Nanophotonics 2021; 10(4): 1319–1327 Research article Guillermo Arregui, Martín F. Colombano, Jeremie Maire, Alessandro Pitanti, Néstor E. Capuj, Amadeu Griol, Alejandro Martínez, Clivia M. Sotomayor-Torres and Daniel Navarro-Urrios* Injection locking in an optomechanical coherent phonon source https://doi.org/10.1515/nanoph-2020-0592 oscillator to a self-sustained state is modulated by a radi- Received November 2, 2020; accepted December 31, 2020; ofrequency tone. We employ a pump-probe phonon published online January 20, 2021 detection scheme based on an independent optical cavity to observe only the mechanical oscillator dynamics. The Abstract: Spontaneous locking of the phase of a coherent lock range of the oscillation frequency, i.e., the Arnold phonon source to an external reference is demonstrated in tongue, is experimentally determined over a range of a deeply sideband-unresolved optomechanical system. external reference strengths, evidencing the possibility to The high-amplitude mechanical oscillations are driven by tune the oscillator frequency for a range up to 350 kHz. The the anharmonic modulation of the radiation pressure force stability of the coherent phonon source is evaluated via its that result from an absorption-mediated free-carrier/tem- phase noise, with a maximum achieved suppression of perature limit cycle, i.e., self-pulsing. Synchronization is 44 dBc/Hz at 1 kHz offset for a 100 MHz mechanical reso- observed when the pump laser driving the mechanical nator. Introducing a weak modulation in the excitation laser reveals as a further knob to trigger, control and sta- bilize the dynamical solutions of self-pulsing based opto- *Corresponding author: Daniel Navarro-Urrios, MIND-IN2UB, mechanical oscillators, thus enhancing their potential as Departament d’Enginyeria Electrònica i Biomèdica, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain, acoustic wave sources in a single-layer silicon platform. E-mail: dnavarro@ub.edu. https://orcid.org/0000-0001-9055-1583 Keywords: injection locking; nonlinear dynamics; Guillermo Arregui, Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science optomechanics; self-sustained oscillator. and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain; and Dept. de Física, Universitat Autonoma de Barcelona, 08193 Enabled by progress in nanofabrication, the enhanced Bellaterra, Spain coupling of optical and mechanical degrees of freedom in Martín F. Colombano, Catalan Institute of Nanoscience and purposely engineered micro-/nanoscale structures has Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science provided physicists with the possibility to observe new and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain; and MIND-IN2UB, Departament d’Enginyeria Electrònica i Biomèdica, physical phenomena in both the classical and quantum Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, 08028 regime [1]. In particular, their interaction in a mechanically Barcelona, Spain compliant and laser-driven optical cavity has led to the Jeremie Maire, Catalan Institute of Nanoscience and Nanotechnology design of cavity-optomechanical systems that enables both (ICN2), CSIC and The Barcelona Institute of Science and Technology, cooling [2] and amplification of mechanical motion [3] by Campus UAB, Bellaterra, 08193 Barcelona, Spain Alessandro Pitanti, NEST Lab., CNR - Istituto di Nanoscienze and mere selection of the driving laser wavelength. This Scuola Normale Superiore, Piazza San Silvestro 12, 56217 Pisa, Italy versatility can be used for tasks spanning the exploration Néstor E. Capuj, Depto. Física, Universidad de La Laguna, 38200 San of the quantum nature of mesoscopic objects to on-chip Cristóbal de La Laguna, Spain; and Instituto Universitario de signal processing. Among the applications are a new Materiales y Nanotecnología, Universidad de La Laguna, 38071 Santa category of self-sustained radio-frequency oscillators Cruz de Tenerife, Spain called optomechanical oscillators (OMOs) [4] which, by Amadeu Griol and Alejandro Martínez, Nanophotonics Technology Center, Universitat Politècnica de Valencia, 46022 Valencia, Spain. construction, also serve as self-sustained coherent phonon https://orcid.org/0000-0001-5448-0140 (A. Martínez) sources and may be used as such. Since any physical Clivia M. Sotomayor-Torres, Catalan Institute of Nanoscience and property in a solid-state system depends on the exact Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science positioning of the atoms, coherent acoustic phonons are and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain; extremely suited to dynamically