Slot-Mode Optomechanical Crystals: A Versatile Platform for Multimode Optomechanics
SSlot-Mode Optomechanical Crystals: A Versatile Platform for MultimodeOptomechanics
Karen E. Grutter, ∗ Marcelo I. Davan¸co, and Kartik Srinivasan † Center for Nanoscale Science and Technology, National Instituteof Standards and Technology, Gaithersburg, MD 20899-6203 USA (Dated: September 18, 2018)We demonstrate slot-mode optomechanical crystals, a class of device in which photonic andphononic crystal nanobeam resonators separated by a narrow slot are coupled through optomechan-ical interactions. In these geometries, nanobeam pairs are patterned so that a mechanical breathingmode is confined at the center of one beam, and a high quality factor ( Q o (cid:38) ) optical mode isconfined in the slot between the beams. Here, we produce slot-mode devices in a stoichiometricSi N platform, with optical modes in the 980 nm band, coupled to breathing mechanical modes at3.4 GHz, 1.8 GHz, and 400 MHz. We exploit the high Si N tensile stress to achieve slot widthsdown to 24 nm, which leads to enhanced optomechanical coupling, sufficient for the observation ofoptomechanical self-oscillations at all studied frequencies. We utilize the slot mode concept to de-velop multimode optomechanical systems with triple-beam geometries, in which two optical modesare coupled to a single mechanical mode, and two mechanical modes are coupled to a single opticalmode. This concept allows great flexibility in the design of multimode chip-scale optomechanicalsystems with large optomechanical coupling at a wide range of mechanical frequencies. I. INTRODUCTION
Sideband-resolved cavity optomechanical systems haverecently demonstrated their potential in a wide varietyof applications, including motion sensing [1, 2], groundstate cooling [3, 4], and optomechanically-induced trans-parency [5, 6]. For these applications, high efficiency re-quires large optomechanical coupling strength in additionto sideband resolution (mechanical frequency (cid:29) opticallinewidth). Additional phenomena have been observed inmultimode cavity optomechanical systems, in which mul-tiple optical and/or mechanical modes interact, includ-ing wavelength conversion [7–9], Raman-ratio thermom-etry [10], energy transfer between mechanical modes [11],and optomechanical mode mixing [12]. Phonon pair gen-eration [13], mechanical mode entanglement [12, 14], andunresolved sideband cooling [15] have also been theoret-ically proposed. In all these systems, improved perfor-mance and broader applicability could be achieved if theoptical and mechanical modes could be independentlytailored to a given application.The slot-mode optomechanical crystal structure, inwhich the optical and mechanical modes are confined inseparate but interacting beams (Fig. 1), is one methodof achieving this flexibility while maintaining large op-tomechanical coupling strength. Simulations [16] haveshown that, in systems in which the optomechanical in-teraction is dominated by moving boundaries, this geom-etry can significantly increase the optomechanical cou-pling strength relative to single nanobeam optomechani-cal crystals. It also provides the desired design flexibilityto enable multimode applications such as optomechanicalwavelength conversion. ∗ [email protected] † [email protected] In this work, we experimentally demonstrate slot-modeoptomechanical crystals implemented in stoichiometricSi N , a material whose broad optical transparency andlarge intrinsic tensile stress make it attractive for manyapplications. In Sec. III, we show how this intrinsic stresscan be exploited to achieve slots with aspect ratios of10:1, and in Sec. IV we demonstrate how tuning thisslot width improves device performance in 3.4 GHz banddevices. Sec. V shows how the mechanical mode fre-quency can be changed while minimally affecting the op-tical mode, with demonstrations of 1.8 GHz and 400 MHzband devices. Finally, in Sec. VI, we extend the slot-mode optomechanical crystal concept to multimode op-tomechanical devices, in which two mechanical modes arecoupled to a single optical mode, and two optical modesare 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 designedto confine the optical mode in the slot. The “mechani-cal beam” is a phononic crystal resonator optimized toconfine the mechanical breathing mode (Fig. 1c) whilemaintaining low optical loss. Both the optical and me-chanical modes are confined along the z -axis by periodicpatterning of holes. In the outer mirror region, the lat-tice spacing is constant, but it varies quadratically inthe cavity region (Fig. 1a). Details on the design ofthis device are outlined in Ref. [16]. The devices in thiswork were designed for optical modes around 980 nmand mechanical breathing modes around 3.4 GHz. Theoptical beam is patterned with identical elliptical holes(188 nm ×
330 nm) along its length, while the mechanicalbeam holes have a constant height (370 nm) and widths a r X i v : . [ phy s i c s . op ti c s ] A ug cavity mirrormirror opticalbeammechanicalbeam slot (a)(b)(c) (d) (e) (f) N o r m . | E |
