PHYSICAL REVIEW RESEARCH 5, L032028 (2023) Letter Intermodulation of optical frequency combs in a multimode optomechanical system Ryan C. Ng ,1,* Paul Nizet,1,* Daniel Navarro-Urrios ,2 Guillermo Arregui ,3 Marcus Albrechtsen ,3 Pedro D. García ,1 Søren Stobbe ,3,4 Clivia M. Sotomayor-Torres ,1,5 and Guilhem Madiot 1,† 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), Campus UAB, Bellaterra, 08193 Barcelona, Spain 2 MIND-IN2UB, Departament d’Electrónica, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, Barcelona 08028, Spain 3 DTU Electro, Department of Electrical and Photonics Engineering, Technical University of Denmark, ørsteds Plads 343, DK-2800 Kgs. Lyngby, Denmark 4 NanoPhoton—Center for Nanophotonics, Technical University of Denmark, Ørsteds Plads 345A, DK-2800 Kgs. Lyngby, Denmark 5 ICREA—Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain (Received 3 March 2023; revised 16 May 2023; accepted 2 August 2023; published 25 August 2023) Phonons offer the possibility to connect the microwave and optical domains while being efficiently transduced with electronic and optical signals. Here, we present a multimodal optomechanical platform, consisting of a mechanical-optical-mechanical resonator configuration. The mechanical modes, with frequencies at 265 MHz and 6.8 GHz, can be simultaneously excited into a phonon lasing regime as supported by a stability analysis of the system. Both the megahertz and gigahertz modes enter a self-sustained oscillation regime, leading to the intermodulation of two frequency combs in the optical field. We characterize this platform experimentally, demonstrating previously unexplored dynamical regimes. These results suggest the possibility to control multiple mechanical degrees of freedom via a single optical mode, with implications in gigahertz phononic devices, signal processing, and optical comb sensing applications. DOI: 10.1103/PhysRevResearch.5.L032028 I. INTRODUCTION systems, which are typically on the order of a few tens of megahertz. Phonons exist over an extremely broad frequency range Intermodulation between mechanical spectra generally oc- from the Hertz to the terahertz regime. The control of phonons curs in optomechanical experiments [22] over a limited range is difficult, especially at higher frequencies, despite the im- of the parameter space and does not involve highly nonlinear plications that such control would enable. For example, in oscillations that produce optomechanical frequency combs. the gigahertz regime, this could enable phonon signal pro- Here, we report an unexplored optomechanical dynamical cessing, nanoacoustic devices, or hybrid quantum systems regime that stems from the simultaneous self-sustained op- such as superconducting qubits [1–3]. Recent advances in tomechanical oscillation of two mechanical modes that are the realization of high-frequency nanoacoustic devices have far apart in frequency, leading to the intermodulation of further motivated the extension of concepts from electron- two coherent processes which spectrally corresponds to the ics and photonics to the realm of phononics [4–9]. In this imbrication of two optical frequency combs. This mechanical- context, cavity optomechanics provides a route toward the optical-mechanical (MOM) configuration is experimentally manipulation and control of phononic states via radiation- realized in a two-dimensional (2D) membrane phononic crys- pressure forces [10–12]. Increasingly complex functionalities tal (PnC) cavity waveguide sustaining both a 7 GHz phononic have been demonstrated in multimode optomechanical sys- waveguide mode and a 265 MHz in-plane mechanical breath- tems, where multiple optical and mechanical degrees of ing mode of the full structure. Although thermal nonlinearities freedom interact [13–17]. Previously, several singular nonlin- also exist in this optomechanical waveguide structure, the ear phenomena have arisen from self-pulsing (SP), involving dynamics of the system is dictated purely by optomechanical collective dynamics between free carriers and temperature interactions, which we confirm theoretically using a mul- in an optical cavity [18]. Such dynamics can produce rich timode optomechanical model. Consequently, the dynamics dynamics, such as chaos [19], injection-locking [20], or fre- are set by mechanical