Slot-Mode Optomechanical Crystals: A Versatile Platform for Multimode Optomechanics

We demonstrate slot-mode optomechanical crystals, a class of device in which photonic and phononic crystal nanobeam resonators separated by a narrow slot are coupled through optomechanical interactions. In these geometries, nanobeam pairs are patterned so that a mechanical breathing mode is confined at the center of one beam, and a high quality factor (Qo>10^5) optical mode is confined in the slot between the beams. Here, we produce slot-mode devices in a stoichiometric Si3N4 platform, with optical modes in the 980 nm band, coupled to breathing mechanical modes at 3.4 GHz, 1.8 GHz, and 400 MHz. We exploit the high Si3N4 tensile stress to achieve slot widths down to 24 nm, which leads to enhanced optomechanical coupling, sufficient for the observation of optomechanical self-oscillations at all studied frequencies. We utilize the slot mode concept to develop multimode optomechanical systems with triple-beam geometries, in which two optical modes are coupled to a single mechanical mode, and two mechanical modes are coupled to a single optical mode. This concept allows great flexibility in the design of multimode chip-scale optomechanical systems with large optomechanical coupling at a wide range of mechanical frequencies.


I. INTRODUCTION
Sideband-resolved cavity optomechanical systems have recently demonstrated their potential in a wide variety of applications, including motion sensing [1,2], ground state cooling [3,4], and optomechanically-induced transparency [5,6]. For these applications, high efficiency requires large optomechanical coupling strength in addition to sideband resolution (mechanical frequency optical linewidth). Additional phenomena have been observed in multimode cavity optomechanical systems, in which multiple optical and/or mechanical modes interact, including wavelength conversion [7][8][9], Raman-ratio thermometry [10], energy transfer between mechanical modes [11], and optomechanical mode mixing [12]. Phonon pair generation [13], mechanical mode entanglement [12,14], and unresolved sideband cooling [15] have also been theoretically proposed. In all these systems, improved performance and broader applicability could be achieved if the optical and mechanical modes could be independently tailored to a given application.
The slot-mode optomechanical crystal structure, in which the optical and mechanical modes are confined in separate but interacting beams (Fig. 1), is one method of achieving this flexibility while maintaining large optomechanical coupling strength. Simulations [16] have shown that, in systems in which the optomechanical interaction is dominated by moving boundaries, this geometry can significantly increase the optomechanical coupling strength relative to single nanobeam optomechanical crystals. It also provides the desired design flexibility to enable multimode applications such as optomechanical wavelength conversion. * karen.grutter@nist.gov † kartik.srinivasan@nist.gov In this work, we experimentally demonstrate slot-mode optomechanical crystals implemented in stoichiometric Si 3 N 4 , a material whose broad optical transparency and large intrinsic tensile stress make it attractive for many applications. In Sec. III, we show how this intrinsic stress can be exploited to achieve slots with aspect ratios of 10:1, and in Sec. IV we demonstrate how tuning this slot width improves device performance in 3.4 GHz band devices. Sec. V shows how the mechanical mode frequency can be changed while minimally affecting the optical mode, with demonstrations of 1.8 GHz and 400 MHz band devices. Finally, in Sec. VI, we extend the slotmode optomechanical crystal concept to multimode optomechanical devices, in which two mechanical modes are coupled to a single optical mode, and two optical modes are coupled to a single mechanical mode.

II. BASIC DEVICE DESIGN
A slot-mode optomechanical crystal, shown in Fig. 1b, consists of two parallel beams separated by a narrow slot. The "optical beam" is a photonic crystal cavity designed to confine the optical mode in the slot. The "mechanical beam" is a phononic crystal resonator optimized to confine the mechanical breathing mode (Fig. 1c) while maintaining low optical loss. Both the optical and mechanical modes are confined along the z-axis by periodic patterning of holes. In the outer mirror region, the lattice spacing is constant, but it varies quadratically in the cavity region (Fig. 1a). Details on the design of this device are outlined in Ref. [16]. The devices in this work were designed for optical modes around 980 nm and mechanical breathing modes around 3.4 GHz. The optical beam is patterned with identical elliptical holes (188 nm×330 nm) along its length, while the mechanical beam holes have a constant height (370 nm) and widths that are varied such that the "ribs" between the holes align with the elliptical holes in the optical beam.
