Oscillators in the gigahertz frequency range are key building blocks for telecommunication and positioning applications. Operating directly in the gigahertz while keeping high frequency stability and compactness is still an up-to-date challenge. Optomechanical crystals have demonstrated gigahertz frequency modes, thus gathering prerequisite features for using them as oscillators. Here, we report on the demonstration, in ambient atmospheric conditions, of an optomechanical crystal based on the concept of bichromatic lattice. It is made of InGaP, a low loss and TPA-free piezoelectric material, which makes it valuable for optomechanics. Self-sustained oscillations directly at 3 GHz are routinely achieved with a low optical power threshold of 40 *μ*W and a short-term linewidth narrowed down to 100 Hz in agreement with phase noise measurements (−110 dBc/Hz at 1 MHz from the carrier) for free running optomechanical oscillators.

## INTRODUCTION

Optomechanical (OM) resonators, exploiting the interaction between light and a moving optical cavity,^{1} have been actively looked into in recent years with impressive demonstrations in the quantum regime.^{2–4} Meanwhile, other important applications have also been found for ultracompact sensors,^{5} microwave to optics transduction,^{6} radiofrequency signal amplification,^{7} or stable microwave oscillators.^{8} Essential features in modern navigation, communication and timing systems, and microwave oscillators at high frequencies are compared in the light of their stability at their natural frequency and their form-factor. With their micrometric size and their mechanical resonance frequency already in the gigahertz range, OM crystals^{9} (OMCs) present a unique potential to reach ultracompact stable microwave oscillators. OM oscillators have been investigated but still lie far from the microwave domain, and spectral purity is, for the moment, an issue which has been scarcely addressed, mainly because of the difficulty in obtaining a mechanical resonator oscillating at a given frequency all while tuning the optical properties of the optical cavity.^{10} Besides, a shared limitation for every application is thermo-optical instabilities that limit the optical power injected inside the resonator. First, OM resonators and, especially, OMCs made of silicon suffer from two-photon absorption preventing the quantum regime in cooling experiments to be achieved. Hence, different materials such as silica,^{11} silicon nitride,^{12} and diamond^{13,14} have been considered as materials of choice, thanks to their large thermal conductivity and low optical absorption. Thus, a high number of intracavity photons have been reached with the diamond OMC.^{13} None of these materials show piezoelectric properties, which could efficiently bridge microwaves to optics. Thus, they are unsuitable for hybrid opto-electro-mechanical devices,^{15} particularly attractive in various contexts, from telecommunications to quantum information and from classical radar to quantum radar.^{16} That is why non-centro-symmetric crystals such as large electronic bandgap III-V semiconductors are appealing for optomechanics and have recently been investigated (gallium phosphide^{17,18} and aluminum nitride^{19}) as they do not suffer from Two Photon Absorption (TPA) when operating in the practical telecom spectral range.

Here, we consider another material Indium Gallium Phosphide (In_{0.5}Ga_{0.5}P) grown on GaAs. Owing to a large electronic forbidden gap (≈1.9 eV), two-photon absorption is suppressed at telecom wavelengths,^{20} which allows reaching soliton pulse compression.^{21} For these reasons, InGaP has been introduced recently in optomechanics,^{22–24} but an OMC has not been realized yet. We introduce a new design concept based on a bichromatic lattice,^{25,30} which has been shown to mitigate the impact of fabrication disorder on the scattering losses, therefore allowing a large quality factor (>10^{5}) in silicon photonics.^{26} We adapt this concept to one dimensional optomechanical crystals and find that high quality factors are achieved despite the fabrication disorder. It is found that the structure is also able to confine mechanical resonances. The self-sustained oscillations have been characterized in detail all the way to the measurement of the phase noise, revealing that our OMC is comparable to much larger microtoroids made of silicon nitride.

## CAVITY DESIGN AND MODELING

The widespread designs introduced in Refs. 27 and 28 rely on tapering the crystal parameters, following a well optimized profile, according to the concept of “gentle confinement.”^{29} Our design is based on a radically different concept, which does not use any tapering at all: all holes are the same with constant radius *r* and period *a*, while the sidewall modulation has a constant depth *y*_{th} = 0.27*a* and is strictly periodic with period *a*′. The two periods are, however, slightly different, *a*′ = 0.98*a*. As shown in Ref. 30 in the context of two dimensional photonic crystals, this creates an effective confining potential, which minimizes radiative leakage but still keeps the mode volume low. The next optical mode is located about 2 THz below in the spectrum (see Appendix D).