properties of the structure that can be quency combs [21], but remain limited in frequency due to engineered over much shorter timescales than that of thermal the relatively low thermalization rates encountered in these effects. The multimodal interaction observed in this system leads to the formation and intermodulation of optical fre- quency combs in the gigahertz domain, with implications for * These authors contributed equally to this work. atomic force and mass sensing [21,23–25]. † Corresponding author Published by the American Physical Society under the terms of the II. MODEL Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) The archetypical representation of an optomechanical cav- and the published article’s title, journal citation, and DOI. ity uses an optical Fabry-Pérot cavity formed by two parallel 2643-1564/2023/5(3)/L032028(7) L032028-1 Published by the American Physical Society RYAN C. NG et al. PHYSICAL REVIEW RESEARCH 5, L032028 (2023) FIG. 1. Mechanical-optical-mechanical (MOM) modal configuration. (a) Generalized and (b) input-output schematic of a MOM multi- modal configuration, where two mechanical modes (M1 in red and M2 in green) couple to a waveguide-coupled single optical mode via their respective optomechanical coupling rates. The optical mode is not explicitly represented in (a) and is highlighted in gray in (b). (c) A scanning electron microscope image of the optomechanical experimental realization using suspended silicon nanomembranes. A zoom-in shows the slotted optical waveguide and a finite-difference time-domain (FDTD) simulation of the electromagnetic energy confined within it. (d) Displacement field distribution of the low-frequency in-plane mechanical mode M1 (top) and of the third-harmonic standing-wave acoustic mode traveling at the interface with the slot (bottom). mirrors, one of which has a mechanical degree of freedom, second mechanical mode, a correction term to each mechan- such as via attachment to a spring. Figure 1(a) presents ical susceptibility can be derived (see Supplemental Material an adaptation of this representation to the MOM configu- [27]). This indirect modification leads to gain competition be- ration, in this case using two moving mirrors. Each mirror tween the two mechanical modes by acting against the direct (or mechanical mode) Mi has its own natural mechanical optomechanical effect and therefore increases the threshold frequency i and damping rate i and couples to the optical powers at which each mode can enter a self-sustained oscil- field with a single-photon optomechanical coupling gi . Their lation regime [28,29]. When 1 ∼ 2 , simultaneous phonon displacement is given by xi (t ). An alternative input-output lasing of both modes generally cannot occur, although it can representation to more clearly observe intermode coupling is be achieved using external modulation of the laser [30]. This shown in Fig. 1(b), where an optical waveguide (represented case of nearly degenerate mechanical modes implies that they here by a fiber loop that is used in the experiment later on in couple to one another through light, leading to hybridization this letter) is side-coupled to an optical mode which interacts of the modes as observed in a SiN membrane placed inside with M1 and M2. The optical mode has a total decay rate an optical fiber cavity [31]. The study of exceptional points κt = κi + κe , where κe is the decay rate associated with the [32], which analyzes how gain and loss within a system can input channel, and κi is the decay rate due to intrinsic losses. be balanced, has also been made possible using such a MOM The linearized classical equations of motion for the cavity configuration [14], although the phonon lasing regime was not amplitude α and for the mechanical displacements x1 and x2 observed. When 1  2 , the coupling between M1 and M2 can be expressed as becomes nonresonant which allows for simultaneous lasing    of the two mechanical modes at a relatively small input power κt κe and leads to the formation and intermodulation of combs that α̇ = j( + g1 x1 + g2 x2 ) − α+ sin , 2 2 will subsequently be described. x¨1 = −1 x˙1 − 21 x1 + 21 g1 |α|2 , III. EXPERIMENTAL REALIZATION x¨2 = −2 x˙2 − 22 x2 + 22 g2 |α|2 , (1) While the underlying physics of such a multimode sys- where  = ω − ω0 