# Cascaded injection locking of optomechanical crystal oscillators

David Alonso-Tomás  MIND-IN2UB, Departament d’Enginyeria Electrónica i
Biomédica, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1,
Barcelona 08028, Spain  Guillermo Arregui  DTU Electro, Department of
Electrical and Photonics Engineering, Technical University of Denmark, Ørsteds
Plads 343, Kgs. Lyngby, DK-2800, Denmark  Laura Mercadé  Nanophotonics
Technology Center, Universitat Politècnica de València, Camino de Vera s/n,
46022 Valencia, Spain  Alejandro Martínez  Nanophotonics Technology Center,
Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Amadeu Griol  Nanophotonics Technology Center, Universitat Politècnica de
València, Camino de Vera s/n, 46022 Valencia, Spain  Néstor E. Capuj  Depto.
Física, Universidad de La Laguna, 38200 San Cristóbal de La Laguna, Spain
Instituto Universitario de Materiales y Nanotecnología, Universidad de La
Laguna, 38071 Santa Cruz de Tenerife, Spain  Daniel Navarro-Urrios
<EMAIL_ADDRESS>MIND-IN2UB, Departament d’Enginyeria Electrónica i Biomédica,
Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, Barcelona
08028, Spain

###### Abstract

Optomechanical oscillators stand out as high-performance and versatile
candidates for serving as reference clocks in sequential photonic integrated
circuits. Indeed, they have the unique capability of simultaneously generating
mechanical tones and optical signal modulations at frequencies determined by
their geometrical design. In this context, the concept of synchronization
introduces a powerful means to precisely coordinate the dynamics of multiple
oscillators in a controlled manner, thus increasing efficiency and preventing
errors in signal processing photonic systems or communication interfaces. In
this work, we demonstrate the cascaded injection locking of a pair of silicon-
based optomechanical crystal cavities to an external reference signal that
subtly modulates the laser driving one of the oscillators. Both cavities
interact solely through a weak mechanical link, making the extension of this
synchronization mechanism to an increased number of optomechanical oscillators
within a common chip more feasible than relying solely on optical
interactions. Thus, the combination of the obtained results, supported by a
numerical model, with remote optical injection locking schemes discussed in
the literature, lays the groundwork for the distribution of reference signals
within large networks of processing elements in future phonon-photon hybrid
circuits.

††preprint: APS/123-QED

Among the diverse array of optomechanical (OM) devices [1], optomechanical
oscillators (OMOs) have attracted significant attention due to their potential
in addressing various scientific and technological challenges. These
oscillators are exceptional platforms, characterized by high-amplitude, self-
sustained and coherent mechanical motion driven and controlled by optical
fields [2]. In this context, although they have already found applications in
precision sensing [3, 4, 5] as well as in frequency generation and conversion
[6, 7, 8, 9] among others, their properties make them excellent candidates for
serving as clock signals in photonic integrated processing circuits. In fact,
with the recent advances in this field and the possibility of combining
optomechanical crystal cavities (OMCs) with routed microwave phonons [10], the
approach could be naturally extended to chip-integrated phonon-photon
platforms where multiple of these oscillators interact with each other.
However, the extension of this system to a large network of OMOs generates the
need of a mechanism for preventing errors related to frequency dispersion and
improving the efficiency of individual OMOs in terms of phase noise and
frequency jitter.
In this context, injection locking, an established process in electronic and
radio-frequency (RF) systems pioneered by the seminal paper of Adler et al.
[11], holds the key to synchronize actions and further improve the overall
response of the system. In an injection locking mechanism, the frequency and
phase of an oscillator become synchronized with those of an external signal.
This external signal is injected into the oscillator and originates from
another local oscillator (LO) that acts as a reference clock. The first
observation of injection locking in an OMO was documented in a silicon
microtoroid subjected to a modulation of the same optical drive that leads to
the optomechanical oscillations [12]. Subsequently, additional works have
demonstrated injection locking in various OMOs using a similar optical
injection scheme where the LO comes from a signal generator [13, 14, 15] or
even from another OMO [16, 17, 18]. Injection locking has also been achieved
through electrical capacitive actuation [19, 20] and by mechanical actuation
with propagating acoustic waves [21].
In order to distribute the reference signal within a chip, a natural strategy
is that of synchronizing several OMOs to the external signal, physically
placed at different locations of the chip, in a cascaded injection locking
(CIL) configuration. To achieve this, supplying a modulated optical pump
concurrently or sequentially to multiple OMOs, either in a parallel or series
arrangement, appears to be a viable approach. However, the practical execution
of this strategy on a reasonably large scale presents a significant challenge
due to the requirement that all these OMOs must share an identical resonant
optical wavelength. In this context a small number of reports have claimed the
spontaneous synchronization of OMOs sharing a common optical mode [22, 23,
24]. In fact, up to date there is only one work reporting the cascaded locking
of individual OMOs [25]. In that study, the OMOs are optically interconnected
by means of a common waveguide in a sequential arrangement and no external
reference was fed to the first OMO. Even though this configuration allows
scalability, its implementation with silicon microdisk difficult the direct
extraction of coherent mechanical signals. Moreover, OMCs suffer from stronger
optical dispersion, making an all-optically mediated synchronization
complicated.

Here, we demonstrate an approach for achieving CIL of the dynamics of a pair
of chip-integrated OMCs acting as OMOs to an external reference signal. In our
platform, the resonators are optically isolated from each other and interact
through a weak mechanical link. Exploiting a mechanical interaction in spite
of an optical one allows a more straightforward means of achieving spontaneous
synchronization of OMCs [26], given that the mechanical resonant frequency
dispersion among different OMCs is usually much smaller than the optical
counterpart. The interaction is considered unidirectional since one of them
(referred to as the ”main” or ”leader”) oscillates with significantly larger
amplitude and injection-locks to the reference tone, while the other one
(designated as the ”secondary” or ”follower”) spontaneously synchronizes to
the main one and/or the external reference only through the weak mechanical
interaction. In this context, we also report that it is possible to injection-
lock the secondary OMO to the external reference, even though the main OMO is
the one receiving the reference tone and oscillating independently.

Fig. 1: Studied configuration and spontaneous synchronization of two
optomechanical crystal oscillators (OMOs) a Conceptual illustration of the
proposed system, where both OMOs, represented as two Fabry-Perot cavities with
one moveable mirror attached to a spring, interact through another spring of
lower elastic constant. Both structures support optical and mechanical modes
with wavelengths and mechanical angular frequencies $\lambda_{i}$ and
$\Omega_{i}$, where i = M,S denotes main and secondary respectively. Laser
light propagating through a tapered fiber is evanescently coupled to each
optical mode with an extrinsic rate $\kappa_{ex,i}$. An external modulation
with frequency $\Omega_{mod}/2\pi$ is applied to the light in resonance with
the main OMO, which is transduced in the secondary channel due to the weak
interaction between both cavities. b SEM micrograph of two close-by silicon
optomechanical crystal nanobeams that are connected by a mechanical link.
White scale bar is 2 $\mu m$. Both beams support optical cavity modes that are
spectrally and spatially isolated, as shown in the optical transmission
spectrum of panel c (optical resonance at room temperature appear at
$\lambda_{S,0}=1535.4$nm and $\lambda_{M,0}=1542.4$nm). d and e Radio-
frequency spectra of the optical transmission associated to the main and the
secondary OMOs respectively, for different values of the wavelength of the
laser driving the main OMO ($\lambda_{M}$). The spectral range in which
spontaneous synchronization occurs is highlighted with green dashed lines. The
wavelength of the laser driving the secondary is fixed at
$\lambda_{S}=1537nm$. A yellow dashed line in panel illustrates the conditions
that are kept in the experiments with an external modulation.

