single-crystal diamond nanobeam waveguide...

Single-Crystal Diamond Nanobeam Waveguide Optomechanics

Behzad Khanaliloo,1,2 Harishankar Jayakumar,1,† Aaron C. Hryciw,2,3 David P. Lake,1

Hamidreza Kaviani,1 and Paul E. Barclay1,2,*1Institute for Quantum Science and Technology, University of Calgary,

Calgary, Alberta, T2N 1N3 Canada2National Institute for Nanotechnology, Edmonton, Alberta, T6G 2M9 Canada

3University of Alberta nanoFAB, Edmonton, Alberta, T6G 2R3 Canada(Received 28 April 2015; revised manuscript received 19 October 2015; published 29 December 2015)

Single-crystal diamond optomechanical devices have the potential to enable fundamental studies andtechnologies coupling mechanical vibrations to both light and electronic quantum systems. Here, wedemonstrate a single-crystal diamond optomechanical system and show that it allows excitation of diamondmechanical resonances into self-oscillations with amplitude >200 nm. The resulting internal stress fieldis predicted to allow driving of electron spin transitions of diamond nitrogen-vacancy centers.The mechanical resonances have a quality factor >7 × 105 and can be tuned via nonlinear frequency

renormalization, while the optomechanical interface has a 150 nm bandwidth and 9.5 fm=ffiffiffiffiffiffiHz

psensitivity.

In combination, these features make this system a promising platform for interfacing light, nanomechanics,and electron spins.

DOI: 10.1103/PhysRevX.5.041051 Subject Areas: Nanophysics, Photonics

I. INTRODUCTION

Nanophotonic optomechanical devices [1–4] enableon-chip optical control of nanomechanical resonators witha precision reaching the standard quantum limit [5–8],enabling tests of quantum nanomechanics [7,9–11], as wellas technologies for sensing [4,12–14] and informationprocessing [15–17]. Single-crystal diamond devices areparticularly exciting for these applications. In addition topossessing desirable mechanical and optical properties,diamond hosts color centers [18] whose electronic andnuclear spins are promising for quantum technologies.Recently, nanomechanical devices with embedded

color centers [19–23] and quantum dots [24,25] have beenused for phononic control [26] of electron spins. Thesehybrid quantum devices [27] harness the vibrations ofnanomechanical resonators to control electron spins via theirelectronic coupling to internal strain. To date, these experi-ments have relied on piezoelectric mechanical actuation.Optomechanical devices offer an attractive alternative formechanical actuation. They use light, the preferred mediumfor transferring quantum information, to create optical forcesthat are sensitive to the mechanical resonator position.In combination with coupling between nanomechanical

resonator strain and electron spins [19–23], this can createan effective interaction between light and spins. Compared topiezoelectric actuation, optomechanical coupling is local-ized, and offers integrated sensitive readout of mechanicaldisplacement. Although optomechanical devices have beenrealized from a broad range of materials [2,6,7,17,28],demonstrating single-crystal diamond optomechanical devi-ces has been delayed by challenges in fabrication. Here, wereport demonstration of single-crystal diamond optomechan-ical devices. These devices allow low-power optical actua-tion of diamond nanomechanical resonators [29–33], andare predicted to enable optical control at wavelengthsdetermined by the device geometry of spin transitions notaccessible using traditional radio or microwave frequencytechniques [19]. They could also enable proposals forcreating hybrid quantum devices for transducing quantuminformation between disparate quantum systems [34–36],creating quantum sensors [37], generating spin squeezingand entanglement [38], and using quantum emitters fornanomechanical cooling [39,40].The single-crystal diamond optomechanical system we

demonstrate here incorporates several features that togethermake it a promising platform for hybrid optomechanicalquantum devices. This system is based on ultrahighmechanical quality factor (Qm > 7.2 × 105) nanobeamwaveguide resonators fabricated from bulk diamond chipsusing a scalable process. The nanobeams are coupledevanescently to a fiber taper waveguide, and can be excitedinto self-oscillation via a combination of optomechanicalcoupling and a stress-enhanced photothermal force that issignificant (>pN) even for sub-μW absorbed power levels.We observe self-oscillations with amplitude >200 nm for

*[email protected]†Present address: Department of Physics, CUNY-City College

of New York, New York, New York 10031, USA.

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW X 5, 041051 (2015)

2160-3308=15=5(4)=041051(21) 041051-1 Published by the American Physical Society

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∼100 nW of absorbed power, and predict that this resultsin an oscillating internal stress field of ∼70 MPa. Thiscorresponds to a strain coupling rate of G=2π ∼ 0.8 MHzto diamond nitrogen-vacancy (NV) center electron spins,indicating that these oscillations can drive coherenttransitions between ms ¼ �1 states [19–23]. The self-oscillation frequency is observed to be renormalized200 kHz by nonlinear nanomechanical effects, providinga tunability that would be beneficial for such experiments,as well as a measure of internal stress and nanobeambuckling. The optomechanical interaction underlying thissystem operates over a 150 nm bandwidth, and is charac-terized by a dissipative optomechanical coupling coeffi-cient that can reach 45 GHz=nm (0.48 MHz per photon),and a displacement sensitivity of 9.5 fm=

ffiffiffiffiffiffiHz

p. This sensi-

tivity is less than a factor of 3 above the quantumuncertaintyin position of the highest Qm nanobeams demonstratedin this work, and can be improved with further deviceoptimization.In combination, these properties make the demonstrated

system promising for realizing optomechanical spin con-trol, as well as fundamental studies of modification ofnanomechanical resonator dynamics by coupling to elec-tronic spins [39,40]. The analysis of the device propertiesand behavior presented here will guide future work tomaximize strain coupling to electronic spins, and to harnessthe large photothermal force for optomechanical coolingand manipulation of nanobeam motion.

II. WAVEGUIDE-OPTOMECHANICS

Experiments in diamond nanophotonics and nanome-chanics have recently been advanced by the availability ofhigh-quality diamond chips grown using chemical vapordeposition. While optomechanical devices can be fabricatedfrom polycrystalline diamond films [32], single-crystaldiamond thin films that retain the desirable combina-tion of optical, mechanical, and quantum electronic proper-ties introduced above are not commercially available andmust be manufactured using ion-implantation [41–43]or wafer-bonding [30,31,44,45] techniques. Fabricationof devices directly from bulk single-crystal diamond chipsis desirable, and until now has relied on either ion beammilling [46] or less damaging and more scalable inductivelycoupled plasma reactive-ion (ICPRIE) Faraday cage angledetching [33,47]. In this work, we demonstrate an ICPRIEprocess that does not require a Faraday cage, and insteadutilizes diamond undercut etching to fabricate nanobeamsfrom bulk single-crystal diamond, such as those shownin Fig. 1(a) that support optical and mechanical modessummarized in Figs. 1(b) and 1(c), respectively. Thisprocess, outlined in Fig. 2 and discussed in more detailin Appendix A, is inspired by earlier bulk silicon nano-fabrication [48] and relies on an oxygen-based ICPRIEprocess operated with zero-rf power, high-ICP power, andan elevated sample temperature (250 °C) to etch diamondquasi-isotropically along diamond crystal planes. The proc-ess is uniform over the diamond chip area, can produce

FIG. 1. Diamond nanobeam waveguide-optomechanical system. (a) SEM images of a single-crystal diamond nanobeam waveguide.Dark high-contrast regions are due to variations in thickness of titanium deposited for imaging purposes. (b) Schematic of fiber taperevanescent waveguide coupling geometry and illustration of the waveguide-optomechanical coupling process. Nanobeam mechanicalresonances, whose typical displacement profiles are shown, modulate the distance between the input waveguide, changing κðhÞ.(c) Dispersion of neff of the waveguide modes of the fiber taper (diameter 1.1 μm) and diamond nanobeam (w × d ¼ 460 × 250 nm2)when they are uncoupled (fiber taper shown by red dashed line, nanobeam shown by blue dashed line) and coupled with h ¼ 300 nm(solid lines). Shaded region indicates the laser scan range used in the experiments. Insets: Mode profiles (x-y plane, dominant electricfield component jEyj) of the symmetric (þ) and antisymmetric (−) coupled modes.