is the detuning between the laser fre- tem are general, one example design in which the MOM quency ω = 2π c/λ and the cavity resonance frequency ω0 , configuration can be experimentally reproduced is in a and sin is the input laser field amplitude, such that Pin = 220-nm-thick suspended silicon membrane fabricated in h̄ω |sin |2 is the laser input power. Note that the displacements silicon-on-insulator with etched shamrock-shaped holes that are normalized by the zero-point fluctuation of each of the form a 2D PnC displaying wide acoustic band gap ∼6 GHz mechanical modes such that the mechanical masses can be [33]. Two PnCs with mirror symmetry are brought together eliminated. In a standard optomechanical cavity, the mechan- and separated by a thin 40-nm-wide air-slot within which ical susceptibility, which describes the mechanical spectral light can be confined. Figure 1(c) shows a scanning electron response to an external perturbation, is altered by the laser microscope (SEM) image of the PnC/air-slot/PnC platform field such that a direct modification of both the mechani- with a zoom-in of the red-shaded rectangle emphasizing cal frequency and damping occurs [26]. When considering a the air-slot provided on the right. A finite-element-method L032028-2 INTERMODULATION OF OPTICAL FREQUENCY COMBS IN … PHYSICAL REVIEW RESEARCH 5, L032028 (2023) FIG. 2. Observed physical regimes. (a) Mechanical RF spectra for the (i), (iii), and (v) megahertz range and (ii), (iv), and (vi) gigahertz range. Three characteristic dynamical regimes are observed: thermal (gray), M1 lasing (red), and M2 lasing (green), and comb intermodulation (overlap of red and green regions), which results from simultaneous M1 and M2 lasing. Insets in (i) and (ii) show averaged spectra (gray) each fitted with a Lorentzian (red). (b) Transmission response measured by driving a localized optical mode with a tunable laser from the blue-detuned side, using Pin = 4.5 mW. (c) Theoretical map highlighting the lasing thresholds for M1 and M2 as a function of the laser power and wavelength. The thresholds are shown with solid lines. The gray, green, and red areas correspond to both modes in the thermal regime, M2 lasing (eff,2 < 0), and M1 lasing (eff,1 < 0), respectively. The experimentally determined thresholds of M2 (green circles) and M1 (red squares) are overlaid. simulation of the optical field energy density confined in the another, as the overlap of the displacement fields is weak due air-slot is also shown. Inherent fabrication imperfections such to the large difference in length scale and frequency. Since the as etched sidewall roughness [34] lead to disorder-induced two plates are identical and independent, each would sustain spatially localized optical modes. From hereon, we are implic- a gigahertz mechanical mode, although they are not readily itly referring to these disorder-induced optical modes when differentiated here. By using a tapered-fiber loop brought into mentioning optical modes in our structure. The localized contact with the optical waveguide, these telecom-wavelength modes provide efficient optomechanical transduction (i.e., the optical modes can be driven to transduce both of these mega- ability to couple light to mechanical motion) due to their high hertz and gigahertz mechanical modes. quality factors and small effective mode volumes, without the need to carefully design a confinement potential [35]. The IV. RESULTS AND DISCUSSION exact geometric parameters of the crystal can be found in the Supplemental Material [27], and further details regarding A localized optical mode is resonantly driven by injecting fabrication and design of the optomechanical structure are light from a tunable laser with input power Pin = 5.0 mW in found in Madiot et al. [36]. We define two mechanical modes the fiber loop. The position of the loop for coupling to the observed in this structure as M1 and M2, referring to the mode and the polarization of the input light are optimized mechanical modes in the megahertz and gigahertz regimes, by maximizing the thermo-optic broadening of the mode, respectively. In the structure explored here, M1 corresponds to observed when scanning the lasing wavelength upward. The the fundamental mechanical breathing mode in the megahertz output field is analyzed with a fast