471 nm700 nm zx rib FIG. 1. (a) Variation of the optomechanical crystal lattice constant along the length of the beams. The period is fixed in themirror regions at the beam ends and varies quadratically in the center cavity region. (b) The slot mode optomechanical crystalis formed by parallel optical and mechanical beams that are separated by a narrow slot. The zoomed-in image of the centershows the finite element method (FEM) simulated electric field amplitude of the optical slot mode around 980 nm. (c) FEMsimulation of the breathing mode of the mechanical beam (around 3.4 GHz). (d), (e), and (f): The width of the slot is variedin an FEM simulation of the (d) resonant wavelength, (e) optical quality factor ( Q o ), and (f) optomechanical coupling g / (2 π ). that are varied such that the “ribs” between the holesalign with the elliptical holes in the optical beam.There have been several demonstrations of sideband-resolved single-nanobeam optomechanical crystals [4, 6,7, 17, 18], in which a GHz frequency mechanical breath-ing mode is coupled to an optical mode localized bythe same physical structure. These geometries are dis-tinguished by the breathing mode’s high frequency (en-abling sideband resolution), isolation from mechanicalsupports due to the phononic mirrors, and strong inter-action with the optical mode. Our goal in this work is toretain these advantageous features while increasing thesystem’s versatility through the slot mode geometry.Optical slot modes have been utilized before to achievelarge optomechanical coupling in microrings/disks [19,20], bilayer photonic crystal slabs [21], and photoniccrystal zipper cavities [22]. These applications werelower frequency ( <
150 MHz) than the 3.4 GHz bandbreathing modes in this work, and, thus operated in theunresolved-sideband regime (mechanical frequency < op-tical linewidth). In addition, previous demonstrationshave not taken full advantage of the flexibility of the slotmode architecture, as the mechanical and optical modeswere supported by the same structural components. Sep-arating the optical and mechanical modes into two beamsenables independent design of these modes. This opensa wide range of frequency combinations that would bedifficult to access with a single optomechanical structure.The slot-mode structure also opens the possibility for ad-ditional interactions with both modes, which can be sep-arately 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 theoptical mode. Adding more optical or mechanical beams,thereby forming more slots, can also increase the devicefunctionality by realizing multimode optomechanical sys-tems, as discussed in Secs. VI and VII. III. STRESS TUNING AND DEVICEFABRICATION
In addition to the design of the mechanical and opticalbeams, device parameters are also strongly dependent onthe width of the slot between the two beams, simulatedin Fig. 1d-f. Given a device with fixed design of the op-tical and mechanical beams, reducing the slot width red-shifts the optical resonance, and reduces Q o somewhat(still above 10 ). The optomechanical coupling rate g increases significantly as the slot width decreases, so theslot between the optical and mechanical beams shouldbe made as small as possible. Lithographically definingsmall spaces and etching high-aspect-ratio trenches areboth challenging in fabrication. This can be mitigatedby taking advantage of the intrinsic film stress of stoi-chiometric Si N ( ≈ F E M MeasuredDisplacementsFilm Stress: 1 GPa optical beam
471 nmslit width = 270 nmslit depth { anchored (b) (a) (c) O p t i c a l B e a m M e c h a n i c a l B e a m Endstress-tuning slit70nmCenter
24 nm
FIG. 2. (a) FEM simulation of a tensile-stressed beam with stress-tuning slits at the ends. (b) Displacement at beam centerwith respect to slit depth. FEM results (line) are for a beam with the same dimensions as the optical beam of the slot-modedevice. Error bars on the measured data are due to the uncertainty in the SEM measurements and are one standard deviationvalues. (c) Scanning electron microscope (SEM) images of a released device. Insets show the slot width at the beam end isabout 70 nm, shrinking to 24 nm at the beam center. displacement of the center of the beam (Fig. 2b). In theslot-mode device, a large initially defined and etched slotwould be reduced post-release to the desired width by in-cluding these stress-tuning slits at the ends of the opticalbeam.Slot mode optomechanical crystal nanobeams werefabricated in 250 nm thick stoichiometric Si N depositedvia low-pressure chemical-vapor deposition on a bare Sisubstrate (tensile stress ≈ ◦ C. The pattern was transferred to the Si N using aCF /CHF reactive ion etch. Devices were released in a45 % KOH solution at 75 ◦ C followed by a dip in a 1:4HCl:H O solution. Finally, the devices were dried on ahotplate.A scanning-electron microscope (SEM) image of a re-leased device is shown in Fig. 2c. The lithographically-defined slots were between 80 nm and 120 nm, and,with the SEM, we measured stress-tuned slots as smallas 24 nm at the center, an aspect ratio of about 10:1that would be difficult to achieve with lithography alone.Fig. 2b graphs the SEM-measured displacements of thebeam centers with respect to the stress-tuning slit depths.The measured trend matches well with the displacementspredicted in the FEM simulations.