## Results

Optical-Mechanical-Optical Configuration. The experiment conducted in this
work closely parallels the scenario depicted in Figure 1a. Two OM cavities,
represented as Fabry-Perot cavities with a mirror attached to a spring, weakly
interacts through another spring of lower elastic constant. Both cavities
support different optical resonances, with red representing the main OMO
($\lambda_{M}$) and blue representing the secondary one ($\lambda_{S}$), which
are resonantly pumped by evanescently coupled laser light propagating through
a tapered fiber. Each oscillator has a resonant mechanical angular frequency
of $\Omega_{i}$, where $i=M,S$ denotes main and secondary respectively. The
oscillation in the cavity length leads to a spectral shift of the optical
resonances. The strength with which these resonant frequencies ($\omega=2\pi
c/\lambda$) are pulled (linearly) in terms of mechanical displacement can be
defined as $G=\partial\omega/\partial x$, which is related to the vacuum
optomechanical coupling rate $g_{0}=Gx_{\mathrm{ZPF}}$. Here,
$x_{\mathrm{ZPF}}=\sqrt{\hbar/2m\Omega}$ represents the mechanical zero-point
fluctuation of the mechanical mode with effective mass m and frequency
$\Omega$. It is worth noting that if the incident laser wavelength is in
resonance with an optical mode, the light transmitted will undergo a
modulation enveloped by the characteristic Lorentzian shape of a Fabry-perot
cavity mode. Hence, input light acts as a probe to observe the mechanical
oscillation of OMOs, but at the same time can play an active role in the
optomechanical system through radiation pressure forces.

Both OM cavities are optically driven into a state of high amplitude, coherent
and self-sustained mechanical motion, referred to as mechanical lasing, in
such a way that one oscillates with significantly larger amplitude than the
other. This configuration effectively emulates a leader-follower arrangement
of oscillators [27]. Subsequently, a periodic external force is applied solely
to the main oscillator, resulting from an external modulation in the amplitude
of the input light in resonance with its supported optical mode. Thus, the
secondary one perceive its actuation only through the weak mechanical
interaction between the systems. If the frequency of the external modulation
is close enough to the resonant frequency of one of the OMOs, its dynamics can
be locked to the external modulation [15]. CIL occurs when the two resonators
synchronize to the external perturbation.

Consider the presented scheme, where the interaction can be understood as a
restoring force emerging from the mechanical link, the equations governing the
dynamics of the leader-follower arrangement of oscillators can be described as
two reactively coupled linear forced harmonic oscillators:

$\ddot{x}_{i}+\Gamma_{i}\dot{x}_{i}+\Omega_{i}^{2}x_{i}=\frac{F_{0,i}(t)}{m_{i}}-2J\Omega_{i}x_{M}\delta_{i,S}\\\
$ (1)

where $\delta$ is the dirac delta, J corresponds to the reactive coupling
strength, $\Gamma_{i}$ is the mechanical decay rate of each resonator and
$F_{0,i}(t)=g_{0,i}\hbar n_{0,i}(t)/x_{ZPF}$, the radiation pressure force
exerted by the temporally dependent intra-cavity photon number ($n_{0,i}$).
The temporal behaviour of this magnitude is governed by the self-induced
modulation associated to the driving mechanism, as well as a periodic external
modulation with frequency ($f_{mod}=\Omega_{mod}/2\pi$) in the case of the
main OMO equation (see [15, 28] and supplementary section S3 for more
details). The coupling term only appears in the equation corresponding to the
follower oscillator since the main cavity oscillates with much higher
amplitude than the secondary one, emulating the unidirectionality of the
system.

Tested Device and Experimental Setup. In this work we use silicon one-
dimensional OMCs, behaving as OMOs. These structures are planarly chip-
integrated suspended nanobeams which behave as photonic crystals with a defect
region, whose design and fabrication methodology are detailed in the Methods
section. The tested device is composed of a pair of nominally identical OMOs,
where the outermost five cells of each nanobeam are anchored to the frame,
restricting the flexural modes of the geometry to its central area. The
specific geometry of the OMO devices, allows positioning the tapered fiber
between both geometries, enabling their simultaneous optical excitation. Here,
the OM coupling is understood as the change in resonant optical frequency
induced by the moving boundaries and photo-elastic effect contributions of the
mechanical mode [29].

The interaction between both OMOs is provided through an engineered mechanical
link (Fig. 1b) between them, placed on the anchored zone, thus ensuring a weak
mechanical coupling. In this platform, the pumping mechanism enabling
mechanical lasing of the OMOs is the anharmonic radiation pressure modulation
$F_{0,i}(t)$ due to the activation of a thermo-optic/free-carrier dispersion
self-pulsing (SP) limit-cycle that emerges in silicon under a certain
threshold of number of photons inside the cavity (see Methods and
supplementary information S2). This driving mechanism is useful when the
system is in the sideband unresolved regime, where the decay rate of the
optical cavity modes is much higher than the frequencies of mechanical
resonators, and has been reported by us and other groups previously [30, 31,
32]. In particular, we have driven to the mechanical lasing regime the three-
antinodes flexural mechanical modes of the secondary and main OMOs, whose room
temperature natural frequencies are ($f_{S}$,$f_{M}$) = ($\Omega_{S}$,
$\Omega_{M}$)/2$\pi$ = (99.57, 99.65) MHz. Here, fabrication disorders break
the symmetry, causing the OMCs to exhibit different mechanical natural
frequencies. These fabrication deviations also lead to an increase in the OM
coupling of these in-plane flexural modes to hundreds of kHz [26], even though
the simulated OM coupling of the nominal geometry is null.

The experimental setup, which is schematically represented in the Methods
section, implements the configuration described in Fig. 1a. The two OMOs are
excited using tunable diode lasers, respectively denoted by L1 and L2, whose
polarization state is set to be transverse electric (i.e., electric field in
the plane of the sample), using fiber polarization controlers (FPC). Light
emitted from L1 is amplitude modulated using a Mach-Zehnder electro-optic
modulator (EOM), with a half-wave voltage $V_{\pi}$ = 3.5 V. A signal
generator (SG) provides the periodic external modulation. Both lasers are then
combined into a single tapered fiber that has been shaped into a microloop at
its thinnest region [33]. The bottom part of the loop acts as a probe that
allows the local excitation of the optical cavity modes of the OMOs if the
cavity regions are in the near field of the fiber and the lasers are in
resonance with the respective optical modes. The signal decoupled from the
resonators is then split and passed through Fabry-Perot wavelength filters
(WF), to afterwards impinge on two photodetectors (PD1 and PD2). The
electrical output can be analyzed in the frequency domain by a spectrum
analyzer (SA) and in the temporal domain with a 4-channel oscilloscope (OSC).
All the electronic equipment owns a response bandwidth of 11 GHz.

The optical transmission spectra of the OMOs displayed in Figure 1c shows that
the first optical resonances of each of the OMOs are separated by about 7nm
(1535.4 nm for the secondary OMO and 1542.4 nm for the main one). This large
separation (0.45%) emerges from fabrication imperfections and the presence of
the fiber, since optical properties are very sensible to small deviations in
the geometry or the surrouding media. In this case, the optical power coupled
fraction for both OMOs is roughly 35% and the loaded optical quality factors
are $Q_{S}=7.3\cdot 10^{3}$ and $Q_{M}=11.3\cdot 10^{3}$, while the intrinsic
values are $9.4\cdot 10^{3}$ and $14.3\cdot 10^{3}$, respectively. This
configuration ensures the absence of direct optical interaction between the
two OMOs.