photoreceiver connected regime, or vibrations along the entire length of the plates, to an electrical spectrum analyzer. As the optical mode is which strongly couples to the optical field in the slot, mak- driven at increasing coupling fractions by varying , different ing it ideal for cavity optomechanical systems [35,37–39]. physical regimes are observed, summarized in their respective Here, M2 corresponds to a stationary wave resulting from power spectral densities in Fig. 2(a) for the megahertz (left the confinement of a mechanical guided mode in the acoustic panels) and the gigahertz (right panels) regimes. By approach- Fabry-Pérot cavity formed on one side of the slot and between ing the optical mode with a blue-detuned laser, the first regime the two acoustic mirrors at the edges [33,36]. The in-plane is a thermal regime in which both modes are thermally excited components of the displacement fields associated with M1 and (gray), although only M2 is transduced at first. As the coupling M2 are shown in Fig. 1(d). Both modes couple to the same op- mode fraction in the driven optical mode increases, a second tical mode, although they are unlikely to couple directly to one regime appears in which a single gigahertz mechanical lasing L032028-3 RYAN C. NG et al. PHYSICAL REVIEW RESEARCH 5, L032028 (2023) peak is observed for M2 with sidebands formed by a ther- mally excited M1 in the megahertz regime (green region). At this point, several other mechanical modes in the megahertz regime are observed. Due to the number of modes, accurate determination of the spectral linewidth of M1 is impeded, especially at coupling fractions which are sufficient to guar- antee the absence of any dynamical backaction. This is the main limiting factor for the determination of the optomechan- ical coupling g1 . For an even greater coupling fraction, M1 also enters a self-sustained oscillation regime (red region). The exponentially growing oscillations that result from an optomechanical instability saturate due to nonlinear effects. This process eventually leads to the emergence of various harmonics when driving the oscillator significantly above its lasing threshold. The resulting spectral feature is an optical frequency comb [40,41]. When both modes lase (region in which the red and green regions overlap), the lasing of M1 FIG. 3. RF spectrum over full spectral range. Power spectral in the megahertz regime modulates the principal lasing peak density of the output optical field in the simultaneous lasing regime of M2 in the gigahertz regime, giving rise to a frequency measured over the 0–13.5 GHz frequency range. Up to 10 pairs comb centered at 2 /2π = 6.764 GHz with spacing equal to of sidebands on each side of 2 are observed, while the second 1 /2π = 265 MHz. The transmission optical spectrum asso- harmonic of this pattern (centered at 22 ) is partially visible near ciated with these measurements is shown in Fig. 2(b) (black the signal analyzer limit. line), accompanied by other scans at different input powers (gray lines). We observe additional broadening of the thermo- optically bistable resonance as Pin increases. shaded region) and when eff,2 < 0 (red shaded region). Each By scanning the laser wavelength λ over the resonance, of these regions is delineated by the lasing threshold of each we collect the lasing thresholds, or wavelengths at which mode, eff,1 = 0 and eff,2 = 0. The experimental results are the amplitude of each mode begins an exponential increase numerically fitted with the theory using only g1 as a fitting or decrease, for both mechanical modes (see Supplemental parameter, with g2 being experimentally calibrated. We find Material [27]). We determine the mechanical lasing thresh- the 95% confidence interval g1 /2π = 480 ± 10 kHz which olds of M1 and M2 at different powers and report them in quantitatively captures the observed experimental values. This Fig. 2(c) with red squares and green circles, respectively. The value is in good agreement with the optomechanical couplings optomechanical interaction alters the mechanical susceptibili- found for the in-plane mechanical modes of similar slotted- ties as computed via Eq. (1) such that the effective mechanical waveguide optomechanical platforms [35]. The red region damping rates eff,i can be extracted to determine