converts the input light into microwave signals
to drive and readout the qubit, and the readout signal is then up-converted
into light for qubit measurement. It has recently been demonstrated that the
optically generated microwave via photodiodes can faithfully serve as both
drive and readout signal for an SC qubit [363]. Another work realized the
optical readout of SC devices using off-the-shelf electro-optic modulators
[365]. Based on these proofs of the concept, the conversion efficiency and
added noise of the photonic link can be further improved using the-state-of-
the-art microwave to optical converters. While most recent works on this topic
have been realized on electro-optical converters [366, 367, 368], the optical
multi-shot readout of SC qubit has been demonstrated on piezo-optomechanical
transducer with high fidelity [369]. In the future, the investigation of heat
load from these converters and the demonstration of optical multiplexing could
enhance the scalability of this technology as a photonic link for SC circuits.

Aside from the classical control and readout of qubits, it has been proposed
that current converter technology can be readily applied in the optically
heralded remote entanglement between distant SC systems [370] and the qubit
states can be transmitted through the quantum teleportation [371], which has
less strict requirement on the conversion efficiency. Following the well-known
DLCZ protocol [372], the diagram of the proposed architecture is illustrated
in Fig. 45(b). In stead of red detuned optical pump, the light is pumped at
the blue side, which generates an entangled microwave-optical photon pair
through an optomechanical parametric down conversion process [373, 374].
Interestingly, the transducer preferably works in low-cooperativity regime
such that the probability of multi-photon events is kept low [368]. In this
way, by passing the optical photons generated from separate SC systems through
a beamsplitter to erase the path information, the detection of a single
optical photon heralds the entanglement between microwave photons located in
each SC circuits. The entangled microwave states can then be transferred to
corresponding SC qubits in each node to build the quantum network. As a
prerequisite, the generation of correlated microwave-optical photon pair and
optically heralded non-classical microwave photon have been demonstrated in
piezo-optomechanical transducers [317, 316]. Further improvement of the
converter’s performance and reducing the losses in the system could increase
the entanglement generation rate and fidelity, paving the way to the quantum
network in a foreseeable future.

###  6.2 Magnetic-Free Optical Isolator

Integrated photonics has witnessed major progresses in recent decades, where
most of the important building blocks have been successfully integrated on
chip, from semiconductor lasers [7, 5], modulators [10], low-loss waveguides
[24], to photodetectors [17]. Despite of these achievements, integration of
optical non-reciprocal devices, such as optical isolators and circulators,
remains as a challenging task, although they are indispensable components for
compact signal routing and protecting lasers from reflection. Traditionally,
these devices are realized using magneto-optical materials based on the
Faraday effect, where non-reciprocal polarization rotation is induced by
external magnetic field. Recently, achievements have been made on the
integration of cerium-substituted yttrium iron garnet (Ce:YIG) on Si photonic
circuits via wafer bonding [375] and on Si3N4 by pulsed laser deposition
[376]. However, the optical loss remains high and a bulky external magnet is
generally required to generate the magnetic field.

Magnetic-free optical non-reciprocity has gained much research efforts, and
many different schemes have been proposed and demonstrated, including optical
nonlinearity [377, 378, 379], optomechanically induced transparency (OMIT)
[380, 381], stimulated Brillouin scattering (SBS) [382, 383, 384, 385], and
dynamic modulation [386, 387, 388, 389]. They all show pros and cons in
practical applications. For example, while the optical nonlinearity can work
passively and all-optically, the isolation depends on the input optical power
and is subject to the dynamic reciprocity [390]. The OMIT and SBS are mostly
realized in a stand-alone sphere or microtoroid that is difficult to integrate
with PICs. Moreover, the isolation bandwidth is limited by the mechanical
linewidth which is generally between kilohertz to megahertz range (see Table
8). In contrast, the dynamic modulation scheme benefits from currently
developed electro-optic and acousto-optic modulators which can be readily
incorporated into existing photonic systems.

The dynamic modulation breaks the Lorentz reciprocity and time-reversal
symmetry through the spatio-temporal modulation of optical waveguides which
induces non-reciprocal interband or intraband photonic transitions. The
intraband transition-based optical isolator has been demonstrated in an
electro-optic phase modulator by exciting traveling microwave that satisfies
the phase matching condition only in one direction [391]. The intraband
transition is hard to implement on AOM since it is nontrivial to match the
group velocity of the optical mode with the acoustic wave. On the contrary,
the interband transition between different optical modes has been widely
studied [388, 389, 57] and is well suited for acoustic waves. Depending on the
specific arrangement of the optical and acoustic modes and the structure
design, different schemes of optical isolator based on dynamic modulation have
been proposed, which are summarized in Ref. [386] in more detail. Here, we
review two main schemes that have been successfully demonstrated in
experiments.

Figure 46:  Optical isolator based on a spatio-temporal modulated optical
resonator. (a) Frequency-momentum diagram illustrating the indirect interband
transition between two optical modes $a$ and $b$ induced by an acoustic wave
$c$. (b) Schematic of the optical transmission in forward and backward
directions. In the forward direction, the strong coupling between the two
optical modes induces Rabi mode splitting, whereas in the backward direction
the transmission remains as original single resonance. (c) Optical isolator
realized in a LiNbO3 optical resonator modulated by an unreleased SAW. (d)
Optical transmission spectrum in the forward (red) and backward (blue)
directions. (c-d) Reprinted from Ref. [140]. (e) False-colored SEM of an
optical isolator realized in a Si3N4 microring resonator modulated by three
HBAR driven at rotational phases. (f) Optical transmission when the actuators
are reconfigured into phase matching (red) and phase mismatch (gray)
conditions. (g) Transmission of optical pulses in the forward (red) and
backward (gray) directions for pulse width of 100 ns (upper) and 10 ns
(lower). (e-g) Reprinted from Ref. [314].

