Spontaneous locking of the phase of a coherent phonon source to an external reference is demonstrated in a deeply sideband-unresolved optomechanical system. The high-amplitude mechanical oscillations are driven by the anharmonic modulation of the radiation pressure force that result from an absorption-mediated free-carrier/temperature limit cycle, i.e., self-pulsing. Synchronization is observed when the pump laser driving the mechanical oscillator to a self-sustained state is modulated by a radiofrequency tone. We employ a pump-probe phonon detection scheme based on an independent optical cavity to observe only the mechanical oscillator dynamics. The lock range of the oscillation frequency, i.e., the Arnold tongue, is experimentally determined over a range of external reference strengths, evidencing the possibility to tune the oscillator frequency for a range up to 350 kHz. The stability of the coherent phonon source is evaluated via its phase noise, with a maximum achieved suppression of 44 dBc/Hz at 1 kHz offset for a 100 MHz mechanical resonator. Introducing a weak modulation in the excitation laser reveals as a further knob to trigger, control and stabilize the dynamical solutions of self-pulsing based optomechanical oscillators, thus enhancing their potential as acoustic wave sources in a single-layer silicon platform.
Enabled by progress in nanofabrication, the enhanced coupling of optical and mechanical degrees of freedom in purposely engineered micro-/nanoscale structures has provided physicists with the possibility to observe new physical phenomena in both the classical and quantum regime . In particular, their interaction in a mechanically compliant and laser-driven optical cavity has led to the design of cavity-optomechanical systems that enables both cooling  and amplification of mechanical motion  by mere selection of the driving laser wavelength. This versatility can be used for tasks spanning the exploration of the quantum nature of mesoscopic objects to on-chip signal processing. Among the applications are a new category of self-sustained radio-frequency oscillators called optomechanical oscillators (OMOs)  which, by construction, also serve as self-sustained coherent phonon sources and may be used as such. Since any physical property in a solid-state system depends on the exact positioning of the atoms, coherent acoustic phonons are extremely suited to dynamically manipulate electric , , optical ,  and magnetic ,  properties of matter. Acoustic wave sources are therefore an enabling technology across physics as well as the first building block for information processing with phonons , .
Even though several strategies to induce mechanical lasing in optomechanical devices have been proposed , , OMOs based on a single optical and mechanical mode are probably the only practical small-footprint implementation , . They can trigger an oscillation from a direct continuous source without needing feedback electronics using mechanisms such as the retarded radiation-pressure force , the back-action induced by the bolometric light force  or by coupling the optomechanical system to carrier/thermal self-sustained cyclic dynamics . In these cases, their all-optical operation, ease of miniaturization, low power consumption and scalability make these oscillators potential candidates as optically driven phonon sources  and a possible replacement to conventional quartz-based oscillators in specific RF-photonic communication and sensing applications such as optical down conversion  or mass sensing . Although the reported output stability of OMOs approaches state-of-the-art optoelectronic oscillators , their performance is often degraded by mechanical effects such as slow frequency drift , intrinsic  or thermomechanical noise  and by instabilities occurring at large displacement amplitudes . Additional postfabrication tunability of their phase and frequency as well as their noise characteristics are therefore critical for their improved performance as both acoustic wave sources and carriers in microwave photonics.
The behavior of a regenerative oscillator driven by an extra external oscillatory signal with a frequency close the oscillation frequency has been explored in a variety of systems such as lasers ,  or masers , electrical tank circuits , , organ pipes , nonlinear mechanical resonators , , superconducting circuits  and human circadian rhythms . These apparently dissimilar systems all share their tendency toward an adjustment of the oscillator’s rhythm to the one externally imposed, as originally proposed by Adler . In recent years, the intrinsic nonlinearity of the optomechanical interaction has been harnessed to explore the spontaneous locking of an oscillator to a reference signal delivered via the driving optical field , , , , via electrical capacitive actuation ,  or by mechanical actuation with propagating acoustic waves . In all these cases, the system is driven to a mechanical lasing state by dynamical back action, which requires a large optomechanical cooperativity . In this article, we demonstrate spontaneous frequency locking of the coherent mechanical motion of a one-dimensional silicon optomechanical crystal (OMC) brought to a self-sustained state using the anharmonic modulation of the radiation pressure force due to the activation of a self-pulsing (SP) mechanism . We use an independent optical cavity the radiofrequency spectrum of which is solely affected by the self-sustained mechanical dynamics. In this way, we overcome a drawback of the transduction schemes in previous power-modulated experiments, which often prevented an unambiguous assessment of the instantaneous amplitude of the mechanical oscillator due to the direct measurement of the reference oscillatory signal.