There have been several demonstrations of sidebandresolved single-nanobeam optomechanical crystals [4,6,7,17,18], in which a GHz frequency mechanical breathing mode is coupled to an optical mode localized by the same physical structure. These geometries are distinguished by the breathing mode's high frequency (enabling sideband resolution), isolation from mechanical supports due to the phononic mirrors, and strong interaction with the optical mode. Our goal in this work is to retain these advantageous features while increasing the system's versatility through the slot mode geometry.
Optical slot modes have been utilized before to achieve large optomechanical coupling in microrings/disks [19,20], bilayer photonic crystal slabs [21], and photonic crystal zipper cavities [22]. These applications were lower frequency (< 150 MHz) than the 3.4 GHz band breathing modes in this work, and, thus operated in the unresolved-sideband regime (mechanical frequency < optical linewidth). In addition, previous demonstrations have not taken full advantage of the flexibility of the slot mode architecture, as the mechanical and optical modes were supported by the same structural components. Separating the optical and mechanical modes into two beams enables independent design of these modes. This opens a wide range of frequency combinations that would be difficult to access with a single optomechanical structure. The slot-mode structure also opens the possibility for additional interactions with both modes, which can be separately accessed from the beam sides opposite the slot. For example, electrodes could be added to the outside of the mechanical beam with minimal perturbation of the optical mode. Adding more optical or mechanical beams, thereby forming more slots, can also increase the device functionality by realizing multimode optomechanical systems, as discussed in Secs. VI and VII.

III. STRESS TUNING AND DEVICE FABRICATION
In addition to the design of the mechanical and optical beams, device parameters are also strongly dependent on the width of the slot between the two beams, simulated in Fig. 1d-f. Given a device with fixed design of the optical and mechanical beams, reducing the slot width redshifts the optical resonance, and reduces Q o somewhat (still above 10 5 ). The optomechanical coupling rate g 0 increases significantly as the slot width decreases, so the slot between the optical and mechanical beams should be made as small as possible. Lithographically defining small spaces and etching high-aspect-ratio trenches are both challenging in fabrication. This can be mitigated by taking advantage of the intrinsic film stress of stoichiometric Si 3 N 4 (≈ 1 GPa). An asymmetric anchoring condition in a doubly-clamped beam induces asymmetry in the stress, thereby causing it to move laterally. Long, thin tethers asymmetrically attached to the ends of parallel nanobeams have been used to shrink gaps to as small as 40 nm after release [23]. To achieve the same effect, we investigated small slits at a beam's ends (Fig. 2a). Finite element method (FEM) simulations show that varying the width and depth of these slits controls the lateral displacement of the center of the beam (Fig. 2b). In the slot-mode device, a large initially defined and etched slot would be reduced post-release to the desired width by including these stress-tuning slits at the ends of the optical beam. Slot mode optomechanical crystal nanobeams were fabricated in 250 nm thick stoichiometric Si 3 N 4 deposited via low-pressure chemical-vapor deposition on a bare Si substrate (tensile stress ≈ 1 GPa). Devices were patterned via electron-beam (E-beam) lithography in a positive E-beam resist and developed in hexyl acetate at 7 • C. The pattern was transferred to the Si 3 N 4 using a CF 4 /CHF 3 reactive ion etch. Devices were released in a 45 % KOH solution at 75 • C followed by a dip in a 1:4 HCl:H 2 O solution. Finally, the devices were dried on a hotplate.
A scanning-electron microscope (SEM) image of a released device is shown in Fig. 2c. The lithographicallydefined slots were between 80 nm and 120 nm, and, with the SEM, we measured stress-tuned slots as small as 24 nm at the center, an aspect ratio of about 10:1 that would be difficult to achieve with lithography alone. Fig. 2b graphs the SEM-measured displacements of the beam centers with respect to the stress-tuning slit depths. The measured trend matches well with the displacements predicted in the FEM simulations.