the phonon is mostly encompassed within the green one, representing a lasing condition of each mode. To allow for direct compar- region in which simultaneous mode lasing occurs. ison with experiment, the theoretical reduced laser detuning Figure 3(a) shows the power spectral density of the output  incorporates a resonant-wavelength λ0 that is corrected optical field measured in the multimode lasing regime with for any thermo-optic shifts of the resonant cavity frequency a resolution bandwidth set to 50 kHz over the full span of using λ0 = λ0 + η|α|2 , where η/h̄ω ≈ 0.35 nm/fJ is a static the electrical spectrum analyzer from 0 to 13.5 GHz. Here, nonlinearity estimated from the experimental data (see Sup- M1 lases at 265 MHz, leading to the emergence of a first plemental Material [27]). The cavity photon number |α|2 as a frequency comb with ∼10 harmonics up to 2.650 GHz. Here, function of λ and Pin is numerically resolved by accounting M2 also lases, but the second harmonic already lies above for the static thermo-optic nonlinearity, and the solutions are the equipment limit. The harmonics of M1 are imprinted in entered into the effective susceptibility to determine the effec- the frame rotating at 2 , such that ∼10 pairs of sidebands tive damping rates. The calibrated experimental parameters are visible around 2 . This frequency pattern is duplicated are λ0 = 1490.6 nm, κi /2π = 5.0 GHz, κe /2π = 4.5 GHz, in all the harmonics of M2, such that the beginning of its 1 /2π = 1.5 MHz, and 2 /2π = 3.4 MHz. Using a phase- second occurrence (centered at 22 ) can be spotted >12 GHz. modulation calibration method [42], we have determined This result is indicative of the observed multimodal inter- the optomechanical coupling associated with M2, g2 /2π = modulation, as the gigahertz sidebands cannot occur in the 260 kHz. From the mechanical frequencies and linewidths, the absence of M1 lasing (see Supplemental Material [27]). The mechanical Q factors are Qm,1 = 1 /1 ≈ 176 and Qm,2 = experimental spectrum is plotted over the 0–20 GHz range for 2 /2 ≈ 1988. Both linewidths are determined by driving direct comparison with Fig. 3(b), which shows a numerical another localized optical mode with λ = 1489.9 nm and Pin = simulation. Both exhibit the same type of spectral structure 0.2 mW. The spectra are acquired with low-resolution band- with slight variations of the harmonic amplitude distribution. width and higher averaging time and are each fitted with Other regimes involving another megahertz mechanical a Lorentzian, as shown in the insets of Figs. 2(a)(i) and mode as well as SP dynamics could also be observed within 2(a)(ii). Similarly, we evaluate the optical Q factor Qopt = this structure for certain experimental input parameters but are ω0 /κt , where ω0 = 2π c/λ0 and κt = κ0 + κe , which leads to ignored by our model. A SP regime with thermal oscillation Qopt ≈ 21172. Figure 2(c) indicates when eff,1 < 0 (green at SP /2π = 13.5 MHz is found in addition to simultaneous L032028-4 INTERMODULATION OF OPTICAL FREQUENCY COMBS IN … PHYSICAL REVIEW RESEARCH 5, L032028 (2023) FIG. 4. Physical regimes in the time domain. Simulated Fourier spectrum of |αout (t )|2 for (a) the M2 lasing regime and (b) the comb intermodulation arising from simultaneous M1 and M2 lasing. The experimental measurement cutoff due of the measurement setup is indicated with a dashed line in (a). For comparison, the frequency range from (b) is highlighted in (a) by a red box. Experimental (top) and simulated (bottom) time traces are shown for the (c) M2 lasing regime and (d) multimode lasing regime. optomechanical lasing of M1 and M2, leading to another level Our platform exhibits the intermodulation of two optical of intermodulation (see Supplemental Material). frequency combs in the gigahertz domain, resulting from self- The regimes where M2 lases independently [Figs. 2(a)(iii)– sustained mechanical oscillations of a gigahertz PnC guided 2(a)(vi)] and where both modes lase simultaneously mode that interacts with a lasing megahertz breathing mode [Figs. 2(a)(v)–2(a)(vi)] are reproduced numerically using via the optical field, which occurs independent of thermal Eq. (1), and are shown in Figs. 4(a) and 4(b), respectively, effects, as supported by our theoretical model. using the output photon flux |αout (t )|2 , where αout = sin −  The modulation frequency (1 ) can also be actively tuned κe 2 α (see Supplemental Material [27]). The experimental even further via an external stimulus, such as an atomic force oscilloscope cutoff frequency is at 8 GHz. The associated microscope tip or other static actuator which acts as a local experimental (top) and simulated (bottom) time traces for perturbation with the dampening leading to an order of magni- each case are shown in Figs. 4(c) and 4(d). Our experimental tude energy loss [43], or with additional electrostatic actuation apparatus is limited to 13.5 GHz [dashed line in Fig. 4(a)], as envisioned with electro-optomechanical systems [44,45]. In which lies slightly below the second harmonic. Both of the as- our case, this dampening would significantly shift M1 and the sociated time traces for this scenario [Fig. 4(c)] show slightly modulation frequency by consequence. Regarding the spectral anharmonic oscillation at 2 . A low-pass filter with cutoff line shape, Miri et al. [40] demonstrated that this distribution frequency at 8 GHz is applied to the simulation time traces can be readily controlled via the optical losses of the cav- (bottom) to reproduce the oscilloscope bandwidth. For the ity, which can also be engineered and experimentally tuned. case where simultaneous lasing occurs [Fig. 4(b)], the fre- Such a modulation scheme further enhances understanding quency is centered at 2 and normalized with 1 , indicating and versatility of high-frequency input-output phononic de- the presence of a comb centered at 2 with spacing given vices [46,47]. Control of the mechanical frequencies could by 1 . This spectrum shows three pairs of sidebands, which be useful to realize harmonic locking [48] and enhance the are also observed experimentally in Fig. 2(a)(vi). These cor- optical frequency comb spectral span. respond to the first four harmonics of M1 that imprint in the More generally, such a process enriches the level of control frame rotating at 2 via the indirect nonlinear coupling of the in multimode optomechanical platforms and could be used for mechanical modes. These harmonics emerge when M1 passes applications such as realizing subthreshold multimode lasing its lasing threshold, like the case of M2. The experimental of two gigahertz modes without the need for an external in- time trace in Fig. 4(d) (top) shows amplitude modulation of termodal modulation of the laser [30]. Within this context, the the gigahertz tone by the megahertz tone, in the output optical extent to which the modulation frequency as well as the spec- field. The simulated trace (bottom) exhibits good qualitative tral distribution of the frequency comb lines can be readily agreement with experiment. controlled has yet to be studied, though such prospects could lead toward the realization of optomechanical logic gates [17]. V. CONCLUSIONS We presented a MOM multimodal optomechanical plat- ACKNOWLEDGMENTS form, connecting spectrally distant phonons and photons in the megahertz, gigahertz, and hundreds of terahertz regimes, We acknowledge the support from the project LEIT funded and demonstrated optical control the mechanical dynamics. by the European Research Council, H2020 Grant Agreement L032028-5 RYAN C. NG et al. PHYSICAL REVIEW RESEARCH 5, L032028 (2023) No. 885689. ICN2 is supported by the Severo Ochoa Marie Sklodowska Curie Grant Agreement (No. 754558). program from the Spanish Research Agency (AEI, Grant No. M.A. and S.S. gratefully acknowledge funding from the SEV-2017-0706) and by the CERCA Programme/Generalitat Villum Foundation Young Investigator Program (Grant No. de Catalunya. R.C.N. acknowledges funding from the EU- 13170), the Danish National Research Foundation (Grant No. H2020 research and innovation programme under the Marie DNRF147—NanoPhoton), Innovation Fund Denmark (Grant Sklodowska Curie Individual Fellowship (Grant No. 897148). No. 0175-00022—NEXUS), and Independent Research Fund O.F. is supported by a BIST Ph.D. fellowship under the Denmark (Grant No. 0135-00315—VAFL). [1] R. V. Laer, B. Kuyken, D. V. Thourhout, and R. Baets, Interac- [17] K. Pelka, G. Madiot, R. Braive, and A. Xuereb, Floquet Control tion between light and highly confined hypersound in a silicon of Optomechanical Bistability in Multimode Systems, Phys. photonic nanowire, Nat. Photon. 