High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide

Optomechanical crystals have attracted great attention recently for their ability to realize strong photon-phonon interaction in cavity optomechanical systems. By far, the operation of cavity optomechanical systems with high mechanical frequency has to employ tapered fibres or one-sided waveguides with circulators to couple the light into and out of the cavities, which hinders their on-chip applications. Here, we demonstrate larger-centre-hole nanobeam structures with on-chip transmission-coupling waveguide. The measured mechanical frequency is up to 4.47 GHz, with a high mechanical Q-factor of 1.4 × 103 in the ambient environment. The corresponding optomechanical coupling rate is calculated and measured to be 836 kHz and 1.2 MHz, respectively, while the effective mass is estimated to be 136 fg. With the transmission waveguide coupled structure and a small footprint of 3.4 μm2, this simple cavity can be directly used as functional components or integrated with other on-chip devices in future practical applications.


S1. Expression for of optomechanical coupling rate measurement
The optomechanical coupling rate for an optomechanical system is defined as where ωo is the optical angular frequency of the cavity, x is the displacement of the mechanical resonator and xzpf, which equals to eff m /2m  , is the displacement of zero-point fluctuation of the mechanical resonator.
Consequently, the intensity of mechanical vibration caused optical frequency shift is related with g. Quantitatively, the square of the optical resonant frequency fluctuation can be expressed as 1 22 oc 2, ng where nc is the phonon occupation number of the interested mechanical mode. In this work, the mechanical motion is mainly excited by thermal environment, thus where T is the temperature of the measurement environment, Ωm is the angular frequency of the mechanical mode and kB and ħ are the Boltzmann and Planck constant, respectively.

S2. Optomechanical coupling rate measurement
The experiments for measuring the optomechanical coupling rate was conducted 1 year later than that only measuring the optical and mechanical frequency and Q-factor. Thus, the performances of the fabricated structure deteriorate, especially the mechanical Q-factor. Consequentially, only the zeroth order mode can be observed.
However, as the optomechanical coupling rate is depended on the profiles of optical and mechanical modes, it should vary little, like the optical and mechanical frequency. The RF source generating a 4.40 GHz signal with root-mean-square (RMS) voltage of 5 mV. The resolution bandwidth (RBW) of the ESA is set to be 300 kHz during the measurement. This RBW is much smaller than the mechanical spectrum width of the OMC cavity but much larger than that of signal generated by the RF source. As the output of the swept-tuned type ESA is the power filtered by a Gaussian bandpass filter, the RF power of the EOM signal detected by the ESA is the peak value (i.e. PEOM = -51.93 dBm) while that of the OMC cavity signal needs integration. Figure S2(b) shows the measurement data of ESA for the OMC cavity in linear coordinate as well as the Lorentzian fitting curve. Consequently, the RF power detected by the ESA for the OMC cavity can be expressed as where Area is the area between the Lorentzian fitting curve and the background noise, i.e. the green area shown in Fig S2(b), and the coefficient of 1.064 is originated from the ratio between the area and the width of the Gaussian curve. For this measurement, POMC = -52.20 dBm is obtained.
To determine the βOMC, we analyse the optical transmission spectrum of the cavity. In frequency domain, the optical transmission spectrum is a Lorentzian curve, i.e.
where T0 is the maximum transmission and κ is the total decay rate of the cavity, which equals to the full width at half maximum (FWHM). The βOMC is maximized to be 2/κ, 44/2π THz -1 for this OMC cavity, when frequency detuning |ω-ωo| equals to κ/2. During the measurement of mechanical spectrum of the OMC cavity, we set the wavelength of the laser at the blue detuning point. The βEOM of the EOM we used at 4.4 GHz is about 0.58 V -1 .
Based on the conditions provided above, the optomechanical coupling rate (g/2π) is estimated to be 1.2 MHz.