The implementation of this concept in the context of optomechanics also requires that the same structure also confines mechanical modes. To the best of our knowledge, the possibility of localizing a mechanical mode using a bichromatic structure has not been considered. The mechanical modes in Fig. 1(c) are computed using the finite element method, implemented in the COMSOL software. The confinement of the mechanical mode oscillating at about 3 GHz [Fig. 1(c)] is explained by the local decrease in the frequency of the breathing modes for increasing misalignment of holes and sidewalls when moving outwards from the center of the cavity. This design ensures the simultaneous localization of photons [Fig. 1(b)], *V*_{opt} = 0.97(*λ*/*n*)^{3}, and phonons [Fig. 1(c)], *V*_{m} = 2.5 × 10^{−19} m^{3} and *m*_{eff} = 1.01*pg*. The largest vacuum optomechanical coupling *g*_{0} involves the fundamental optical and mechanical modes, while any other combination of modes results in a much smaller coupling. The calculated photoelastic^{31} and moving boundary contributions^{32} are *g*_{0,MB}/2*π* = −117 kHz and *g*_{0,PE}/2*π* = 494 kHz; hence, *g*_{0}/2*π* = 377 kHz (see Appendixes B and C for details on *g*_{0} computation and the values of the photoelastic tensor). The cavity is coupled to the input waveguide by removing *n*_{lh} holes and sidewall corrugation on one side (out of the 51 holes in total), and the waveguide is coupled to a lensed fiber using an inverse taper.^{33}

The device is fabricated on an InGaP membrane grown by metalorganic chemical vapor deposition (MOCVD) lattice-matched to GaAs. The OM crystal is processed following the same recipe as for two dimensional photonic crystals.^{20,25}

## OPTICAL CHARACTERIZATION

The optical resonances are probed in a reflection geometry using a high resolution optical heterodyne technique^{25} (see Appendix F). Figure 2(a) shows the power reflection spectrum around the fundamental order mode. The loaded quality factor *Q*_{L} decreases by a factor of 0.6 for each period removed as coupling to the waveguide is increased; the intrinsic quality factor *Q*_{0}, extracted from the fit of the measured complex amplitude, is 2.2 ± 0.2 × 10^{5} [Fig. 2(b)].^{34} We measured an intrinsic quality factor over 10^{5} in 9 out of 12 nominally identical cavities. The whole spectrum is shown in Fig. 7 of Appendix D where the second order mode is found at lower frequency.

Absorption, at room temperature, is extracted from the normalized reflectivity as a function of the laser detuning *ν*_{L} − *ν*_{0} swept from blue to red such that the resonance is thermally pulled^{35} until the bistable transition occurs. This transition is represented by the black circles in Fig. 2(c). This, to a very good approximation, corresponds to the detuned resonance *ν*′ (see Appendix G). When plotted against the on-chip power (i.e., the incident power), *ν*′ reveals a linear dependence [Fig. 2(d)], hence suggesting linear absorption, likely due to defects at the surface. Following the same procedure as in Refs. 36 and 37, the dissipated power is extracted based on the calculated thermal resistance and the measured dependence of the resonance with temperature. This leads to an estimate of the absorption rate Γ_{abs}/2*π* = 8 MHz, which is much smaller than the total intrinsic losses Γ_{0}/2*π* ≈ 1 GHz. Correspondingly, the fraction of the dissipated on-chip power is *α* = 4Γ_{abs}(*κ* − Γ_{0})/*κ*^{2} ≈ 0.4%, with *κ* being the photon cavity decay rate. Absorption could be interpreted in terms of an effective imaginary refractive index^{38} through *n*′(InGaP) = *n*(InGaP)Γ_{abs}/2*πν* ≈ 10^{−7}, which is substantially lower than the estimate in Ref. 23 at *λ* = 1064 nm and consistent with the measurement of intrinsic *Q* > 2 × 10^{5}, still limited by elastic scattering.^{25}