BEHZAD KHANALILOO et al. PHYS. REV. X 5, 041051 (2015)

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horizontal undercut surfaces, and is fully compatiblewith standard commercial etching tools. Eliminating theneed for a Faraday cage allows the process to be employedat a wider range of nanofabrication facilities, while theflat undercutting property allows fabrication of structuressuch as the high-quality factor microdisk optical cavitiesin Ref. [49].Using this diamond undercutting process, nanobeams

are fabricated with widths w ¼ 300 − 540 and 750 nmand lengths L ¼ 50 − 80 μm on the same chip. Nanobeamthickness d is adjusted by controlling the etching timestogetherwith the nanobeamwidthw. For the etch parametersused here, the nanobeam thickness is d ∼ 250–350 nm,depending on w. Because of the crystal-plane sensitivenature of the undercut etch, for the etch parameters usedhere the narrow nanobeams have a flat bottom surface,while the w ¼ 750 nm nanobeams have a triangular bottomsurface mimicking the ridge visible in Fig. 1(a), as shownin Appendix A. These bottom surfaces can be flattenedby longer undercut etching, consistent with observedbehavior of microdisks undercut using this process [49].The fabricated nanobeams support mechanical resonan-

ces, the lowest order of which are illustrated in Fig. 1(b),with frequencies and effective mass in the MHz andpg range, respectively. The nanobeams also support wave-guide modes, which can be evanescently coupled withhigh efficiency to an optical fiber taper [50], as shownschematically in Fig. 1(b), by tuning the device geometry tomatch the nanobeam mode phase velocity with that of afiber taper mode. Despite the refractive index differenceof the SiO2 fiber taper and diamond, phase-matching isrealized in nanobeams with subwavelength cross section.This is illustrated in Fig. 1(c), which shows the effectiverefractive index dispersion, neffðλÞ, of the fundamental TEmodes of a fiber taper and of a diamond nanobeam,together with that of the “supermodes” of the evanescentlycoupled waveguides (n�eff ) [51]. Phase-matching occurs atλ0 ∼ 1570 nm, where an anticrossing in n�effðλÞ indicatesthat the waveguide modes are coupled.Evanescent waveguide coupling, which has been well

studied in the context of photonic and optoelectronicdevices [51,52], is shown here to provide sensitive

optomechanical read-out of mechanical fluctuations thatmodulate the waveguide separation h. In contrast to cavity-optomechanical systems, which exploit dispersive couplingbetween mechanical fluctuations and narrow-band opticalresonances [53], waveguide-optomechanical couplingutilizes wideband dissipative optical transduction of thenanobeam mechanical position. The sensitivity of theevanescent waveguide-optomechanical coupling can bederived from coupled-mode theory (Appendix C) and isdetermined by the dependence of the normalized fibertaper transmission T on the separation h between thewaveguides:

Tðh; λÞ ¼ cos2ðsLcÞ þ�Δβ2

�2 sin2ðsLcÞ

s2: ð1Þ

Here, Lc is the coupler interaction length determined by thefiber taper geometry, ΔβðλÞ ¼ ΔneffðλÞ2π=λ is the propa-gation constant mismatch of the uncoupled waveguidemodes, s2 ¼ κ2 þ Δβ2=4. κðhÞ is the per unit lengthamplitude coupling coefficient, which depends on theoverlap of the evanescent fields of the coupled waveguides.The theoretical displacement sensitivity achievable bymonitoring fluctuations in T is

soxðfÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Tðh; λÞPdℏωo=ηqe þ SðdetÞP

P2dð∂T=∂κÞ2ð∂κ=∂hÞ2ð∂h=∂xÞ2

s; ð2Þ

where SðdetÞP is the single-sided noise equivalent opticalpower spectral density (units of W2=Hz) of the detector andtechnical noise, and Pd is the detected output power in theabsence of coupling (T ¼ 1). The first term in the numeratoraccounts for photon shot noise, where ηqe is the detectorquantum efficiency. This expression neglects radiationpressure backaction, which is not significant for the mea-surements we present here, but will ultimately limit sox .The impact of coupler geometry, operating condition, andwaveguide design on detection sensitivity is described bythe denominator of Eq. (2). Sensitivity is maximized formechanical resonances whose displacement x efficientlymodulates h, i.e., j∂h=∂xj ∼ 1. Strong evanescent overlap

FIG. 2. Process flow for creating diamond nanobeams using quasi-isotropic reactive-ion undercut etching.

SINGLE-CRYSTAL DIAMOND NANOBEAM WAVEGUIDE … PHYS. REV. X 5, 041051 (2015)

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enhances j∂κ=∂hj, while phase-matching and operationnear κðhÞLc ∼ π=4; 3π=4;… maximizes j∂T=∂κj.Experimental observation of the optomechanical proper-

ties of the diamond waveguide-optomechanical system isperformed by measuring TðtÞ ¼ T þ δTðtÞ of a dimpledoptical fiber taper [50] positioned in the nanobeam near–field for varying h and λ. The efficiency of the evanescentcoupling determines the time-averaged transmission T.Fluctuations of the nanobeam position, together with othernoise, are imprinted on δTðtÞ. Figure 3(a) shows TðλÞwhenthe fiber taper is positioned at h ∼ 200 nm above the centerof a nanobeam (w × d ¼ 460 × 250 nm2). Minimum trans-mission To ¼ 0.05, corresponding to a coupling efficiencyof 1 − To ¼ 0.95 assuming negligible insertion loss, isobserved near λ0 ¼ 1560 nm. The 3 dB bandwidth Δλ >150 nm is consistent with predictions from the n�effðλÞanticrossing in Fig. 1(c), and increasing w is observedto increase λ0 [Fig. 3(a), inset], consistent with expectedbehavior.The coherent nature of the waveguide coupling is

revealed by ToðhÞ. As shown in Fig. 3(b), ToðhÞ isminimized at h ∼ 200 nm, where κLc ¼ π=2. Forh < 200 nm, To increases with decreasing h due to thecodirectional coupling undergoing more than a half “flop”and light coupled into the nanobeam being outcoupled back

into the fiber taper [52]. In contrast, incoherent scatteringloss increases monotonically with decreasing h [54] and issmall in the system studied here. As shown in Fig. 3(b),Eq. (1) fits ToðhÞ well with an exponentially decaying κðhÞas a fitting parameter, which in turn agrees closely withκðhÞ predicted from coupled-mode theory for interactionlength Lc ∼ 7 μm. This Lc is consistent with the observedcurvature in optical images of the fiber taper dimple.The optomechanical properties of the coupled wave-

guides are observed from the power spectral density SvðfÞof the photodetected signal generated by fluctuations inoutput power PdδTðtÞ. Figure 4(a) shows the equivalentdisplacement spectral density sxðfÞ (units m=


p) of the

nanobeam motion when the fiber taper is positionedh ∼ 100 nm above the nanobeam. Peaks from thermallydriven nanobeam resonances are clearly visible at frequen-cies fm ¼ ½1.3; 3.1; 4.2� MHz, corresponding to the funda-mental out-of-plane (v1), fundamental in-plane (h1), andsecond-order out-of-plane (v2) resonances, whose simu-lated displacement profiles are shown in Fig. 1(b).Resonance labels are determined by comparison of fmwith simulations and by measuring transduction sensitivityas a function of in-plane taper position. Assuming a 35MPainternal compressive stress, consistent with studies of thenanobeam response described in the following section, wefind simulated values of 1.3, 3.2, and 4.2 MHz, in goodagreement with observed values. sxðfÞ is obtained bycalibrating SvðfÞ to the theoretical thermomechanicalpower spectral density of the v1 resonance (m ¼ 7.6 pg),as described in Appendix E. The observed technical noisefloor of sox ¼ 9.5� 1.0 fm=Hz1=2 is below that of otherbroadband integrated waveguide measurements [32,54,55],and is more than an order of magnitude more sensitive thansingle-pass free-space reflection measurement techniquesused with diamond nanomechanical structures [33]. Fiberinterferometers incorporating low-finesse optical cavities[56,57] have reached similar sensitivity. This sensitivity canbe further improved to the sub-fm=Hz1=2 range by increas-ing Lc and Pd, or by terminating the nanobeams withphotonic crystal mirrors to introduce optical feedback at theexpense of optical bandwidth, as in the multipass cavity-fiber-interferometer system of Ref. [58]. Uncertainty in soxis derived from uncertainty in m resulting from possiblevariations from the nominal device geometry and materialparameters.The h dependence of SvðfmÞ1=2 and sox for the v1

resonance, shown in Fig. 4(b), provides a direct measureof the optomechanical coupling. The on-resonance signalis directly related to the slope of TðhÞ: Svðh; fmÞ1=2 ∝jdT=dhj, with a distinct minimum when κLc ¼ π=2.Highest sensitivity is observed at h ¼ 100 nm (κLc ¼3π=4), and soxðhÞ agrees well with predictions from theexperimentally characterized TðhÞ and Eq. (2). Asdescribed in Appendix E, the predicted sensitivity assumesa shot-noise-limited laser source over the measurement

FIG. 3. Efficient evanescent waveguide coupling. (a) Fibertaper transmission TðλÞ versus wavelength for h ¼ 200 nm.Inset: Variation of T with nanobeam width w and λ forapproximately constant h. (b) Dependence of fiber taper trans-mission on fiber height. Experimentally measured values for To(red points) are fit with the model described by Eq. (1) (red line).The corresponding κðhÞ extracted from the fit (blue line) andpredicted from simulation (blue points) are also shown.