The first scheme is based on an optical resonator, and the working principle
is illustrated in Fig. 46(a-b). The optical resonator supports a pair of two
closely spaced optical modes as shown in the frequency-momentum ($\omega-k$)
space diagram in Fig. 46(a), where the sign of the momentum indicates the
propagation direction in the optical waveguide. Under the modulation of an
acoustic wave that carries a non-zero momentum, the light in one mode can be
scattered to the other if the energy and momentum conservation conditions are
satisfied, where the frequency and momentum of the acoustic wave equal to the
corresponding difference between the two optical modes. This is also known as
the phase matching condition. In that sense, the momentum of the acoustic wave
prescribes the direction to which the light can be scattered. From Fig. 46(a),
it can be seen the phase matching can only be fulfilled in the forward
direction, where the two optical modes can be coupled strongly. The same
equations of motion developed for the microwave-to-optical converter can be
used to describe this three wave mixing process. Instead of an optical pump,
the acoustic drive serves as the pump to induce the optical coupling. In the
strong coupling regime where the coupling rate is much larger than the loss
rate, the optical modes start showing Rabi-splitting [57], and a transparency
window appears at the original optical resonance position. In the backward
direction, the two optical modes are not coupled due to phase mismatch and
show a single resonance. If the optical resonator is critically coupled to the
waveguide, most of the light is dissipated into the environment without
transmission. With this non-reciprocal transmission, an optical isolator can
be built.

Initial experimental attempts have been made on a released AlN optical
resonator modulated by SAW, where a fundamental TE00 and a second order TE10
are coupled [134, 392]. However, limited by the modulation efficiency and
power handling capability, only non-reciprocal modulation sideband generation
is demonstrated without entering into the strong coupling regime. Recently,
the same group achieved strong coupling using an unreleased LiNbO3 SAW which
has a larger piezoelectric response and better thermal conductance (see Fig.
46(c)) [140]. The finger of the IDT has a non-zero angle with the optical
waveguide in order to generate momentum parallel to the waveguide. Non-
reciprocal transmission of a TE10 mode at 1556 nm has been realized, as shown
in Fig. 46(d). One of the most important metrics for an optical isolator is
the isolation ratio defined as the transmission ratio between forward and
backward directions. A maximum of 12.75 dB isolation ratio is achieved at
optical resonance with insertion loss of 1.13 dB [140]. The isolation ratio is
primarily determined by how critically the optical resonator is coupled to the
bus waveguide, which is sensitive to the fabrication disorder. We should also
note that, in the forward direction, there will be non-zero modulation
sideband generated in the other optical mode with a frequency separated by the
acoustic frequency. This sideband can be filtered by either a frequency filter
or a mode multiplexer. Not only for telecommunication band, the same design is
also suitable for visible light and similar result has been demonstrated
[140].

Apart from the SAW AOM, HBAR has also been utilized in demonstrating an
optical isolator on the Si3N4 photonic circuit [228], as shown in Fig. 46(e).
Differently from the SAW, HBAR excites the out-of-plane acoustic waves, which
carry momentum perpendicular to the optical waveguide. To generate the in-
plane momentum, it is proposed to include three discrete HBAR modulators that
are driven by relative phases rotating in a circle. In this way, an effective
rotational wave can be synthesized along the optical microring resonator. The
resonator supports a pair of TE and TM modes with orthogonal polarization,
which are then coupled through the modulation. Despite of the entirely
different material systems and the choice of optical and mechanical modes,
similar optical isolation performance is achieved with 10 dB isolation and 0.8
dB insertion loss, which demonstrates the universality of the dynamic
modulation approach. Furthermore, the actuator phases can be reconfigured to
switch the rotation direction such that the isolation direction can be
controlled in situ, as illustrated in Fig. 46(f). The modulation sideband in
the forward direction can be easily separated by a polarizing beamsplitter.
The isolation of optical pulses is also demonstrated in Fig. 46(g). Despite of
the compact size, the isolation bandwidth is limited by the optical linewidth
to hundreds of megahertz to gigahertz. Although bandwidth can be increased by
adding extra optical losses, the required mode splitting also increases, which
leads to larger RF power. Nevertheless, the narrow bandwidth is sufficient in
protecting laser systems and suitable for rejecting back-reflection noise for
microwave-to-optical quantum transducers.

Figure 47:  Optical isolator based on a spatio-temporal modulated optical
waveguide. (a) Frequency-momentum diagram showing the phase matching condition
between two optical waveguide modes ($a$ and $b$) and the acoustic wave $c$.
(b) Schematics illustrating the mode transmission in forward and backward
directions. In the forward direction, the light transmits without mode
conversion, while it completely converts to another mode due to phase matched
acoustic modulation. (c) Schematic of the experimental realization of a
waveguide-based optical isolator through SAW excited by a series of cascaded
IDTs. The mode multiplexers at the input and output ports combine and separate
the two waveguide modes. (d) Optical transmission in the forward and backward
directions showing a wide isolation bandwidth (shaded area). (c-d) Reprinted
from Ref. [300].

The broadband optical isolator can be realized in a dynamically modulated
optical waveguide. Similar to the case of the resonator, indirect interband
transition can be achieved between two optical waveguide modes (e.g. TE00 and
TE10) under a proper spatio-temporal modulation [388, 386]. As can be seen
from Fig. 47(a), the phase matching condition can only be satisfied in the
backward direction, such that under sufficient modulation strength, one mode
can be completely converted into the other mode (see Fig. 47(b)). By
separating and rejecting the converted mode, the backward transmission can be
blocked. On the contrary, in the forward direction the original mode remains
intact and gets fully transmitted. The proof of concept is initially
demonstrated on an electrically modulated doped Si waveguide [69]. However,
the insertion loss is as high as 70 dB. Modulation through SAW has been
theoretically proposed [393] and experimentally studied on an unreleased SI
waveguide modulated by AlN SAW [135]. Due to the inefficient AOM, only non-
reciprocal modulation sideband is demonstrated. Recently, the same group
improved the modulation efficiency by releasing the Si waveguide and cascading
more IDTs as shown in Fig. 47(c) [300]. Full mode conversion was achieved
under 22 dBm RF power, which led to an optical isolation of 14 dB and 0.74 dB
insertion loss, as shown in Fig. 47(d). The isolation is mainly limited by the
unwanted interband crosstalk and evanescent coupling between waveguides. It is
interesting to notice that there is a second band that shows isolation in the
opposite direction, where the phase is matched in the forward direction. The
maximum bandwidth for isolation bigger than 10 dB is 59 GHz which is suitable
for isolating high speed signals. The bandwidth is primarily limited by the
difference of group velocity of the two optical modes, which can be further
improved through the dispersion engineering of waveguide bands.