The particular system investigated here is a pair of close-by one-dimensional OMCs fabricated in a silicon-on-insulator wafer, as shown in the central SEM micrograph of Figure 1. We investigate the fundamental optical cavity mode of each OMC at wavelengths of 1531 nm (right-R) and 1550 nm (left-L), respectively, whose eigenfields are shown in Figure 1(c). Fabrication disorder distorts the nominal y-symmetry of the single OMC optical field profiles and naturally provides the spectral detuning between the two optical cavities, preventing any optical crosstalk as shown in the schematic of Figure 1. In addition, the same distortion leads to a high vacuum optomechanical coupling rate go between the optical resonances and the mechanical flexural modes with three antinodes along the x direction, that would otherwise be optomechanically dark. For the right OMC, this mechanical mode (Figure 1(b)) displays a frequency of fR = 100.37 MHz and a calculated value of go,RR = 2π ⋅ 514 kHz. The precise geometry and reasoning behind the clamping and interconnect structure has been described elsewhere , but it essentially allows the simultaneous optical excitation of the two optical modes as well as controlled phonon leakage from one beam to its neighbor. The mechanical eigenstructure of identical nanobeams would exhibit the symmetry of the system and would induce a perfect hybridization of the two mechanical modes into a symmetric and antisymmetric coupled modes. The mentioned fabrication disorder breaks the as-designed symmetry and the original string-like modes weakly hybridize as a result. However, the two-mode picture with in-phase and anti-phase oscillation still applies (see Supplementary Information). As a result of this weak coupling, a small part of the energy of the mechanical mode supported by the right beam (imperceptible in Figure 1(b)) resides in the left beam, which leads to a small calculated cross-coupling term of go,LR = 2π ⋅ 8 kHz between the mechanical mode of the right OMC and the optical cavity of the left OMC, two orders of magnitude smaller than the direct term go,RR. The mechanical dynamics of the right OMC are controlled via a strong pump laser (dark blue in Figure 1) that couples light into the right optical cavity, eventually driving the mechanical mode into a free-running phonon lasing state at frequency . In this coherent state, the mechanical dynamics are accessed by inspecting the optomechanical transduction of a weak probe optical signal (dark red in Figure 1) addressed on the left cavity, a signal that originates from the cross coupling term go,LR. When the pump laser is modulated with a weak RF tone at frequency fmod far from the bare oscillation frequency , this imprints an additional peak in the power spectral density (PSD) of the pump laser, but does not change the PSD of the probe laser, which, by construction, only shows the peak at fOMO. Bringing fmod sufficiently close to can lock the OMO to fmod, which would lead to a single peak . However, the direct spectral observation of the modulation imposed onto the pump when reading-out the pump after interaction with the sample complicates the unambiguous attribution of the presence of a single peak to injection locking at a particular modulation strength and frequency, as represented via the blue box in Figure 1. To discard the possibility that the external modulation is strong enough to quench the initially prepared lasing dynamical state, a direct determination of the mechanical dynamics is required. To this end the probe cavity is used. When the probe laser is analyzed at the output, the presence of a peak at of similar magnitude to that observed at in the unlocked case is an unambiguous spectral evidence of injection locking, as depicted in the red box of Figure 1.
Measurements are performed at ambient conditions by bringing a tapered microloop optical fiber between the two structures, coupling light propagating in the fiber to the fundamental optical resonance of both beams (Figure 2(b)). The measured optical resonances have quality factors of and , respectively, which are much lower than their calculated values due to the presence of surface roughness. Simultaneous coupling to both cavities is achieved via two tunable near-infrared lasers (Yenista TUNICS T100S-HP) and an in-fiber combiner. The transmitted/reflected power is split and a Fabry–Pérot bandpass filter (BPF) in each arm is used to measure independently the light coming from the chosen laser, allowing the dynamics associated to both optical resonances to be read-out separately. The signals are fed to a spectrum analyzer (SPA), to a 5 GHz oscilloscope and/or to a vector network analyzer (VNA). The latter provides an AC voltage VAC to an electro-optic modulator (EOM) and modulates the intensity of the input laser light, as well as measuring the magnitude and phase of the scattering response (S21) of the system at the excitation frequency. The EOM is set at the quadrature point to suppress higher harmonics, with the half-wave voltage given by . The full measurement setup is shown in Figure 2(a).