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bandwidth of interest (i.e., the MHz range) and a photo-

detector noise equivalent power SðdetÞP ¼ 2.5 pW=ffiffiffiffiffiffiHz

pcorresponding to the nominal specification for the detectorused here. The minimum soxðhÞ in Fig. 4(b) is degradedcompared to the sensitivity of the measurement in Fig. 4(a)due to operating at lower Pd.As discussed in Appendix D, the optomechanical trans-

duction dT=dh is related to the dissipative optomechanical

coupling coefficient gðeÞγ , defined by the optomechanicalmodulation of tunneling rate γe of photons from thenanobeam to the forward propagating fiber taper mode.Assuming the coupler demonstrated here is integrated

into a cavity of length L ¼ Lc, we predict gðeÞγ =2π >45 GHz=nm, where c=ng is the group velocity of light inthe coupler. The corresponding per photon optomechanicalcoupling rate is go=2π ∼ 0.48 MHz for the v1 resonance of ananobeam of length Lc and cross section of the device inFig. 4(b) (fm ∼ 66 MHz, zero-point motion xzpf ¼ 11 fm).In addition to being tunable via adjustment of h, thisdissipative coupling is significantly larger than in previouslystudied systems [59–62]. However, note that when thedissipative coupling is maximized, it is accompanied bylarge γe. Also note that the coupler introduces dispersive

coupling, discussed in Appendix D, which is small com-

pared to gðeÞγ ðhÞ for h > 200 nm.Unlike dispersive optomechanical coupling, dissipative

optomechanics [59] can be used to cool nanomechanicalresonances in cavity-optomechanical systems operating inthe sideband-unresolved regime [63,64]. This is a poten-tially useful tool for optomechanical cooling of relativelylow-frequency nanomechanical resonances. Realizing dis-sipative cooling of the devices studied here would requirefurther device fabrication development in order to integratethe fiber-coupled nanobeam into a cavity. Furthermore, asdiscussed in Appendix D, it is necessary that the cavitymode can be coupled unidirectionally [53] to the fiber taperwaveguide via the nanobeam [64] and have internal lossrate γi ≪ γe. Oval racetrack resonators similar to thosedemonstrated by Burek et al. [47], modified to incorporateclamped diamond nanobeams with L ¼ Lc ¼ 21 μm asupporting traveling-wave resonance, are promising in thisregard. We predict in Appendix D that over a narrow range

of h, which balances small γe and large gðeÞγ , optomechan-ical reduction of the fundamental nanobeam normal-modephonon population to close to 102 quanta, limited byoptomechanical backaction, may be possible in such a

FIG. 4. Nanomechanical and optomechanical properties. (a) Measured sxðfÞ when the fiber taper is coupled to a nanobeam(L × w × d ¼ 50 × 0.46 × 0.25 μm3) in ambient conditions, with Pd ∼ 100 μW. The vertical axis and noise floor (dashed line) ofsox ¼ 9.5 fm=


pare calibrated from the fit to the f ¼ 1.3 MHz mechanical resonance thermomechanical displacement spectrum

(red solid line). (b) Top: Svðf1Þ1=2 observed (points) and fit with a function ∝ jdT=dhj (solid line). Bottom: Displacement sensitivitysoxðhÞ of the fundamental out-of-plane resonance from measurement (blue points) and predicted from the measured TðhÞ for varyinginput power (solid lines). The starred point corresponds to the sensitivity and operating condition of the measurement in (a). MeasuredSvðfÞ for a nanobeam with L × w × d ¼ 60 × 0.75 × 0.3 μm3 (c) in vacuum (d) at 5 K. Insets: Fits to observed mechanical resonances,and corresponding Qm. Unlabeled peaks in (a), (c), and (d) below 1 MHz are related to fiber vibrations.


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device if an internal loss rate of γi=2π ¼ 1 GHz and aninternal backscattering rate less than γi can be achieved.Nanobeam mechanical dissipation in the measurements

discussed above is dominated by damping from theambient air environment. To assess the nanobeam mechani-cal properties, measurements are performed in vacuum andlow-temperature conditions. Generally, after reducing airdamping, Qm is observed to increase above 104 in alldevice geometries, with the lowest dissipation (Qm > 105)observed in the larger cross-section (w × d ¼ 750×300 nm2) devices. Figures 4(c) and 4(d) show mechanicalresonances of a high-Qm nanobeam measured in vacuum(80 μTorr) and in cryogenic conditions (T ¼ 5 K and5 μTorr). In vacuum, the v1 and h1 resonances have Qm ≈1.4 × 105 and 2.0 × 105, respectively. At low temperatures,dissipation is further reduced, such that Qm ≈ 1.7 × 105

and 7.2 × 105 for the v1 and h1 modes, respectively. Thesevalues of Qm are higher than previous reports of nano-mechanical devices fabricated from single-crystal opticalgrade diamond [30,31,33].Allmeasurements aremade at lowpower to avoid inducing optomechanical linewidth narrow-ing, and spectral diffusion is not observed on the few-secondmeasurement time scale. The observed increase in Qm withincreasing nanobeam mode order and type (v versus h),together with data presented below for a larger L, smallercross-section nanobeam with lower Qm than the device inFigs. 4(c) and 4(d), indicates that clamping loss is not thedominant dissipation mechanism for these structures [65].This behavior, and other possible sources of loss includingsurface dissipation [31], requires further investigation inorder to understand how to increase Qm in future devices.Two possible approaches include fabricating devices fromelectronic grade material or with larger dimensions [31].Given the demonstrated device performance, the meas-

urement precision required to reach the standard quantumlimit is sSQLx ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiℏQm=ðmω2

mÞp

¼ 3.1 fm=ffiffiffiffiffiffiHz

p, where

m ¼ 23 pg is the effective mass of the h1 resonance [9].While this is ∼3 times smaller than the measurementsensitivity demonstrated here, promising approaches forreducing sox below sSQLx include improving the optome-chanical coupling by increasing Lc, and fabricating higher-Qm devices.

III. TUNABLE NONLINEAR DYNAMICS

Backaction from waveguide-optomechanical couplingcan dramatically modify the nanobeam dynamics, and isshown here to amplify the nanobeam motion and revealnonlinear nanomechanical properties of the device. Theseeffects can be tuned by adjusting the waveguide position,and depend critically on the presence of internal stress inthe nanobeam. Demonstration of optomechanically modi-fied nanobeam dynamics is shown in Fig. 5(a), wheretime-resolved TðtÞ of a fiber taper coupled to a high-aspect-ratio nanobeam (L × w × d ¼ 80 × 0.48 × 0.25 μm3) in a

vacuum environment is recorded while h is discretelystepped (30 nm/step, 0.8 steps/s, Pi ∼ 300 μW). Forlarge h, ToðhÞ behaves similarly to the ambient conditionmeasurements in Fig. 4(a). However, at h ∼ 400 nm fluc-tuations δTðtÞ become large-amplitude self-oscillationswith a peak-to-peak change ΔT > 0.3, corresponding tonanobeam displacement exceeding 200 nm, as illustrated inthe inset of Fig. 5(a). To the best of our knowledge, thisamplitude is larger than in previous reports of on-chipexternally driven single-crystal [66] and polycrystalline[32] diamond nanobeams. Note that calibration of theobserved nanobeam mechanical displacement is enabledby the tunable nature of the optical coupling and the use ofencoded nanopositioning of the fiber taper. Finite elementsimulations predict a variation of axial stress at the centerof the nanobeam on the order of 70 MPa for the v1resonance of the compressed nanobeam oscillating withthis amplitude, corresponding to a strain to NV electronicspin-coupling rate of G=2π ∼ 0.8 MHz, enabling optome-chanical control of their quantum state [20,21]. Whenh < 225 nm, the self-oscillations stop. Figure 5(b) showsa spectrograph of these data, where the v1 resonancenear fm ¼ 430 kHz with Qm ¼ 2.5 × 104 is observedto increase in amplitude, shift to lower frequency, andgenerate harmonics.Unlike previously observed diamond nanobeam oscil-

lations driven by electrostatic [66] or optical modulation[32], the observed oscillations are regenerative in nature.These self-oscillations are driven by the dynamic inter-action between a delayed photothermal force [67,68] andthe waveguide-optomechanical coupling. They have notbeen previously observed in waveguides, and can possesslarge amplitudes owing to the large dynamic range andbandwidth of the waveguide-optomechanical system. Theresulting optomechanical backaction renormalizes themechanical dissipation rate from γm to γ0m,

γ0mγm

¼ 1 −QmFk

ωmτ

1þ ω2mτ

2

dTdx

PiζLi; ð3Þ

where τ is the photothermal force response time and F isthe photothermal force strength per unit absorbed powerPabs ¼ ð1 − TÞPiζLi, defined by the corresponding nano-beam deflection and spring constant k ¼ mω2

m of the modeof interest. Li is the distance over which input lightpropagates in the nanobeam before being outcoupledand ζ is the waveguide absorption coefficient per unitlength. Figure 5(c) compares experimentally observedγ0mðhÞ with theoretical values obtained from Eq. (3) inputwith measured TðhÞ as well as parameters discussed below.Predicted γ0mðhÞ agrees well with experiment, particularlyin reproducing the range of h over which the optomechan-ical coupling (jdT=dxj) is sufficiently large such thatγ0m < 0, resulting in self-oscillations.