Borrowing the concept from the solid state physics, the aforementioned two
schemes are both based on indirect interband transition where the momentum is
involved in the scattering process. Another scheme exploring the direct
interband transition has also been proposed where the phase of modulation
induces an effective gauge potential for photons [394, 395]. In analogy to the
Aharonov-Bohm effect for electrons [396], the effective gauge potential can be
used to construct an effective magnetic field for photons and break the
reciprocity by creating a direction dependent phase [397]. Although the
concept has been demonstrated using an electrically modulated doped Si
waveguide, the isolation is small and the insertion loss is unacceptable
[397]. While the implementation using AOM has been realized based on off-the-
shelf bulk AOM devices [398], it will be of great interest to integrate the
system on chip using SAW or BAW AOM in the future for a new type of magnetic-
free chip-scale optical isolator with broad bandwidth.

Table 8:  Comparison of optical non-reciprocal devices realized using different schemes: MO (magneto-optic), NL (nonlinear optics), OM (optomechanical), SBS (stimulated Brillouin scattering), AB (Aharonov-Bohm effect), SM (synthetic magnetic field), and ST (spatio-temporal modulation). WG: waveguide. IL: Insertion loss. $a$: Non-reciprocal sideband modulation. $b$: Optical pump power.  Ref. |  Year |  Scheme |  Structure |  Material |  Isolation |  IL (dB) |  Bandwidth |  Power |  CMOS
---|---|---|---|---|---|---|---|---|---
[375] |  2016 |  MO |  Ring |  Si+Ce:YIG |  32 dB |  2.3 |  15 GHz |  10 mW |  No
[376] |  2020 |  MO |  Ring |  Si3N4 +Ce:YIG |  28 dB |  1 |  15 GHz |  |  No
[377] |  2018 |  NL |  Toroid |  SiO2 |  24 dB |  7 |  1 MHz |  No drive |  No
[378] |  2020 |  NL |  Ring |  Si |  20 dB |  1.3 |  20 GHz |  No drive |  Yes
[380] |  2016 |  OM |  Toroid |  SiO2 |  10 dB |  14 |  250 kHz |  17 $\upmu$Wb |  No
[399] |  2018 |  OM |  Toroid |  SiO2 |  10 dB |  3 |  60 kHz |  60 $\upmu$Wb |  No
[400] |  2018 |  OM |  Sphere |  SiO2 |  17 dB |  2 |  200 kHz |  7.8 mWb |  No
[382] |  2017 |  SBS |  Sphere |  SiO2 |  11 dB |  0.14 |  400 kHz |  235 $\upmu$Wb |  No
[397] |  2014 |  AB |  WG |  Doped Si |  2.4 dB |  |  20 nm |  34 dBm |  Yes
[401] |  2021 |  SM |  Ring |  AlN |  3 dB |  9 |  4 GHz |  16 dBm |  Yes
[69] |  2012 |  ST |  WG |  Doped Si |  3 dB |  70 |  200 GHz |  25 dBm |  Yes
[402] |  2018 |  STa |  WG |  Si |  39 dB |  NA |  125 GHz |  90 mWb |  Yes
[134] |  2018 |  STa |  Ring |  AlN |  15 dB |  NA |  1 GHz |  18 dBm |  Yes
[135] |  2021 |  STa |  WG |  AlN+Si |  16 dB |  NA |  100 GHz |  21 dBm |  Yes
[314] |  2021 |  ST |  Ring |  AlN+Si3N4 |  10 dB |  0.8 |  0.7 GHz |  25 dBm |  Yes
[140] |  2021 |  ST |  Ring |  LiNbO3 |  12.75 dB |  1.13 |  0.199 GHz |  29 dBm |  Yes
[300] |  2022 |  ST |  WG |  AlN+Si |  14.37 dB |  0.74 |  59 GHz |  22 dBm |  Yes
[391] |  2023 |  ST |  WG |  LiNbO3 |  48 dB |  0.5 |  13 GHz |  21 dBm |  Yes

Table 8 summarizes and compares the performances of optical isolator realized
on different platforms. It can be seen that the scheme of spatio-temporal
modulation has become one of the most promising methods in terms of
compatibility with current photonic circuits, and the development of
integrated AOM technology facilitates the recent demonstration of optical
isolator on Si3N4 [228], Si [300], and LiNbO3 [140] photonics. While the ring-
based structure is more compact, the waveguide structure supports wider
bandwidth, which can be chosen according to specific application. Thus far,
the widest bandwidth is achieved on the SAW modulated Si waveguides [300], and
the highest isolation is demonstrated on the LiNbO3 electro-optic phase
modulator [391]. The required RF power is on the order of 100 mW. These
performances can be further improved by engineering the dispersion of optical
waveguides, studying different pair of modes (e.g. TE and TM [403]), and
exploiting new piezoelectric material.

Figure 48:  Working principle of traditional AOM based on Bragg diffraction.
(a) Schematic of the AOM where the incident light is diffracted by the
traveling SAW excited by IDT. The Bragg angle is labeled. (b) Phase matching
condition for anti-Stokes process (left) and Stokes process (right), showing
the frequency shifting (top) and momentum conservation (bottom).