Spontaneous Synchronization. First, we focus on the case where no external
modulation is provided. As previously explored in [26], the weak interaction
between the two self-sustained OMOs in the mechanical lasing regime can induce
spontaneous synchronization of the follower oscillator to the leader within a
specific range of detuning between their natural frequencies. In the present
study, the oscillation frequencies of both OMOs just above the self-
oscillation threshold are too far apart to enable spontaneous synchronization
of their dynamics. To control the frequency difference, we increased the
wavelength of the laser driving the OMO with the larger mechanical frequency,
which in this case is the main cavity. This leads to an increase in the time-
averaged number of intracavity photons, causing more photons to be absorbed
and the geometry to heat up. Figure 1d and 1e illustrate this experiment. The
top panels show, in a contour plot representation, the RF spectrum of the
signal detected by PD1 and PD2 as the wavelength of L1 is modified. The bottom
panels illustrate the RF spectrum at the initial wavelength. The spectra
corresponding to the main OMO (Figure 1d) show that $f_{M}$ reduces gradually
as the wavelength increases, in an almost linear way due to a relaxation of
the elastic constants of the material [34]. Here, the only signs of the
mechanical feedback coming from the secondary OMO appear as RF peaks at
$f_{S}$ and $2f_{M}-f_{S}$. These sidebands slightly approach $f_{S}$ and
disappear within a range of wavelengths between 1544.8 nm and 1546.1 nm, which
corresponds to an OMO frequency difference of $|f_{M}-f_{S}|<$ 25 KHz.

The secondary OMO dynamics displayed in Figure 1e reveals a similar behaviour.
At the initial condition, a high-amplitude RF peak is placed at $f_{S}$,
surrounded by two principal sidebands at $f_{M}$ and $2f_{S}-f_{M}$ of much
larger amplitude than in the previous case, which confirms the unidirectional
character of the interaction. Here, it is clear that the range in which the
sidebands disappear matches with an abrupt jump of $f_{S}$ becoming now equal
to $f_{M}$, which is evidence of spontaneous synchronization of the secondary
OMO dynamics to that of the main OMO. The magnitude of the main RF peak
associated to the secondary OMO oscillation keeps the same value along the
whole spectral range analyzed in the experiment, which rules out the
possibility of resonant forcing. The wide range of tunability of $f_{M}$,
expanded when compared to the case reported in Ref. [26], allows displaying
the whole synchronization range, i.e., the whole Arnold tongue. Under these
conditions, the Arnold tongue appears to be symmetric with respect to the
natural value of $f_{S}$.

Fig. 2: Colormaps of the power spectral density (PSD) of the optical
transmission associated to the secondary OMO when an external harmonic tone
modulates the main one. a Explanation of the observed RF peaks at low
amplitude of external modulation. The vertical axis is associated to the
modulation frequency ($f_{\mathrm{mod}}$) and is highlighted with a green
dashed arrow. The RF peaks associated to the natural oscillation of the
secondary resonator and the main one are highlighted with blue and red dots.
The rest of the peaks are sidebands appearing at different frequency
combinations of the mentioned peaks. The amplitude of the voltage applied to
the modulator ($V_{\mathrm{max}}$) is 0.0042$V_{\pi}$. b-d Set of panels
representing PSD versus $f_{\mathrm{mod}}$ for $V_{\mathrm{max}}$ varying from
$V_{max}$= 0.0051$V_{\pi}$ to $V_{\mathrm{max}}$ = 0.0081$V_{\pi}$. The range
of $f_{mod}$ in which secondary OMO, main OMO, and cascaded injection locking
are observed are highlighted with blue, red and light green arrows. e
Representative RF spectra of each regime observed in the case of higher
amplitude of modulation ($V_{\mathrm{max}}$ = 0.0081$V_{\pi}$). Grey, blue,
green and red areas denote free-running, secondary, cascaded and main lock-in
respectively. These colors are used also to highlight the position of the RF
external tone, and the main and secondary oscillating frequencies (green, red
and blue, respectively). f-i Numerical simulations showcasing similar regimes
to the ones observed in experimental measurements. Selected voltage
($V_{\mathrm{max}}$) is (0.0127, 0.0196, 0.0254, 0.350) $V_{\pi}$,
respectively.

Cascaded Injection Locking. The previous experiment demonstrates the
spontaneous synchronization of the two OMOs in a leader-follower scheme. In
the upcoming studies, we keep the wavelength of L1 at 1544 nm (see the yellow
dashed line in Figure 1e), where $|f_{M}-f_{S}|=$ 45 KHz, placing the two OMOs
outside the Arnold tongue, i.e., they do not synchronize spontaneously. We now
introduce a harmonic signal of frequency $f_{\mathrm{mod}}$ and amplitude
$V_{\mathrm{max}}$ generated by the SG, which modulates the amplitude of the
output of L1. Figure 2 shows the result of monitoring the RF signal of the
secondary OMO optical channel as a function of $f_{\mathrm{mod}}$ for
different $V_{\mathrm{max}}$ values. When $f_{\mathrm{mod}}$ is sufficiently
detuned from the natural oscillation frequencies, the RF spectrum consists of
the peaks reported in Figure 1e, with the main peak at $f_{S}$ surrounded by
two main sidebands at $f_{M}$ and $2f_{M}-f_{S}$ (see Figure 2a or the grey
spectra in Figure 2e). Additionally, there is the contribution of the
modulation tone (green dashed arrow in Figure 2a). Given that the modulation
is applied to L1, the appearance of this tone in the signal corresponding to
the secondary OMO is a consequence of the interaction with the main OMO.
Indeed, it follows the spectral response of the flexural in-plane mechanical
modes, which have intrinsic natural linewidths of typically 0.2 MHz at
atmospheric pressure.

Now, as $f_{\mathrm{mod}}$ increases and approaches to $f_{S}$ we observe
sidebands corresponding to the beat between the external modulation and higher
harmonics of the secondary OMO signal. When the frequency of the external
modulation is close enough to $f_{S}$, the latter suffer a frequency pulling
effect towards $f_{\mathrm{mod}}$ which ends up with a narrow injection
locking region of the secondary OMO dynamics to the external tone. By further
increasing $f_{\mathrm{mod}}$, so that we analyze the region around $f_{M}$,
the same effects are displayed but in a stronger extent. This is consistent
with the fact that the main OMO is directly experiencing the modulation of the
radiation pressure force while the secondary OMO senses a much weaker
mechanical perturbation through the mechanical link. It is worth noting that
the frequency pulling effect in both injection locking ranges is the expected
behaviour in a phase locking mechanism [15].

If we focus now on panels 2b-d, we observe that the two locking regions
described before, denoted with blue and red arrows for the secondary and main
OMO injection locking ranges respectively, widen as $V_{\mathrm{max}}$
increases. Above $V_{\mathrm{max}}$ = 0.0051$V_{\pi}$, CIL of the two
oscillators to the external drive is achieved, which is highlighted with a
light green arrow. The CIL range starts on the lower-frequency end of the main
OMO injection locking range, which is where the frequencies of the two OMOs
are closer. Within this regime, the RF spectrum collapses into a single tone
at $f_{\mathrm{mod}}$ (see green spectra of Fig. 2e). The frequency range of
CIL widens as $V_{\mathrm{max}}$ increases, achieving few tens of kHz in the
extreme case (see Fig. 2d). For large enough $f_{mod}$ values, the two OMO
frequencies become too far apart to keep the CIL and it is lost, however, the
main OMO remains locked to the external reference signal through a larger
bandwidth. In this later situation, the secondary cavity oscillates
independently again (red spectra of figure 2e) and sidebands corresponding to
the beat between the external modulation and secondary OMO harmonics appear at
a combination of their frequencies. It is important to mention that figure 2b
and 2c-d represent different accesses to the CIL regime. In the first case,
the secondary oscillator spontaneously synchronizes to the perturbation
generated by the main one when it is locked to the external frequency. This
occurs because the detuning between mechanical frequencies after the
injection-locking of the main OMO, falls below the threshold for spontaneous
synchronization (Fig. 1e). On the other hand, when the external modulation
generates a mechanical perturbation large enough to maintain the secondary OMO
locked until the main Arnold tongue, the CIL is accessed by the locking of
both oscillators to the external feedback.