## PROBING OF BROWNIAN MOTION OF THE OSCILLATOR

The optomechanical crystal considered in this section has an optical quality factor of *Q* = 3 × 10^{4}. The noise spectrum of the mechanical resonator reveals several peaks. The one with the largest frequency (*f*_{m} = 2.924 GHz [see the inset of Fig. 3(a)] is identified as the fundamental mode (see the spectrum in Fig. 8 of Appendix E). From the Lorentzian fit in the inset of Fig. 3(a), the mechanical linewidth is equal to Γ_{m}/2*π* = 1.2 MHz and the mechanical Q factor at room temperature and atmospheric pressure is *Q*_{m} = Ω_{m}/Γ_{m} = 2300 ± 150, which corresponds to the measurement at zero detuning. The vacuum optomechanical coupling is measured at room temperature and standard pressure with the technique discussed in Ref. 39. The reflected optical power is detected by a fast avalanche photodiode that is amplified by a 40 dB low noise amplifier before going to an electric spectrum analyzer (ESA). The electric power spectra corresponding to the mechanical motion of the resonator are compared to a calibration tone with spectrum *S*_{mod} generated by a phase modulator in the input optical path. This calibration tone induces frequency fluctuations that can be quantified, allowing the measurement of the power spectral density of the frequency fluctuations of the optical resonance due to the optomechanical interaction *S*_{νν}, as shown in the inset of Fig. 3(a) (details in Appendix I). In our case, it was not possible to operate the OM resonator at low enough power to avoid dynamical backaction while maintaining the detection level well above noise. Thus, what we measure and plot in Fig. 3(a) is the amplitude of the frequency fluctuations $G=\u222bS\nu \nu (f)dfnth$ that corresponds to *g*_{0} at vanishing laser-cavity detuning *ν*_{L} − *ν*′, which is corrected for the thermally induced spectral shift (see Appendix G). Considering the uncertainty on the photoelastic coefficients, the measured *g*_{0}/2*π* = 380 kHz is very close to the calculations solely including the photoelastic and moving boundary contributions. This is consistent with the fact that the thermomechanical term^{24} is negligible in our system (see discussion in Appendix K).

The corresponding mechanical linewidth [Fig. 3(b)] and optical spring (Fig. 10 of Appendix P) are measured and compared to theory^{40} (see Appendix P). The parameters used in the model (gathered in Table I of Appendix J) are measured: *κ*/2*π* = 6.5 GHz, Γ_{0}/2*π* = 0.9 GHz, Ω_{m}/2*π* = 2.92 GHz, and *g*_{0}/2*π* = 380 kHz. Only the on-chip laser power levels used in the model, *P*_{c} = 43.5, 47.9, and 51 *μ*W, have been adjusted within 20% of the experimental values indicated in Fig. 3(a).

## SELF-SUSTAINED OSCILLATIONS

We routinely observe self-sustained oscillations^{40} on devices with different loaded Q factors. We focus on the cavity with loaded quality factor *Q* = 30 000. As the power is increased, the resonator eventually undergoes regenerative oscillations. The threshold is predicted by the condition that the mechanical loss equates the optical antidamping calculated above: Γ_{m} + Γ_{om} = 0. Using the measured parameters above yields *P*_{c,tr} = 47 *μ*W, which is close to the measured value 40 *μ*W.

The knowledge of the number of phonons allows the calculation of the limit to the short-term linewidth, following Refs. 8, 41, and 42, similarly to the Schawlow-Townes limit for lasers,

Equation (1) is valid above threshold. To verify it, we record several spectra as the laser wavelength is swept toward the red across the resonance and the on-chip power is increased. The mechanical resonance drifts by 700 kHz for an on-chip power of 53 *μ*W [Fig. 4(a)]. We note that the spectra are very well fitted by a Voigt function [Fig. 4(b)], which is the convolution of a Gaussian function in which Full Width at Half Maximum (FWHM) is equal to *σ*_{G} = 5047 ± 929 Hz [which corresponds to the Resolution Bandwidth (RBW) used to record the different spectra], and a Lorentzian function. The Lorentzian linewidth, corresponding to the short-term linewidth, decreases from 1.2 ± 0.08 MHz to Γ_{eff,L}/2*π* = 80 ± 20 Hz for an on-chip power of 53 *μ*W.

In Fig. 4(c), the short-term linewidth is plotted against the RF integrated power (see Appendix O for the measurement of RF power). Assuming the optomechanical transduction to be linear, which is only an approximation, the number of phonons $n\xaf$ can be deduced from $n\xafth/n\xaf=PRF,th/PRF$, where $n\xafth$ is the number of phonons at thermal equilibrium, given by $n\xafth=kBT/\u0127\Omega m$, and *P*_{RF,th} is the RF integrated power at thermal equilibrium, when there is no dynamical backaction.