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For h < 200 nm, photothermal damping broadens γm.However, coupling between ωm and the fiber taper positionfor this range of h (see Appendix G), combined withfluctuations in fiber position due to external vibrations,artificially broadens γm and makes accurate measurementof the cooling rate challenging. If Pi were increased to3 mW, γ0m=γm ∼ 10 is theoretically possible. If applied to ahigher-Qm device, such as the structures in Fig. 4(d), largerphotothermal normal-mode cooling may be possible at thispower level.In addition to exhibiting high Qm and strong dissipative

optomechanical coupling, the nanobeams have two proper-ties that make them sensitive to optomechanical photo-thermal actuation. The nanobeam thermal time constantτ ∼ 0.7 μs, calculated using finite element simulations, ison the same time scale as ωm. As seen from Eq. (3), thisis a necessary condition for significant feedback from thephotothermal force [67–70]. A more subtle but equallyimportant property is the presence of compressive stressand accompanying buckling in the nanobeam, which asdiscussed below, dramatically enhances F .For the nanobeam geometry studied here, finite element

analysis presented in Appendix G indicates that F can beenhanced by over 2 orders of magnitude in a compressed

and buckled nanobeam. The level of compressive stress in ananobeam can be estimated from the deviation of ωm fromthe nominal value expected for an uncompressed device[33]. The nanobeam we study in Fig. 5 is observed to be ina postbuckled state with measured ωm=2π ∼ 430 Hz, lowerthan the nominal value expected (680 kHz) for an uncom-pressed nanobeam. Matching finite element simulated andobserved ωm predicts an internal compressive stress of∼37 MPa and an accompanying buckling amplitude ofx ¼ −122 nm. For this value of internal stress and buck-ling, a photothermal force of F ¼ −26 pN=μW is pre-dicted, which is over 100 times larger than predicted for anuncompressed device, and as a result is significant even forlow optical absorption within the nanobeam. This enhance-ment originates from more efficient conversion of axialstress to transverse deflection in buckled nanobeams(see Eq. (G7)), and is also present to a lesser degree incompressed prebuckled nanobeams with small amounts ofbending. This effect may be effective for photothermalactuation of compressed optomechanical crystal [2] andring or racetrack devices [47]. The negative signs of F andx indicate that the photothermal force and bucklingamplitude are downward, consistent with all of the self-oscillation behavior discussed in this section. Note that the

FIG. 5. Nanobeam self-oscillations. (a) Time-resolved TðhðtÞÞ of a phase-matched fiber taper nanobeam (L × w × d ¼80 × 0.50 × 0.25 μm3) system as h is reduced in discrete steps (visible as sharp steps in T). Inset: Large-amplitude oscillation ofthe nanobeam position. (b) Spectrograph showing the power spectral density of the data in (a). (c) Predicted renormalized mechanicaldissipation rate compared with measured values. (d) Scatter plot of shift in nanobeam resonance frequency versus oscillation amplitudesquared for data in the inset regions from (b) at the onset of self-oscillation. The solid line is a linear guide to the eye.


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buckled nanobeam shape and resulting variation of h alongthe nanobeam length is not included in waveguide couplingcalculations since this curvature is small compared withthat of the dimpled fiber taper.Given the above device parameters (summarized in

Table I of Appendix G), the only free-fitting parameterneeded to match Eq. (3) with experiment is the waveguideabsorption coefficient, which is found to be ζ ¼ 0.12 cm−1,corresponding to an optical loss rate effective quality factorof Qo ∼ 6.6 × 105. This absorption rate is consistent withloss in other diamond nanophotonic devices fabricatedwith both this process [49] and other techniques [47], andindicates that only a fraction (∼100 nW) of the input poweris absorbed and responsible for driving the self-oscillations.This absorbed power level corresponds to an average ofapproximately 0.04 photons absorbed per transit time ofthe coupler region. While small, this absorption rate ishigher than expected in bulk diamond. This is possibly aresult of etching-related surface state absorption or impu-rities in the diamond, and requires further investigation.Nonlinear coupling between nanobeam oscillation ampli-

tude and ωm provides an additional probe of the internalstress and buckled state of the device. The high sensitivityand the time-resolved absolute measurement of nanobeamposition provided by thewaveguide-optomechanical systemallows nanobeam resonance frequency to be studied as afunction of oscillation amplitude. Nonlinear softening of themechanical resonance frequency by ΔωmðhÞ is observed atthe onset of self-oscillations, as highlighted in Fig. 5(b).Nanobeam softening and hardening is an indicator ofinternal stress [32,66], and the softening observed here isfound to be closely related to the buckled nanobeam state.Nanobeam buckling breaks the device vertical symmetryand introduces a nonlinear softening term to the nanobeamdynamics, which counteracts the hardening effect describedby the intrinsic Duffing nonlinearity [71]. This competitionbetween nonlinear effects is given by

Δωm ¼ v2

ωm

�3

8α3 − x2

15

4ω2mα23

�; ð4Þ

where v is the oscillation amplitude and α3 is the Duffingcoefficient of the unbuckled nanobeam, as described inAppendix G. Equation (4) clearly shows the softeninginfluence of x. The optomechanical system we study hereprovides a direct measurement of Δωm and v for varying h,allowing x to be estimated experimentally. Figure 5(d)shows a scatter plot of measured Δωm as a function of v2

for varying fiber taper position. As predicted from Eq. (4),Δωm and v2 are found to be linearly related, and from thesedata and Eq. (4), x ¼ −98 nm is estimated, which is inexcellent agreement with finite element simulation predic-tions of x given above. This approach for simultaneouslyactuating nanomechanical motion and directly monitoringthe nonlinear response of nanomechanical devices can easily

be applied to other material systems. For example, fibertaper coupling to silicon and other semiconductor nanowire[72] and photonic crystal [52] waveguides is possible.During the self-oscillation limit cycle, a low fundamental

self-oscillation frequency of ∼180 kHz is measured, con-sistent with the behavior of a nanomechanical resonatoroscillating between buckled states, as observed by Bagheriet al. [15]. Further evidence of this behavior is found inFig. 5(a), which shows that during self-oscillations Tdecreases, indicating that the nanobeam, initially in abuckled down state, moves on average closer to the fibertaper while self-oscillating. Note that harmonics whichemerge during self-oscillation, becoming stronger andmore numerous as v increases, result from both the non-linear response of TðhÞ and nonlinearities of the nano-beam [73].

IV. DISCUSSION AND CONCLUSION

We have demonstrated an optomechanical system fromsingle-crystal diamond and have shown that it possesses aunique combination of high sensitivity, broad bandwidth,high-quality single-crystal diamond material, high Qm,and low-power self-oscillation threshold. It has potentialto enable measurement of quantum motion of nanobeamresonances and fundamental studies and technologiesbased on hybrid quantum devices. Excitation of nano-mechanical self-oscillations with nW absorbed powerillustrates the sensitivity of the diamond nanobeams tosmall driving forces, and their nonlinear dynamical soft-ening provides a glimpse of the changing stress withinthe nanobeam during large-amplitude oscillations. Thisdemonstration of optomechanical excitation of the diamondnanomechanical environment is a step towards control ofquantum electronic systems such as nitrogen vacanciesusing optomechanical actuation of classical nanomechan-ical oscillators [19–21,26,38]. Future devices may bedesigned to operate at visible (637 nm) wavelengthsresonant with NV center optical transitions, creating addi-tional opportunities for realizing hybrid quantum devices.The detailed analysis and understanding of the devices

presented here allows prediction of the internal straindynamics and self-oscillation threshold of nanobeams usedfor creating hybrid quantum systems. This analysis revealsthat the threshold can be surprisingly low, and that photo-thermal forces do not require large absorption or an opticalcavity to introduce significant backaction in the waveguide-optomechanical system. This understanding will guidefurther optimization of device geometry and operatingconditions needed to improve the measurement sensitivityin order to resolve quantum motion of the nanobeam and toharness the photothermal forces for optomechanical cool-ing. Reducing the nanobeam length may allow coupling tonanomehanical modes with frequencies sideband resolvedfrom spin transitions or resonant with higher-energy dia-mond nitrogen-vacancy spin transitions tuned by external


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magnetic fields [19,34,38]. If the nanobeams studied inFigs. 4(c) and 4(d) were shortened to L ¼ Lc ¼ 7 μm, thethree lowest-order resonances would shift to fm ∼ 135, 275,and 366 MHz, and the absolute optomechanical read-outsensitivity would remain constant. However, note thatthe photothermal actuation would become less efficient[see Eq. (G7)], and resonant optical modulation maybe required to drive large-amplitude oscillations of thesenanobeams.This system is also promising for implementations of

optically actuated classical logic based on buckled nano-mechanical states [15] and can be extended to use theoptical gradient force for optomechanical nanobeam actua-tion, enabling excitation of higher-frequency resonances ofsmaller structures [2]. Finally, the scalable nanofabricationtechnique demonstrated here is widely applicable to dia-mond nanophotonic devices for sensing, nonlinear optics,and quantum information processing, and can be easilyadopted by researchers with access to standard nanofabri-cation tools. Extending this approach to integrate thewaveguide-optomechanical system into an optical cavity,for example, a racetrack resonator [47], may allowdissipative optomechanical cooling of the nanobeamresonances [64]. Similarly, replacing the fiber taper withan on-chip waveguide optomechanically coupled to thenanobeam may be possible; such a system will retain thefeatures demonstrated here, trading stability inherent tomonolithic systems in exchange for tunability provide byfiber taper coupling.

ACKNOWLEDGEMENTS

We thank Charles Santori and David Fattal for usefulinitial discussions related to the fabrication approach usedhere. We would like to acknowledge support for this workfrom NSERC, iCore/AITF, CFI, NRC.