manipulate electric [5, 6], and ICREA - Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain optical [7, 8] and magnetic [9, 10] properties of matter. Open Access. © 2021 Guillermo Arregui et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 1320 G. Arregui et al.: Injection locking of a phonon source Acoustic wave sources are therefore an enabling technol- using the anharmonic modulation of the radiation pressure ogy across physics as well as the first building block for force due to the activation of a self-pulsing (SP) mechanism information processing with phonons [11, 12]. [44]. We use an independent optical cavity the radio- Even though several strategies to induce mechanical frequency spectrum of which is solely affected by the self- lasing in optomechanical devices have been proposed sustained mechanical dynamics. In this way, we overcome a [13, 14], OMOs based on a single optical and mechanical drawback of the transduction schemes in previous power- mode are probably the only practical small-footprint modulated experiments, which often prevented an unam- implementation [15, 16]. They can trigger an oscillation biguous assessment of the instantaneous amplitude of the from a direct continuous source without needing feedback mechanical oscillator due to the direct measurement of the electronics using mechanisms such as the retarded reference oscillatory signal. radiation-pressure force [13], the back-action induced by The particular system investigated here is a pair of the bolometric light force [17] or by coupling the opto- close-by one-dimensional OMCs fabricated in a silicon-on- mechanical system to carrier/thermal self-sustained cyclic insulator wafer, as shown in the central SEM micrograph of dynamics [18]. In these cases, their all-optical operation, Figure 1. We investigate the fundamental optical cavity ease of miniaturization, low power consumption and mode of each OMC at wavelengths of 1531 nm (right-R) and scalability make these oscillators potential candidates as 1550 nm (left-L), respectively, whose eigenfields are shown optically driven phonon sources [19] and a possible in Figure 1(c). Fabrication disorder distorts the nominal replacement to conventional quartz-based oscillators in y-symmetry of the single OMC optical field profiles and specific RF-photonic communication and sensing appli- naturally provides the spectral detuning between the two cations such as optical down conversion [20] or mass optical cavities, preventing any optical crosstalk as shown sensing [21]. Although the reported output stability of in the schematic of Figure 1. In addition, the same distortion OMOs approaches state-of-the-art optoelectronic oscilla- leads to a high vacuum optomechanical coupling rate go tors [22], their performance is often degraded by mechan- between the optical resonances and the mechanical flexural ical effects such as slow frequency drift [23], intrinsic [24] or modes with three antinodes along the x direction, that thermomechanical noise [25] and by instabilities occurring would otherwise be optomechanically dark. For the right at large displacement amplitudes [26]. Additional post- OMC, this mechanical mode (Figure 1(b)) displays a fabrication tunability of their phase and frequency as well frequency of fR = 100.37 MHz and a calculated value of as their noise characteristics are therefore critical for their go,RR = 2π ⋅ 514 kHz. The precise geometry and reasoning improved performance as both acoustic wave sources and behind the clamping and interconnect structure has been carriers in microwave photonics. described elsewhere [45], but it essentially allows the The behavior of a regenerative oscillator driven by an simultaneous optical excitation of the two optical modes as extra external oscillatory signal with a frequency close the well as controlled phonon leakage from one beam to its oscillation frequency has been explored in a variety of sys- neighbor. The mechanical eigenstructure of identical tems such as lasers [27, 28] or masers [29], electrical tank nanobeams would exhibit the symmetry of the system and circuits [30, 31], organ pipes [32], nonlinear mechanical would induce a perfect hybridization of the two mechanical resonators [33, 34], superconducting circuits [35] and human modes into a symmetric and antisymmetric coupled modes. circadian rhythms [36]. These apparently dissimilar systems The mentioned fabrication disorder breaks the as-designed all share their tendency toward an adjustment of the oscil- symmetry and the original string-like modes weakly hy- lator’s rhythm to the one externally imposed, as originally