Equation (1) is plotted in black in Fig. 4(c). As the measurements are performed at room temperature, $n\xafth+1\u2248n\xafth$ and in that case, as pointed out in Ref. 8, the short-term linewidth is limited by thermal noise. As the experimental points obtained by fitting the spectra with the Voigt function follow the limit given by Eq. (1), we can conclude that the short-term linewidth of the self-sustained oscillations is limited by Brownian motion and this should be improved by lowering the temperature bath.

A deeper insight into the noise properties of the oscillator^{8} is gained by examining the spectral density of the phase noise $L(f)$ (Fig. 5), measured when the device is oscillating at its maximum amplitude. The cavity considered for this measurement has slightly different parameters (in particular, a lower optical quality factor *Q* = 2.5 × 10^{4}), and a stronger signal-to-noise ratio is obtained through optical heterodyning (see Appendix M). The coupled power used here is 52 *μ*W. From 5 kHz to 2 MHz, the phase noise spectral power density follows the slope *PSD* = Γ_{eff,L}/*f*^{2}, which is associated with the phase random walk. The Lorentzian linewidth Γ_{eff,L}/2*π* = 120 Hz is extracted, which is consistent with the direct measurement on the signal spectral power [Fig. 4(b)]. While white phase noise, due to thermal noise in the photodetector, dominates at higher frequencies, technical noise (1/*f*^{3}) dominates below 5 kHz, which is typical of a free running oscillator.

## CONCLUSION

In conclusion, an optomechanical crystal based on InGaP, a III-V piezoelectric semiconductor, has been developed based on a novel design. The typical intrinsic optical Q factor is about 2 × 10^{5}, whereas the loaded Q is controlled by removing holes. While nonlinear absorption is absent in the telecom spectral range, owing to the large electronic bandgap, the linear absorption is very small (Γ_{abs}/2*π* = 8 MHz), which, combined to a long thermal relaxation rate compared to the oscillation frequency, implies a negligible contribution of thermomechanical forces to damping Γ_{om}. The measured vacuum coupling constant is *g*_{0}/2*π* ≈ 380 kHz, which is in good agreement with modeling. At room temperature and standard pressure, the mechanical damping is *Q*_{m} = 2300, with a corresponding figure of merit *Q* × *f* = 6 × 10^{12}, which is of the same order of magnitude as in Ref. 23. Self-sustained oscillations are achieved routinely with a loaded optical *Q*_{L} > 2.5 × 10^{4}, with an on-chip optical power level of about 40 *μ*W. The measured mechanical short-term linewidth narrows down to about 100 Hz, limited by classical Brownian noise, and would decrease with temperature. Compared to other optomechanical oscillators, the 1/*f*^{2} term of the phase noise is basically the same as in silicon nitride microtoroids,^{44} which is also a low loss material once corrected for the carrier frequency to allow a fair comparison.^{43} We note that Micro-Electro-Mechanical Systems (MEMS)^{45} are about 10 dB below, but our OMC provides an optical output, convenient for the distribution of the signal on-chip. Completed with piezoelectric transducers and hybridized on a silicon photonic circuit,^{46} this device could be used for microwave to optical conversion and more elaborate miniaturized optoelectronic oscillators. We note that self-stabilization schemes have been proposed for OM resonators.^{47} Further improvement could be achieved by inducing tensile stress in the membrane.^{22,48} In perspective, this technology could be suitable for the investigation of complex nonlinear phenomena,^{49} quantum experiments, or synchronization of several oscillators.^{50} A hurdle to the synchronization of several oscillators is the difficulty in realizing two resonators oscillating at the same frequency due to fabrication errors. An encouraging result in this respect is the achievement of phonon routing between two optomechanical crystals connected by a phononic waveguide.^{51}

## ACKNOWLEDGMENTS

This work was supported by the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 732894 (FET Proactive HOT).

This work was also partly supported by the RENATECH network. We acknowledge support from a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program: Labex GANEX (Grant No. ANR-11-LABX-0014) and Labex NanoSaclay (Reference No. ANR-10-LABX-0035) with Flagship CONDOR. The authors declare no competing interests.