IV. DEMONSTRATION OF SLOT-MODE CONCEPT
The experimental setup used to characterized the Si 3 N 4 slot-mode optomechanical crystals is shown in Fig. 3a, and was previously described in [17]. All measurements were taken at room temperature and pressure. Devices were characterized with a 980 nm external cavity tunable diode laser, which was coupled evanescently to the devices via a dimpled optical fiber taper waveguide (FTW) with a minimum diameter of ≈ 1 µm.
Among the measured devices, a device with a 50 nm stress-tuned slot had the highest intrinsic optical quality factor Q o at (1.65 ± 0.09) × 10 5 (linewidth of 2.0 GHz ± 0.1 GHz) [24], as shown in Fig. 3b. Q o s up to ≈ 2.4 times higher have been demonstrated in Si 3 N 4 single-nanobeam optomechanical crystals [25], but it is expected that the slot mode would have lower Q o s because the geometry has more scattering sites near the optical mode. In these devices, narrower slots generally resulted in lower Q o s, with 20 nm slot devices having the lowest Q o s around 2.5 × 10 4 . With further optimization, improving Q o in smaller slot designs is feasible.
We also used optical characterization to more precisely determine the effect of the stress tuning. Iterations of PR = photoreceiver devices were made with the same optical and mechanical design but stress-tuning slits of varying depth, so that the only difference among these devices would be the final, stress-tuned slot width. An example is shown in Fig. 3b. Three devices with the same optical and mechanical design show a red shift of the optical resonance as the designed stress-tuned slot width decreases (the stress-tuning slit depth increases). This trend is expected from simulation (Fig. 1d), and was consistent in 24 of 27 unique device designs, indicating that varying this stresstuning slit depth is a reliable technique for tuning the slot width.  For mechanical mode spectroscopy, the signal was detected with a high-bandwidth (8 GHz) photoreceiver, the output of which was sent to a real-time electronic spectrum analyzer. Optomechanical characterization required longer-term stability of the coupling, so the FTW was positioned a few hundred nanometers to the side of the device and affixed via van der Waals forces to nearby protruding parts of the Si 3 N 4 film. The coupling distance was chosen for a transmission minimum around 70 %. The blue detuning of the laser further increased the measurement stability by enabling access to the thermally self-stable regime [27] so that the laser did not have to be externally locked to the cavity.
We used a calibration signal from a phase modulator to measure g 0 in a few devices [28,29], as shown in Fig. 3c, where the phase modulator calibration tone is shown along with the thermal noise spectrum of the 3.49 GHz mechanical breathing mode (quality factor Q m ≈ 3900). For a device with a designed, stress-tuned slot width of 60 nm, we measured g 0 /2π = 184 kHz ± 2 kHz, where the uncertainty comes from the uncertainty in the thermal noise spectrum fit and the measurement of the phase modulator V π = 2.78 V ± 0.01 V (Sec. A). This value matches well with the FEM-simulated g 0 values (Fig. 1f). Another device, having a designed, stresstuned slot width of 20 nm, had a phase-modulatorcalibrated g 0 /2π = 317 kHz ± 3 kHz, which also aligns with FEM simulations and confirms the significant improvement in coupling achieved by producing narrow slots. We note that such slot mode geometries are of particular importance for materials such as Si 3 N 4 , the low refractive index (vs. Si or GaAs) of which limits the achievable coupling strength in single nanobeam geometries.
In addition, with the laser blue-detuned (∆ > 0), optomechanical back-action coherently amplifies the mechanical mode, increasing the detected amplitude while decreasing the effective mechanical linewidth γ m,eff . Assuming only optomechanical damping changes with input power, the effective linewidth is related to the optical power at the coupling point to the device P in as follows, where κ is the intrinsic optical loss rate, κ ex is the external coupling rate, ω o is the optical resonant frequency, and Ω m is the intrinsic mechanical frequency [30,31]: Thus, the effective mechanical linewidth should change linearly with respect to optical power, with the intercept indicating the intrinsic mechanical linewidth γ m and the slope proportional to g 2 0 . For a blue-detuned laser, this slope is negative, and when the optomechanical amplification cancels out γ m , the device reaches the regime of regenerative self-oscillation. The P in at which this occurs is the threshold power.