IV. DEMONSTRATION OF SLOT-MODECONCEPT
The experimental setup used to characterized theSi N slot-mode optomechanical crystals is shown inFig. 3a, and was previously described in [17]. All mea-surements were taken at room temperature and pressure.Devices were characterized with a 980 nm external cavitytunable diode laser, which was coupled evanescently tothe devices via a dimpled optical fiber taper waveguide(FTW) with a minimum diameter of ≈ µ m.Among the measured devices, a device with a 50 nmstress-tuned slot had the highest intrinsic optical quality factor Q o at (1 . ± . × (linewidth of 2 . ± . Q o s up to ≈ . N single-nanobeamoptomechanical crystals [25], but it is expected that theslot mode would have lower Q o s because the geometryhas more scattering sites near the optical mode. In thesedevices, narrower slots generally resulted in lower Q o s,with 20 nm slot devices having the lowest Q o s around2 . × . With further optimization, improving Q o insmaller slot designs is feasible.We also used optical characterization to more preciselydetermine the effect of the stress tuning. Iterations of PR = photoreceiver nanobeam
DAQ fiber taperwaveguide
ESA polarizationcontroller
980 nmECDL
ESA = electronic spectrum analyzer (a) (c)
EOM
RF gen.
DAQ = data acqusition cardEOM = electro-optic phase modulator polarizationcontroller
ECDL = external cavity diode laser (b)
Detuning (pm) -15 150 N o r m a li z ed P o w e r FIG. 3. (a) Optical modes are detected by swept-wavelengthspectroscopy, while mechanical modes are measured when thelaser is on the blue-detuned shoulder of the optical mode. For g calibration, the laser is phase-modulated. (b) Optical res-onant wavelength of three devices with different stress-tunedslot widths. (inset) Optical spectrum and fit of highest mea-sured Q o among these devices, having designed gap of 50 nmand Q o = (1 . ± . × [24] (c) Example mechanicalspectrum, including phase modulator calibration peak. Thispower spectral density plot is referenced to a power of 1 mW= 0 dB. Lorentzian fit of thermal noise spectrum is in red. devices were made with the same optical and mechan-ical design but stress-tuning slits of varying depth, sothat the only difference among these devices would bethe final, stress-tuned slot width. An example is shownin Fig. 3b. Three devices with the same optical and me-chanical design show a red shift of the optical resonanceas the designed stress-tuned slot width decreases (thestress-tuning slit depth increases). This trend is expectedfrom simulation (Fig. 1d), and was consistent in 24 of 27unique device designs, indicating that varying this stress-tuning slit depth is a reliable technique for tuning the slotwidth. (a)(b) (c)
70 nm slot20 nm slot 20 nmslot 70 nm slot Ω m /2 π ≈ Ω m /2 π ≈ FIG. 4. (a) Mechanical spectra at different input opticalpowers ( P in ) for a device with a designed stress-tuned slotwidth of 70 nm, intrinsic Q o = (3 . ± . × , intrinsic Q m = 2380 ±
90 [26], and Ω m / (2 π ) ≈ .
31 GHz. (b) Mechan-ical spectra at different P in for a device with 20 nm designedstress-tuned slot width, intrinsic Q o = (3 . ± . × , in-trinsic Q m = 2400 ± m / (2 π ) ≈ .
49 GHz. (c)Measured γ m, eff / (2 π ) of the devices from (a) (blue) and (b)(red). Error bars represent the uncertainty in the fit of the me-chanical spectra to a Lorentzian. Dashed lines show weightedlinear fits of the subthreshold γ m, eff / (2 π ). The power spectraldensity plots in (a) and (b) are referenced to a power of 1 mW= 0 dB. For mechanical mode spectroscopy, the signal was de-tected with a high-bandwidth (8 GHz) photoreceiver,the output of which was sent to a real-time electronicspectrum analyzer. Optomechanical characterization re-quired longer-term stability of the coupling, so the FTWwas positioned a few hundred nanometers to the side ofthe device and affixed via van der Waals forces to nearbyprotruding parts of the Si N film. The coupling dis-tance was chosen for a transmission minimum around70 %. The blue detuning of the laser further increasedthe measurement stability by enabling access to the ther-mally self-stable regime [27] so that the laser did not haveto be externally locked to the cavity.We used a calibration signal from a phase modulator tomeasure g in a few devices [28, 29], as shown in Fig. 3c,where the phase modulator calibration tone is shownalong with the thermal noise spectrum of the 3.49 GHzmechanical breathing mode (quality factor Q m ≈ g / π = 184 kHz ± V π = 2 .