### APPENDIX A: DEPENDENCE OF *g*_{0} WITH THE THICKNESS OF THE MEMBRANE

Figure 6 shows the evolution of the optomechanical coupling and the optical and mechanical volumes with the thickness of the membranes.

### APPENDIX B: COMPUTING VACUUM OM COUPLING

The optomechanical coupling is calculated using the expressions from Ref. 52 as follows:

### APPENDIX C: PHOTOELASTIC PARAMETERS

The following photoelastic parameters for the computation of *g*_{0} are taken from Ref. 53:

We note that there is an uncertainty of about 10% in the above values. These values are given for a null angle with the (001) axis. As the injection axis of our cavities is along (110), a rotation must be applied to the photoelastic tensor.^{52}

### APPENDIX D: OPTICAL SPECTRUM

The spectrum in Fig. 7 is recorded by the Optical Coherence Tomography (OCT) method.^{25} Two resonances can be seen in this spectrum: the resonance with the highest frequency (the fundamental mode) is around 194 THz and the next resonance (the first order mode) is around 192.25 THz. Interferences can also be seen in the reflection spectrum: the low frequency signature is attributed to the interference between the input of the waveguide and the fiber facet, whereas the high frequency feature is linked to an interference between the input of the waveguide and the input of the photonic crystal. Further analysis of the reflection spectrum of such a cavity can be found in Ref. 54.

### APPENDIX E: MECHANICAL SPECTRUM

As shown in Fig. 8, the fundamental mode at 2.92 GHz is indeed the mode with the highest mechanical frequency. The first order mode cannot be seen as it has an odd symmetry, whereas the fundamental optical mode has an even symmetry. According to calculation, *g*_{0} between the fundamental optical mode and the second order mechanical mode is equal to *g*_{0}/2*π* = 23 kHz, which is much smaller than the *g*_{0}/2*π* = 380 kHz for the coupling between the fundamental optical and mechanical modes.

### APPENDIX F: EXTRACTION OF THE COMPLEX AMPLITUDE SPECTRUM

The interferogram *s*(*ν*) that is measured with the OCT system is related to the complex amplitude of the optical field from the sample *Ẽ*(*ν*) = *r*(*ν*)*E*_{r} through $s=E\u0303Er*+c.c.=rEr2+c.c.$, where *E*_{r} is the reference field. The transfer function (here, the complex reflectivity) *r*(*ν*) can be retrieved using the Hilbert transform, as shown, for instance, in Refs. 55 and 56, to extract a complex spectrum from the time interferogram *r*(*t*) measured by continuously changing the length of one of the arms of an unbalanced Michelson interferometer and a partially coherent light source. Here, the Hilbert transform is applied to a signal in the frequency domain, but the procedure is formally identical. This is achieved by taking the inverse Fourier transform *S* of the interferogram *s* and then calculating *R*(*t*) = *S*(*t*) + *sign*(*t*)*S*(*t*) and finally by Fourier transforming again to obtain *r*(*ν*).

From *r*(*ν*), the intrinsic losses Γ_{0} and the coupling losses *γ* can be deduced as^{25}

with *z* and *p* equal to

### APPENDIX G: THERMO-OPTIC INDUCED SPECTRAL SHIFT

For a linear evolution of the resonance frequency with the on-chip power, the time-domain coupled mode theory^{57} yields the following equation for the true value of the detuning:

where $\Delta \u2032=2\pi (\nu L\u2212\nu 0\u2032)$ and Δ = 2*π*(*ν*_{L} − *ν*_{0}), with $\nu 0\u2032$ being the current resonance frequency and *ν*_{0} being the “cold” cavity resonance frequency.

From Eq. (5) of Ref. 35, the maximum temperature change the system can undergo occurs when *ν*_{L} = *ν*_{0}′ or equivalently, when Δ = Δ*ν*_{bist} and Δ′ = 0. Therefore, the true detuning can be written as a function of *κ*,

Equation (G2) is, therefore, solved to obtain the real detuning.