We use this relationship to determine the intrinsic Q m of these devices by looking at the detected mechanical spectrum with respect to power. To compensate for the cavity's power-dependent thermo-optic shift, for each input power, we adjust the laser wavelength to the optimal detuning value, which corresponds to the point at which the mechanical peak is maximized. We then linearly fit the subthreshold γ m,eff with respect to P in to find γ m . This same procedure is used to compare devices with similar optical and mechanical parameters; in this case, the slope is an indicator of the relative effective g 0 . Fig. 4a and b show measurements of two such devices with similar optical and mechanical performance but different stress tuning. One device, which had a designed stress-tuned slot width of 70 nm (stress-tuning slit depth of 220 nm), had an intrinsic Q o = (3.7 ± 0.1) × 10 4 and an intrinsic Q m = 2380 ± 90 [26]. The data correspond-ing to this device are shown in Fig. 4a and the blue data in Fig. 4c. The other device had a designed stresstuned slot width of 20 nm (stress-tuning slit depth of 295 nm), an intrinsic Q o = (3.2 ± 0.1) × 10 4 , and an intrinsic Q m = 2400 ± 300. The data corresponding to this narrower-slot device are shown in Fig. 4b and the red data in Fig. 4c.
Comparing measurements of these two devices, the mechanical mode in the 20 nm slot device is more amplified than in the 70 nm slot device. For P in ≈ 1.2 mW, the detected mechanical peak in the 20 nm slot device is ≈ 65 dB above the noise floor, while the 70 nm slot device's mechanical peak is only ≈ 19 dB above the noise floor. The measurements of the optomechanical narrowing of the effective mechanical linewidth (Fig. 4c), show that the slope of the line for the narrower-slot device (red) is much steeper than for the wider-slot device (blue). Because they have similar optical and mechanical Qs, this suggests that the device with the 20 nm slot has a higher effective g 0 . It is also noteworthy that the narrower stress-tuned slot enhances the back-action enough that it reaches self-oscillation above a threshold power of 900 µW.
Among all the devices measured, devices with more aggressive stress tuning (narrower slots) generally had steeper linewidth-narrowing slopes, implying higher ef- fective g 0 s, as expected from simulation (Fig. 1f) and confirmed by the aforementioned phase modulator calibration measurements. The most aggressively tuned devices, with slots designed to be 20 nm, had high enough effective g 0 s that all but one of them reached the threshold for self-oscillation within the power range of the laser. This indicates that narrowing the slot via stress-tuning is an effective way to enhance the optomechanical coupling in slot-mode optomechanical crystal devices.
We note that the measured threshold powers are much lower, and the mechanical-linewidth-narrowing slopes much steeper, than would be expected with the g 0 values obtained from the phase modulator calibration method. This suggests other factors in the system are contributing to the effective optomechanical back-action. These could include DC optical gradient forces acting to pull the beams closer [22] and interaction of the breathing mode with the oscillating flexural beam modes.
In particular, although we designed these devices for optimal coupling to the mechanical breathing mode, and focused our measurements on characterizing it, there are other mechanical modes that couple to the optical mode. Defects in the fabricated device give rise to additional breathing-type mechanical modes [32], but the most well-coupled modes tend to be the lateral flexural beam modes. An FEM simulation of the fundamental lateral flexural beam mode (11.4 MHz) of the mechanical beam is shown in Fig. 5a. Because it is well-coupled to the optical mode and has a much lower frequency than the mechanical breathing mode, its threshold power for self-oscillation is very low; we measure it to be at P in ≈ 150 µW. We also note that, upon detection of the optical signal modulated by self-oscillating breathing and flexural modes we observed mixing tones as sidebands of the breathing mode, as shown in Fig. 5c.