APPENDIX A: FABRICATION PROCESS

The fabrication process flow is outlined in Fig. 2. Achemical-vapor-deposition-grown, h100i-oriented single-crystal diamond optical grade substrate (Element Six) iscleaned in boiling piranha (3∶1H2SO4∶H2O2) and coatedwith a 300 nm thick layer of plasma-enhanced chemicalvapor deposition (PECVD) Si3N4 as a hard mask. Next,2–5 nm of titanium is deposited as an anticharging layer.Nanobeam structures with axes aligned along the h110idirection are patterned in ZEP520A electron-beam lithog-raphy resist and developed at −15 °C in ZED-N50. Theresulting ZEON electron-beam positive (ZEP) resist patternis transferred to the nitride hard mask using an inductively-coupled plasma reactive-ion etch (ICPRIE) process withC4F8=SF6 chemistry. An anisotropic oxygen plasmaICPRIE step transfers the pattern to the diamond, followedby a 160 nm thick conformal coating of PECVD Si3N4 toprotect the vertical diamond sidewalls. A short anisotropicC4F8=SF6 ICPRIE step clears the nitride from the bottom ofthe windows while keeping the side and top surface of thedevices protected. To create suspended nanobeams, a quasi-isotropic oxygen plasma etch [74] is performed at 2500 WICP power, 0 W rf power, and elevated wafer temperatureof 250 °C. This etch step is relatively slow, requiring ∼5 hto undercut the nanobeams studied here. Figure 6(a) showscross sections of fabricated devices with varying nominalwidths. The undercut etch rate is observed to increase athigher ICP power; however, this option is not available forthe devices fabricated for this report. An Oxford Plasmalabis used for all plasma etch steps. Finally, the titanium andnitride layers are removed by wet-etching in 49% HF, andthe sample is cleaned a second time in boiling piranha. Notethat adding a second vertical diamond etching step immedi-ately prior to the quasi-isotropic etch is expected to reducethe necessary undercut time, as in the related single-crystal

FIG. 6. (a) Nanobeam cross sections, created by focused ion beam (FIB) milling, of two nanobeams with different w patterned on thesame chip. Note that the FIB milling process rounds the nanobeam edges, resulting in some apparent curvature in the nanobeam profiles.(b) SEM image of typical nanobeams observed to self-oscillate. Dashed straight lines are guides to emphasize the direction of nanobeambuckling.


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reactive etching and metallization (SCREAM) siliconprocess [48].

APPENDIX B: MEASUREMENT SETUP

The optomechanical properties of the diamond nanobeamwaveguides are studied by monitoring the optical trans-mission of a dimpled optical fiber taper positioned in thenear field of devices of interest. The dimpled fiber taper isfabricated by modifying the procedure presented in Ref. [50]to use a ceramic mold for creating a dimple. Measurementsperformed in ambient conditions use high-resolution (50 nm)dc stepper motors to position the fiber taper. Vacuum (roomtemperature) and cryogenic measurements are performedin a closed-cycle cryostat (Montana Systems NanoscaleWorkstation) whose sample chamber is configured withstick-slip and piezo-positioning stages (Attocube) for con-trolling the sample and fiber taper positions. Before cooling, aturbopump is used to evacuate the chamber to pressures in the10−5 Torr range. Room-temperature vacuum measurementsare performed in these conditions. At 5 K, cryopumpingreduces the chamber pressure to the 10−6 Torr range. Duringthe low-temperature measurements, the fiber taper is posi-tioned in contact with lithographically defined supportson the diamond chip to reduce coupling of vibration fromthe cryostat cooling stages to low-frequency resonances ofthe optical fiber taper. These supports, visible in Fig. 6(c),allow the fiber taper to be positioned in the nanobeam nearfield without contacting the nanobeam.Two external cavity tunable diode lasers (New Focus

Velocity 6700) are used to probe the fiber taper trans-mission T over a wavelength range from 1475 to 1625 nm.A New Focus 1811 photodetector (PD1) with a noise

equivalent power sðdetÞP ¼ffiffiffiffiffiffiffiffiffiffiSðdetÞP

q¼ 2.5 pW=


pis used

to monitor the average (Pd) and fluctuating (δPdðtÞ) outputpower from the fiber taper. A New Focus 1623 detector(PD2) is also used in some measurements of Pd.A Tektronix RSA5106A real-time spectrum analyzer

(RSA) allows fast spectral analysis during the experimentsand recording of in-phase and quadrature (IQ) time series ofthe PD1 output voltage VðtÞ. All of the SvðfÞ data wepresent here are generated in postprocessing from VðtÞdata. By choosing a low center (demodulation) frequency(typically 0 or 2.5 MHz), and sampling VðtÞ with abandwidth exceeding the nanobeam resonance frequencies(typically 5 MHz), both δTðtÞ and T can be recorded by theRSA. To avoid damaging the RSAwith a large dc input, apair of bias tees (Minicircuits ZFBT-6GWB+) togetherwith electrical attenuators are used to reduce the low-frequency (<100 kHz) components of VðtÞ. The self-oscillation data in Fig. 5 is then acquired in a single45 s time series while the fiber height above the sample isstepped in 30 nm increments. Attenuation of the low-frequency signal is compensated for in postprocessing.

APPENDIX C: WAVEGUIDECOUPLED-MODE THEORY

Numerical prediction of the coupling coefficient κðhÞdescribing the interaction between the nanobeam and fibertaper waveguide modes can be obtained from the opticaldispersion of the eigenmodes of the coupled waveguides(“supermodes”). This process is described below.The field propagating through the coupled waveguide

system can be represented as a superposition of modes ofthe uncoupled optical fiber taper and diamond nanobeam.In the case of waveguides with two nearly phase-matchedcopropagating (positive group velocity) modes, this evo-lution can be approximately described by coupled-modeequations describing the field amplitude af;nðzÞ in the fiberand nanobeam waveguides, respectively, as a function ofpropagation distance z through the coupling region:

dafdz

¼ −jðβf þ κffÞaf − jκan; ðC1Þ

dandz

¼ −jðβn þ κnnÞan − jκaf; ðC2Þ

where βf and βn are the λ-dependent propagation constantsof the uncoupled fiber and nanobeam modes, respectively.Also included in this model are “self-term” correctionsκff;nnðhÞ to βf;n resulting from the modification of the localdielectric environment by the coupled waveguides. Thecoupler response can also be predicted by solving Eqs. (C1)and (C2) with boundary condition corresponding to unityinput power to the fiber taper (jafð0Þj2 ¼ 1, janð0Þj2 ¼ 0),resulting in the taper transmission Tðλ; hÞ given by Eq. (1)in the main text.Numerically, prediction of κðhÞ and κaa;bbðhÞ can be

realized from the dispersion of the supermodes of thecoupled waveguides. These modes can be modeled assuperpositions of the uncoupled modes of waveguidea and b. Their z dependence is of the form e−iβ�z, whichsatisfies Eqs. (C1) and (C2) for

β� ¼~βf þ ~βn

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ~βf − ~βn2

�2

þ κðhÞ2s

; ðC3Þ

with ~βf;n ¼ βf;nðλÞ þ κff;nnðhÞ [51].Intrawaveguide and self-coupling coefficients κðhÞ and

κff;nnðhÞ are determined by fitting β�ðh; λÞ with Eq. (C3).β�ðh; λÞ of the coupled waveguides and βf;nðλÞ of theuncoupled waveguides are calculated numerically with amode solver (Lumerical MODE Solutions) and are shownin Figs. 1(c) and 7. The resulting values for κðhÞ andκff;nnðhÞ, shown in Fig. 8, are found to decay exponentiallywith h. For simplicity, it is assumed that κff ¼ κnn duringthe fitting process, and that λ dependence is assumed to besmall over the coupling bandwidth of interest. Note that


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while κðhÞ decays over a length-scale comparable to thewaveguide evanescent field decay, κff;nnðhÞ are near-fieldterms that decay much more quickly and do not signifi-cantly impact the predicted values of κðhÞ for h > 100 nm.For a given h, β�ðλÞ can be converted to an effective

index dispersion curve, n�ðλÞ ¼ β�ðλÞλ=2π, as shown inFig. 1(c), which clearly exhibits an anticrossing near βf ¼βn whose width scales with jκj. The excellent agreementbetween the numerically calculated β� and the semianalyticmodel described by Eq. (C3) indicates that this model issuitable for predicting the waveguide coupling.

APPENDIX D: OPTOMECHANICALCOUPLING AND BACKACTION

A cavity optomechanical system with both dissipativeand dispersive optomechanical coupling can be realized byincorporating the fiber-coupled diamond nanobeam withinan optical resonator. Although not demonstrated in thispaper, such a system could be achieved by either terminat-ing the nanobeam waveguide with photonic crystal mirrorsor fabricating a racetrack structure incorporating a nano-beam along each side. Both approaches require furtherdevelopment of the fabrication process. In this appendix,dissipative and dispersive coupling coefficients are derivedfor such a system, and the potential for using this system foroptomechanical cooling is analyzed.