### APPENDIX H: ESTIMATION OF THE ON-CHIP POWER

Optical power is coupled to the photonic crystal cavity using a fiber-collimator and a microscope objective. When the pump is out of resonance, these two optical components are the main sources of losses. Therefore, the on-chip power is estimated by taking into account losses coming from the collimator and the objective,

where *α*_{c} corresponds to the loss due to the fiber-collimator and *α*_{m} represents the loss due to the microscope objective. Therefore, the on-chip power is found using the following formula:

### APPENDIX I: MEASUREMENT OF THE VACUUM OM COUPLING

The optical source is a Keysight tunable laser. The laser is then modulated by a MPZ LN 10 phase modulator from Photline Technologies. After coupling into the cavity, the reflected light is detected by an Optilab APR-10-M APD photodetector and analyzed by a Rhode and Schwarz FSV 40 Electrical Spectrum Analyzer (ESA). All optical fibers are entirely polarization maintaining.

The measurement of *g*_{0} is carried out according to the method described in Ref. 39. The optomechanical coupling corresponds to the optical frequency shift resulting from the smallest displacement that can be detected using the mechanical resonator; therefore, the method consists in measuring the power density spectrum of the frequency shift *S*_{νν}(*f*) at thermal equilibrium, where the average amplitude of the thermal mechanical fluctuation is known and corresponds to $nth=kBT\u0127\Omega m$ phonons.

The vacuum coupling constant is therefore defined as

where the integral^{58} is computed about the mechanical resonance *f*_{m}. The unknown transduction coefficient relating *S*_{νν} to the measured electric power spectrum *S* is determined using a calibration tone generated by a phase modulator, which is inserted in the input path between the light source and the cavity,

where *S*_{cal} is the spectral power density in the phase modulation peak and $\varphi 0=\pi VcalV\pi $, with *V*_{π} = 6.11 V.

As the ESA measures the electrical power $S\u0303$ within the selected resolution bandwidth RBW, it follows that $\u222bScal(f)df=S\u0303cal$, as the calibration tone is spectrally narrower than RBW. In contrast, the spectrum of the frequency fluctuations of the OM oscillator is broader and, following,^{39} its integral is evaluated from the fitted Lorentzian line shape with FWHM Γ_{m} as $\u222bS(f)df=max(S\u0303)\Gamma m/RBW$. This leads to the known formula,

The experiment is carried out by setting the calibration tone away from the resonance but still close enough such that the transduction function can still be considered constant.

### APPENDIX J: PARAMETERS USED IN THE MODEL

Optical properties | Coupled quality factor | Q | 30 000 |

Intrinsic quality factor | Q_{0} | 200 000 | |

Resonance frequency (THz) | ν_{0} | 193.79 | |

Mechanical properties | Mechanical frequency (GHz) | f_{m} | 2.92 |

Quality factor | Q_{m} | 2300 | |

Zero point fluctuation (fm) | x_{ZPF} | 1.6 | |

Effective mass (pg) | m_{eff} | 1.07 | |

Thermal properties | Relaxation time (μs) | τ_{th} | 18 |

Linear thermal expansion (10^{−6} K) | α | 5.3 | |

Thermomechanical force (nN) | F_{th} | 0.6 | |

Frequency shift per displacement (Hz/m^{−1}) | G | 1.51 × 10^{21} | |

Optomechanical coupling (kHz) | g_{0} | 2π.380 |

Optical properties | Coupled quality factor | Q | 30 000 |

Intrinsic quality factor | Q_{0} | 200 000 | |

Resonance frequency (THz) | ν_{0} | 193.79 | |

Mechanical properties | Mechanical frequency (GHz) | f_{m} | 2.92 |

Quality factor | Q_{m} | 2300 | |

Zero point fluctuation (fm) | x_{ZPF} | 1.6 | |

Effective mass (pg) | m_{eff} | 1.07 | |

Thermal properties | Relaxation time (μs) | τ_{th} | 18 |

Linear thermal expansion (10^{−6} K) | α | 5.3 | |

Thermomechanical force (nN) | F_{th} | 0.6 | |

Frequency shift per displacement (Hz/m^{−1}) | G | 1.51 × 10^{21} | |

Optomechanical coupling (kHz) | g_{0} | 2π.380 |

### APPENDIX K: INFLUENCE OF PHOTOTHERMAL FORCES ON ANTIDAMPING AND OPTICAL SPRING

To quantify the influence of photothermal forces, we use the model developed in Ref. 24, which takes into account the dynamical effects of the evolution of the temperature in the OMC,

where Γ_{th} = *R*_{th}*ħω*_{L}*κ*_{abs}. *τ*_{th} is the thermal relaxation time, which is found by numerical simulation. *F*_{th} is the photothermal force, found by considering the influence of a linear expansion of the OMC due to one photon absorbed. By linearizing around an equilibrium point, we find the expressions for the optical spring and antidamping as a function of normalized detuning $x=\Delta \kappa $,

From Eqs. (K4) and (K5), one can surmise that the influence of photothermal forces is negligible when the relaxation time is slow compared to the oscillation dynamics. Indeed, when plotting the contribution of photothermal forces to antidamping and comparing it to the antidamping due to radiation pressure (Fig. 9), a difference of 4 orders of magnitude is clear between the two contributions.