78 V ± .
01 V (Sec. A). Thisvalue matches well with the FEM-simulated g values(Fig. 1f). Another device, having a designed, stress-tuned slot width of 20 nm, had a phase-modulator-calibrated g / π = 317 kHz ± N , thelow refractive index (vs. Si or GaAs) of which limits theachievable coupling strength in single nanobeam geome-tries.In addition, with the laser blue-detuned (∆ > γ m, eff . As-suming only optomechanical damping changes with inputpower, the effective linewidth is related to the opticalpower at the coupling point to the device P in as follows,where κ is the intrinsic optical loss rate, κ ex is the exter-nal coupling rate, ω o is the optical resonant frequency,and Ω m is the intrinsic mechanical frequency [30, 31]: γ m, eff = γ m + g ω o (cid:126) κ ex P in ∆ + ( κ/ (cid:32) κ/ m ) + ( κ/ − κ/ − Ω m ) + ( κ/ (cid:33) (1a)= γ m + g S( κ, κ ex , ω o , ∆ , Ω m ) P in (1b)Thus, the effective mechanical linewidth should changelinearly with respect to optical power, with the interceptindicating the intrinsic mechanical linewidth γ m and theslope proportional to g . For a blue-detuned laser, thisslope is negative, and when the optomechanical amplifi-cation cancels out γ m , the device reaches the regime ofregenerative self-oscillation. The P in at which this occursis the threshold power.We use this relationship to determine the intrinsic Q m of these devices by looking at the detected mechanicalspectrum with respect to power. To compensate for thecavity’s power-dependent thermo-optic shift, for each in-put power, we adjust the laser wavelength to the optimaldetuning value, which corresponds to the point at whichthe mechanical peak is maximized. We then linearly fitthe subthreshold γ m, eff with respect to P in to find γ m .This same procedure is used to compare devices withsimilar optical and mechanical parameters; in this case,the slope is an indicator of the relative effective g .Fig. 4a and b show measurements of two such deviceswith similar optical and mechanical performance but dif-ferent stress tuning. One device, which had a designedstress-tuned slot width of 70 nm (stress-tuning slit depthof 220 nm), had an intrinsic Q o = (3 . ± . × andan intrinsic Q m = 2380 ±
90 [26]. The data correspond-ing to this device are shown in Fig. 4a and the bluedata in Fig. 4c. The other device had a designed stress-tuned slot width of 20 nm (stress-tuning slit depth of295 nm), an intrinsic Q o = (3 . ± . × , and an in-trinsic Q m = 2400 ± P in ≈ . ≈
65 dB above the noise floor, while the 70 nm slotdevice’s mechanical peak is only ≈
19 dB above the noisefloor. The measurements of the optomechanical narrow-ing of the effective mechanical linewidth (Fig. 4c), showthat 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 Q s,this suggests that the device with the 20 nm slot has ahigher effective g . It is also noteworthy that the nar-rower stress-tuned slot enhances the back-action enoughthat it reaches self-oscillation above a threshold power of900 µ W.Among all the devices measured, devices with moreaggressive stress tuning (narrower slots) generally hadsteeper linewidth-narrowing slopes, implying higher ef- (a) (b)(c) (b)
FEM: 11.4 MHz
Breathing Mode(3.5 GHz)FundamentalMode (11.4 MHz) −5 − FIG. 5. (a) FEM simulation of the fundamental lateral flex-ural beam mode. (b) Measured 3 dB linewidth of the fun-damental flexural beam mode (blue) and the breathing mode(red) as a function of P in . Error bars represent the uncer-tainty in the fit of the mechanical spectra to a Lorentzian.The fundamental mode self-oscillates at P in ≈ µ W, whilethe breathing mode self-oscillates at P in ≈ µ W. (c) Side-bands on the breathing mode (red) and the spectrum of har-monics of the lower-frequency flexural beam modes (blue) lineup, indicating a mixing between the two. (inset) The full,double-sided spectrum around the self-oscillating breathingmode. All power spectral density plots in (c) are referencedto a power of 1 mW = 0 dB. fective g s, as expected from simulation (Fig. 1f) andconfirmed by the aforementioned phase modulator cali-bration measurements. The most aggressively tuned de-vices, with slots designed to be 20 nm, had high enougheffective g s that all but one of them reached the thresh-old for self-oscillation within the power range of the laser.This indicates that narrowing the slot via stress-tuning