1. Dissipative optomechanical coupling coefficient

The rate of “external” coupling of light between ananobeam and the forward propagating mode of a fibertaper waveguide γe can be determined from T and theround-trip time of light circulating in the hypotheticalcavity in which the nanobeam is integrated. A standing-wave cavity formed from a nanobeam terminated by endmirrors couples bidirectionally to the fiber taper, as lightpasses through the coupling region twice per round-trip:once propagating in the forward direction and oncepropagating in the backward direction. In this system,

γe ¼vg2L

ln ð1=TÞ ∼ vg2L

1 − TffiffiffiffiT

p ; ðD1Þ

whereL is the total nanobeam length, vg is the group velocityof light in the nanobeam, and the right-hand expressionis a good approximation when T ∼ 1 [75]. The “parasitic”coupling rate into the backward propagating fiber tapermode is γp ¼ γe, and the total output coupling rate is 2γe.Parasitic loss can limit the effectiveness of dissipativeoptomechanical cooling, as discussed below.The corresponding dissipative optomechanical coupling

coefficient into the forward propagating fiber taper mode isgiven by

gðeÞγ ¼ ∂γe∂h χ ∼ − vg

2L1ffiffiffiffiT

p 1þ T

2T

∂T∂h χ: ðD2Þ

Here, 0 ≤ χ ≤ 1 is a dimensionless parameter accountingfor imperfect correspondence between nanobeam displace-ment and change in waveguide separation h; i.e., χ is theaverage value of dh=dx across the coupler region for themechanical resonance of interest. The parasitic dissipativecoupling coefficient to the backward propagating fiber

mode for a standing-wave cavity is gðpÞγ ¼ gðeÞγ .A nanobeam integrated within a traveling-wave cavity,

e.g., a racetrack resonator, can be coupled unidirectionallyto the fiber taper, assuming that backscattering within theresonator is small compared to internal loss. In this

FIG. 7. Propagation constants (β�) of the even and oddsupermodes of the coupled nanobeam and fiber taper wave-guides, as a function of waveguide separation h, for varying λ.Points indicated by open squares and circles where calculatedusing Lumerical MODE Solutions. Solid lines are fits using thecoupled mode theory model described in Appendix C.

FIG. 8. Waveguide coupling coefficients as a function of fibertaper and nanobeam waveguide separation h. Each data point isobtained by fitting β�ðλÞ with the coupled-mode theory model.Solid lines are single exponential fits to the data points.


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configuration light passes through the coupling regiononce per round-trip, 2L represents the total round-trip

path length, and both γp ¼ 0 and gðpÞγ ¼ 0, while gðeÞγ isunchanged from the standing-wave configuration.

2. Dispersive optomechanical coupling coefficient

The dispersive optomechanical coupling coefficient gω isdetermined by the h dependence of the round-trip phaseϕðhÞ accumulated by light circulating in the nanobeam.In general, ϕ depends on the waveguide mode mixingwithin the coupling region. However, at phase-matchingit can be shown that the h dependence of ϕ arisessolely from κnn and κff discussed in Appendix C. For astanding-wave cavity geometry,

ϕ

2¼ βoLþ κff þ κnn

2Lc: ðD3Þ

The contributions to ϕ from κff and κnn are due to arenormalization of the dielectric environment of eachwaveguide due to its neighbor. The corresponding dis-persive optomechanical coupling coefficient is given by

gω ¼ vg2L

∂ϕ∂h χ ¼ vg

Lc

L∂∂h

�κff þ κnn

2

�χ: ðD4Þ

Note that for κ2 ≫ ðκff − κnnÞ2, we can use Eq. (C3) towrite

gω ¼ ω

ng

Lc

L∂∂h

�nþ þ n−

2

�χ; ðD5Þ

where ng ¼ c=vg. This expression provides a convenientmeans to calculate gω from n�ðhÞ of the coupled wave-guide supermodes at phase-matching. As discussed inAppendix F, Eq. (D5) also shows that gω is proportionalto the optical gradient force, as in standard cavity opto-mechanical systems [53].Equations (D3)–(D5) can be modified for the case of a

traveling-wave cavity, where circulating photons interactwith the coupling region once per round-trip, by changingLc → Lc=2.

3. Predicted optomechanical coupling

The optomechanical coupling coefficients of the systemstudied in the main text can be predicted from Eqs. (D2)and (D5) using measured values of TðhÞ and simulated

n�ðhÞ. Figure 9(a) shows gðeÞγ ðhÞ, gωðhÞ, and γeðhÞ pre-dicted for the device with the fit of TðhÞ used in Fig. 4(b),assuming L ¼ Lc ¼ 7 μm and χ ¼ 1.0. This shows that gωis small compared to gðeÞγ for h > 200 nm. For the devicegeometry we study here, dispersive coupling is predicted tovanish at a critical value of h ∼ 450 nm. The maximumdissipative coupling into the forward propagating fiber

mode is gðeÞγ =2π ¼ 45 GHz=nm.

For cavity-optomechanics applications, it is desirable to

operate in a regime of small γe and large gðeÞγ . This may beachieved in future studies by increasingLc. Experimentally,this requires modifying the fiber taper dimple profile orusing a straight fiber taper as in Ref. [52]. Figure 9(b)shows the predicted optomechanical coupling when Lc ¼L ¼ 21 μm, assuming the same κðhÞ and slightly imperfectphase-matching extracted from fits to the experimental dataused in Fig. 4(b). The corresponding simulated TðhÞ is

shown in Fig. 9(c). For this system, low-loss and gðeÞγ ≫ gωare possible, for example, near h ∼ 370 nm.

4. Considerations for dissipative optomechanicalcooling using a cavity

Dissipative optomechanical coupling has been showntheoretically [64] and experimentally [63] to allow coolingof a mechanical resonator in an unresolved sideband cavity-optomechanical system. Unresolved sideband cooling

requires tuning of the relative strength of gðeÞγ , gω, and γe.This is possible in the system studied here by adjusting h,

FIG. 9. Predicted optomechanical coupling coefficients forvarying fiber height when (a) Lc ¼ 7 μm, as in the experiment,and (b) Lc ¼ 21 μm, for a nanobeam of length L ¼ Lc.Also shown is the coupling rate between the nanobeam andthe forward propagating fiber taper waveguide mode γeðhÞ.(c) Corresponding transmission functions for Lc ¼ 7 μm andLc ¼ 21 μm.


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as shown in Figs. 9(a) and 9(b). However, dissipativeoptomechanical cooling performance is limited by opticalloss γi þ γp into channels other than the input or outputchannel, and cooling to the quantum ground state requiresthat γe ≫ γi, γp [64].Waveguide absorption, scattering loss, and leaky mirrors

typically determine γi in nanophotonic cavities; theselosses are <10 GHz in state-of-the-art diamond devices[47]. Eliminating γp is possible through development oftraveling-wave optical cavity devices incorporating thediamond nanobeam, such as an oval racetrack resonator[47]. Note that such cavities would additionally require lowbackscattering rates to satisfy this criteria. Alternatively,implementing dissipative waveguide coupling betweenon-chip waveguides may allow replacing the fiber taperwaveguide with a “single-sided” input-output couplingwaveguide, as in Ref. [72], preventing leakage into aparasitic backwards propagating waveguide mode at theexpense of the tunable nature of the coupling demon-strated here.To evaluate the potential for dissipative optomechanical

cooling using the demonstrated interface integrated withina cavity, we use the formalism in Refs. [64]. We assumethat the nanobeam discussed above is incorporated into onehalf of an oval racetrack optical resonator with sides oflength Lc ¼ 21 μm and semicircular ends of radius 10 μm,and is unidirectionally coupled to the fiber taper with nobackscattering. Figure 10 shows the predicted minimumphonon number n achievable as a function of h, assumingγi=2π ¼ 1 GHz and γp ¼ 0, Pi ¼ 2 mW of power input tothe fiber taper, and a bath temperature of 5 K. Other deviceparameters are given in the figure caption. For this system,

dissipative optomechanical cooling from a phonon occu-pancy of nph ∼ 15000 to nph ∼ 200 is predicted when thefiber taper is positioned on either side of the transmission“recovery” point h ∼ 370 nm, where T → 1. Reducingγi=2π to 0.1 GHz, a technically challenging but potentiallyattainable value, would allow n ¼ 16 to be reached.For extremely small γi=2π ¼ 1 MHz, n < 1 is possible.These results highlight the key role of γi, and also show thatfor the purposes of optomechanical cooling, operating the

waveguide coupler near maximum gðeÞγ is not optimal dueto accompanying large γe.This calculation is performed using nph ¼ ðnthγmþ

nbaγomÞ=ðγm þ γomÞ, where nth is the thermal phononoccupancy, nba ¼ SFFð−ωmÞ=½SFFðωmÞ − SFFð−ωmÞ� isthe phonon population created by optomechanical back-action, and γom ¼ x2zpf ½SFFðωmÞ − SFFð−ωmÞ� is the opto-mechanical damping rate. xzpf is the zero-point fluctuationamplitude of the resonator, and SFFðωÞ is the totaloptomechanical force power spectral density acting onthe nanobeam, given in Ref. [64] as

SFFðωÞ ¼~B2N4x2zpf

�γe

½ωþ 2Δ − 2γt~A~B�2

ðωþ ΔÞ2 þ γ2t =4

þ ðγi þ γpÞ½ðΔ − 2

~A~BγtÞ2 þ γ2t

4�

ðωþ ΔÞ2 þ γ2t =4

�: ðD6Þ

Here, NðhÞ is the intracavity photon number, ~BðhÞ ¼gðeÞγ xzpf=γe and ~AðhÞ ¼ gωxzpf=γe are the normalized opto-mechanical coupling coefficients, and γt ¼ γe þ γp þ γi.For the calculations shown here, we set the detuningbetween the cavity mode and input power toΔ ¼ Δopt ¼ ωm=2þ ~A= ~Bγt, which causes cancellationbetween the dissipative and dispersive backaction at theinput port of the cavity.