### APPENDIX L: MEASUREMENT OF THE RF RESONANCE

The RF spectra are fitted using the Voigt line shape. This function is defined as the convolution of a Lorentzian line shape $L(x)=\gamma \pi \u22121(x2+\gamma 2)\u22121$ and a Gaussian broadening function $G(x)=exp(\u2212x2/2\sigma 2)/2\pi \sigma $, namely,

The Voigt function is calculated efficiently through the Faddeeva function *w*(*z*) (implemented in Ref. 59) and through the following relations: $V(x;\gamma ,\sigma )=R[w(z)]/\sigma 2\pi $ and $z=(x+\u0131\gamma )/\sigma 2$.

The fit is taken considering data points above the noise level estimated at −55 dB below the peak.

### APPENDIX M: MEASUREMENT OF THE PHASE NOISE

Phase noise is measured through the heterodyne technique^{60} using a frequency synthesizer as a local oscillator at *f*_{LO}. First, the optical signal extracted from the cavity is mixed with a continuous wave strong optical carrier. The optical signal obtained after mixing is sent to a balanced photodetector (Discovery Semiconductors). The electrical signal is amplified using a 20 dB Mini-Circuits amplifier and then further amplified by another 40 dB Femto amplifier before mixing. The low frequency signal *v*(*t*) is digitized using a 12 bit real time sampling oscilloscope (Lecroy HDO), sampling time 1.25 × 10^{6} samples/s, with *N* = 2.5 × 10^{6} samples. Then, the signal *v*_{n} = *v*(*n*Δ*t*) is processed as in Ref. 61. First, *v*_{n} is multiplied by exp(−2*πıf*_{0}Δ*tn*) and Fourier transformed using Fast Fourier Transform (FFT) (denoted as $F$). Then, the low-frequency part of the spectrum (|*f*| < *BW*) is transformed back in the time domain, which gives the analytic signal *v*_{a} around the carrier frequency *f*_{0} = *f*_{OM} − *f*_{LO}. The phase is obtained by taking the argument of each sample *ϕ*_{n} = *arg*(*v*_{a}(*t*_{n})). Then, the power spectral density of the phase is evaluated within a certain spectral band *f* ∈ [*f*_{i}, *f*_{i+1}] using the standard procedure. This defines a time span 1/2*f*_{i} long enough to resolve *f*_{i}. Consequently, *ϕ*_{n} is distributed in *N*_{i} consecutive windows Δ*W*_{j} with duration 1/2*f*_{i} and containing *M*_{i} samples such that *M*_{i}*N*_{i} = *N*. In each time window, the nonstationary contributions (trend and average) are removed and then a suitable window function (Hanning *h*_{n}) is applied to the signal $\varphi \u0303n,j,i$, before Fourier transform (FFT). Finally, the power spectra $Sj,i(fk,i)=|F(\varphi \u0303n,j,ihn)|2$ are averaged over the windows, being *f*_{k,i} ∈ [*f*_{i}, *f*_{i+1}]. More precisely,

Then, half of the spectral power density *S*_{φ}(*f*) is taken in order to get the phase noise $L(f)$in dBc/Hz.

### APPENDIX N: COMPUTING ERROR BARS

To compute the error bars, the fitting parameters are varied by a small quantity. We then calculate the residue (equal here to the sum of $\u22111\u2212yfit,iyi2$) with these new fitting parameters. The error bar is then equal to the maximum shift, which yields the same residue as the original parameters within a 2% error bar.

In the case where there are several experimental values for a single parameter (for example, in the measurement of the optomechanical coupling *g*_{0} and the mechanical quality factor *Q*_{m}), the error bar is simply equal to the standard deviation obtained from the different experimental values.