Fig. 3: Temporal traces, Poincaré map and phase noise of the cascaded
injection locking state. a Temporal traces of the reference tone (dark green)
and of the optical transmission of the secondary OMO (blue) as recorded
simultaneously in two channels of the oscilloscope. b Enlargement of a single
period of the two signals. c Time trace of the secondary OMO transmission
signal in a polar representation, where the distance to the centre is the
transmission amplitude value and the angle is the phase difference with
respect to the maximum of the modulation signal. The total temporal register
used for this representation is 1 $\mu s$. d Phase noise measurements of the
free-running secondary OMO (blue), radio-frequency reference tone (dark green)
and of the secondary (light green) in the CIL regime.

The system of coupled differential equations corresponding to the mechanical
resonators (Eq. 1) and the SP mechanism (see Methods and supplementary S2) can
be solved numerically to compute the temporal dynamics of intra-cavity photons
and, hence, the transmission temporal traces for the main and secondary OMO
(see Supplementary Material S3). In order to compare numerical simulations
with experimental data, a Fast Fourier Transform (FFT) is performed on the
computed transmission temporal trace of the secondary OMO as
$f_{\mathrm{mod}}$ is swept over a range between the natural frequencies.
Figures 2g-i show the same contour RF representation used to visualize
experimental measurements for different amplitudes of the external modulation.
Here, simulations clearly reproduce the different regimes observed, including
the injection locking range induced in the secondary OMO by the mechanical
perturbation, as well as the CIL of both OMOs for higher amplitudes of
modulation.

Temporal Traces and Phase Noise. Temporal measurements of the dynamics of the
two OMOs in the CIL regime ($f_{\mathrm{mod}}$ = 99.61 MHz) were performed
using the oscilloscope. Figure 3a represents the time traces of the external
modulation signal and the secondary OMO when it is locked to the external tone
by means of CIL. Both traces are taken simultaneously using the first one as a
trigger signal in the oscilloscope. A zoom of a single period of the two
traces is shown in Figure 3b, where the secondary OMO transmission is now
represented in a color scale. Figure 3c displays the transmission signal in a
polar representation suitable to avoid self-intersections and to superimpose
all the full cycles recorded in the transmission temporal trace, which spans a
total of 1 $\mu s$, i.e., about 100 cycles. This is a specific type of
Poincaré map in which the transmission of the secondary OMO is sampled at a
frequency equal to $f_{\mathrm{mod}}$ over all the cycles. The radius is
associated to the limit-cycle optical transmission trace and the angle to the
relative phase of the harmonic modulation signal with respect to its maximum,
i.e., we express the reference signal as $\cos(2\pi f_{\mathrm{mod}}t)$, where
$2\pi f_{\mathrm{mod}}t$ is the polar angle of the plot. To better illustrate
this, the transmission curves have been plotted in both sub-panels with a
common color scale linked to each phase point. If there is no frequency
locking between the two signals, the resulting curve will slightly rotate with
time, filling the whole polar space if the temporal acquisition is long
enough. However, the trajectory drawn in Figure 3c repeats itself in every
cycle and no drift is observed regardless of the acquisition time.

Finally, high-precision linewidth measurements were registered using a phase
noise analyzer integrated in the SA. In Figure 3d we compare the phase noise
of the free-running secondary OMO (blue), with that of the signal generator
(dark green) and that of the secondary OMO (light green) within the CIL
regime. The part of the phase noise associated to the secondary OMO that is
greatly improved is that involving the long-term stability of the OMOs, that
is, below offset frequencies of some kHz. We associated this effect to the
weak nature of the synchronization mechanism. Indeed, in Ref [26] we
demonstrated that it takes several hundreds of oscillations to restore the
spontaneous synchronization dynamics when the system is perturbed externally.
The performance of the secondary OMO remains the same for frequencies above 10
kHz, which involve faster noise mechanisms [35] to which spontaneous
synchronization does not provide any improvement.

## Discussion

We have studied the dynamics of a pair of Si-based OMCs acting as OMOs, that
are weakly coupled mechanically. In the absence of an external signal, the
system behaves as a leader-follower configuration, where the main resonator
exhibits a larger oscillation amplitude. Additionally, we observed regions of
injection-locking in both OMO dynamics when introducing an external harmonic
modulation on the laser driving the main oscillator at frequencies close to
their natural ones. This behavior is anticipated in the case of the main OMO,
given that the modulation is directly applied to it [15], but it is not
straightforward to observe in the case of the secondary OMO. The origin of
this latter injection-locking mechanism lies in the mechanical interaction
between both integrated OMOs, which remains effective even if the main one is
not injection-locked to the reference. As a result of the study, we have
unequivocally demonstrated, both in the frequency and temporal domains, the
cascaded injection-locking of the OMOs to an external reference signal,
whereby the main OMO is injection-locked to the reference and the secondary
OMO is spontaneously synchronized to the mechanical perturbation coming from
the link. All these results were qualitatively reproduced with numerical
simulations modeling the non-linear dynamics of the coupled system of
oscillators.

The reported cascaded injection-locking mechanism in chip-integrated Si-based
OMOs, where the interaction is provided through a mechanical link, offers an
alternative for distributing signals among OMOs that do not share a common
optical mode. This scheme enables the use of multiple optical channels and
avoids issues related to optical frequency dispersion among nominally
equivalent chip-integrated OMCs due to fabrication disorder. In this context,
these geometries not only allow for the extraction of coherent mechanical
signals through the substrate, leading to photon-phonon hybrid circuits [10],
but also can be optically excited by common on-chip bus waveguides [36].
Hence, the proposed platform could be extended to multiple oscillators
sequentially locked to the one receiving a reference signal, with the
adaptation time of the OMOs to the external reference being an important
factor to consider. Combining the presented configuration with an optical
injection locking scheme [18] opens a path for distributing reference signals
along distant clusters of processing elements interacting remotely. The
complex nonlinear dynamics emerging in this type of platform could be
leveraged not only for signal distribution but also for advanced applications,
including enhanced frequency generation or conversion, as well as
ultrasensitive measurements by monitoring the collective dynamics of the
system [37].

Overall, this study represents a necessary step towards achieving full dynamic
control of phonon lasers through synchronization, which is a critical element
in the development of phonon-photon hybrid circuits.

## Methods

OM crystal design. The experimental structures employed in this study consist
of one-dimensional silicon-based OM crystal cavities. These are standalone
nanobeams simultaneously functioning as photonic and phononic crystals. The
unit cell features a central mass with a pitch (a) of 500 nm, a central hole
with a radius (r) of 150 nm, and stubs extending from the top and bottom of
the central mass with a length (d) of 250 nm (see Figure 5). The incorporation
of holes and stubs in the same geometry enables independent modification of
the photonic or phononic bandgap [38]. Parameters are specifically design so
that the geometry acts as a full photonic and phononic mirror in the frequency
ranges of 200 THz and 4 GHz. The adiabatic reduction of these parameters at
the center of the crystal, to $\Gamma$ = 85% of their original values, forms a
defect region giving rise to confined modes (which are not used in this work).
The OM crystal comprises 32 unit cells, collectively spanning a length of
approximately 16 $\mu$m.