V. FLEXIBLE MECHANICAL RESONATOR DESIGN
Separating the mechanical and optical modes into two beams in the slot-mode architecture adds flexibility in designing these modes compared to a single nanobeam. By modifying the design of the mechanical beam, a wide range of mechanical frequencies can be accessed without significantly affecting the optical mode. To that end, we demonstrate lower-frequency designs around 1.8 GHz and 400 MHz. Implementing highly-localized breathing modes in various RF bands (here, the IEEE-defined UHF and L) broadens the potential application space. One way to change the mechanical frequency is simply to change the full width of the mechanical beam while keeping the same lattice variation.
For the 1.8 GHz band design, shown in Fig. 6a, the mechanical beam width was increased from 700 nm to 1.55 µm. Measurements of a fabricated device (Fig. 6b) having a designed, stress-tuned slot width of 80 nm are shown in Fig. 6c- (1.01 ± 0.03) × 10 5 , and the measured intrinsic Q m = 2130 ± 50, as derived from the weighted linear fit shown in Fig. 6e. These values are comparable to the 3.4 GHz band devices. The optomechanical coupling of this device was strong enough that we observed self-oscillation for laser powers above ≈ 2 mW (P in is ≈ 20 % of the laser power at the FTW input). Above threshold, we also observed harmonics on the breathing mode (Fig. 6d). These arise from nonlinear modulation of the optical field due to the Lorentzian optical mode shape, as reported in other systems [33][34][35].
For the 400 MHz design, the mechanical beam width was increased to 4 µm. At this width, the mechanical mode is not well-confined for the same lattice parameters, but the "ribs" still contribute to the optical confinement. Thus, we kept the ribs to maintain high Q o , but increased the effective mechanical lattice constants by "breaking" two-thirds of the ribs, as shown in Fig. 6f. Measurements of a fabricated device (Fig. 6g) having a slot width of 80 nm are shown in Fig. 6h-j. The measured intrinsic Q o = (1.02 ± 0.02) × 10 5 , and the measured intrinsic Q m = 800 ± 300, as derived from the weighted linear fit shown in Fig. 6j. The Q o is comparable to that of the 3.4 GHz band devices, indicating that the "brokenrib" geometry minimally perturbs the optical mode. The Q m , however, is much lower than in the 3.4 GHz band devices. This is likely due to an increase in air damping with decreased frequency [36] and an increase in anchor loss from the effective two-thirds decrease in the number of lattice periods in the mirror region of the mechanical beam. As with the other devices in this work, the mechanical spectra (Fig. 6h) include other, less-well-coupled peaks that correspond to additional breathing-type mechanical modes from defects in the fabricated device [32] or harmonics of lower-frequency flexural modes. The optomechanical coupling of the 414 MHz breathing mode of this device was strong enough that it reached selfoscillation for laser powers above ≈ 1.2 mW. As with the 1.8 GHz band device, we observed harmonics on the breathing mode above threshold (Fig. 6i).

VI. MULTIMODE OPTOMECHANICAL DEVICES
In addition to increasing flexibility in the available mechanical frequencies, the slot-mode device architecture enables new functionality in that it is straightforward to add another separate optical and/or mechanical mode. In this work, we demonstrate two cases: a single optical mode simultaneously coupled to two different mechan- state laser cooling of an unresolved-sideband mechanical resonator [15]. Moreover, recent progress has been made in studying M-O-M devices in other platforms experimentally, including recent investigations of Bogoliubov mechanical modes [13], as well as systems showing synchronization of mechanical resonators via a travelling optical mode [37,38]. An O-M-O slot-mode device provides a new platform for optical frequency conversion, as proposed in Ref. [16]. Unlike in previous demonstrations of optomechanically-enabled optical frequency conversion [7][8][9], the O-M-O device enables quasi-independent optical mode selection and independent optimization of the coupling into each optical mode.