### APPENDIX O: MEASUREMENT OF RF POWER

This measurement is done with a SIR5 Thorlabs photodiode and a HSA-Y-2-40 Femto amplifier. The spectrum is recorded, thanks to FSV-40 Rohde & Schwarz electric spectrum analyzer. If the signal is narrower than the Resolution Bandwidth (RBW; here, equal to 5 kHz), then the RF power is simply equal to the maximum of the mechanical spectrum. If the signal is larger than the RBW, then the power below the peak is integrated.

### APPENDIX P: COMPUTING EFFECTIVE MECHANICAL LINEWIDTH AND OPTICAL SPRING

For the effective mechanical linewidth, the derivation is straightforward as its calculation can be found in Ref. 40. The narrowing due to the dynamical backaction Γ_{om}, when Δ = *ν*_{L} − *ν*′ > 0,

with the number of photons in the cavity given by the usual coupled mode theory.

For the effective mechanical frequency, the evolution of the mechanical frequency with the temperature needs to be taken into account. Knowing the evolution of the elasticity constants of GaAs (as data were missing for InGaP and GaP), we first computed the evolution of the mechanical mode frequency with temperature using the Comsol software. We found it to be linear with temperature, Δ*ω*_{th} = *α*_{th}Δ*T*, where −198 kHz K^{−1}.

Then in Eq. (K1), we neglected the thermodynamical term and added a thermal force *F*_{th} = *k*_{th}*x* = 2Δ*ω*_{m,th}*ω*_{m}*m*_{eff}*x*, where Δ*ω*_{m,th} is the shift in mechanical frequency due to a temperature change. Thus, the evolution of the mechanical frequency which takes into account both the optical spring effect and the evolution of the mechanical frequency with the temperature is

In Fig. 10, the power was adjusted within 20% of the experimental value.

### APPENDIX Q: EVOLUTION OF THE RF POWER WITH CAVITY PHOTON NUMBER

Figure 11 shows the characteristic threshold behavior of the optomechanical oscillator in self-sustained oscillations.

### APPENDIX R: COMPARISON TO STATE OF THE ART GHz OSCILLATORS

A brief summary of the state of the art of GHz oscillators is given above in Table II.

. | Phase noise at . | Phase noise at . | Phase noise at . | . | . |
---|---|---|---|---|---|

Oscillator . | 100 Hz (dBc/Hz) . | 1 kHz (dBc/Hz) . | 100 kHz (dBc/Hz) . | Frequency (GHz) . | Size . |

This work | −20 | −50 | −85 | 2.92 | 20 × 0.62 μm^{2} |

Single loop optoelectronic | −95 | −120 | −160 | 10 | Several tens of centimeters |

oscillator (OEO) from Ref. 62 | |||||

Integrated OEO^{63} | −50 | −55 | −80 | 8.87 | 5 × 6 cm^{2} |

Oscillator based on Kerr comb | −60 | −115 | −130 | 10 | 1 mm |

generation^{64} | |||||

Film bulk acoustic resonator^{65} | −50 | −76 | −120 | 2 | 130 × 60 μm^{2} |

. | Phase noise at . | Phase noise at . | Phase noise at . | . | . |
---|---|---|---|---|---|

Oscillator . | 100 Hz (dBc/Hz) . | 1 kHz (dBc/Hz) . | 100 kHz (dBc/Hz) . | Frequency (GHz) . | Size . |

This work | −20 | −50 | −85 | 2.92 | 20 × 0.62 μm^{2} |

Single loop optoelectronic | −95 | −120 | −160 | 10 | Several tens of centimeters |

oscillator (OEO) from Ref. 62 | |||||

Integrated OEO^{63} | −50 | −55 | −80 | 8.87 | 5 × 6 cm^{2} |

Oscillator based on Kerr comb | −60 | −115 | −130 | 10 | 1 mm |

generation^{64} | |||||

Film bulk acoustic resonator^{65} | −50 | −76 | −120 | 2 | 130 × 60 μm^{2} |

## REFERENCES

As data for InGaP are absent in the literature, we have used the parameters of GaP as in Ref. 25.

The procedure used here does not operate in extreme conditions such as overcoupled cavity.

Which should depend on the geometry since it represents absorption due to surface defects.

$20\u2061logN+L(f)$, where N is the ratio between the higher and lower operation frequency.

Here, we consider the single-sided spectrum.