isan effective way to enhance the optomechanical couplingin slot-mode optomechanical crystal devices.We note that the measured threshold powers are muchlower, and the mechanical-linewidth-narrowing slopesmuch steeper, than would be expected with the g valuesobtained from the phase modulator calibration method.This suggests other factors in the system are contribut-ing to the effective optomechanical back-action. Thesecould include DC optical gradient forces acting to pullthe beams closer [22] and interaction of the breathingmode with the oscillating flexural beam modes.In particular, although we designed these devices foroptimal coupling to the mechanical breathing mode, andfocused our measurements on characterizing it, thereare other mechanical modes that couple to the opticalmode. Defects in the fabricated device give rise to ad-ditional breathing-type mechanical modes [32], but themost well-coupled modes tend to be the lateral flexuralbeam modes. An FEM simulation of the fundamentallateral flexural beam mode (11.4 MHz) of the mechani-cal beam is shown in Fig. 5a. Because it is well-coupledto the optical mode and has a much lower frequencythan the mechanical breathing mode, its threshold powerfor self-oscillation is very low; we measure it to be at P in ≈ µ W. We also note that, upon detection of theoptical signal modulated by self-oscillating breathing andflexural modes we observed mixing tones as sidebands ofthe breathing mode, as shown in Fig. 5c.
V. FLEXIBLE MECHANICAL RESONATORDESIGN
Separating the mechanical and optical modes into twobeams in the slot-mode architecture adds flexibility indesigning these modes compared to a single nanobeam.By modifying the design of the mechanical beam, a widerange of mechanical frequencies can be accessed with-out significantly affecting the optical mode. To that end,we demonstrate lower-frequency designs around 1.8 GHzand 400 MHz. Implementing highly-localized breath-ing modes in various RF bands (here, the IEEE-definedUHF and L) broadens the potential application space.One way to change the mechanical frequency is simplyto change the full width of the mechanical beam whilekeeping the same lattice variation.For the 1.8 GHz band design, shown in Fig. 6a, themechanical beam width was increased from 700 nm to1.55 µ m. Measurements of a fabricated device (Fig. 6b)having a designed, stress-tuned slot width of 80 nmare shown in Fig. 6c-e. The measured intrinsic Q o = (b) (d)(g) (h) (i) (j) (b)(a) (c) (e)(f) . μ m . μ m FIG. 6. (a) FEM simulation of the 1.8 GHz band mechanical breathing mode of the 1.55 µ m wide mechanical beam. (b) SEMimage of a fabricated 1.8 GHz band device. (c) Detected mechanical spectra at different FTW input optical powers. (d) At aFTW input optical power of 4.7 mW, harmonics of the 1.895 GHz mechanical mode are visible. (e) Measured γ m, eff / (2 π ) ofthe 1.895 GHz mechanical mode. Error bars represent the uncertainty in the fit of the mechanical spectra to a Lorentzian. Thedashed line shows the weighted linear fit of the subthreshold γ m, eff / (2 π ). (f) FEM simulation of the 400 MHz band mechanicalbreathing mode of the 4 µ m wide mechanical beam. (g) SEM image of a fabricated 400 MHz band device. (h) Mechanicalspectra measured at different FTW input optical powers. (i) At a FTW input optical power of ≈ . γ m, eff / (2 π ) of the 414 MHz mechanical mode. Error bars representthe uncertainty in the fit of the mechanical spectra to a Lorentzian. The dashed line shows the weighted linear fit of thesubthreshold γ m, eff / (2 π ). The power spectral density plots in (c), (d), (h) and (i) are referenced to a power of 1 mW = 0 dB. (1 . ± . × , and the measured intrinsic Q m =2130 ±
50, as derived from the weighted linear fit shownin Fig. 6e. These values are comparable to the 3.4 GHzband devices. The optomechanical coupling of this de-vice was strong enough that we observed self-oscillationfor laser powers above ≈ P in is ≈