Sweeping the laser (1) wavelength at high CW power from the blue-detuned side of the resonance at λR = 1531 nm to λ = 1538 nm leads to multiple dynamical states of the right OMC. For large detunings and relatively low intracavity photon numbers nc, the incoherent mechanical motion activated by the thermal Langevin force imprints a modulation around the transmission DC value. The measured PSD is highlighted with dashed line 1 in the left colormap of Figure 2(c) and evidences the presence of several mechanical modes. In particular, the motion of the aforementioned mechanical mode at is transduced, with a measured natural linewidth of , i.e., . As the wavelength, and therefore nc, increases, the absorption in the tightly confined optical mode volume leads to the generation of free-carriers and to heating of the lattice. Since the refractive index of silicon depends on both temperature (ΔT) via the thermo-optic (TO) effect and free-carrier density (Ne) via free-carrier dispersion (FCD), so does the cavity wavelength. The cavity dynamics are therefore coupled to these two variables via the laser detuning, forming a system of nonlinear coupled differential equations with laser wavelength λL and power Pin as the free parameters. For particular values of these two, the system of equations exhibits a dynamical solution in the form of a anharmonic limit cycle, i.e. self-pulsing (SP), occurring at a frequency νSP. This results in a self-sustained oscillation of the cavity resonance around the laser line at a frequency νSP and coherently modulates the cavity photon number nc. Since νSP depends on the driving parameters, further sweeping the wavelength tunes νSP. Whenever a harmonic of the SP approaches the mechanical frequency fm, the photon number is modulated and the mechanical resonator is pumped via the radiation-pressure force , with . When this happens, the mechanical oscillation generally entrains the SP, forcing νSP to be a simple fraction M of fm . The system therefore undergoes self-sustained, coherent, high-amplitude mechanical oscillations, i.e., mechanical lasing. This can be observed for M = 2 and M = 1 in the white dashed lines 2 and 3 of Figure 2(c). Additional information on the SP and the induced phonon lasing is given in the Supplementary Information and elsewhere , . The interest of using the neigboring OMC can be foreseen on the right colorplot of Figure 2(c), where the spectrum of laser (2) probing cavity L is shown as the wavelength of laser (1) is swept. They evidence that the amount of leaked mechanical energy coming from the oscillation of the right nanobeam and the resulting cross coupling term go,LR enables reading the mechanical dynamics while filtering out other phenomena. Indeed, such a scheme disentangles the read-out of the mechanics from the cyclic dynamics of the free-carriers and temperature, the spatially diffusive nature and characteristic length-scales of which minimize their contribution to the neighbor cavity. It is observed that the modulation peak at 100.37 MHz is present for both an M1 and M2 phonon lasing states—dashed lines 2 and 3 in Figure 2(c)—without the presence of previous harmonics associated to the SP dynamics. The read-out laser (2) power is set to a value sufficiently high to transduce the incoming mechanical signal, but below its own SP Hopf bifurcation . The broad spectral feature spanning 20–40 MHz across the right colormap of Figure 2(c) is a fingerprint of proximity to such bifurcation and is used as a sign of stability in the optical coupling conditions throughout the measurements. We therefore experimentally satisfy the detection scheme depicted in Figure 1, which opens the door to act upon the dynamical state of the complex self-pulsing/lasing state of the right cavity and observing the effect on the mechanical resonator dynamics via the left cavity. This experimental scheme was previously employed in the study by Colombano et al. , where both mechanical modes were brought to lasing and their motion synchronized. Further details are found in the Supplementary Information.
We focus now on the particular situation represented by dashed line 3 in Figure 2(c), where the right beam is pumped by laser (1) to the M = 1 mechanical lasing state at and we modulate the power of the laser at a frequency fmod slightly detuned from the mechanical frequency (, ). Note that the free-running self-sustained oscillation frequency in the absence of external modulation is usually smaller than the natural frequency fR due to thermomechanical softening . In the transmitted signal, RF spectrum two distinct peaks at the oscillator frequency and at the modulation frequency fmod are observed, as shown by the top black curve of Figure 2(d). For a smaller frequency difference, i.e., , only one peak at fmod is observed (blue curve), which is a first fingerprint of injection locking . However, this is unambiguously confirmed by employing the phonon detection scheme previously described. With the laser (2) as the probe set to be in resonance with the optical mode of the left beam and perfectly transmitted by the BPF, we are able to observe the mechanical lasing state of the neighbor cavity. When the modulation to laser (1) is on (bottom of Figure 2(d)), a peak at the modulation frequency fmod becomes prominent only when the mechanical lasing is injection locked to the reference (blue curve) and . For a larger-frequency difference (black curve), only the transduction of the mechanical lasing at fOMO is strong. Note that fOMO in the bottom panel is slightly smaller than in the top panel due to temporal drift associated to thermal and/or alignment variations. In these modulation conditions, the presence of the small peak at the modulation frequency evidences the onset of nontrivial mechanical dynamics of the lasing cavity. Indeed, a possible contribution from free-carrier/temperature dynamics at fmod has already been discarded by the fact that only the mechanical peak is detected in that configuration when the lasing cavity is in M = 2 (Figure 2(c)).
