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We demonstrate a highly efficient optomechanical oscillator based upon a small silicon microdisk resonator with a 2-μm radius. The device exhibits a strong optomechanical coupling of 115 GHz/nm and a large intrinsic mechanical frequency-Q product of 4.32 × 1012 Hz. It is able to operate at a high frequency of 1.294 GHz with an ultralow threshold of 3.56 μW while working in the air environment. The high efficiency, high frequency together with the structural compactness and CMOS compatibility of our device enables great potential for broad applications in photonic-phononic signal processing, sensing, and metrology.
© 2012 Optical Society of America
Self-sustaining mechanical oscillators have broad application potential from optical/wireless communication, bio-molecule sensing, to frequency metrology [1–3]. Recently, there emerged an intriguing approach by taking advantage of optical forces in micro-/nano-optomechanical cavities [4–6], where the strong dynamic back-action is able to excite regenerative mechanical oscillation with only a continuous-wave laser input [7–22]. For practical applications, a high-frequency operation above gigahertz while simultaneously maintaining a high efficiency is essential for a variety of applications which rely critically on the oscillation frequency and efficiency for enhancing the capacity of information processing or the probing resolution of bio-sensing [1–3]. However, among current demonstrated optomechanical oscillators, only two types are able to work beyond VHF frequencies, either depending on delicate photonic crystals [11] or intrinsic Brillouin scattering of materials [13, 14]. In practice, it is highly desirable to have a device platform easy for fabrication and integration while with flexible engineering capability of oscillation frequency [1–3]. In this paper, we demonstrate such an optomechanical oscillator on the silicon-on-insulator platform by combining strong optomechanical coupling, tiny effective motional mass, high optical and mechanical qualities into a single compact device. The optomechanical oscillator is able to operate at a frequency as high as 1.294 GHz with a power threshold as small as 3.56 μW while operating in the air.
The employed device structure is a compact silicon microdisk resonator with a radius of 2-μm sitting on a silica pedestal (Fig. 1(a)). It is fabricated from a silicon-on-insulator wafer by use of e-beam lithography to define the device pattern, fluorine-based plasma to etch the silicon layer, and hydrofluoric acid to undercut the silica pedestal. In general, the optomechanical coupling in a whispering-gallery cavity scales inversely with the device radius R as , leading to a small optomechanical coupling in most devices with diameters of tens of microns [7, 13–15, 17–20]. In contrast, by taking advantage of strong mode confinement enabled by the high refractive index of silicon, we are able to shrink the device radius down to only 2 μm while maintaining a high optical quality factor. Such a small radius results in an optomechanical coupling coefficient of |gOM|/(2π) = 100 GHz/nm for a telecom-band wavelength, which corresponds to a strong per-photon force of about 66 fN.
Fig. 1 (a) Scanning electron microscopic (SEM) image of the device. The silicon microdisk has a layer thickness of 260 nm, sitting on a 2-μm high silica pedestal with an undercut width of ∼ 1.9 μm. (b) & (c) The FEM simulations of fundamental and second-order radial-stretching mechanical mode, respectively. The color map indicates the amplitude of mechanical displacement. (d) Simulated mechanical frequencies as a function of microdisk radius for the fundamental (red) and second-order (blue) radial-stretching mode with a undercut-to-radius ratio of 95%. The dashed line indicates that for our fabricated device.
Radiation pressure in the device couples strongly to the radial-stretching mechanical modes (Figs. 1(b) and 1(c)) whose dominant motion is along the radial direction of the microdisk. Simulation by the finite-element method (FEM) indicates the frequencies of these modes scale inversely with the device radius (Fig. 1(d)). In particular, shrinking the device radius down to 2 μm is able to significantly increase the mechanical frequency up to and 3.328 GHz for the fundamental and second-order mode, respectively, with a small effective motional mass of meff =5.7 and 13.0 picograms, leading to a strong vacuum optomechanical coupling rate of 108 kHz for the fundamental mode. We are able to achieve a large undercut-to-radius ratio of 95% (Fig. 1(a)) which dramatically suppresses the clamping loss [2] and thus enables a high mechanical quality factor even at a high mechanical frequency.