OM crystal fabrication. In this work, the studied device comprises two
equivalent one-dimensional OM crystals joined by one of its stubs at one end.
These crystal pairs were fabricated on conventional silicon-on-insulator (SOI)
SOITEC wafers, featuring a silicon layer with a thickness of 220 nm, a
resistivity of approximately 1 to 10 $\Omega$ cm-1, and p-doping at around
$10^{15}$ cm-3. The SOI wafer included a buried oxide layer with a thickness
of 2 $\mu$m. The design was etched onto a poly-methyl-methacrylate (PMMA)
resist film with a thickness of 100 nm using electron beam technology.
Subsequently, the pattern was transferred into the silicon layer using
Reactive Ion Etching (RIE). Buffered Hydrofluoric Acid (BHF) was then applied
to remove the buried oxide layer and release the fabricated beam structures.

Experimental Setup.

Fig. 4: Scheme of the experimental setup. Left panel shows a SEM image of the
chip-integrated OM device, where main and secondary OMOs are highlighted in
red and blue respectively. Right panel shows an schematic representation of
the experimental layout described in the main text. L1 and L2, tunable lasers;
EOM, electro-optic modulator; SG, signal generator; OSC, oscilloscope; SA,
spectrum analyzer; PD1 and PD2, photodetectors; L1 FILT and L2 FILT, Fabry-
Perot wavelength filters; VOA, variable optical attenuator.

Equations of motion of free-carrier, temperature and optomechanical systems.
In this section, we report the mechanism used to drive optomechanical cavities
to a coherent, high-amplitude and self-sustained state of mechanical
oscillation in the sideband unresolved regime. It is based on the interplay
between the thermo-optic (TO) effect and free carrier dispersion (FCD) that
emerges in silicon crystals. Both effects have an impact on the refractive
index of the material and hence, on the optical resonance of the cavity. The
dynamical evolution of free carriers ($N$) and increase in temperature
($\Delta T$) can be expressed as two coupled differential macroscopic
equations:

$\displaystyle\dot{N}$
$\displaystyle=-\frac{1}{\tau_{FC}}N+\alpha_{SPA}n_{0}(N_{0}-N)$ (2)
$\displaystyle\dot{\Delta T}=-\frac{1}{\tau_{T}}\Delta T+\alpha_{FC}n_{0}N$

where the coupling magnitude is the intracavity photons,
$n_{0}=n_{0,m}\Delta\lambda_{0}^{2}/(\Delta\lambda_{0}^{2}+4(\lambda_{r}-\lambda_{l})^{2})$,
with $n_{0,m}=\frac{2P_{l}\kappa_{e}\lambda_{0}}{\kappa^{2}hc}$. Here,
$\Delta\lambda_{0}$ is the linewidth of the optical resonance at room
temperature, $P_{l}$ and $\lambda_{l}$ the power and wavelength of the
incident laser light, respectively; $\kappa$ the overall damping rate and
$\kappa_{e}$ the extrinsic one. Note that the position of the resonance is
modified by the TO and FCD contributions and can be written in first-order
approximation as
$\lambda_{r}\approx\lambda_{0}-\frac{\partial\lambda_{r}}{\partial
N}N+\frac{\partial\lambda_{r}}{\partial T}\Delta T$. Regarding the meaning of
the coupled system, the first equation takes into account the single-photon
absorption (SPA) through $N_{0}$ intragap states per unit volume and a
recombination time of $\tau_{FC}$. The second one considers the fraction of
photons that are absorbed and transformed into heat through free carrier
absorption (FCA). In that way, $\alpha_{SPA}$ represents the rate of free-
carrier density increase per photon and unit of density of available intragap
states, while $\alpha_{FC}$ the rate of temperature increase per photon and
unit of free-carrier density. It is worth mentioning that the response of
$n_{0}$ to the different contributions its adiabatic since the system is in
the regime where $\kappa$ is much larger than the characteristic rates
(1/$\tau_{FC}$, 1/$\tau_{SPA}$).

Under specific conditions of the driving laser, the whole system can enter in
a dynamical regime described by a self-sustained limit cycle, which we refer
as self-pulsing. The modulation on intracavity photons is then transduced
linearly to the radiation pressure optical force that drive the oscillator
($F_{0}(t)\propto n_{0}$). Hence, the mechanical modes of the optomechanical
crystal can be described as damped linear harmonic oscillators driven by an
anharmonic force:

$\frac{d^{2}x(t)}{dt^{2}}+\Gamma\frac{dx(t)}{dt}+\Omega^{2}x(t)=\frac{F_{0}(t)}{m_{eff}}$
(3)

The anharmonic modulation of intra-cavity photons induced by the self-pulsing
limit cycle at a certain frequency $\nu_{SP}$ can do resonant driving of the
mechanical modes, providing amplification and achieving a self-sustained
motion of large amplitude (refered as mechanical lasing for the similarity
with the optical counterpart). This effect can occur for any of the different
M harmonics present in the non-linear modulation if the optomechanical
coupling and the input power are sufficiently high (see supplementary material
S2). It is worth mentioning that once the mechanical lasing regime is
achieved, its contribution to the perturbation generated in $\lambda_{r}$
through the optomechanical coupling can no longer be neglected:

$\lambda_{r}\approx\lambda_{0}-\frac{\partial\lambda_{r}}{\partial
N}N+\frac{\partial\lambda_{r}}{\partial T}\Delta
T+\frac{\lambda_{0}^{2}g_{0}}{2\pi cx_{ZPF}}x$ (4)

## Aknowledgements

This work was supported by the MICINN projects ALLEGRO (Grants No.
PID2021-124618NB-C22 and PID2021-124618NB-C21) and MOCCASIN-2D (Grant No.
TED2021-132040B-C21).