In the example M-O-M device of this work, we surround an optical beam with a 1.8 GHz band mechanical beam (Sec. V) and a 3.4 GHz band mechanical beam, with 80 nm slots between the beams (Fig. 7b). The resultant optical mode is concentrated in both slots simultaneously (Fig. 7a). We couple to the optical mode by hovering the FTW a few hundred nanometers above the optical beam. The measured intrinsic Q o = (1.26 ± 0.02) × 10 5 (Fig. 7c), and we simultaneously detect modulation from both the 1.8 GHz band and 3.4 GHz band mechanical modes (Fig. 7d). For the same input optical power, the detected 1.8 GHz band mode has a larger amplitude than the 3.4 GHz band mode primarily because a lower frequency mode has a larger thermal noise motional amplitude for the same temperature. With the optical quality factor in excess of 10 5 (linewidth ≈ 2.4 GHz), this device is in the range of sideband-resolved operation for both the 3.4 GHz and 1.8 GHz band modes, suggesting that this device is a candidate for high-frequency Bogoliubov mechanical mode studies. We have also measured M-O-M devices with 1.8 GHz band and 400 MHz band mechanical breathing modes coupled to the same 981.85 nm optical mode (Sec. B).
The O-M-O device demonstrated here comprises a mechanical beam with a 1.8 GHz band mechanical breathing mode coupled to an optical beam on each side, with 80 nm slots between the beams (Fig. 8a). The top nanobeam was made slightly wider, resulting in a redshifted optical resonance. The top and bottom optical modes were characterized separately by repositioning the FTW, and the measured optical spectra are shown in Fig. 8b. The top mode had a measured intrinsic Q o = (1.05 ± 0.02) × 10 5 at 973.21 nm, and the bottom mode had a measured intrinsic Q o = (1.1 ± 0.1) × 10 5 at 947.34 nm.
The mechanical breathing mode at ≈ 1.835 GHz was detected when coupled both to the top and to the bottom optical modes (Fig. 8c). (There is another, less-wellcoupled peak at ≈ 1.831 GHz that corresponds to either an additional breathing-type mechanical mode from defects in the fabricated device [32] or a harmonic of a lower-frequency flexural mode.) We also measured the effective mechanical linewidth as a function of power for both optical modes (Fig. 8d). A weighted linear fit of these measurements indicates that the intrinsic Q m as measured via each optical mode is in good agreement: from the top mode, Q m = 1800 ± 100, and from the bottom mode, Q m = 1800 ± 200. The resonant frequency also matches, as shown in Fig. 8c, implying that these two optical modes are in fact coupled to the same mechanical mode. The difference in detected mechanical peak heights and the difference in the slopes of mechanical linewidth with respect to optical power stem from the fact that the bottom optical mode at 947 nm couples more strongly to the mechanical mode.

VII. DISCUSSION
We have demonstrated slot-mode optomechanical devices in which the mechanical breathing mode of a patterned nanobeam is coupled to an optical mode that is laterally confined by a second patterned nanobeam and resides within the slot between the two beams. Along with large optomechanical coupling rates in excess of 300 kHz (as measured via phase-modulator calibration) enabled in part by narrow slot widths that can be achieved by taking advantage of the tensile film stress in Si 3 N 4 , this platform allows for flexible design of the optical and mechanical modes, with mechanical beams tailored to support breathing modes ranging from 400 MHz to 3.5 GHz. Moreover, this geometry can naturally be extended to multimode systems; we have shown triplenanobeam devices with two different mechanical modes coupled to a single optical mode, as well as a triplenanobeam device in which two different optical modes are coupled to a single mechanical mode.
Future work will focus on the use of these multimode geometries in applications such as optical wavelength conversion and Bogoliubov mechanical mode formation for phonon pair generation. Though some of the current devices are already weakly in the sideband-resolved limit (κ/2π ≈ 2 GHz < Ω m /2π ≈ 3.4 GHz), additional improvements in Q o would enable sideband resolution for all of the mechanical frequencies studied. Finally, the implementation of on-chip waveguides will likely be necessary to achieve long-term, stable coupling to multimoded systems.

Supplementary Material
This section provides supplementary information for this work. It describes the procedure for determining the phase modulator V π , presents data from an additional triple-nanobeam device, and explores how higherorder optical slot modes couple to various mechanical modes. In Sec. IV, we use the phase modulator calibration method [1,2] to measure the optomechanical coupling g 0 of slot-mode optomechanical crystals. The V π of the electro-optic phase modulator must be accurately known in order to do this calibration.