###  6.3 Integrated Acousto-Optic Frequency Shifter

Traditional AOM has long been an indispensable component in laser systems and
applications. It provides a convenient way to control the intensity,
frequency, and direction of the light. The working principle of traditional
AOM is illustrated in Fig. 48(a), where an incident light undergoes a Bragg
diffraction after interaction with a traveling (or resonant) acoustic wave
excited by an IDT. For most widely used AOM, it works at the Bragg regime,
where the light is mostly scattered into the first diffraction order. The
frequency of the deflected light is shifted by the acoustic frequency. For
traveling acoustic wave, it naturally carries momentum, such that, to satisfy
the phase matching condition, the light is only scattered into a single
sideband. As shown in Fig. 48(b), the light can be scattered to the Stokes
(lower frequency) or anti-Stokes (higher frequency) sideband, depending on its
incident direction relative to the acoustic wave. The diffraction efficiency
is maximized when the light is incident at the Bragg angle that is dictated by
the phase matching condition as [137]:

$\text{sin}(\theta_{B})=\frac{K}{2k}=\frac{\Omega v_{\text{opt}}}{2\omega
v_{\text{ac}}}$ (36)

where $K$ ($k$), $\Omega$ ($\omega$), and $v_{\text{ac}}$ ($v_{\text{opt}}$)
are the momentum, angular frequency, and velocity of acoustic (optical) wave,
respectively. The intensity of the diffracted light can be controlled by
modulating the diffraction efficiency which is related with the power and
frequency of the RF drive [404]. This capability is conventionally utilized in
the Q-switch laser [405] and in the generation of optical pulses with ultra-
high on-off contrast [355]. The frequency shifting of the acoustically
scattered light naturally leads to the desirable single sideband suppressed
carrier (SSB-SC) modulation, which is complicated to implement using electro-
optical modulators [406]. This unique feature of AOM has been translated into
an indispensable optical device, acousto-optic frequency shifter (AOFS), which
is widely used in the heterodyne detection [134, 407].

Figure 49:  Experimental demonstrations of integrated AOFS based on Bragg
diffraction. (a, c) Schematic of AOFS using SAW excited on thin film LiNbO3.
The input and output light is confined in etched LiNbO3 rib waveguides for (a)
and in BIC mode formed by polymer for (c). (b, d) Optical spectrum of the
deflected light for device in (a) and (c), respectively. (a-b) Reprinted from
Ref. [137]. (c-d) Reprinted from Ref. [404].

Despite of the ubiquitous application of AOM, current commercially available
AOM is fabricated on a stand-alone bulk piezoelectric substrate and the
acoustic frequency is limited to a few hundreds of megahertz. The required RF
drive power is at Watts scale due to the large acoustic and optical mode
volume. The implementation of Bragg diffraction on integrated planar photonics
has been demonstrated on a released AlN membrane for the first time [408].
Despite the low scattering efficiency, asymmetric modulation with 11 GHz
frequency shift is achieved with only 100 mW RF power. Recently, the
efficiency is improved by exciting SAW on an unreleased LiNbO3 thin film,
which has higher piezoelectric response and power handling capability [137,
404], as shown in Fig. 49. The input and output light is guided through either
an etched LiNbO3 rib waveguide [137] or a bound-in-continuum (BIC) waveguide
mode formed by patterned polymer on LiNbO3 film [404]. The light from the
waveguide transits into a two-dimensional Gaussian beam confined in the slab
formed by the LiNbO3 thin film, which interacts with the SAW in the same
region. An ideal AOFS would require most of the input light to be shifted to
the desired frequency with zero power remaining in the original frequency or
scattered to other sidebands. This behavior can be characterized by the
modulation efficiency (the fraction of input light that is scattered to the
target sideband), and carrier and sideband suppression ratios. The optical
spectrum of the shifted light is shown in Fig. 49(b, d). It can be found that
most optical power is in the anti-Stokes sideband with around 3 GHz higher
frequency, and the center carrier and other sideband is 30 dB and $>$40 dB
smaller, respectively. The efficiency is measured to be around 3$\%$ per 1 W
RF drive. The frequency shifting to the Stokes sideband can be realized by
changing to the other input port [137, 404]. The optical bandwidth is
proportional to $\lambda\Lambda/W$, where $\lambda$, $\Lambda$, and $W$ are
the optical wavelength, period of IDT fingers, and width of the IDT,
respectively [137]. Optical bandwidths of 14 nm and 35 nm are achieved for the
works in Ref. [137] and [404], respectively. These integrated AOFS have
compact size, low power consumption, and gigahertz operation, which can be
used for on-chip heterodyne detection, microwave signal processing, and
optical signal routing.

Figure 50:  Experimental realizations of integrated AOFS through interband
transition. (a, b) AOFS realized using optical microring resonator, the same
device presented in Fig. 46(e). (a) Optical transmission of TE and TM modes
when TE (TM) is converted to TM (TE) in upper (lower) panel. The x-axis is the
laser frequency relative to the corresponding optical resonance. (b) Optical
spectrum of output light in TM mode when RF drive is turned on (green) and
light in TE mode when RF is off (red). The upper panel is in dB scale while
the lower is linear. (c-e) AOFS using integrated Si waveguide, the same device
presented in Fig. 47(c). (c) Energy exchange between symmetric (red) and anti-
symmetric (blue) modes as a function of RF power. (d) Optical spectrum of
output symmetric mode when input anti-symmetric mode into the waveguide. The
light is mainly converted into Stokes sideband (red), while some power remains
in carrier (blue) and is scattered to anti-Stokes sideband (purple). (e)
Optical power in different frequency components when sweeping the RF drive
frequency around the acoustic resonance. (c-e) Reprinted from Ref. [300].

To further improve the modulation efficiency, the acoustic and optical
interaction length should be increased. However, this is hard to implement in
the scheme presented in Fig. 49. This is because longer interaction length
would require wider acoustic width and IDT aperture, which leads to broad
deflected light beam with lower collection efficiency by integrated waveguide.
Also, wider IDT inevitably compromises the optical bandwidth as mentioned
before. This issue can be circumvented by confining light in optical
waveguides which also helps with higher interaction strength due to better
mode confinement [142]. AOFS based on this scheme has been demonstrated in
both microring and straight waveguide structures which all enhance the
interaction length [314, 300]. It is interesting to find that the AOFS and the
isolator discussed before share a similar working principle, where one optical
mode is completely converted into another mode through an indirect interband
transition [389, 388]. Therefore, the same device can function as both an AOFS
and an optical isolator.