20 % of the laserpower at the FTW input). Above threshold, we also ob-served harmonics on the breathing mode (Fig. 6d). Thesearise from nonlinear modulation of the optical field due tothe Lorentzian optical mode shape, as reported in othersystems [33–35].For the 400 MHz design, the mechanical beam widthwas increased to 4 µ m. At this width, the mechanicalmode is not well-confined for the same lattice parame-ters, but the “ribs” still contribute to the optical con-finement. Thus, we kept the ribs to maintain high Q o ,but increased the effective mechanical lattice constantsby “breaking” two-thirds of the ribs, as shown in Fig. 6f.Measurements of a fabricated device (Fig. 6g) having aslot width of 80 nm are shown in Fig. 6h-j. The measuredintrinsic Q o = (1 . ± . × , and the measured in-trinsic Q m = 800 ± Q o is comparable to thatof the 3.4 GHz band devices, indicating that the “broken-rib” geometry minimally perturbs the optical mode. The Q m , however, is much lower than in the 3.4 GHz banddevices. This is likely due to an increase in air dampingwith decreased frequency [36] and an increase in anchor loss from the effective two-thirds decrease in the numberof lattice periods in the mirror region of the mechanicalbeam. As with the other devices in this work, the me-chanical spectra (Fig. 6h) include other, less-well-coupledpeaks that correspond to additional breathing-type me-chanical modes from defects in the fabricated device [32]or harmonics of lower-frequency flexural modes. The op-tomechanical coupling of the 414 MHz breathing modeof this device was strong enough that it reached self-oscillation for laser powers above ≈ . VI. MULTIMODE OPTOMECHANICALDEVICES
In addition to increasing flexibility in the available me-chanical frequencies, the slot-mode device architectureenables new functionality in that it is straightforward toadd another separate optical and/or mechanical mode.In this work, we demonstrate two cases: a single opticalmode simultaneously coupled to two different mechan-ical beams (“M-O-M”) and a single mechanical modecoupled to two different optical modes (“O-M-O”). M-O-M devices have a variety of possible applications, withtheoretical proposals including mechanical mode entan-glement and phonon pair generation [12, 14] and ground- (d)(c)
Norm. |E| (a) (b) O p t i c a l B ea m FIG. 7. (a) FEM simulation of the optical mode of an M-O-M device designed for coupling to 3.4 GHz band (bottom beam)and 1.8 GHz band (top beam) mechanical breathing modes. The optical mode is in both slots simultaneously. (b) SEM imageof a fabricated M-O-M device. (c) Optical spectrum of M-O-M device. Measurement is in gray, and the Lorentzian fit is in red.Measured intrinsic Q o = (1 . ± . × (d) Both mechanical modes measured simultaneously, FTW input power ≈ Q m = 3175 ± Q m = 3350 ±
10, where uncertainty comes from 95 % confidence interval of fit. (insets)FEM eigenmode simulations of corresponding mechanical breathing modes. state laser cooling of an unresolved-sideband mechanicalresonator [15]. Moreover, recent progress has been madein studying M-O-M devices in other platforms experi-mentally, including recent investigations of Bogoliubovmechanical modes [13], as well as systems showing syn-chronization of mechanical resonators via a travelling op-tical mode [37, 38]. An O-M-O slot-mode device providesa new platform for optical frequency conversion, as pro-posed in Ref. [16]. Unlike in previous demonstrationsof optomechanically-enabled optical frequency conver-sion [7–9], the O-M-O device enables quasi-independentoptical mode selection and independent optimization ofthe coupling into each optical mode.In the example M-O-M device of this work, we sur-round an optical beam with a 1.8 GHz band mechanicalbeam (Sec. V) and a 3.4 GHz band mechanical beam,with 80 nm slots between the beams (Fig. 7b). The resul-tant optical mode is concentrated in both slots simultane-ously (Fig. 7a). We couple to the optical mode by hover-ing the FTW a few hundred nanometers above the opticalbeam. The measured intrinsic Q o = (1 . ± . × (Fig. 7c), and we simultaneously detect modulation fromboth the 1.8 GHz band and 3.4 GHz band mechanicalmodes (Fig. 7d). For the same input optical power, thedetected 1.8 GHz band mode has a larger amplitude thanthe 3.4 GHz band mode primarily because a lower fre-quency mode has a larger thermal noise motional ampli-tude for the same temperature. With the optical qualityfactor in excess of 10 (linewidth ≈ . Q o = (1 . ± . × at 973.21 nm, and the bottommode had a measured intrinsic Q o = (1 . ± . × at947.34 nm.The mechanical breathing mode at ≈ .
835 GHz wasdetected when coupled both to the top and to the bot-tom optical modes (Fig. 8c). (There is another, less-well-coupled peak at ≈ .