bridize as a result. However, the two-mode picture with in- proposed by Adler [37]. In recent years, the intrinsic phase and anti-phase oscillation still applies (see Supple- nonlinearity of the optomechanical interaction has been mentary Information). As a result of this weak coupling, a harnessed to explore the spontaneous locking of an oscil- small part of the energy of the mechanical mode supported lator to a reference signal delivered via the driving optical by the right beam (imperceptible in Figure 1(b)) resides in field [22, 38–40], via electrical capacitive actuation [41, 42] or the left beam, which leads to a small calculated cross- by mechanical actuation with propagating acoustic waves coupling term of go,LR = 2π ⋅ 8 kHz between the mechanical [43]. In all these cases, the system is driven to a mechanical mode of the right OMC and the optical cavity of the left OMC, lasing state by dynamical back action, which requires a two orders of magnitude smaller than the direct term go,RR. large optomechanical cooperativity [1]. In this article, we The mechanical dynamics of the right OMC are controlled demonstrate spontaneous frequency locking of the coherent via a strong pump laser (dark blue in Figure 1) that couples mechanical motion of a one-dimensional silicon opto- light into the right optical cavity, eventually driving the mechanical crystal (OMC) brought to a self-sustained state mechanical mode into a free-running phonon lasing state at G. Arregui et al.: Injection locking of a phonon source 1321 frequency fOMO, 0 . 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 fOMO, 0 , 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 fOMO, 0 can lock the OMO to fmod, which would lead to a single peak [38]. 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 Figure 1: Optomechanical system and detection scheme. (a) Illustration of the optomechanical setting explored. Two optical modulation strength and frequency, as represented via the cavities of frequencies ωR (blue) and ωL (red), with ωR − ωL ≫ κ L, R , blue box in Figure 1. To discard the possibility that the can be driven via a common bus waveguide. A single mechanical external modulation is strong enough to quench the initially mode of interest (green) is coupled optomechanically to both optical prepared lasing dynamical state, a direct determination of cavities, although the coupling strength is highly asymmetric, the mechanical dynamics is required. To this end the probe i.e., gRR ≫ gLR . Cavity R is optically driven, leading to cavity is used. When the probe laser is analyzed at the optomechanical self-oscillation of the mechanical mode at frequency fOMO, 0 . This system is realized with two close-by silicon output, the presence of a peak at fmod = fOMO of similar optomechanical crystal nanobeams that are connected by a magnitude to that observed at fOMO, 0 in the unlocked case is mechanical link. Both beams support optical cavity modes that are an unambiguous spectral evidence of injection locking, as spectrally and spatially isolated (c), as shown in the central SEM depicted in the red box of Figure 1. image. A flexural mode of the right beam (b) is selected by pumping Measurements are performed at ambient conditions by (blue path) the right optical cavity mode at a wavelength and power where phonon lasing at frequency fOMO, 0 ∼ fm ∼ 100 MHz occurs. Due bringing a tapered microloop optical fiber between the two to the link, the mechanical mode also couples weakly to the left structures, coupling light propagating in the fiber to the optical cavity, which is addressed via a probe optical signal (red fundamental optical resonance of both beams (Figure 2(b)). path). The optomechanical transduction of the right (left) cavity The measured optical resonances have quality factors of exhibits a strong (weak) peak at the oscillation frequency fOMO, 0 . If QR = 1.6 ⋅ 104 and QL = 8.1 ⋅ 103 , respectively, which are the pump laser is intensity-modulated at frequency fmod, the PSD of transmitted pump (probe) light has (does not have) an additional much lower than their calculated values Qcalc = 6 ⋅ 105 due to peak at fmod. Tuning fmod can lead to injection locking. While the the presence of surface roughness. Simultaneous coupling to presence of a single peak is a necessary condition for injection both cavities is achieved via two tunable near-infrared lasers locking, it is not a sufficient one in the pump PSD (blue box). The (Yenista TUNICS T100S-HP) and an in-fiber combiner. The presence of a peak at fmod in the PSD of the probe laser is however an transmitted/reflected power is split and a