For the same device in Fig. 46(e), where a pair of TE and TM modes of a Si3N4
microring resonator are coupled by HBAR, AOFS can be realized by designing the
optical resonator in overcoupled regime. The overcoupling is required to have
high extraction efficiency of the frequency shifted light. The strong coupling
between the two modes leads to complete optical conversion when cooperativity
equals one. The optical transmission when injecting TE light into the
waveguide and under 20 dBm RF drive is shown in the upper panel of Fig. 50(a).
At the optical resonance, 50% of the input light is converted into TM mode,
which has a higher frequency. Since the device is not in the strong
overcoupled regime, the rest 50% is dissipated as the intrinsic loss of the
optical resonator. The conversion is also bidirectional, where the input TM
mode can be converted into TE mode (see the lower panel of Fig. 50(a)). The
asymmetry of the modulation sidebands can be seen from the output optical
spectrum as shown in Fig. 50(b). A single frequency laser at TE mode is input
and modulated with a fixed frequency microwave drive. The spectrum is broad
because of the limited resolution of the spectrum analyzer. The main peak of
the output TM light shifts to a lower wavelength. The light remaining in the
carrier and other sideband is more than 10 dB smaller. The frequency shift can
be seen more clearly on the linear scale. The choice of TE and TM mode pair
rotates the polarization in addition to the frequency shift, which also makes
it convenient to separate the converted light from the carrier.

In addition to the microring resonator, AOFS is also performed on a straight
Si waveguide, the same device shown in Fig. 47(c). A pair of symmetric and
anti-symmetric waveguide modes is coupled by a SAW excited from the AlN IDT.
As can be seen in Fig. 50(c), a Rabi-like oscillation between the two modes is
observed while the RF drive power is swept, demonstrating complete mode
conversion. The optical spectrum of the output light at different frequencies
is illustrated in Fig. 50(d). The Stokes sideband dominates and is more than
30 dB larger than the carrier and anti-Stokes sideband. The Si waveguide
supports a broad optical bandwidth of 1.7 nm, in stark contrast to the
microring resonator. The RF bandwidth is limited by the bandwidth of the IDT
to a few MHz (see Fig. 50(e)), which can be improved in the future by adopting
chirped IDT design [409].

While the integrated optical waveguides help to extend the interaction length
between photon and phonon on chip, the confining of acoustic wave could
further improve the propagation length and interaction strength. In that
sense, the development of acoustic waveguide and phononic circuits has drawn
much attention recently, and major progresses have been made in both released
[324] and unreleased [302, 301] waveguiding. Furthermore, the co-propagation
of phonon and photon in the same waveguide has been demonstrated successfully,
paving the path for furture optomechanical circuits [410, 411].

Apart from the classical application in heterodyne detection, AOFS will find
application in future frequency-domain optical quantum information processing
and quantum computing based on frequency-bin optical qubits [412, 413]. As
demonstrated in Fig. 50(c), the conversion between the two optical modes can
be controlled by the RF power such that the output power is equally split into
the two modes. Thus, a frequency domain beam splitter can be built which is
the building block for optical quantum computing [414, 415]. The frequency
beam splitter is conventionally constructed by cascading electro-optical
modulators and optical pulse shaper [416, 417], which is bulky and power
consuming, making it difficulty for scaling. Therefore, it can be foreseen
that the integrated AOFS will become an indispensable component for future
large scale optical quantum computing. So far, the experiments of AOFS are
done in the classical domain, it will be necessary to explore the quantum
behavior of it, such as the operation with non-classical light, quantum
coherence and interference [418], and quantum fidelity [416].

##  7 Future Developments

In the previous chapters, we have reviewed the design of integrated
piezoelectric actuators and their applications in tunable laser and
microcombs, acousto-optic modulators, and non-reciprocal devices. In this
chapter, we will give our perspectives on potential developments of
piezoelectric actuators in future integrated photonic and quantum systems.

Figure 51:  Schematic of the interaction between photon, phonon, and SC qubit
enabled by piezoelectric materials. The transduction between each node is
bidirectional. Reprinted from Ref. [217].

###  7.1 Hybrid integrated quantum systems

Hybrid integrated quantum systems have attracted extensive efforts recently,
as they take advantages of distinct quantum objects for high-performance
quantum applications [419, 420]. For example, the superconducting (SC) qubit
has become one of the mature platforms for quantum computing. However, the
operation at cryogenic temperature limits its scalability. On the other hand,
optical photon is good at quantum information communication with low quantum
noise, which makes it possible for scalable quantum network [372]. In contrast
to the electromagnetic wave, the low speed of acoustic wave endows it with
sub-micron wavelength at microwave frequencies, which allows compact device
size. Moreover, the long coherence time (low losses) of the mechanical
resonator is suitable to store quantum information. A hardware-efficient
Quantum Random Access Memory (QRAM) has been theoretically proposed [421].
Therefore, hybrid quantum system built from these different modules paves the
way for a multi-functional and scalable quantum processor. However, achieving
the transduction between different domains with high quantum efficiency and
low added noise is still challenging.

Figure 52:  Diagram showing the coupling between optical, mechanical, and SC
qubit devices with different optical and mechanical structures. The black link
shows the work demonstrated in the literature, while the black dashed link
hasn’t been achieved yet. Reprinted from Ref. [217].

Figure 51 illustrates a diagram in which the piezoelectric material plays an
important role in mutually coupling SC qubit, optical photon, and phonon. The
coupling between SC qubit and mechanics through the piezoelectric effect has
been initially demonstrated by Andrew N Cleland et al.in 2010 using an AlN BAW
resonator [190]. As nanofabrication technologies advance, major progress has
been made in coupling SC qubit with various mechanical structures, such as SAW
[191, 422, 423], HBAR [172, 173], and phononic crystal [192, 424, 425], as
shown in Fig. 52. It has grown as a new field of the so-called circuit quantum
acousto-dynamics (cQAD). The cQAD has recently witnessed promising advances,
from phonon-number-resolved readout by SC qubit [192] to phonon-mediated
remote communication and entanglement between SC qubits [423, 191]. The
further development of cQAD will lead to indispensable building blocks in the
future SC quantum computing system, featuring a long coherence time and a
compact size.