APPENDIX E: OPTOMECHANICALTRANSDUCTION SENSITIVITY

This appendix describes the procedure for predicting thetheoretical measurement sensitivity of the optomechanicalwaveguide read-out and for extracting the experimentallyobserved measurement sensitivity from measured thermo-mechanical signals. Single-sided power spectral densitiesare used throughout.

1. Theoretical sensitivity

The displacement sensitivity of the waveguide-optomechanical system is determined by the minimummechanical motion to actuate a signal larger than the noisefloor of the measurement apparatus. For direct photo-detection of the optical power transmitted by the wave-guide, the power spectral density SðsÞv ðfÞ of the transduced

FIG. 10. Predicted achievable phonon number via optomechan-ical cooling of a fiber taper coupled nanobeam incorporatedinto an oval racetrack resonator (γp ¼ 0). Device parameters areγi=2π¼ 1GHz, Lc ¼L¼ 21 μm, ωm=2π¼ 7.3MHz,m ¼ 3.1pg,Qm ¼ 106, 5 K bath temperature. For the oval racetrack cavity,2L ¼ 2Lc þ 2πRc, where Rc ¼ 10 μm is the radius of curvatureof the cavity end.


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signal from a mechanical displacement described byspectral density SxðfÞ is

SðsÞv ðfÞ ¼ SxðfÞ�gtiPd

∂T∂x

�2

; ðE1Þ

where gti is the detector transimpedance gain, Pd is thedetected power in the absence of coupling (T ¼ 1), andj∂T=∂xj describes the optomechanical actuation of thecoupler. Similarly, the measurement noise can be written as

SðnÞv ðfÞ ¼ SðSNÞv ðfÞ þ SðdetÞv ðfÞ: ðE2Þ

SðdetÞv describes noise intrinsic to the detector and is related

to the detector’s noise equivalent power figure (SðdetÞP ) by

SðdetÞv ¼ g2tiSðdetÞP . The contribution from photon shot noise

is given by

SðSNÞv ðfÞ ¼ g2ti2ℏωoTPd

ηqe; ðE3Þ

where ηqe is the detector quantum efficiency. Note that theimpact of shot noise is affected by the operating point T ofthe coupler. This analysis does not consider optomechan-ical backaction, which is small for the optical powers usedin the high-sensitivity measurements presented here.To calculate the minimum detectable spectral density

Sox , we require unity signal-to-noise ratio, SðsÞv ¼ SðnÞv ,resulting in

Sox ¼2ℏωoTPd=ηqe þ SðdetÞP

ðPd∂T∂xÞ2

: ðE4Þ

Sox has units of nm2=Hz and is related to the minimumdetection sensitivity given in the text by sox ¼

ffiffiffiffiffiSox

p. In the

case of a codirectional evanescent coupler, ∂T=∂x can becalculated from Eq. (1) in the main text. For perfect phase-matching (Δβ ¼ 0),

∂T∂x ¼ − sinð2κLÞ ∂κ∂h

∂h∂x : ðE5Þ

Alternately, ∂T=∂x can be measured experimentally byrecording TðhÞ and determining dT=dh. This approach isused together with Eq. (E4) to generate the predictedmeasurement sensitivity in Fig. 4, assuming that∂h=∂x ¼ −1.

2. Thermomechanical calibration

The observed thermal nanobeam motion is used tocalibrate the measurement noise floor using a standardprocedure described in, for example, Refs. [61,76].Thermomechanical resonances in SvðωÞ are fit using

SvðωÞ ¼ SðnÞv þGSðthÞx ðωÞ, where SðthÞx ðωÞ is the single-sided power spectral density of a thermal oscillator,

SðthÞx ðωÞ ¼ 4kBTeωm

Qm

1

m½ðω − ω2mÞ2 þ ðωωm=QmÞ2�

; ðE6Þ

and G is a constant determined by the transduction gain ofthe measurement, as described theoretically by Eq. (E1),and treated as a fitting parameter for the purpose of thecalibration procedure. Here, kB is Boltzmann’s constant, Teis the operating temperature, and m is the effective massof the resonance, as defined in Ref. [77]. From the fitvalues, spectra can be converted from electrical (W=Hz) todisplacement (m2=Hz) power spectral density: Sx ¼ Sv=G.

APPENDIX F: OPTICAL GRADIENT FORCE

Dielectric objects in an evanescent field experience anoptical gradient force. The optical gradient force betweencoupled waveguides can be predicted from n�ðhÞ usingthe formalism of Ma and Povinelli [78]. Following thisformalism, the force induced by power P� in each of thesupermodes is given by

F� ¼ P�Lc

c∂n�∂h

��ωo

¼ −P�Lc

c∂n�∂x

��ωo

: ðF1Þ

Operating at phase-matching with power Pi input into thefiber taper, the power in the even and odd supermodes isP� ¼ Pi=2, and the total optical gradient force on thenanobeam is

Fg ¼PiLc

2c

�∂nþ∂h þ ∂n−

∂h�; ðF2Þ

where positive (negative) Fg indicates a repulsive (attrac-tive) force. Figure 11 shows the predicted FgðhÞ=LcPi forvarying alignment of the fiber taper with the center axis ofthe nanobeam. Corrections due to the curvature of the

FIG. 11. Optical gradient force as a function of h, when thefiber taper is aligned along the nanobeam axis, and offset 500 nmlaterally, approximating the position in the self-oscillationmeasurements.


041051-14

dimpled fiber taper are not considered explicitly and areassumed to be accounted for by the effective coupler lengthLc extracted from the experimental measurements. Giventhis approximation, Fg is constant over the interactionlength of the ideal two-port waveguide coupler andvanishes outside of the coupling region.

APPENDIX G: NANOBEAM PHOTOTHERMALAND NONLINEAR DYNAMICS

Waveguide-optomechanical coupling is observed toinduce nanobeam self-oscillations, as shown in Fig. 5 ofthe main text. Here, we analyze this effect by modeling thenanobeam as a nonlinear harmonic oscillator interactingwith the optical field of the coupled waveguides through adynamic photothermal force and an instantaneous opticalgradient force [1,70,79,80].

1. Nanobeam equation of motion

The dynamics of the nanobeam resonance driven by astochastic thermal force FsðtÞ and coupled to the opticalfiber taper are approximately described by the followingequation of motion:

xþ ωm

Qm_xþ ðω2

mðxÞ þ α3x2 þ α2xÞx

¼ 1

mðFgðxÞ þ Fptðx; tÞ þ FsðtÞÞ; ðG1Þ

where xðtÞ is the displacement amplitude of the nanobeammechanical resonance of interest, defined relative to theposition of the undriven nanobeam. Changes in x modulatethe nanobeam and the fiber taper spacing h, and for thecase of the v1 resonance, hðxÞ ¼ ho − xðtÞ if the fiber taperis positioned ho above the center of the nanobeam.Optomechanical coupling arises from the dependence ofthe optical gradient force, FgðhðxÞÞ and the photothermalforce FptðhðxÞÞ on nanobeam position. In addition, ωm

varies due to internal strain resulting from optically inducedchanges in static deflection x and local nanobeam temper-ature relative to the environment Θ. For sufficiently largejxðtÞj, the nanobeam response becomes nonlinear, asdescribed by α2 and α3.

2. Thermal effects

Local heating of the nanobeam by waveguide opticalabsorption occurs on a time scale determined by thenanobeam geometry and material properties. The thermaldynamics of the nanobeam, defined by the maximumtemperature change Θ, relative to the operating temper-ature, are assumed to follow

dΘdt

¼ −κtΘþ RLiζPn; ðG2Þ

where τ ¼ 1=κt is the nanobeam thermal time-constant andPn is the optical power coupled into the nanobeam. In theabsence of delayed optical feedback (i.e., a cavity), Pninstantaneously follows x and is given by

Pn ¼ ½1 − TðhðxÞÞ�Pi; ðG3Þ

with TðhÞ described by Eq. (1) in the main text. Here, ζ isthe per unit length absorption coefficient of the nanobeamwaveguide and R is the heating power per unit of absorbedoptical power. Ideally, ζ is determined by the materialproperties of diamond, but in nanophotonic devices itcan be modified by imperfect surfaces. The effectivenanobeam waveguide optical interaction length is givenby LiðhÞ ¼

RL0 janðzÞj2dz=jafð0Þj2 and represents the dis-

tance over which light propagates in the nanobeam. It canbe calculated from Eqs. (C1) and (C2).Local heating of nanobeam waveguides induces deflec-

tions in the nanobeam position. We write the correspondingphotothermal force as

Fptðx; tÞ ¼ FΘðx; tÞ κtR; ðG4Þ

where F is the force per unit absorbed power (LiζPn) insteady state. The resulting change in nanobeam deflectionis δxpt ¼ Fpt=k, where k ¼ ω2

mm is the nanobeam springconstant.Local heating and accompanying thermal expansion

of the device also modifies ωm. This effect is representedby

dωm

dΘ¼ Ct

κtR; ðG5Þ

where Ct is a constant related to the elastic properties of thenanobeam and has units rad s−1W−1.