Figure 2(a) shows our experimental setup, where the tunable laser is launched into the device through a fiber taper which is docked onto the two nanoforks for stable operation. The optical wave transmitted from the cavity carrying the information of mechanical motion is detected by a high-speed detector with a bandwidth of 1.3 GHz whose output is characterized by a spectrum analyzer. The device is tested in the air environment. Figure 2(b) shows the recorded cavity transmission for an optical mode at 1502.03 nm polarized in the disk plane (transverse electric (TE) mode. Fig. 2(b), inset), which exhibits an intrinsic optical Q factor of 3.5 × 105. To enhance the optical transduction of mechanical motion, we increase the coupling depth of cavity transmission to 40%, corresponding to a Q factor of 3.1 × 105 for the loaded cavity. By locking laser frequency half way into the cavity resonance (cavity transmission 80%) at the blue-detuned side, we observe the RF spectrum of the cavity transmission as shown in Fig. 3(a). It shows that the fundamental radial-stretching mechanical mode at 1.294 GHz is clearly visible due to its strong coupling to the optical mode. The recorded mechanical frequency is very close to the simulated value, with a slight discrepancy primarily due to the uncertainty in measuring the undercut width. The second-order stretching mode was not observed simply because its frequency is beyond our detector bandwidth.
Fig. 2 (a) Experimental setup. VOA: variable optical attenuator. MZI: Mach-Zehnder interferometer. (b) Experimentally recorded (blue) cavity transmission for an optical mode at 1502.03 nm, with a theoretical fitting (red). The inset shows the simulated mode profile.
Fig. 3 (a) RF spectrum of cavity transmission. The irregular background is due to the spectrum analyzer which has a capture bandwidth of 36 MHz. (b) The spectral density of mechanical displacement for the fundamental radial stretching mode. The experimental data (blue) is compared directly with the theory (red). The experimental data were recorded with an input optical power low enough not to introduce noticeable dynamic back-action. In both figures, the gray traces show the noise background of the detector.
Figure 3(b) shows the detailed transduced spectral intensity of mechanical displacement for the fundamental radial-stretching mode (blue curve), which is compared directly with the cavity-optomechanical theory (red) [23, 24]. The theoretical curve takes into account the intrinsic thermal mechanical vibration at room temperature as well as the detector noise and the shot noise from the optical wave. The comparison indicates that the actual optomechanical coupling coefficient in our device is |gOM|/(2π) = 115 GHz/nm, which is about 15% higher than the simulated value. As shown in Fig. 3(b), the strong optomechanical coupling in our device enables a displacement probing sensitivity of . As this sensitivity is dominantly limited by the detector noise, it can be further improved by boosting the cavity transmitted signal (i.e., through a low-noise optical amplifier). The theoretical noise background is slightly smaller than the experimental value possibly due to the frequency noise of laser which is not included in the theory. The detailed analysis of Fig. 3(b) shows that the mechanical mode has an intrinsic linewidth of only 388 kHz, which corresponds to a mechanical Q factor of 3.3 × 103 while the device resides in the air. Consequently, the mechanical mode exhibits a frequency-quality product of 4.32×1012 Hz, close to the largest values reported [2,3,11,22,25].
Such a high-quality optomechanical cavity readily implies its application for an efficient optomechanical oscillator. To enhance the dynamic back-action for exciting optomechanical oscillation, we increase the laser-cavity detuning at the blue-detuned side to about 1.6 times the linewidth of the loaded cavity (cavity transmission of 96.4%), so that the created Stokes sideband falls into the cavity resonance. Figure 4(a) shows the RF spectra of cavity transmission at two levels of optical powers dropped into the cavity. When the dropped optical power is well below the oscillation threshold, the mechanical spectrum (blue) is close to that of thermal mechanical vibration (Fig. 3(b)) with a slightly narrowed linewidth due to amplification by dynamic back-action. In contrast, by increasing the dropped optical power to 4 μW, the mechanical mode is excited into coherent oscillation and the peak value of the mechanical spectral intensity is dramatically enhanced by more than 50 dB. Detailed analysis of the mechanical spectrum (Fig. 4(a), inset) shows that the mechanical linewidth is drastically suppressed down to 854 Hz, corresponding to an effective mechanical Q factor as high as 1.52 × 106.
Fig. 4 (a) RF Spectrum of the oscillator, with a dropped optical power of 0.63 (blue) and 4.0 (red) μW, respectively. The inset shows the detailed spectrum at the pump power of 4.0 μW, with a Lorentzian fitting (red). (b) The mechanical energy as a function of the dropped optical power. The mechanical energy is measured by integrating the spectral area of the transduced mechanical spectrum, normalized by the intrinsic thermal mechanical energy at room temperature. The red curve is a linear fit to data above threshold (except the last three points showing saturation). The inset shows the theoretical power threshold for our device as a function of laser-cavity frequency detuning normalized by the mechanical frequency Ωm. The dashed line indicates the detuning used in our experiment.