On the other hand, piezoelectric materials also exhibit an electro-optic
effect which allows for direct coupling between SC qubit and optical photon.
Although the quantum transduction efficiency from microwave to optics is still
low compared with the transduction mediated by mechanics, classic optical
control and readout of SC qubit has been demonstrated recently [368, 367,
366], which can serve as the photonic link for SC circuits between cryogenic
and room temperatures. The optomechanical interaction between photon and
phonon is the main focus of this review, and its quantum application was
discussed in the previous chapter. As shown in Fig. 52, the coupling between
various optical and mechanical modes has been demonstrated except for the
modulation of photonic crystal using BAW resonator. It is worth exploring this
combination in the future since it supports higher optical $Q$ and thermal
conductance than the OMC, and smaller mode volume than the optical ring.

From the above discussion, one can envision an architecture of the hybrid
quantum system where different functional modules based on distinct quantum
objects are coordinated together to produce a high-level quantum processor
that is capable of computing (SC qubit), storing (phonon) and communicating
(photon) quantum information with high fidelity and scalability. Additionally,
piezoelectric materials (which are also electro-optic materials) serve to
electrically connect these modules, allowing hybrid integration through wire
bonding or flip-chip techniques [426]. The materials required for each module
are compatible with currently standard CMOS processes, such as Si, Si3N4, and
AlN, which enables chip-scale integration and massive production in the
future.

Figure 53:  Programmable photonic circuits enabled by piezoelectric actuators.
(a) Schematic of the Mach-Zehnder Mesh (MZM) for arbitrary linear optical
transformation, which consists of cascaded Mach-Zehnder Interferometers (MZI).
(b) Example of one MZI with the optical phases of each arm controlled by the
piezoelectric phase shifter. (c) Experimental realization of an MZI with four
piezoelectric phase shifters. (d) Scanning electron microscopy (SEM) image of
the cross-section of the released AlN piezoelectric actuator. (c-d) Reprinted
from Ref. [221].

###  7.2 Piezoelectric programmable photonic circuits

Programmable photonic circuits [230, 231, 229] have evolved as the core
processor in today’s advanced classical and quantum optical computing
applications, such as optical neural networks (ONN) for deep learning [55] and
the photonic quantum simulator [56]. The main building block of the circuit is
a mesh of Mach-Zehnder Interferometers as shown in Fig. 53(a). In each MZI
unit, the light output can be programmed by controlling the relative phases
between the top and bottom arms (see Fig. 53(b)). By encoding the optical
information in each input channel (1, 2, 3, …), the circuit is capable of
implementing a unitary linear-optical transformation at the output, which is
one of the fundamental operations in complex computation tasks such as optical
quantum information processing and machine learning.

The key component of the programmable MZI is the optical phase shifter. For
better performance of the MZI, an ideal phase shifter would require a high-
speed response (large bandwidth), low power consumption, and compact size. The
latter two metrics together determine the scalability of the Mach-Zehnder mesh
(MZM). To achieve the unitary transformation of $N$ input channels, total of
$N(N-1)/2$ MZI will be needed [231]. This $O(N^{2})$ scaling places a restrict
requirement on the power consumption and size of the designed phase shifter.
Conventionally, the MZI is thermally tuned to build the programmable circuit,
from Si ONN [55] to the implementation of quantum algorithms on a Si3N4
photonic circuits [56]. Although the thermal phase shifter is easy to design
and fabricate, its actuation speed is slow (millisecond) and power consumption
is large (milliwatt). Moreover, the thermal tuning is incompatible with
cryogenic experiments, such as co-integration with SC circuits, single photon
detector, or trapped ion. Modulation of the Si waveguide through free carrier
injection has been demonstrated with low power and high speed [427]. However,
the light propagation loss is too high for large-scale integration. Although
the electrostatic phase shifter provides a solution for low power and compact
size, its bandwidth is limited by the mechanical resonance to 10-100 kHz [100,
87, 104, 428]. Therefore, a lower power, low optical loss, high-speed, and
compact-size phase shifter will become essential for future programmable
photonic circuits.

Piezoelectric actuators could become a good candidate for programming MZI in
terms of power and speed. As shown in Fig.53(c, d), the piezoelectric tuning
of a Si3N4 MZI using AlN actuators has recently been demonstrated with a
dynamic and static power consumption of 200 $\mu$W and 6 nW, respectively,
allowing operation at a cryogenic temperature of 5 K [221]. The actuator shows
a 3-dB bandwidth of up to 120 MHz. Implementing a $4\times 4$ unitary
transformation is achieved using 6 MZI. The main factor that currently limits
the scalability of the piezoelectric programmed is the relatively high voltage
length product $V_{\pi}L$, which trades off between drive voltage and device
length. For the above work, the $V_{\pi}L$ is 50 V$\cdot$cm. To have a low
voltage that is compatible with CMOS circuits, the length of the phase shifter
must be on the order of centimeters. Although folding the waveguide shrinks
the footprint of an MZI into 1.3$\times$0.6 mm2 [221], the long propagation
length and bending loss lead to a high insertion loss which also prevents
large-scale integration. Therefore, it is essential to optimize the design of
the piezoelectric actuator to have smaller $V_{\pi}L$. If we recall from Eq.
10, the change of refractive index due to the photoelastic effect is
proportional to $n^{3}$ of the waveguide. Thus, Si waveguide with higher
refractive index is worth of reconsidering, in addition to its mature CMOS-
compatible fabrication. On the other hand, the choice of materials with higher
piezoelectric coefficients, such as PZT and AlScN, could further improve the
tuning efficiency by five to ten times. With these improvements, the building
of scalable and high-speed programmable photonic circuits will be within reach
in the near future.

###  7.3 Synthetic frequency dimensions in photonics

Synthetic dimension has drawn much research interest in recent decades, as it
provides a way to study high-dimensional physics beyond the intuitive three
spatial dimensions. By exploring the coupling between degrees of freedom other
than the physical space, a high-dimensional lattice can be synthesized in a
low-dimensional structure. This allows the investigation of rich topological
phenomena that are otherwise hard or impossible to implement in real space.
For instance, the magnitude and phase of the coupling between sites in the
synthesized lattice can be controlled easily, such that the long-range hopping
can be induced, where novel topological and condensed matter phenomena can be
studied [58]. Various platforms have been proposed and developed to realize
synthetic dimensions, such as cold atoms [429, 430], superconducting circuits
[431, 432], optomechanics [433, 434], and photonics [435, 59].