Fabry–Pérot unambiguous evidence of injection locking (red box). bandpass filter (BPF) in each arm is used to measure inde- pendently the light coming from the chosen laser, allowing Sweeping the laser (1) wavelength at high CW power the dynamics associated to both optical resonances to be from the blue-detuned side of the resonance at λR = 1531 nm to read-out separately. The signals are fed to a spectrum λ = 1538 nm leads to multiple dynamical states of the right analyzer (SPA), to a 5 GHz oscilloscope and/or to a vector OMC. For large detunings and relatively low intracavity network analyzer (VNA). The latter provides an AC voltage photon numbers nc, the incoherent mechanical motion acti- VAC to an electro-optic modulator (EOM) and modulates the vated by the thermal Langevin force imprints a modulation intensity of the input laser light, as well as measuring the around the transmission DC value. The measured PSD is magnitude and phase of the scattering response (S21) of the highlighted with dashed line 1 in the left colormap of system at the excitation frequency. The EOM is set at the Figure 2(c) and evidences the presence of several mechanical quadrature point VDC = 0.5Vπ to suppress higher harmonics, modes. In particular, the motion of the aforementioned me- with the half-wave voltage given by Vπ = 6.7 V. The full chanical mode at fR = 100.37 MHz is transduced, with a measurement setup is shown in Figure 2(a). measured natural linewidth of Γm = 0.15 MHz, i.e., Qm ∼ 670. 1322 G. Arregui et al.: Injection locking of a phonon source Figure 2: Experimental setup and device characterization. (a) The main optical signals are derived from two external-cavity diode laser and sent into a tapered microloop optical fiber to evanescently couple light into the silicon optomechanical device. The optical signal from the OMC is collected by the same fiber taper and equally split in two, either in transmission or in reflection using an in-fiber circulator. The optical signals in both arms are band-pass filtered by narrow wavelength filters (λ-BPF) and impinge on two fast photodetectors (PDs). The electrical signals are sent to a spectrum analyzer (SPA), a vector network analyzer (VNA) or an oscilloscope (OSC) for characterization. EOM stands for electro-optic modulator and FPC for fiber polarization controller. (b) Optical transmission spectrum, exhibiting two optical modes at λR = 1531 nm and λL = 1550 nm. (c) Colormaps of the power spectral density (PSD) of the transmitted light when sweeping the laser (1) wavelength driving the optical resonance R at high power while another laser (2) weakly probes the resonance L; (left) λBPF = λ1 and (right) λBPF = λ2. The dashed white lines highlight (1) the thermal transduction, (2) M = 2 mechanical lasing and (3) M = 1 mechanical lasing states. (d) Within the configuration achieved in dashed line 3, adding a weak intensity-modulation via the EOM results in a peak in the PSD when its frequency fmod is far from the phonon lasing frequency fOMO (black signal), but locks the oscillator frequency for small detunings (blue signal). When the addressed signal is the probe, the spectra reproduce the ones schematically given in Figure 1. As the wavelength, and therefore nc, increases, the absorp- parameters, further sweeping the wavelength tunes νSP. tion in the tightly confined optical mode volume leads to the Whenever a harmonic of the SP approaches the mechanical generation of free-carriers and to heating of the lattice. Since frequency fm, the photon number is modulated and the me- the refractive index of silicon depends on both temperature chanical resonator is pumped via the radiation-pressure force (ΔT) via the thermo-optic (TO) effect and free-carrier density Frp = ℏGnc , with G = g o /xzpf . When this happens, the me- (Ne) via free-carrier dispersion (FCD), so does the cavity chanical oscillation generally entrains the SP, forcing νSP to wavelength. The cavity dynamics are therefore coupled to be a simple fraction M of fm [46]. The system therefore un- these two variables via the laser detuning, forming a system dergoes self-sustained, coherent, high-amplitude mechanical of nonlinear coupled differential equations with laser wave- oscillations, i.e., mechanical lasing. This can be observed for length λL and power Pin as the free parameters. For particular M = 2 and M = 1 in the white dashed lines 2 and 3 of Figure 2(c). values of these two, the system of equations exhibits a Additional information on the SP and the induced phonon dynamical solution in the form of a anharmonic limit cycle, lasing is given in the Supplementary