9, 199 (2015). Rev. Lett. 129, 123603 (2022). [2] E. A. Kittlaus, W. M. Jones, P. T. Rakich, N. T. Otterstrom, [18] D. Navarro-Urrios, N. E. Capuj, J. Gomis-Bresco, F. Alzina, A. R. E. Muller, and M. Rais-Zadeh, Electrically driven acousto- Pitanti, A. Griol, A. Martínez, and S. Torres, A self-stabilized optics and broadband non-reciprocity in silicon photonics, Nat. coherent phonon source driven by optical forces, Sci. Rep. 5, Photon. 15, 43 (2021). 15733 (2015). [3] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, [19] D. Navarro-Urrios, N. E. Capuj, M. F. Colombano, P. D. and W. D. Oliver, A quantum engineer’s guide to superconduct- García, M. Sledzinska, F. Alzina, A. Griol, A. Martínez, ing qubits, Appl. Phys. Rev. 6, 021318 (2019). and C. M. Sotomayor-Torres, Nonlinear dynamics and [4] K. Fang, M. H. Matheny, X. Luan, and O. Painter, Optical chaos in an optomechanical beam, Nat. Commun. 8, 14965 transduction and routing of microwave phonons in cavity- (2017). optomechanical circuits, Nat. Photon. 10, 489 (2016). [20] G. Arregui, M. F. Colombano, J. Maire, A. Pitanti, N. E. [5] L. Mercier de Lépinay, C. F. Ockeloen-Korppi, D. Malz, and Capuj, A. Griol, A. Martínez, C. M. Sotomayor-Torres, and D. M. A. Sillanpää, Nonreciprocal Transport Based on Cavity Flo- Navarro-Urrios, Injection locking in an optomechanical coher- quet Modes in Optomechanics, Phys. Rev. Lett. 125, 023603 ent phonon source, Nanophotonics 10, 1319 (2021). (2020). [21] P. E. Allain, B. Guha, C. Baker, D. Parrain, A. Lemaître, G. Leo, [6] S. Volz, J. Ordonez-Miranda, A. Shchepetov, M. Prunnila, J. and I. Favero, Electro-Optomechanical Modulation Instability Ahopelto, T. Pezeril, G. Vaudel, V. Gusev, P. Ruello, E. M. Weig in a Semiconductor Resonator, Phys. Rev. Lett. 126, 243901 et al., Nanophononics: State of the art and perspectives, Eur. (2021). Phys. J. B 89, 15 (2016). [22] K. E. Grutter, M. I. Davanço, and K. Srinivasan, Slot-mode [7] G. Ma, M. Xiao, and C. Chan, Topological phases in acoustic optomechanical crystals: A versatile platform for multimode and mechanical systems, Nat. Rev. Phys. 1, 289 (2019). ptomechanics, Optica 2, 994 (2015). [8] X. Zhang, M. Xiao, Y. Cheng, M.-H. Lu, and J. Christensen, [23] P. E. Allain, L. Schwab, C. Mismer, M. Gely, E. Mairiaux, Topological sound, Commun. Phys. 1, 97 (2018). M. Hermouet, B. Walter, G. Leo, S. Hentz, M. Faucher [9] R. C. Ng, A. El Sachat, F. Cespedes, M. Poblet, G. Madiot, et al., Optomechanical resonating probe for very high frequency J. Jaramillo-Fernandez, O. Florez, P. Xiao, M. Sledzinska, sensing of atomic forces, Nanoscale 12, 2939 (2020). C. M. Sotomayor-Torres et al., Excitation and detection of [24] M. Sansa, M. Defoort, A. Brenac, M. Hermouet, L. Banniard, acoustic phonons in nanoscale systems, Nanoscale 14, 13428 A. Fafin, M. Gely, C. Masselon, I. Favero, G. Jourdan et al., (2022). Optomechanical mass spectrometry, Nat. Commun. 11, 3781 [10] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity (2020). optomechanics, Rev. Mod. Phys. 86, 1391 (2014). [25] W. Yu, W. C. Jiang, Q. Lin, and T. Lu, Cavity optomechanical [11] C. Sanavio, V. Peano, and A. Xuereb, Nonreciprocal topological spring sensing of single molecules, Nat. Commun. 7, 12311 phononics in optomechanical arrays, Phys. Rev. B 101, 085108 (2016). (2020). [26] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, [12] S. Barzanjeh, A. Xuereb, S. Gröblacher, M. Paternostro, C. A. Quantum Theory of Cavity-Assisted Sideband Cooling of Me- Regal, and E. M. Weig, Optomechanics for quantum technolo- chanical Motion, Phys. Rev. Lett. 99, 093902 (2007). gies, Nat. Phys. 18, 15 (2022). [27] See Supplemental Material at http://link.aps.org/supplemental/ [13] M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. 10.1103/PhysRevResearch.5.L032028 for more details about McEuen, and M. Lipson, Synchronization of Micromechanical theory, design, and for experimental and numerical methods. Oscillators Using Light, Phys. Rev. Lett. 109, 233906 (2012). [28] T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and [14] H. Xu, D. Mason, L. Jiang, and J. Harris, Topological energy K. J. Vahala, Analysis of Radiation-Pressure Induced Mechan- transfer in an optomechanical system with exceptional points, ical Oscillation of an Optical Microcavity, Phys. Rev. Lett. 95, Nature (London) 537, 80 (2016). 