Using optical photons as the media to build the synthetic dimension has been
attractive in recent research, benefiting from the versatile internal degrees
of freedom of photons, such as frequency, polarization, orbital angular
momentum (OAM), waveguide modes, and arrival time of pulses. These degrees
have long been utilized in signal multiplexing for high-speed optical
telecommunication. Here, optical lattices synthesized in these new dimensions
paves the way to simulating high-dimensional lattices for studying fundamental
physics and exploring nontrivial topological effects [436, 435], such as the
four-dimensional quantum Hall effect [437], photonic gauge potential [438,
439], quantum walks [440, 441, 442], Bloch oscillation [443], and photonic
topological insulator [444]. The exploration of these effects in turn helps
the development of novel photonic devices that provide new ways to manipulate
the frequency and propagation of photons. For example, the synthetic quantum
Hall lattice enables unidirectional frequency translation that is
topologically protected [60], and nonreciprocal optical transmission has been
demonstrated [445].

Figure 54:  Synthetic frequency dimension formed in dynamically modulated
optical resonator. Schematics of the synthesized optical frequency lattice in
(a) 1D, (b) 2D, and (c) 3D by inducing the coupling of adjacent modes (red),
the next-nearest neighbour (green), and the next-next-nearest neighbour
(yellow). (a-c) Reprinted from Ref. [217]. (d) Schematic of the experimental
realization of the optical computing in frequency dimension using acousto-
optically modulated photonic crystal resonator. (e) Diagram illustrating the
matrix vector multiplications in frequency lattice by programming the mutual
coupling among lattice sites through optical modulation. (d-e) Reprinted from
Ref. [446].

Synthetic dimension in frequency is of particular interest because of the
ultra-wide optical bandwidth which provides a large computational space. In
addition, the optical phase modulator has been well developed, making it
convenient to couple different frequency components. An optical ring resonator
consists of a series of equidistant optical resonances, which naturally forms
an optical lattice in frequency. Phase modulation of the resonator at the
frequency equal to the FSR of the resonator introduces coupling between each
two adjacent modes, as shown in Fig. 54(a). The dynamics of the coupling can
be well described by the tight-binding model, which was originally applied in
solid-state physics to model the band structure of the electron in a crystal
lattice. The corresponding Hamiltonian can be written as [58]:

$H_{\text{TB}}=\sum_{k}\omega_{k}a_{k}^{{\dagger}}a_{k}+\sum_{k}ga_{(k+1)}^{{\dagger}}a_{k}e^{i\phi}+h.c.$
(37)

where $\omega_{k}$ is the frequency of each optical mode which serves as an
on-site potential. $g$ and $\phi$ are the coupling amplitude and phase between
two adjacent modes, which are determined by the microwave drive. $\hbar$ is
omitted for simplicity. Therefore, it can be seen the dynamics of the photon
in the synthesized frequency lattice mimics the hopping of electron in an atom
lattice, indicating the potential of optical simulation. Since the reciprocal
variable of frequency is time, the band structure can be easily obtained by
measuring the time resolved spectrum of the modulated resonator [59].

Beyond 1D lattice, higher dimensional frequency lattice can be constructed by
adding microwave drives at multiple FSRs, as shown in Fig. 54(b, c). These
drives induce long-range hopping terms in the Hamiltonian as
$\sum_{k}g_{N}a_{(k+N)}^{{\dagger}}a_{k}e^{i\phi_{N}}$, which allows us to
fold the 1D lattice into more complex structure [447, 436], such as 2D, 3D, or
beyond. Very rich topological phenomena can be studied by engineering the
magnitude and phase of each coupling term $g_{N}e^{i\phi_{N}}$. For instance,
the modulation phase $\phi_{N}$ serves as an effective gauge potential for the
photon in the frequency dimension [58]. An effective magnetic field can be
generated, depending on the phase difference between coupling terms, as shown
in Fig. 54(b). Additionally, if the drive is detuned from the FSR, an
effective electric force will be formed, making it possible to study the Bloch
oscillation of photon in the frequency lattice [443]. The frequency dimension
can further combine with the spatial dimension by coupling multiple optical
resonators, enabling unique control of the photon in frequency and real space
[437, 439, 445].

The experimental realization of the synthetic frequency dimension in photonics
has witnessed promising progress recently. The initial proves of the concept
were demonstrated using fiber optics and off-the-shelf electro-optic
modulators, including synthetic gauge potential [59], synthetic Hall ladder
[60], effective electric field [448], and high-dimensional lattices [436].
Although this approach is simple to implement, the large size of the fiber
optical ring resonator limits its scalability and application in making novel
optical devices. A few recent works have achieved the synthetic frequency
dimension in integrated photonic platforms, such as thin film lithium niobate
microresonator that is electro-optically modulated [442, 449], and Si photonic
resonator modulated by a p-n junction [450]. Since the synthetic frequency
dimension only requires narrow band frequency modulation, acousto-optic
modulator could play an important role, benefiting from its high modulation
efficiency at mechanical resonances. A synthetic Hall effect has been
demonstrated by modulating two coupled AlN photonic resonators through SAW,
and nonreciprocal optical transmission is achieved by engineering the
modulation phases [445]. The main challenge for AOM is to match the acoustic
resonance with the FSR, especially for on-chip optical resonators with FSR
typically from 10 GHz to THz range. Low-FSR Si3N4 ring resonators have been
demonstrated with FSR of 2.5 GHz and optical $Q$ over 5 million [52, 26],
which allows for on-chip synthetic frequency dimension enabled by AOM.