The range of modulation frequencies for which injection locking occurs, i.e., the Arnold tongue , is investigated by performing measurements such as those shown in Figure 2(d) while step-changing the excitation frequency from lower to higher frequencies. The results for , corresponding to a VNA tone of −5 dBm, are depicted as a colormap in Figure 3(a and b), with the BPFs centered in λ = 1536 nm and λ = 1551 nm, respectively. Note that during the acquisition of Figure 3(a and b), fOMO = 100.322 MHz in the absence of modulation is slightly lower than previously shown, which is due to a change in the optical coupling condition. Panel (a) clearly evidences the presence of an injection-locked region spanning 23 kHz, confirmed by the measurement performed on the other cavity (panel (b)). The absence of the directly transmitted and modulated light allows the evolution of the dynamics of the mechanical oscillator to be observed. The first remarkable observation is that prior to injection locking, the mechanical response exhibits a peak at the modulation frequency fmod, indicating that the trajectory of the oscillator becomes non sinusoidal, as was already pointed out in Figure 2(d). Moreover, approaching the edges of the lock range, the peak at the mechanical lasing is dragged toward fmod, i.e., frequency pulling ,  occurs, which is a typical feature of self-sustained oscillators driven by an external source , . At the borders of the lock range, the quasi-locked regime , , where the oscillator periodically oscillates between a locked and unlocked state , is observed (see Supplementary Information). In addition to acquiring the PSD signal of the transmitted light, the complex scattering parameter S21 of the system is measured by the VNA. Figure 3(c and d) shows its magnitude and phase when measured with the filter pass-band in resonance with either laser. The signal obtained by measuring at the pump laser wavelength is complex since both the modulation from the directly transmitted light and the modulation from the effect of the frequency tone on the cavity dynamics can interfere. However, the amplitude response observed in Figure 3(d) is simpler and shows that the oscillation amplitude of the mechanical oscillator remains constant throughout the lock range, at least for the corresponding modulation amplitude, and that the RF tone appearing at the borders outside that range follows the characteristic lineshape of a driven harmonic oscillator. Furthermore, the phase response Δϕ is characterized by an evolution of approximately across the lock range, which is another standard feature of injection-locked electric tank oscillators  observed in power-modulated OMOs . The PSD signal acquired for several modulation amplitudes VAC, now only measuring with the filter transmitting the laser light outcoupled from the probe cavity, is depicted in Figure 4. This set of measurements clearly demonstrates the increasing size of the Arnold tongue with modulation amplitude and its asymmetric growth, i.e., a larger lock range at . This asymmetry, which is not reported in other injection-locked systems , , is simulated via numerical integration of the full system of equations (see Supplementary Information) and is not due to a hysteretic behavior of the system. It originates from high-order synchronization peaks  which are specially relevant here due to the specific anharmonic nature of the self-pulsing dynamics. For the modulation at excitation frequencies fmod close to becomes too strong and the initial phonon lasing dynamics are lost, leading to a driven harmonic oscillator with a Lorentzian-like amplitude response in those regions and to an overall response that goes beyond pure locking.