033901 (2005). [15] F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, [29] U. Kemiktarak, M. Durand, M. Metcalfe, and J. Lawall, Mode Optical circulation in a multimode optomechanical resonator, Competition and Anomalous Cooling in a Multimode Phonon Nat. Commun. 9, 1798 (2018). Laser, Phys. Rev. Lett. 113, 030802 (2014). [16] H. Xu, L. Jiang, A. Clerk, and J. Harris, Nonreciprocal control [30] L. Mercadé, K. Pelka, R. Burgwal, A. Xuereb, A. Martínez, and and cooling of phonon modes in an optomechanical system, E. Verhagen, Floquet Phonon Lasing in Multimode Optome- Nature (London) 568, 65 (2019). chanical Systems, Phys. Rev. Lett. 127, 073601 (2021). L032028-6 INTERMODULATION OF OPTICAL FREQUENCY COMBS IN … PHYSICAL REVIEW RESEARCH 5, L032028 (2023) [31] A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D. photonic crystal cavity, Appl. Phys. Lett. 97, 181106 Kashkanova, C. Deutsch, J. Reichel, and J. G. E. Harris, (2010). Optically Mediated Hybridization between Two Mechanical [40] M.-A. Miri, G. D’Aguanno, and A. Alù, Optomechanical fre- Modes, Phys. Rev. Lett. 112, 013602 (2014). quency combs, New J. Phys. 20, 043013 (2018). [32] M.-A. Miri and A. Alù, Exceptional points in optics and pho- [41] L. Mercadé, L. L. Martín, A. Griol, D. Navarro-Urrios, and tonics, Science 363, eaar7709 (2019). A. Martínez, Microwave oscillator and frequency comb in a [33] O. Florez, G. Arregui, M. Albrechtsen, R. C. Ng, J. Gomis- silicon optomechanical cavity with a full phononic bandgap, Bresco, S. Stobbe, C. M. Sotomayor-Torres, and P. D. García, Nanophotonics 9, 3535 (2020). Engineering nanoscale hypersonic phonon transport, Nat. [42] M. L. Gorodetksy, A. Schliesser, G. Anetsberger, S. Deleglise, Nanotechnol. 17, 947 (2022). and T. J. Kippenberg, Determination of the vacuum optome- [34] M. Albrechtsen, B. Vosoughi Lahijani, R. E. Christiansen, chanical coupling rate using frequency noise calibration, Opt. V. T. H. Nguyen, L. N. Casses, S. E. Hansen, N. Stenger, Express 18, 23236 (2010). O. Sigmund, H. Jansen, J. Mørk et al., Nanometer-scale pho- [43] J. Rieger, A. Isacsson, M. J. Seitner, J. P. Kotthaus, and E. M. ton confinement in topology-optimized dielectric cavities, Nat. Weig, Energy losses of nanomechanical resonators induced Commun. 13, 6281 (2022). by atomic force microscopy-controlled mechanical impedance [35] G. Arregui, R. C. Ng, M. Albrechtsen, S. Stobbe, C. M. mismatching, Nat. Commun. 5, 3345 (2014). Sotomayor-Torres, and P. D. García, Cavity Optomechanics [44] L. Midolo, A. Schliesser, and A. Fiore, Nano-opto-electro- with Anderson-Localized Optical Modes, Phys. Rev. Lett. 130, mechanical systems, Nat. Nanotechnol. 13, 11 (2018). 043802 (2023). [45] A. Pitanti, J. M. Fink, A. H. Safavi-Naeini, J. T. Hill, C. U. Lei, [36] G. Madiot, R. C. Ng, G. Arregui, O. Florez, M. Albrechtsen, A. Tredicucci, and O. Painter, Strong opto-electro-mechanical S. Stobbe, P. D. García, and C. M. Sotomayor-Torres, Optome- coupling in a silicon photonic crystal cavity, Opt. Express 23, chanical Generation of Coherent GHz Vibrations in a Phononic 3196 (2015). Waveguide, Phys. Rev. Lett. 130, 106903 (2023). [46] S.-Y. Yu, X.-C. Sun, X. Ni, Q. Wang, X.-J. Yan, C. He, [37] Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, X.-P. Liu, L. Feng, M.-H. Lu, and Y.-F. Chen, Surface phononic Design of dispersive optomechanical coupling and cooling in graphene, Nat. Mater. 15, 1243 (2016). ultrahigh-Q/V slot-type photonic crystal cavities, Opt. Express [47] H. Pirie, S. Sadhuka, J. Wang, R. Andrei, and J. E. Hoffman, 18, 23844 (2010). Topological Phononic Logic, Phys. Rev. Lett. 128, 015501 [38] X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. (2022). Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati et al., An [48] J. Zheng, Y. Li, N. Goldberg, M. McDonald, X. Luan, A. Hati, integrated low phase noise radiation-pressure-driven optome- M. Lu, S. Strauf, T. Zelevinsky, D. A. Howe et al., Feedback chanical oscillator chipset, Sci. Rep. 4, 6842 (2014). and harmonic locking of slot-type optomechanical oscillators [39] A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. to external low-noise reference clocks, Appl. Phys. Lett. 102, Painter, Optomechanics in an ultrahigh-Q two-dimensional 141117 (2013). L032028-7