In addition to studying the topological effects, the synthetic frequency
dimension has been utilized to implement an arbitrary linear transformation by
engineering the coupling between frequency modes [451]. Similar to the MZM
presented in the last section, the input optical states can be arbitrarily
mapped to the output but in the frequency space. The advantages of frequency
domain implementation include the compact device size and scalability, since
all optical channels transmit in the same optical waveguide. Recently, matrix
vector multiplication has been demonstrated by acousto-optic modulating a
single Si photonic crystal resonator [446], as shown in Fig. 54(d, e). In the
future, linear optical quantum information processing in the frequency
dimension can be achieved by incorporating non-classical light. Although
experimental realizations using bulk EOM and pulse shaper have been achieved
[417, 416], chip-scale photonic circuit and AOM could enable large-scale
optical computing with high-fidelity frequency-bin logic gates [451, 446].
Additionally, a non-unitary transformation can be realized by introducing the
modulation of gain or loss of the optical resonator, which is generally the
case in neural networks [451]. Finally, combining with the optical
nonlinearity of the photonic waveguide, such as Si3N$4$, Si, and LiNbO$3$,
novel dynamics of the soliton generation has been theoretically predicted
[452]. The nonlinearity is also required for implementing deep learning
algorithms, a step for fully optical neural networks [55]. Therefore, we
believe the integrated AOM will play an important role in both classical and
quantum optical computing in the synthetic frequency dimension through
efficient modulation and frequency conversion.

##  8 Conclusion

In this review, the role of integrated piezoelectric actuators in controlling
and modulating photonic devices via optomechanical interactions, including
photoelastic and moving boundary effects, was explored. Compared to electro-
optic modulation, piezoelectric actuation works with optical materials devoid
of the Pockels effect, like Si and Si3N4. This review thoroughly examines the
piezoelectric tuning of optical ring resonators and waveguides, focusing on
piezoelectric materials (mainly AlN and PZT) and the design of mechanical
structures (either released or unreleased). Achievements include low power
consumption (tens of nanowatts) and rapid actuation speeds (sub-microsecond),
offering benefits over traditional thermal tuning methods. Although the tuning
range of piezoelectric actuators remains less than that of integrated heaters,
their quick response facilitates a broad range of applications in integrated
lasers and frequency comb sources, including frequency-agile narrow-linewidth
lasers, the initiation and control of Kerr microcombs, and the stabilization
of frequency comb lines.

Beyond quasi-DC tuning, the piezoelectric actuator also facilitates acousto-
optic modulation at microwave frequencies, exhibiting higher efficiency
compared to EOMs at the expense of narrow bandwidth. We explored various
integrated AOMs utilizing distinct acoustic modes, ranging from surface and
bulk acoustic waves to optomechanical crystals, each offering unique benefits.
For instance, surface acoustic waves provide high modulation efficiency,
whereas bulk acoustic wave resonators are noted for their small size and
compatibility with thick optical cladding to achieve high optical $Q$.
Moreover, optomechanical crystals have shown the greatest optomechanical
coupling strength, enhancing their utility in quantum transduction. These
miniature AOMs are extensively used in a range of classical and quantum
optical devices, including optical isolators, microwave to optical converters,
and optical frequency shifters, leading the way in controlling light
propagation and frequency on a chip-scale. This is crucial for the development
of future hybrid integrated systems for optical classical and quantum
information processing.

Equipped with DC tuning and AOM, the concept of a multifunctional photonic
processor is conceivable, as depicted in Fig. 1 at the outset of this review.
Ranging from tunable laser sources and programmable beam splitters to optical
modulators, the Piezo-on-Photonic platform facilitates the integration of
these elements on a single chip, albeit requiring moderate adjustments in
device design and fabrication processes to suit various materials, etching
selectivity, and release compatibility. Looking ahead, piezoelectric actuators
are poised to significantly contribute to the development of programmable
photonic circuits across spatial and frequency domains, enabling sophisticated
optical simulation and computation. Furthermore, investigating new
piezoelectric materials like AlScN [453, 454], BaTiO3 [121], and HZO [455]
could enhance tuning and modulation efficiency, thus reducing the required
drive voltage $V_{\pi}$ and shrinking the device size. Additionally, exploring
innovative fabrication techniques for the monolithic integration of III-V gain
components on the same piezoelectric-actuated photonic chip could provide
tunable lasers with reduced noise and enable large-scale production [246,
456]. Consequently, we anticipate that piezoelectric actuation will evolve
into an essential component in future photonic circuit designs, offering
distinctive tuning and modulation capabilities with high efficiency, minimal
power requirements, and cost-effectiveness.

##  9 Acknowledgement

We thank Scott E. Kenning and Jiahao Sun for valuable discussion and critical
reading of this review. We thank Bin Dong, Michael Zervas, Anat Siddharth, Rui
Ning Wang, Terence Blésin, Jijun He, Rui Ning Wang, Erwan Lucas, Arslan S.
Raja, Grigory Lihachev, Tianyi Liu, Miles H. Anderson, Wenle Weng, Ben Yu,
Mert Torunbalci, Alaina Attanasio, Wil Kao, Andrea Bancora, Viacheslav
Snigirev, Johann Riemensberger, Vladimir Shadymov, Diego Visani, Andrey
Voloshin for contributions to the fabrication and experimental realization of
piezoelectric actuated photonic devices. We also thank the collaborations from
Christian Koos, Yung Chen, and Huanfa Peng. We acknowledge the support from
the Defense Advanced Research Projects Agency (DARPA) Microsystems Technology
Office (MTO) under contract HR0011-15-C-055 (DODOS), the Air Force Office of
Scientific Research under award no. FA9550-19-1-0250 and no. FA8655-20-1-7009,
US National Science Foundation’s RAISE TAQS program under grant no. PHY
18-39164, the EU H2020 research and innovation program under grant agreement
no. 732894 (HOT), the Swiss National Science Foundation under grant agreement
no. 176563 (BRIDGE), and the Marie Sklodowska-Curie IF grant agreement No.
101033663 (RaMSoM). H.T. acknowledges the support from NSF QISE-Net under
grant no. DMR 17-47426. We thank the support from the EPFL centre of
MicroNanoTechnology (CMi) and the Birck Nanotechnology Center at Purdue
University for the fabrication of integrated Si3N4 photonic chips and AlN
actuators. We also thank Plasma-Therm LLC. and Advanced Modular Systems Inc.
for AlN deposition, and Radiant Technologies Inc. for fabricating PZT
actuators.

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