As seen in Figure 4, injection locking controls the oscillation frequency over bandwidths as high as several hundreds of linewidths of the bare OMO. Locking to a low-noise reference also improves its stability, gauged over short times via its phase noise. The phase noise analysis was carried out with a different fiber taper, slightly changing the external coupling condition and resulting in an even wider Arnold tongue at a fixed modulation amplitude and exhibiting injection locking physics for voltages up to (PVNA = −10 dBm). The phase noise measurements are carried out using the dedicated module of the SPA and setting the frequency carrier to the oscillator frequency , which matches fmod inside the Arnold tongue. Figure 5(a) depicts the phase noise as a function of frequency offset for different levels of modulation power, starting from the free-running phonon lasing (black dashed line). We first see that for powers up to the phase noise increases for frequency offsets below 20 kHz. We attribute this to an unstable behavior of the system, giving rise to hopping between the locked and the unlocked state due to environmental noise. At larger excitation powers the OMO locks to the external reference and, from that modulation amplitude, a strong and progressive phase noise reduction can be observed, especially at frequency offsets below 30 kHz but spanning the full range of the phase noise measurement setup up to 1 MHz. The broad peak around 7 kHz arises from an unknown noise source in the optical path and is disregarded. The phase noise level at frequency offsets of 1, 10 and 100 kHz is given as a function of modulation amplitude in the inset to Figure 5. It shows both the increased low-offset phase noise prior to synchronization and an approximately linear phase noise reduction in the power lock range. A reduction of 1.1 dBc/Hz per excitation dBm is observed both at 1 kHz (blue) and 10 kHz (black), while the reduction is least pronounced at 100 kHz offset (red) and above. Given those values, the maximum achieved phase noise suppression is 44 dBc/Hz (1 kHz), 20 dBc/Hz (10 kHz) and 8.5 dBc/Hz (100 kHz). Last, the behavior of the phase noise at as a function of the modulation frequency fmod across the full Arnold tongue is assessed. When the modulation tone is swept from 100 MHz, outside the lock range, the modulation tone appears as a noise peak at the frequency difference (see Figure 5(b)), as do all sidebands at frequency offsets . Close to the onset of injection locking, at 100.22 MHz, where these additional peaks disappear, a considerable increase in the low-frequency offset phase noise is observed, that is also attributed to hopping between the locked and unlocked state . This phase noise behavior is probably also associated to the quasi-locked regime described in the Supplementary Information. These considerations also apply in the opposite side of the lock range, where the increased phase noise is both more pronounced and extends throughout a broader range. Inside the Arnold tongue, the phase noise characteristics remain practically constant, evidencing the potential of injection locking to generate low-noise oscillatory references and coherent phonon sources with controlled frequency and purity.
In conclusion, we have reported the observation of injection locking of self-pulsing-driven regenerative oscillations of an optomechanical system to an external optical reference and isolated the mechanical signal by engineering the system geometry and read-out scheme. The lock range achieved can go up to 350 kHz, enabling a large tuning range over the oscillator frequency. Besides the direct consequences for acoustic wave self-sustained sources of the results shown here, applications using distributed arrays of oscillators within a single small-footprint chip, like robust time-keeping with standard optoelectronic oscillators , or neuromorphic computing , may benefit from the reported physics of injection locking from a variety of perspectives. Variations in natural mechanical frequencies due to unwanted fabrication variations make mutual synchronization of weakly coupled beams as in the study by Colombano et al.  relatively hard to observe. Due to the wide lock range achieved, injection locking can be used to presynchronize one of the coupled OMOs to an external drive and use it as a lead oscillator to which the others can synchronize via weak mechanical links, a configuration under exploration. Moreover, in cases where mechanical lasing of a particular mode cannot be achieved with an adiabatic laser parameter sweep, usually because the attained self-pulsing frequency νSP is still far from fm, injection locking of the SP limit cycle to the external tone can be used to increase νSP. With this additional degree of freedom mechanical lasing can be reached, leading to a state that persists after the modulation is switched off, i.e., it enables exploration of usually inaccessible dynamical attractors of the system. Pulsed or modulated operation in this configuration could enable switching between different dynamical states, as reported in the study by Maire et al.  with a top pumping scheme. Furthermore, the feed forward nature of the technique stabilizes the oscillator without the need of feedback circuits, achieving a phase noise suppression up to 44 dBc/Hz at 1 kHz and 8.5 dBc/Hz at 100 kHz, resulting in an oscillator operated at atmospheric conditions with phase noise levels close to state of the art silicon optomechanical oscillators , , .
Funding source: European Comission
Award Identifier / Grant number: PHENOMEN (GA: 713450)
Funding source: CERCA Programme / Generalitat de Catalunya
Funding source: Ministerio de Ciencia, Innovación y Universidades
Award Identifier / Grant number: PGC2018-094490-B-C22PGC2018-101743-B-I00RYC-2014-15392
Funding source: Spanish Research Agency
Award Identifier / Grant number: AEI, grant no. SEV-2017-0706
The authors thank Dr. E. Chavez for his careful and critical reading of the manuscript.
Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This research was funded by EU FET Open project PHENOMEN (GA: 713450). ICN2 is supported by the Severo Ochoa program from the Spanish Research Agency (AEI, grant no. SEV-2017-0706) and by the CERCA Programme/Generalitat de Catalunya. G. A. and C. M. S.-T. acknowledge the support from the Spanish MICINN project SIP (PGC2018-101743-B-I00). D. N. U., G. A. and M. F. C. gratefully acknowledge the support of a Ramón y Cajal postdoctoral fellowship (RYC-2014-15392), a BIST studentship, and a Severo Ochoa studentship, respectively. D. N. U. acknowledges the funding through the Ministry of Science, Innovation and Universities (PGC2018-094490-B-C22).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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