## References

  * [1] Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F.  Cavity optomechanics.  _Rev. Mod. Phys._ 86, 1391–1452 (2014).
  * [2] Kippenberg, T. J., Rokhsari, H., Carmon, T., Scherer, A. & Vahala, K. J.  Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity.  _Phys. Rev. Lett._ 95, 033901 (2005).
  * [3] Yu, W., Jiang, W. C., Lin, Q. & Lu, T.  Cavity optoemchanical spring sensing of single molecules.  _Nat. Commun._ 7, 12311 (2016).
  * [4] Yu, W., Jiang, W. C., Lin, Q. & Lu, T.  Radiation-pressure-antidamping enhanced optomechanical spring sensing.  _ACS Photonics_ 5, 4164–4169 (2018).
  * [5] Liu, F., Alaie, S., Leseman, Z. C. & Hossein-Zadeh, M.  Sub-pg mass sensing and measurement with an optomechanical oscillator.  _Opt. Express_ 21, 19555–19567 (2013).
  * [6] Hossein-Zadeh, M. & Vahala, K.  Photonic rf down-converter based on optomechanical oscillation.  _IEEE Photonics Technology Letters_ 20, 234–236 (2008).
  * [7] Mercadé, L., Morant, M., Griol, A., Lorente, R. & Martínez, A.  Photonic frequency conversion of ofdm microwave signals in a wavelength-scale optomechanical cavity.  _Laser Photon Rev_ 15, 2100175 (2021).
  * [8] Mercadé, L. _et al._ Testing optomechanical microwave oscillators for satcom application.  _Journal of Lightwave Technology_ 40, 4539–4547 (2022).
  * [9] Ghorbel, I., Swiadek, F., Zhu, R. & et al.  Optomechanical gigahertz oscillator made of a two photon absorption free piezoelectric iii/v semiconductor.  _APL Photonics_ 4, 116103 (2019).
  * [10] Fang, K., Matheny, M. H., Luan, X. & Painter, O.  Optical transduction and routing of microwave phonons in cavity-optomechanical circuits.  _Nature Photon._ 10, 489–496 (2016).
  * [11] Adler, R.  A study of locking phenomena in oscillators.  _Proceedings of the IEEE_ 61, 1380–1385 (1973).
  * [12] Hossein-Zadeh, M. & Vahala, K.  Observation of injection locling in an optoemchanical rf oscillator.  _Appl. Phys. Lett._ 93, 191115 (2008).
  * [13] Luan, X., Huang, Y., Y.Li & et al.  An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset.  _Sci Rep_ 4, 6842 (2015).
  * [14] Shlomi, K. _et al._ Synchronization in an optomechanical cavity.  _Phys. Rev. E_ 91, 032910 (2015).
  * [15] Arregui, G. _et al._ Injection locking in an optomechanical coherent phonon source.  _Nanophotonics_ 10, 1319–1327 (2021).
  * [16] Shah, S. Y., Zhang, M., Rand, R. & Lipson, M.  Master-slave locking of optomechanical oscillators over a long distance.  _Phys. Rev. Lett._ 114, 113602 (2015).
  * [17] Li, J. _et al._ All-optical synchronization of remote optomechanical systems.  _Phys. Rev. Lett._ 129, 063605 (2022).
  * [18] Alonso-Tomás, D. _et al._ Unidirectional synchronization of silicon optomechanical nanobeam oscillators by external feedback.  _ACS Photonics_ 11, 7–12 (2023).
  * [19] Pitanti, A. _et al._ Strong opto-electro-mechanical coupling in a silicon photonic crystal cavity.  _Opt. Express._ 23, 3196–3208 (2015).
  * [20] Bekker, C., Kalra, R., Baker, C. & Bowen, W.  Injection locking of an electro-optomechanical device.  _Optica_ 4, 1196–1204 (2017).
  * [21] Huang, K. & Hossein-Zadeh, M.  Injection locking of optomechanical oscillators via acoustic waves.  _Opt. Express._ 26, 8275–8288 (2018).
  * [22] Bagheri, M., Poot, M., Fan, L., Marquardt, F. & Tang, H. X.  Photonic cavity synchronization of nanomechanical oscillators.  _Phys. Rev. Lett._ 111, 213902 (2013).
  * [23] Zhang, M. _et al._ Synchronization of micromechanical oscillators using light.  _Phys. Rev. Lett._ 109, 233906 (2012).
  * [24] Zhang, M., Shah, S., Cardenas, J. & Lipson, M.  Synchronization and phase noise reduction in micromechanical oscillator arrays coupled through light.  _Phys. Rev. Lett._ 115, 163902 (2015).
  * [25] Gil-Santos, E. _et al._ Light-mediated cascaded locking of multiple nano-optomechanical oscillators.  _Phys. Rev. Lett._ 118, 063605 (2017).
  * [26] Colombano, M. F. _et al._ Synchronization of optomechanical nanobeams by mechanical interaction.  _Phys. Rev. Lett._ 123, 017402 (2019).
  * [27] Pykovsky, A., Rosenblum, M. & Kurths, J.  _Synchronization: A Universal Concept in Nonlinear Sciences_ (Cambridge University Press, Cambridge, England, 2003).
  * [28] Navarro-Urrios, D., Arregui, G., Colombano, M. F. & et al.  Giant injection-locking bandwidth of a self-pulsing limit-cycle in an optomechanical cavity.  _Commun. Phys_ 5, 330 (2022).
  * [29] Eichenfield, M., Chan, J., Camacho, R. M., Vahala, K. J. & Painter, O.  Optomechanical crystals.  _Nature_ 462, 78–82 (2009).
  * [30] Johnson, T. J., Borselli, M. & Painter, O.  Self-induced optical modulation of the transmission through a high-q silicon microdisk resonator.  _Opt. Express._ 14, 817–831 (2006).
  * [31] Navarro-Urrios, D., Capuj, N., Gomis-Bresco, J. & et al.  A self-stabilized coherernt phonon source driven by optical forces.  _Sci. Rep._ 5, 15733 (2015).
  * [32] de O. Luiz, G., Rodrigues, C. C., Alegre, T. & Wiederhecker, G. S.  Synchronization of silicon thermal free-carrier oscillators.  _J. Opt. Soc._ 40, 1779–1785 (2023).
  * [33] Ding, L., Belacel, C., Ducci, S., Leo, G. & Fabero, I.  Ultralow loss single-mode silica tapers manufactured by a microheater.  _Appl. Opt._ 49, 2441–2445 (2010).
  * [34] Navarro-Urrios, D. _et al._ Optical and mechanical mode tuning in an optomechanical crystal with light-induced thermal effects.  _J. Appl. Phys._ 116, 093506 (2014).
  * [35] Rokhsari, H., Hossein-Zadeh, M., Hajimiri, A. & Vahala, K.  Brownian noise in radiation-pressure-driven micromechanical oscillators.  _Appl. Phys. Lett_ 89, 261109 (2006).
  * [36] Urrios, D. N. _et al._ Room-temperature silicon platform for ghz-frequency nanoelectro-opto-mechanical systems.  _ACS Photonics_ 9, 413–419 (2022).
  * [37] Lamberti, F.-R., Palanchoke, U., Geurts, T. P. J. & et al.  Real-time sensing with multiplexed optomechanical resonators.  _Nano Lett._ 22, 1866–1873 (2022).
  * [38] Gomis-Bresco, J., Navarro-Urrios, D., Oudich, M. & et al.  A one-dimensional optomechanical crystal with a complete phononic band gap.  _Nat. Commun_ 5, 4452 (2014).

## S1 One-dimensional optomechanical crystal cavities

Fig. 5: Design of the optomechanical (OM) cavity. a SEM image of a fabricated
nominal geometry. The cavity region is highlighted in red, where the
parameters of the unit cell composing the mirror (blue regions), are
adiabatically modified to generate a defect. b Unit cell of the OM crystal
with parameters (a,e,r,d,h) = (500, 220, 150, 200, 1500) nm. c Optical band
dispersion diagram showing a bandgap for TE modes in the range of 200 THz. Red
dashed line represents the frequency of the optical cavity mode, which is
constructed from the lower band edge through the adiabatic modification
towards the center.

## S2 Numerical simulations of the self-pulsing and mechanical lasing
dynamics.

The methods section of the main text presents a model for the description of
the anharmonic modulation of the intra-cavity photon number that emerges in
silicon due to free carrier dispersion (FCD) and thermo-optic (TO) effects.
Here, the system is solved numerically in MATLAB using standard ordinary
differential equation (ODE) solving methods. Parameters of the model are
extracted from previous works and experimental measurements [2, 3, 4, 5].
Initial conditions are settled to [$N$(0), $\Delta T$(0), $x(0)$,
$\dot{x}(0)$] = [$1.5\cdot 10^{17}cm^{-3}$,
$(\lambda_{l}-\lambda_{0})/(\partial\lambda_{r}/\partial T)$, 0, 0]. We select
a temporal step of $\Delta t=2\cdot 10^{-11}s$ to characterize properly the
fast dynamics of free carriers and a time span $\Delta t=1\cdot 10^{-4}s$ for
assuring that the system achieves a stationary dynamical regime. Afterwards, a
Fast Fourier Transform (FFT) is performed to the temporal trace of the
computed transmission (extracted from the intra-cavity photon number) in the
stationary solution. This process is repeated while sweeping the wavelength of
the incident laser light from lower to higher values. Initial conditions are
replaced after the first iteration with the stationary case of previous
solutions. In that way, each iteration depends on the previous one, in a
similar way to the real experiment. The evolution of the obtained RF spectra
is shown in Fig. 6a. Initially, when the laser light have just entered
resonance (which in this case corresponds to 1540 nm), it is possible to
observe a signal oscillating at the frequency of the mechanical mode. Here
light is playing a passive role as a probe of the mechanical oscillation, but
is not providing amplification of the mechanical motion. After a certain
threshold, the self-pulsing (SP) mechanism is activated, whose harmonics can
be visualized as the red curves. When a certain harmonic M resonates with the
mechanical mode, it provides amplification and frequency-locks to it
(plateaus). Note that the increase of photons in the cavity involves heating,
leading to a shift of the optical resonance to longer wavelengths. At some
point, the nonlinear system can no longer display a state of self-sustained
oscillation and the resonance blue-detune from the laser, going back to its
room temperature spectral position.