To measure V π , we send the 980 nm laser signal through the phase modulator, modulated at 3.5 GHz by an RF signal generator, and into a scanning Fabry-Pérot interferometer. The detected signal traces out the carrier and phase-modulator-induced sidebands in the optical signal, and we view them on an oscilloscope, as shown in Fig. S1a. Changing the power applied by the RF signal generator to the phase modulator changes the magnitude of the carrier and sidebands. The RF power P RF is related to the signal voltage V sig = √ 2ZP RF , where the phase modulator input impedance Z = 50 Ω. Knowing this, we can graph the peak magnitudes with respect to V sig , as shown in Fig S1b. For a phase modulator, the carrier peak magnitude should follow the curve A (J 0 (πV sig /V π )) 2 , and the first sideband magnitude should follow the curve A (J 1 (πV sig /V π )) 2 , where A scales the amplitude of the Bessel functions of the first kind J 0 and J 1 . We fit data from the carrier and first sideband to these functions in Fig. S1b, and both fits result in V π = 2.78±0.01 V, where the uncertainty comes from the fit and is one standard deviation. This value corresponds well with the vendorspecified value for the phase modulator.  Fig. S2b. The resultant optical mode is concentrated in both slots simultaneously (Fig. S2a). We couple to the optical mode by hovering the FTW a few hundred nanometers above the optical beam. The measured intrinsic Q o was (5.80 ± 0.06)×10 4 (Fig. S2c), and we simultaneously detect modulation of the transmitted optical signal from both the 400 MHz band and 1.8 GHz band mechanical modes (Fig. S2d). For the same input optical power, the detected 400 MHz band mode has a larger amplitude than the 1.8 GHz band mode primarily because a lower frequency mode has a larger thermal noise motional amplitude for the same temperature. As with the other devices in this work, the mechanical spectrum (Fig. S2d) includes other, less-wellcoupled peaks that correspond to additional breathingtype mechanical modes from defects in the fabricated device [3] or harmonics of lower-frequency flexural modes.

Appendix C: Coupling to Higher-Order Slot Modes
These slot-mode optomechanical crystals confine multiple optical modes in the slot in addition to the fundamental mode for which they were designed. We observed some of these higher-order modes, as shown in Fig. S3a and b. Because they are distributed more widely along the slot, these modes couple less strongly to the highly-localized breathing mode and more strongly to other mechanical modes in the device, such as higher- Higher-order optical slot mode in the same device at ≈ 971 nm with intrinsic Qo = (3.57 ± 0.08) × 10 4 . (c) Mechanical spectra measured while coupled to the fundamental optical mode (red) and higherorder optical mode (black). The optical power input to the fiber taper waveguide was ≈ 2.2 mW at the 949 nm mode and ≈ 3.5 mW at the 971 nm mode. This power spectral density plot is referenced to a power of 1 mW = 0 dB. order breathing-type mechanical modes arising from fabrication defects [3].
In the example of Fig. S3, a device with a designed, stress-tuned slot width of 20 nm has an optical mode at ≈ 949 nm with an intrinsic optical Q = (3.96 ± 0.09) × 10 4 as well as an optical mode at ≈ 971 nm with an intrinsic optical Q = (3.57 ± 0.08) × 10 4 . When pumped at the 949 nm fundamental optical mode, it self-oscillates at the ≈ 3.52 GHz mechanical breathing mode, and we begin to see sidebands due to mixing with the modulation from the low-frequency flexural beam modes, as described in Sec. 4 of the main text. However, even with 50 % more optical power, the mechanical breathing mode does not self-oscillate when pumped at the 971 nm optical mode. In addition, the mechanical spectrum reveals another peak at ≈ 3.48 GHz with about the same optomechanical coupling to the 971 nm mode as the ≈ 3.52 GHz mode, suggesting that this higher-order optical mode is also coupling to some higher-order, less-well-confined mechanical mode.