831 GHz that corresponds to eitheran additional breathing-type mechanical mode from de-fects in the fabricated device [32] or a harmonic of alower-frequency flexural mode.) We also measured theeffective mechanical linewidth as a function of power forboth optical modes (Fig. 8d). A weighted linear fit ofthese measurements indicates that the intrinsic Q m asmeasured via each optical mode is in good agreement:from the top mode, Q m = 1800 ± Q m = 1800 ± VII. DISCUSSION
We have demonstrated slot-mode optomechanical de-vices in which the mechanical breathing mode of a pat-terned nanobeam is coupled to an optical mode thatis laterally confined by a second patterned nanobeamand resides within the slot between the two beams.Along with large optomechanical coupling rates in ex-cess of 300 kHz (as measured via phase-modulator cali-bration) enabled in part by narrow slot widths that can λ ≈ 973 nm λ ≈ 947 nm (d)(a) (c) λ ≈ 973 nm λ ≈ 947 nm (b) Norm. |E| O p t i c a l B e a m O p t i c a l B e a m MechanicalBeam
FIG. 8. (a) SEM image of fabricated O-M-O device. (b) Separately-measured optical spectra of O-M-O device. Data are ingray, and Lorentzian fits are in red. The 947.34 nm mode (“bottom” beam) has intrinsic Q o = (1 . ± . × , and the973.21 nm mode (“top” beam) has intrinsic Q o = (1 . ± . × . (insets) FEM simulations of the optical slot modesassociated with bottom and top optical beams. (c) Mechanical spectra measured at different FTW input optical powers. Topspectra were acquired while optically coupled to the top beam, and bottom spectra were acquired while optically coupled tothe bottom beam. (d) γ m, eff / (2 π ) as measured via the top optical mode (red) and the bottom optical mode (blue) with respectto FTW input power. Dashed lines show weighted linear fits of γ m, eff / (2 π ). Error bars represent the uncertainty in the fit ofthe mechanical spectra to a Lorentzian. be achieved by taking advantage of the tensile film stressin Si N , this platform allows for flexible design of the op-tical and mechanical modes, with mechanical beams tai-lored to support breathing modes ranging from 400 MHzto 3.5 GHz. Moreover, this geometry can naturally beextended to multimode systems; we have shown triple-nanobeam devices with two different mechanical modescoupled to a single optical mode, as well as a triple-nanobeam device in which two different optical modesare coupled to a single mechanical mode.Future work will focus on the use of these multimodegeometries in applications such as optical wavelengthconversion and Bogoliubov mechanical mode formation for phonon pair generation. Though some of the currentdevices are already weakly in the sideband-resolved limit( κ/ π ≈ < Ω m / π ≈ . Q o would enable sideband resolution forall of the mechanical frequencies studied. Finally, the im-plementation of on-chip waveguides will likely be neces-sary to achieve long-term, stable coupling to multimodedsystems. FUNDING INFORMATION
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This section provides supplementary information for this work. It describes the procedure for determiningthe phase modulator V π , presents data from an additional triple-nanobeam device, and explores how higher-order optical slot modes couple to various mechanical modes. Appendix A: Measuring Phase Modulator V π In Sec. IV, we use the phase modulator calibrationmethod [1, 2] to measure the optomechanical coupling g of slot-mode optomechanical crystals. The V π of theelectro-optic phase modulator must be accurately knownin order to do this calibration.To measure V π , we send the 980 nm laser signalthrough the phase modulator, modulated at 3.5 GHz byan RF signal generator, and into a scanning Fabry-P´erotinterferometer. The detected signal traces out the car-rier and phase-modulator-induced sidebands in the opti-cal signal, and we view them on an oscilloscope, as shownin Fig. S1a. carrier first sidebandcarrier first sidebandsecondsideband (a) (b) FIG. S1. (a) An example of the output of the scanning Fabry-P´erot interferometer for a phase-modulated optical input sig-nal, as read by an oscilloscope. Spectrum shows the carrierpeak and the first and second sidebands. (b) Carrier (blue)and first sideband (red) peak heights with respect to RF sig-nal voltage applied to the phase modulator. Points are themeasured values, with error bars indicating the voltage reso-lution of the oscilloscope. Lines are the fits of the data.
Changing the power applied by the RF signal genera-tor to the phase modulator changes the magnitude of thecarrier and sidebands. The RF power P RF is related tothe signal voltage V sig = √ ZP RF , where the phase mod-ulator input impedance Z = 50 Ω. Knowing this, we cangraph the peak magnitudes with respect to V sig , as shownin Fig S1b. For a phase modulator, the carrier peak magnitude should follow the curve A ( J ( πV sig /V π )) ,and the first sideband magnitude should follow the curve A ( J ( πV sig /V π )) , where A scales the amplitude of theBessel functions of the first kind J and J . We fit datafrom the carrier and first sideband to these functions inFig. S1b, and both fits result in V π = 2 . ± .