Information and else- i.e. self-pulsing (SP), occurring at a frequency νSP. This results where [18, 46]. The interest of using the neigboring OMC can in a self-sustained oscillation of the cavity resonance around be foreseen on the right colorplot of Figure 2(c), where the the laser line at a frequency νSP and coherently modulates the spectrum of laser (2) probing cavity L is shown as the wave- cavity photon number nc. Since νSP depends on the driving length of laser (1) is swept. They evidence that the amount of G. Arregui et al.: Injection locking of a phonon source 1323 leaked mechanical energy coming from the oscillation of the only when the mechanical lasing is injection locked to the right nanobeam and the resulting cross coupling term go,LR reference (blue curve) and fOMO = fmod . For a larger- enables reading the mechanical dynamics while filtering out frequency difference (black curve), only the transduction other phenomena. Indeed, such a scheme disentangles the of the mechanical lasing at fOMO is strong. Note that fOMO in read-out of the mechanics from the cyclic dynamics of the the bottom panel is slightly smaller than in the top panel free-carriers and temperature, the spatially diffusive nature due to temporal drift associated to thermal and/or align- and characteristic length-scales of which minimize their ment variations. In these modulation conditions, the pres- contribution to the neighbor cavity. It is observed that the ence of the small peak at the modulation frequency modulation peak at 100.37 MHz is present for both an M1 and evidences the onset of nontrivial mechanical dynamics of M2 phonon lasing states—dashed lines 2 and 3 in Figure 2(c)— the lasing cavity. Indeed, a possible contribution from free- without the presence of previous harmonics associated to the carrier/temperature dynamics at fmod has already been SP dynamics. The read-out laser (2) power is set to a value discarded by the fact that only the mechanical peak is sufficiently high to transduce the incoming mechanical detected in that configuration when the lasing cavity is in signal, but below its own SP Hopf bifurcation [47]. The broad M = 2 (Figure 2(c)). spectral feature spanning 20–40 MHz across the right color- The range of modulation frequencies for which injection map of Figure 2(c) is a fingerprint of proximity to such locking occurs, i.e., the Arnold tongue [49], is investigated by bifurcation and is used as a sign of stability in the optical performing measurements such as those shown in Figure 2(d) coupling conditions throughout the measurements. We while step-changing the excitation frequency from lower to therefore experimentally satisfy the detection scheme depic- higher frequencies. The results for VAC = 0.004Vπ , corre- ted in Figure 1, which opens the door to act upon the sponding to a VNA tone of −5 dBm, are depicted as a color- dynamical state of the complex self-pulsing/lasing state of map in Figure 3(a and b), with the BPFs centered in the right cavity and observing the effect on the mechanical λ = 1536 nm and λ = 1551 nm, respectively. Note that during the resonator dynamics via the left cavity. This experimental acquisition of Figure 3(a and b), fOMO = 100.322 MHz in the scheme was previously employed in the study by Colombano absence of modulation is slightly lower than previously et al. [45], where both mechanical modes were brought to shown, which is due to a change in the optical coupling lasing and their motion synchronized. Further details are condition. Panel (a) clearly evidences the presence of an found in the Supplementary Information. injection-locked region spanning 23 kHz, confirmed by the We focus now on the particular situation represented measurement performed on the other cavity (panel (b)). The by dashed line 3 in Figure 2(c), where the right beam is absence of the directly transmitted and modulated light al- pumped by laser (1) to the M = 1 mechanical lasing state at lows the evolution of the dynamics of the mechanical oscil- fOMO, 0 = 100.36 MHz and we modulate the power of the lator to be observed. The first remarkable observation is that laser at a frequency fmod slightly detuned from the me- prior to injection locking, the mechanical response exhibits a chanical frequency (VAC = 0.004Vπ , fmod = 100.215 MHz). peak at the modulation frequency fmod, indicating that the Note that the free-running self-sustained oscillation fre- trajectory of the oscillator becomes non sinusoidal, as was quency fOMO, 0 in the absence of external modulation is already pointed out in Figure 2(d). Moreover, approaching the usually smaller than the natural