The particular non-linear dynamics at a certain $\lambda_{l}$ corresponding to
the regime of M = 3 is shown in figures 6 b-e, where the third harmonic of the
SP is the one providing amplification to the mechanical mode. Here, the self-
limit cycle formed by temperature and free-carrier population is depicted (b),
as well as the RF spectra associated with the predicted transmission computed
from $n_{0}$ (c). It is worth noting that in this regime, the amplitude of the
mechanical oscillation is around 0.05 nm (d), extending to an order of
magnitude higher for the M = 1 plateau. Output transmitted light exhibits a
clear nonlinear behaviour (e).

Lastly, in Figs. 6g-f we have compared the simulated RF spectrum using a
larger power value than that used in Fig. 6a with our experimental results,
demonstrating significant agreement. In this case, the dynamical solutions of
the system do not include isolated SP so that different M mechanical lasing
regimes are continuously accessed during the sweep of the laser wavelength,
leading to a staircase-like curve.

Fig. 6: Numerical simulations of the macroscopic model for the self-pulsing
mechanism. a RF spectra obtained when performing a Fast Fourier Transform to
the computed transmission for different values of the incident laser
wavelength. The sweep is done from left (low wavelengths) to right and initial
conditions for each iterations are selected as the solution of the previous
one. The region where the mechanical modes do not experience amplification is
highlighted in green, while the part where the SP is active, in red. b-e
Different magnitudes associated to the solution corresponding to the
mechanical lasing regime (M = 3), where the third harmonic of the SP is the
one providing amplification to the mechanical mode. In particular, the self-
limit cycle drawn by free carriers and temperature, RF spectra, mechanical
oscillator parameters and computed normalized transmission are shown
respectively. f-g Comparison between a simulated contour RF plot (f) obtained
in a similar way to the one shown in (a) with experimental data (g). The
parameters of the simulation as incident power, damping rate or linewidth are
selected to match the ones observed in that measurements.

## S3 Numerical simulations of mechanically coupled OM oscillators

Fig. 7: Schematic representation of the OM system. The tapered fiber is
highlighted in yellow, while red and blue colors denote each oscillator (main
and secondary, respectively). The mechanical part of each OMO is represented
as a mass ($m_{i}$) connected to a spring of elastic constant ($K_{i}$). Each
mechanical subsystem have a certain mechanical angular frequency
($\Omega_{i}$) and dissipation rate ($\Gamma_{i}$). The interaction is
represented with an spring of lower elastic constant $K_{INT}$ $<<$ $K_{M}$ &
$K_{S}$, which result in a coupling rate of
$\Omega_{INT}\sim\sqrt{2J\Omega_{S}}$.

The experiment of the main text considers two mechanically coupled OM crystal
oscillators which are self-sustained by the anharmonic mechanism described
before. The whole picture can be understood through Figure 7, where the main
elements of the system are schematically represented. Two tunable lasers
excite both optical modes which support separated optical resonances, having
extrinsinc ($\kappa_{e,i}$) and intrinsic ($\kappa_{in,i}$) optical decay
rates, respectively. At the same time, the resonant wavelengths are modulated
through the TO and FCD effects following the dynamics of the self-pulsing
mechanism. The optomechanical coupling communicates the optical and mechanical
modes so that this intrinsic modulation of the radiation pressure force can
drive the resonators to the mechanical lasing regime (M = 1). Once this regime
is achieved, the large amplitude of the mechanical oscillation becomes a non-
negligible source of modulation of the intra-cavity photon number.

At this point there are two optical channels undergoing a modulation given by
the mechanical oscillation of each OM crystal. The mechanical interaction is
represented as a spring with low elastic constant ($K_{INT}$ $<<$ $K_{M,S}$)
that communicates both oscillators. The coupling is considered to be reactive
and can be understood as a restoring force that emerges from the linking
tether as it is pulled of its relaxed condition.

On the other hand, an external modulation is also applied to the incident
power that arrives to the main cavity. Here, the applied voltage to the Mach-
Zender electro-optic modulator (EOM) is:

$V(t)=V_{max}sin(2\pi f_{mod}t)+V_{DC}$ (5)

where $f_{mod}$ is the frequency of the external modulation, $V_{DC}$ a
voltage that can be applied to operate in the quadrature point. Thus, the
power that enters in the main cavity can be read as

$P_{input}\propto 1+\cos\left(\pi\frac{sin(2\pi
f_{mod}t)+V_{DC}}{V_{\pi}}\right)$ (6)

where $V_{\pi}$ is the characteristic voltage to change the phase $\pi$
radians in the EOM. The offset voltage ($V_{DC}$) is set at the quadrature
point $V_{DC}$ = 0.5$V_{\pi}$ to minimize larger harmonics in the perturbation
generated by the signal generator (SG) so that, if the maximum voltage of the
RF modulation signal is small, the output light power responds linearly.
Considering that the intra-cavity photon number is proportional to the
incident power, it is arrived to:

$n_{0}^{\prime}(t)=n_{0,m}\left(1-\sin\left(\pi\frac{V_{max}}{V_{\pi}}\sin(2\pi
f_{mod}t)\right)\right)n_{0}(t)$ (7)

where there self-induced modulation will be enveloped by the external one.

Fig. 8: Numerical simulation of the spontaneous synchronization between both
OMOs depending on their separation in mechanical natural frequency ($\Delta$).
The coupling strength (J/2$\pi$) is set to 18 KHz. a Contour radio-frequency
(RF) plot of the normalized Fourier transform of the computed optical
transmission in the second oscillator as a function of $\Delta$. The green
arrow highlight the range where spontaneous synchronization occurs, while the
red dashed one, the sideband associated to the main OMO dynamics. b - d
Representations of the phase space covered by the mechanical displacement
($X_{i}$) of both OMOs for the different regimes observed in the contour plot.