The performance of the optomechanical oscillator can be more accurately quantified by measuring the mechanical energy as a function of pumping power. By integrating the spectral area of the transduced mechanical spectrum at various dropped optical powers, we obtained the curve shown in Fig. 4(b), where the mechanical energy is normalized by the room-temperature thermal mechanical energy in the absence of dynamic back-action. Figure 4(b) shows a clear lasing behavior in which the mechanical energy remains rather small when the dropped optical power is below a certain level but starts to increase dramatically and depends linearly on the pump power when the dropped optical power is above the threshold value. By fitting the above-threshold data with a linear curve, we obtain a threshold power of 3.56 μW, which is the lowest value measured up to date for such a high oscillation frequency [11,13,14,18]. The fitted slope is 2.38 × 103 μW−1, corresponding to a slope efficiency of optical-to-mechanical energy conversion about 3.54% and that of photon-to-phonon number scattering about 5.46×103. The slight saturation of mechanical energy at high pump power is likely due to the perturbation to laser-frequency locking induced by the thermal-optical effect in the cavity.
The power threshold of an optomechanical oscillator is given by [7, 12, 23, 24]
where Pd is the threshold optical power dropped into the cavity, Γ0 and Γt are the photon decay rate for the intrinsic and loaded optical cavity, respectively. Δ is the laser-cavity detuning. Γm is the energy decay rate of the mechanical mode. Equation (1) together with our cavity parameters gives theoretical values of power threshold which is plotted in the inset of Fig. 4(b). For a Δ = 1.6Γt used in our experiment, the theoretical expectation of power threshold is 3.4 μW (dashed line in the inset of Fig. 4(b)), which agrees very well with our experiment. Note that this power threshold is not the optimal value due to the non-optimized laser-cavity detuning used in the experiment. As the mechanical frequency is nearly twice of the photon decay rate, Ωm ≈ 2Γt, our device is close to the sideband-resolved regime in which mechanical amplification is maximized when Δ ≈ Ωm [7] (see the inset of Fig. 4(b)). Therefore, further optimization of laser-cavity detuning would enable reaching the optimal power threshold of 2.0 μW.Interestingly, Fig. 4(b) shows that the normalized mechanical energy is about 885 at the pumping power of 4 μW. This value is about 1.9 times as that inferred from the mechanical linewidth shown in Fig. 4(a). The discrepancy is very likely due to the mechanical frequency fluctuations induced by the optical spring, which depends on the intracavity photon number Nph as [7, 12, 23, 24]
The slight perturbation to the laser-cavity detuning by the thermal-optical fluctuations in the cavity would result in a temporal variation of mechanical frequency, thus broadening the averaged mechanical spectrum to a certain extent and leading to an overestimate of the mechanical linewidth. The same reason is likely to be responsible for the slight asymmetry on the wings of the mechanical spectrum (Fig. 4(a), red curve). As the optical spring is rather small in the sideband-resolved regime which has negligible effect on the mechanical energy, the mechanical energy is thus a more accurate approach to interpret the oscillation performance. A normalized mechanical energy of 885 implies that the real linewidth of mechanical oscillation is likely to be only about 438 Hz. Equation (2) indicates that the optical spring disappears at , which, in the sideband-resolved regime, nearly coincides with maximal mechanical amplification. Therefore, locking laser frequency at this detuning would eliminate the impact of optical spring while maintaining the maximal pumping efficiency.In summary, We have demonstrated a highly efficient optomechanical oscillator based upon a silicon microdisk resonator with a radius of only 2 μm. The optomechanical resonator exhibits a strong optomechanical coupling coefficient of 115 GHz/nm. Its fundamental radial-stretching mechanical mode exhibits a frequency of 1.294 GHz and an intrinsic mechanical Q factor of 3.3 × 103, corresponding to a large frequency-quality product of 4.32 × 1012 Hz. We are able to excite coherent optomechanical oscillation with a threshold dropped power as low as 3.56 μW which is the lowest measured up to date for such a high oscillation frequency. A high-quality high-frequency optomechanical oscillator is essential for a variety of applications ranging from frequency metrology, clock distribution, force/mass/displacement/bio-sensing, to signal processing in microwave photonics. The high efficiency and high frequency nature of our demonstrated device, together with its structural compactness and its compatibility with CMOS fabrication thus exhibits great potential for future application along these directions.
Acknowledgments
This work was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant ECS-0335765).
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