frequency fR due to edges of the lock range, the peak at the mechanical lasing is thermomechanical softening [48]. In the transmitted dragged toward fmod, i.e., frequency pulling [37, 50] occurs, signal, RF spectrum two distinct peaks at the oscillator which is a typical feature of self-sustained oscillators driven frequency fOMO ∼ fOMO, 0 and at the modulation frequency by an external source [22, 39]. At the borders of the lock range, fmod are observed, as shown by the top black curve the quasi-locked regime [38, 42], where the oscillator peri- of Figure 2(d). For a smaller frequency difference, odically oscillates between a locked and unlocked state [30], i.e., fOMO, 0 − fmod = 0.015 MHz, only one peak at fmod is is observed (see Supplementary Information). In addition to observed (blue curve), which is a first fingerprint of in- acquiring the PSD signal of the transmitted light, the complex jection locking [40]. However, this is unambiguously scattering parameter S21 of the system is measured by the confirmed by employing the phonon detection scheme VNA. Figure 3(c and d) shows its magnitude and phase when previously described. With the laser (2) as the probe set measured with the filter pass-band in resonance with either to be in resonance with the optical mode of the left beam laser. The signal obtained by measuring at the pump laser and perfectly transmitted by the BPF, we are able to observe wavelength is complex since both the modulation from the the mechanical lasing state of the neighbor cavity. When directly transmitted light and the modulation from the effect the modulation to laser (1) is on (bottom of Figure 2(d)), a of the frequency tone on the cavity dynamics can interfere. peak at the modulation frequency fmod becomes prominent However, the amplitude response observed in Figure 3(d) is 1324 G. Arregui et al.: Injection locking of a phonon source simpler and shows that the oscillation amplitude of the me- a hysteretic behavior of the system. It originates from high- chanical oscillator remains constant throughout the lock order synchronization peaks [51] which are specially relevant range, at least for the corresponding modulation amplitude, here due to the specific anharmonic nature of the self-pulsing and that the RF tone appearing at the borders outside that dynamics. For VAC > 0.007Vπ the modulation at excitation range follows the characteristic lineshape of a driven har- frequencies fmod close to fOMO, 0 becomes too strong and the monic oscillator. Furthermore, the phase response Δϕ is initial phonon lasing dynamics are lost, leading to a driven characterized by an evolution of approximately π/2 across the harmonic oscillator with a Lorentzian-like amplitude lock range, which is another standard feature of injection- response in those regions and to an overall response that goes locked electric tank oscillators [30] observed in power- beyond pure locking. modulated OMOs [38]. The PSD signal acquired for several As seen in Figure 4, injection locking controls the oscil- modulation amplitudes VAC, now only measuring with the lation frequency over bandwidths as high as several hun- filter transmitting the laser light outcoupled from the probe dreds of linewidths of the bare OMO. Locking to a low-noise cavity, is depicted in Figure 4. This set of measurements reference also improves its stability, gauged over short times clearly demonstrates the increasing size of the Arnold tongue via its phase noise. The phase noise analysis was carried out with modulation amplitude and its asymmetric growth, i.e., a with a different fiber taper, slightly changing the external larger lock range at fmod > fOMO, 0 . This asymmetry, which is coupling condition and resulting in an even wider Arnold not reported in other injection-locked systems [38, 42], is tongue at a fixed modulation amplitude and exhibiting simulated via numerical integration of the full system of injection locking physics for voltages up to VAC = 0.015Vπ equations (see Supplementary Information) and is not due to (PVNA = −10 dBm). The phase noise measurements are carried Figure 3: Spectral-domain evidence of injection locking to an external modulation. (Color online) (a) Color plot of the power spectral density (PSD) of the transmitted light of the laser driving the right beam fundamental optical resonance at λ1 = 1536 nm with Pin = 3 mW when changing the modulation frequency in steps of 100 Hz. In a given range, the phonon lasing peak synchronizes to the modulation tone. (b) Figure 4: Phonon lasing oscillation frequency. (Color online) Equivalent measurement by observing light from the