Considering both, the mechanical interaction and the external modulation in
the intracavity photon number, the system of differential equations that
describes the dynamics of both oscillators read as:

$\displaystyle\dot{N_{i}}$
$\displaystyle=-\frac{1}{\tau_{FC}}N_{i}+\alpha_{SPA}n_{0,i}(N_{0}-N_{i})$
(8a) $\displaystyle\dot{\Delta T_{i}}=-\frac{1}{\tau_{T}}\Delta
T_{i}+\alpha_{FC}n_{0,i}N_{i}$ (8b) $\displaystyle\ddot{x_{i}}+$
$\displaystyle\Gamma_{i}\dot{x_{i}}+\Omega_{i}^{2}x_{i}+\delta_{i,S}2Jx_{M}=\frac{\hbar
g_{0,i}}{m_{eff,i}X_{ZPF}}n_{0,i}$ (8c)

where the sub-index i = M,S denotes main and secondary respectively. As
mentioned in the main text, we consider an unidirectional system where the
main OMO is oscillating with much larger mechanical amplitude. Hence, the
interaction is only present in the secondary OMO dynamics since the
contribution of $x_{s}$ has been neglected 2J($x_{M}$ \- $x_{S}$) $\approx$
2J$x_{M}$. Regarding the intracavity photons, the first oscillator is the one
that receives the external modulation, thus $n_{0,M}$ corresponds to the
modulated case (Eq. 7) while the temporal evolution of $n_{0,S}=n_{0}$ is only
governed by the modulation on the optical resonance of the cavity due to the
TO and FCD effects as well as the mechanical motion. This means that the
secondary oscillator is only aware of the external modulation through the
mechanical coupling term. The intrinsic magnitudes of the model (dissipation
rates, intragap-states…) have been considered the same for both oscillators.

Even if Eqs. 8a, 8b and 8c represent a system of nonlinear coupled
differential equations and do not have analytical solution, it is possible to
solve it numerically as a problem of eight first order differential equations.
First, it is necessary to set the incident laser wavelength so that both
oscillators are in the mechanical lasing regime in the absence of mechanical
coupling and external modulation. To do that, parameters J and $V_{max}$ are
settled to 0 and an analysis is performed to find the properly detuning, which
is chosen to be $\lambda_{l,i}-\lambda_{0,i}=4.5$ nm. Given that in the
experiment both mechanical oscillators are relatively close in frequency
(around tens of KHz), simulations require a high-resolution to be able to
resolve that close-by peaks. This can be achieved by increasing the time span
to 4e-4, which provides a resolution of 3KHz after the Fourier transform.

Spontaneous synchronization

Fig. 9: Numerical simulations of the injection locking of the external
modulation signal to the main OMO dynamics. Each panel shows the normalized
fast Fourier transform of the optical transmission in the main cavity when
performing a sweep in the modulation frequency from lower to higher values.
The selected amplitude of modulation is highlighted above each contour plot.
White dashed line indicates the starting value for the locking regime in the
case of low power modulation.

This section deals with the case in which no external modulation is applied.
Once both oscillators are in the mechanical lasing regime, it is possible to
perform a similar analysis to the first experiment of the main work (Figure
1e). We have varied the reactive coupling strength to select the proper value
that reproduces the spontaneous synchronization range of the experiment. Fig.
8a shows the contour RF plot of the computed spectra for the secondary OMO
intra-cavity photons dynamics, when varying the mechanical frequency of the
main OMO and hence, the detuning between both oscillators. The main OMO
dynamics is observed as sidebands that approach to the center peak as the
mechanical detuning between the frequencies both resonators decreases. At the
beginning, since both oscillator are not synchronized, their mechanical
displacements cover most of the phase space when plotted against each other
(Fig. 8b). When their natural frequencies are close enough, spontaneous
synchronization occur and their displacements follow a closed trajectory (Fig.
8c). Finally, when the system exits the Arnold tongue, and the mechanical
displacements plotted in the phase space fill the graph again (Fig. 8d).

Here J/2$\pi$ is set to 18 KHz, providing a spontaneous synchronization range
of 50 KHz, similar to that observed in the experiment. It is important to
mention that in the simulation, the unidirectional character of the
interaction is introduced just by including the interaction term on the
secondary OMO equation and neglecting $x_{S}$ with respect to $x_{M}$.
However, in contrast to the real experiment, both oscillators exhibit similar
amplitudes. Since the spontaneous synchronization range depends on the
amplitude of the mechanical perturbation that the secondary OMO is receiving,
the amplitude of the mechanical oscillation of the main OMO plays an important
role. Hence, the actual interaction strength could be one order of magnitude
lower, confirming the weak character of the coupling and indicating that
fabrication disorders are the main source of splitting between the mechanical
resonant frequencies.

Cascaded injection locking

In this section we include the external modulation with a certain power
$P_{mod}=100Sin(\pi V_{max}/V_{\pi})$, aiming to simulate the cascade
injection locking experiment. In the simulations we establish a frequency
difference of 45 KHz between both OMOs, thus assuring no spontaneous
synchronization occurs in the absence of an external input. Figure 9 focuses
in the main oscillator (the one that directly receives the modulated signal),
where it is represented the fast Fourier transform of transmitted light.
Spectra are represented in a normalized linear scale with arbitrary units.
Here, a sweep in $f_{mod}$ is done from low to high frequencies to observe the
range of injection locking depending on the power of modulation. The step
taken for performing the sweep in external modulation frequency is set to 1
KHz.

Fig. 10: Countour RF plots in a normalized linear scale of the fast Fourier
transform of the optical transmission in the second cavity when performing a
sweep on the external modulation frequency. Panels from left to right have
been computed using different amplitudes of external modulation. The most
intense RF signal corresponds to the secondary OMO natural frequency, while
the sidebands corresponds to the transduced motion from the main oscillator
due to mechanical coupling. The CIL regime is highlighted with a green thick
line.

Regarding the range for injection locking, we observe an increasing on its
value with $P_{mod}$ as expected from previous simulations [6]. The locking
range expands particularly in the region after the initial locking (white
dashed line), which corresponds to the range where the frequency of external
modulation is above the natural one of the oscillator. Finally, it is worth to
mention that the natural frequency of the second oscillator is not present in
these graphs since the coupling term has not been added to the system of the
main oscillator (Eq 8c).

The next step is to analyze the transmission of the second oscillator (the one
that receives the modulation indirectly through the mechanical link). Figure
10 shows the results of this analysis As we expected, the mechanical
oscillation of the main OMO is now observed in the secondary oscillator
dynamics. Even if this modulation arrives to it in an indirect manner, the
mechanical perturbation is enough to generate injection locking of the
secondary OMO dynamics to the external signal, reproducing the experimental
observations. Now, as we increase the modulation power, the locking ranges
increase and start to appear a region where both, the main and the secondary
oscillators are locked to the external modulation, what we call cascade
injection locking (CIL). Simulations also predict the expansion of this range
when increasing the modulation amplitude.

## References

  * [1] Colombano M F, Arregui G, Capuj N E, Pitanti A, Maire J, Griol A, Garrido B, Martinez A, Sotomayor-Torres C M, Navarro-Urrios D. (2019) Synchronization of optomechanical nanobeams by mechanical interaction. $\it{Phys.Rev.Lett}$ 123 017402\.
  * [2] Navarro-Urrios D, Colombano M F, Maire J et al. (2020). Properties of nanocrystalline silicon probed by optomechanics. $\it{Nanophotonics}$, 9(16)
  * [3] Maire J, Chávez-Ángel E, Arregui G et al. (2022). Thermal Properties of Nanocrystalline Silicon Nanobeams. $\it{Adv.Funct.Mater.}$ 32 2105767\.
  * [4] Navarro-Urrios D, Capuj N E, Maire J, Colombano M F et al. (2018). Nanocrystalline silicon optomechanical cavities. $\it{Opt.Express}$ 26 9829-9839
  * [5] Gorodetksy M L, Schliesser A, Anetsberger G, Deleglise S and Kippenberg T G. (2010). Determination of the vacuum optomechanical coupling rate using frequency noise calibration. $\it{Opt.Express}$, 18 23236-23246
  * [6] Arregui G, Colombano M, Maire J, Pitanti A, Capuj N, Griol A, Martínez A, Sotomayor-Torres C, Navarro-Urrios D. (2021). Injection locking in an optomechanical coherent phonon source. $\it{Nanophotonics}$. 10(4) 1319-1327.