01 V, wherethe uncertainty comes from the fit and is one standarddeviation. This value corresponds well with the vendor-specified value for the phase modulator.
Appendix B: Additional M-O-M Device
In addition to the example M-O-M device in Sec. 7, wefabricated and characterized an M-O-M device in whicha 400 MHz band mechanical beam and a 1.8 GHz bandmechanical beam surround an optical beam, with 80 nmslots between the beams, shown in Fig. S2b. The resul-tant optical mode is concentrated in both slots simultane-ously (Fig. S2a). We couple to the optical mode by hover-ing the FTW a few hundred nanometers above the opticalbeam. The measured intrinsic Q o was (5 . ± . × (Fig. S2c), and we simultaneously detect modulation ofthe transmitted optical signal from both the 400 MHzband and 1.8 GHz band mechanical modes (Fig. S2d).For the same input optical power, the detected 400 MHzband mode has a larger amplitude than the 1.8 GHz bandmode primarily because a lower frequency mode has alarger thermal noise motional amplitude for the sametemperature. As with the other devices in this work, themechanical spectrum (Fig. S2d) includes other, less-well-coupled peaks that correspond to additional breathing-type mechanical modes from defects in the fabricated de-vice [3] or harmonics of lower-frequency flexural modes. Appendix C: Coupling to Higher-Order Slot Modes
These slot-mode optomechanical crystals confine mul-tiple optical modes in the slot in addition to the fun-damental mode for which they were designed. We ob-served some of these higher-order modes, as shown inFig. S3a and b. Because they are distributed more widelyalong the slot, these modes couple less strongly to thehighly-localized breathing mode and more strongly toother mechanical modes in the device, such as higher-1 (d)(c)
Norm. |E| (a) (b)
MechanicalBeams O p t i c a l B ea m FIG. S2. (a) FEM simulation of the optical mode of an M-O-M device designed for coupling to 1.8 GHz band (bottom beam)and 400 MHz band (top beam) mechanical breathing modes. The optical mode is in both slots simultaneously. (b) SEM imageof a fabricated M-O-M device. (c) Optical spectrum of M-O-M device. Measurement is in gray, and the Lorentzian fit is in red.Measured intrinsic Q o = (5 . ± . × (d) Both mechanical modes measured simultaneously, FTW input power ≈ Q m = 1030 ±
20, and1.927 GHz mode has effective Q m = 1450 ±
20, where uncertainty comes from 95 % confidence interval of fit. (insets) FEMeigenmode simulations of corresponding mechanical breathing modes. (a) (b) λ ≈ 949 nm λ ≈ 971 nm (c) FIG. S3. (a) Fundamental optical slot mode at ≈
949 nm withintrinsic Q o = (3 . ± . × . (b) Higher-order opticalslot mode in the same device at ≈
971 nm with intrinsic Q o =(3 . ± . × . (c) Mechanical spectra measured whilecoupled to the fundamental optical mode (red) and higher-order optical mode (black). The optical power input to thefiber taper waveguide was ≈ . ≈ . order breathing-type mechanical modes arising from fab-rication 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 . ± . × as well as an optical mode at ≈
971 nm with an intrin-sic optical Q = (3 . ± . × . When pumped atthe 949 nm fundamental optical mode, it self-oscillatesat the ≈ .
52 GHz mechanical breathing mode, and webegin to see sidebands due to mixing with the modu-lation from the low-frequency flexural beam modes, asdescribed in Sec. 4 of the main text. However, even with50 % more optical power, the mechanical breathing modedoes not self-oscillate when pumped at the 971 nm opticalmode. In addition, the mechanical spectrum reveals an-other peak at ≈ .
48 GHz with about the same optome-chanical coupling to the 971 nm mode as the ≈ .
52 GHzmode, suggesting that this higher-order optical mode isalso coupling to some higher-order, less-well-confined me-chanical mode. [1] M. L. Gorodetsky, A. Schliesser, G. Anetsberger,S. Deleglise, and T. J. Kippenberg, “Determination ofthe vacuum optomechanical coupling rate using frequencynoise calibration,” Opt. Express , 23236 (2010).[2] K. C. Balram, M. Davan¸co, J. Y. Lim, J. D. Song, andK. Srinivasan, “Moving boundary and photoeleastic cou- pling in GaAs optomechanical resonators,” Optica , 414-420 (2014).[3] M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala,and O. Painter, “Optomechanical crystals,” Nature462