second laser Colormaps of the power spectral density (PSD) of light transmitted at that weakly probes the optical mode at λ2 = 1550 nm of the left beam, the wavelength of the probe laser for modulation amplitudes from allowing only the observation of the mechanical dynamics of the VAC = 0.0019Vπ to VAC = 0.0061 Vπ. The dashed white lines are guides right beam. The coherent complex scattering response of the system to the eye and help to visualize the asymmetric growth of the Arnold measuring S21 in the VNA is depicted for the filter aligned with the tongue with respect to the free-running oscillation frequency at laser driving the right (c) and the left (d) optical modes. fOMO,0 = 100.322 MHz. G. Arregui et al.: Injection locking of a phonon source 1325 out using the dedicated module of the SPA and setting the frequency carrier to the oscillator frequency fOMO, 0 , 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 P VNA = −30 dbm the phase noise increases for frequency offsets below 20 kHz. We attribute this to an un- stable 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 dis- regarded. 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 Figure 5: Phase noise characteristics. (Color online) (a) Phase noise measurements of the phonon lasing reduction of 1.1 dBc/Hz per excitation dBm is observed state with a radio-frequency external tone at fmod ≈ fOMO, 0 . The both at 1 kHz (blue) and 10 kHz (black), while the reduc- locked (unlocked) states are represented with black to red curves tion is least pronounced at 100 kHz offset (red) and above. (dashed black line). The evolution of the phase noise at offsets 1, 10 Given those values, the maximum achieved phase noise and 100 kHz as a function of the modulation power PVNA are depicted suppression is 44 dBc/Hz (1 kHz), 20 dBc/Hz (10 kHz) and in the inset, with the lock power range in shaded gray. (b) Color plot 8.5 dBc/Hz (100 kHz). Last, the behavior of the phase of the phase noise as a function of the frequency offset for multiple driving frequencies fmod. The chosen range for fmod covers the full noise at P VNA = −10 dbm as a function of the modulation Arnold tongue for a drive power of P VNA = −10 dbm. 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 oscillations of an optomechanical system to an external peak at the frequency difference (see Figure 5(b)), as do all optical reference and isolated the mechanical signal by ⃒⃒ ⃒⃒ sidebands at frequency offsets f = n⃒⃒fOMO − fmod ⃒⃒. Close to engineering the system geometry and read-out scheme. the onset of injection locking, at 100.22 MHz, where these The lock range achieved can go up to 350 kHz, enabling a additional peaks disappear, a considerable increase in the large tuning range over the oscillator frequency. Besides low-frequency offset phase noise is observed, that is also the direct consequences for acoustic wave self-sustained attributed to hopping between the locked and unlocked sources of the results shown here, applications using state [50]. This phase noise behavior is probably also distributed arrays of oscillators within a single small- associated to the quasi-locked regime described in the footprint chip, like robust time-keeping with standard op- Supplementary Information. These considerations also toelectronic oscillators [52], or neuromorphic computing apply in the opposite side of the lock range, where the [53], may benefit from the reported physics of injection increased phase noise is both more pronounced and ex- locking from a variety of perspectives. Variations in natural tends throughout a broader range. Inside the Arnold mechanical frequencies due to unwanted fabrication var- tongue, the phase noise characteristics remain practically iations make mutual synchronization of weakly coupled constant, evidencing the potential of injection locking to beams as in the study by Colombano et al. [45] relatively generate low-noise oscillatory references and coherent hard to observe. Due to the wide lock range achieved, in- phonon sources with controlled frequency and purity. jection locking can be used to presynchronize one of the In conclusion, we have reported the observation of coupled OMOs to an external drive and use it as a lead injection locking of self-pulsing-driven regenerative oscillator to which the others can synchronize via weak 1326 G. Arregui et al.: Injection locking of a phonon source mechanical links, a configuration under exploration. [3] T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and Moreover, in cases where mechanical lasing of a particular K. J. 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