3. Static response of compressed nanobeams

In general, Ct and F sensitively depend on both thenanobeam elastic properties and geometry and the internalresidual stress acting on the device. The nanobeams used inthe self-oscillation studies have significant internal stress,manifesting in smaller ωm than expected from theirnominal dimensions, and can be in a buckled geometry,as shown in the SEM image in Fig. 6(b). To modelnanobeam behavior in the presence of compressive stressand buckling, together with imperfect nanobeam shape andclamping points, we consider both an approximate analyticmodel and finite element simulations.The behavior of an ideal beam under axial compressive

loads has been widely analyzed [81]. For compressiveaxial load Fi, ωm of a nanobeam with maximum deflectionx can be expressed as


041051-15

ω2m ¼ ðωi

mÞ2�1 − Fi

Fcþ 3x2

AEFc

π2

ð2LÞ2�; ðG6Þ

where ωim is the resonance frequency of an ideal unloaded

nanobeam, A is the nanobeam cross-sectional area, E isYoung’s modulus, and Fc is the critical buckling load.Equation (G6) is equivalent to the model presented inRef. [71] and reveals the interplay between deflection,axial load, and stiffness. For example, we see thatdωm=dx ∝ x, which is a result of more efficient conversionof transverse actuation to axial strain with increasingnanobeam curvature. Locally heating the nanobeam modi-fies Fi → Fi þ ηϵEΘ, where ϵ is the thermal expansioncoefficient of the nanobeam and 0 < η ≤ 1 is a geometricfactor that accounts for nonuniform heating distribution ofthe nanobeam. In general, Fi and x are not independent.However, in an ideally buckled nanobeam [81], we canshow that

dxdΘ

¼ FkκtR¼ x

ηϵwdE2mω2

mL; ðG7Þ

indicating that photothermal deflection can be enhanced inbuckled nanobeams with jxj > 0.Although the ideal nanobeam buckling model is instruc-

tive, it fails to reproduce experimental features such asdeflection for axial load below the critical buckling loadand effects related to imperfect elastic clamping pointsand beam deformation [82]. To predict F and Ct whileincluding nanobeam nonidealities, we use finite elementANSYS software to simulate ωm and x as a function of axialand transverse loads and for specified absorbed power. Theresults, summarized in Fig. 12 and Table I, indicate thatsignificant enhancement of F compared to an unloadednanobeam is expected.The simulations are conducted as follows. The simulated

structure consists of a nanobeam with dimensions L × w ×d ¼ 80 × 0.48 × 0.25 μm3 and includes the surroundingdiamond chip. Clamping-point geometry is found tosignificantly affect the simulated nanobeam properties.In fabricated nanobeams, the undercut process results inrelatively complex clamping-point geometry. Here, theclamping points are modeled with a triangular verticalprofile roughly approximating that of fabricated structures,extending 0.5 μm on the nanobeam bottom surface and1.0 μm vertically along the undercut sidewall, as shown inFig. 12(a). ωm and x are then calculated as a function of Fi,as shown in Fig. 12(b). Agreement between simulated andexperimental ωm is realized at two values of Fi, corre-sponding to prebuckled and postbuckled nanobeam states,with x ¼ −9 and −123 nm, respectively, where negative xindicates buckling down.F and Ct are estimated by simulating changes to x and

ωm when power Pabs is uniformly absorbed across half ofthe nanobeam. During these simulations the bottom surface

of the diamond substrate is fixed at constant temperature,and only conductive heat loss is considered. The corre-sponding temperature distribution is used to predict Rand κt, as summarized in Table I. The photothermal resultsare shown in Fig. 12(c), which plots the changes Δxand Δωm in deflection and frequency, respectively, forPabs ¼ 1 μW. These results clearly illustrate the sensitivityof photothermal effects on compressive stress and deflec-tion and indicate that F varies by over 2 orders ofmagnitude depending on the compressive axial loadingof the nanobeam.Simulations of Δωm when a vertical transverse load

mimicking the optical gradient force is distributed along thecoupling region of the nanobeam (length Lc) are alsoperformed and are shown in Fig. 12(d). These results showthat dωm=dFg varies by nearly 3 orders of magnitudedepending on the compressive axial load.

FIG. 12. (a) Longitudinal cross section of nanobeam used inANSYS finite element simulations, showing the “arched” clamping-point geometry. (b) Finite element simulations of resonancefrequency and nanobeam deflection as a function of axial com-pressive stress. Dashed lines indicate value of Fi where simulatedωm approximately matches the experimentally observed value.(c) Simulated change in resonance frequency and nanobeamdeflection for Pabs ¼ 1 μW of absorbed optical power. (d) Simu-lated change in resonance frequency from a Fg ¼ 1 pN transverseload applied across the coupling region of the nanobeam (lengthLc). Nanobeam dimensions are L×w×d¼ 80×0.48×0.25 μm3,as in the self-oscillation measurements.


041051-16

4. Dynamics: Small-amplitude oscillations

For small mechanical oscillation amplitude, the powerin the nanobeam waveguide can be approximated byPn ¼ PnðhoÞ − PiðdT=dxÞjho ½xðtÞ − x�. Inserting this intoEq. (G2) and retaining only terms that are linear in x allowsthe Laplace transform of Eqs. (G1) and (G2) to becombined into a single linear equation:

− ω2xðωÞ − iωmω

QmxðωÞ þ ω2

mxðωÞ

¼ FsðωÞm

þ�dFg

dxþ dFpt

dxiωτ þ 1

ω2τ2 þ 1

�xðωÞm

; ðG8Þ

where constant force terms resulting in changes tostatic nanobeam deflection x have been left out for claritybut are included implicitly in ωmðxÞ. Rearranging termsreveals the optomechanical renormalization of the nano-beam dynamics,

−ω2xðωÞ − iγ0mωxðωÞ þ ω02mxðωÞ ¼

FsðωÞm

; ðG9Þ

with

ω02m

ω2m¼ 1 − 1

1þ ω2τ2dFptðhoÞ

dx1

k− dFgðhoÞ

dx1

k; ðG10Þ

γ0mγm

¼ 1þQmωmτ

1þ ω2τ2dFptðhoÞ

dx1

k; ðG11Þ

where dFpt=dx ¼ −FζLiPidT=dx. Static effects modifythe unperturbed resonance frequency ωo

m according to

ωm ¼ ωom þ CtζLiPnðhÞ þ

dωm

dFgFgðhÞ: ðG12Þ

Equations (G10)–(G12) illustrate that combinations ofmechanical softening or hardening and amplification ordamping are possible. Higher-order thermoelastic effects[70] are predicted to be small and are not included in thisanalysis.

5. Nonlinear dynamics: Large-amplitude oscillations

When γ0m approaches zero, the amplitude of mechanicaloscillations about the static (i.e., buckled) position growsand nonlinear contributions to the system dynamicsbecome significant. These nonlinear modifications origi-nate from mechanisms intrinsic to the nanomechanicaldevice geometry, or to the optomechanical response of thesystem, and are characterized here by nonzero α2;3 anddnT=dxn (n > 1), respectively. Two nonlinear featuresobserved in Fig. 5 are a softening in ωm at the onset ofself-oscillation and frequency harmonics in the self-oscillation region. While the latter effect is significantlyaffected by the nonlinear response of the optomechanicalsystem, the analysis of Zaitsev et al. [70] shows that theoptomechanical nonlinearity plays a negligible role insoftening ωm for the optomechanical system studied here.Rather, this softening is dominantly due to geometricnonlinearities of the deflected nanobeam.Nonlinear coefficients α2 and α3 in Eq. (G1) can

be derived from the Euler-Bernoulli equation for ananobeam with static deflection x, as in Refs. [83,84].Using an approximate ansatz for the static nanobeam shape,expressions for α2 and α3 can be derived [71]. Foroscillations about x, if we group the nanobeam deforma-tions into time-dependent and time-independent partsaccording to

ϕðl; tÞ ¼ ðxðtÞ þ xÞ 12ð1 − cosð2πl=LÞÞ; ðG13Þ

where l is the coordinate running the length of theundeflected nanobeam, the Euler-Bernoulli equation yieldsEq. (G1), with nonlinear coefficients

α2 ¼ xE6ρ

�2π

L

�4 3

8; ðG14Þ

TABLE I. Parameters input to model for γ0mðhÞ and ω0mðhÞ.

Values in brackets are predictions from simulations and areincluded for comparison with values determined from experi-mental fits.

ParameterInputvalue Units Source

κt 1.4 μs−1 Finite elementmethod (FEM)simulation

R 0.11 K=μW μs FEM simulation

ωim=2π 680 kHz FEM simulation

m 10 pg FEM simulation

x −122 nm FEM simulation

α=2π −16ð−28Þ Hz=nm2 Fit (FEM)

Ct 1.0 kHz=μW FEM simulation

ζ 0.12 cm−1 Fit

F=k −0.36 nm=μW FEM simulation

F −26 pN=μW FEM simulation

Fg Figure 11 Optical modesolver

dωm=dFg −7163 (−377) rad=s pN−1 Fit (FEM)

Pi 300 μW Experimentalparameter

Qm 25 000 Experimentalparameter


041051-17

α3 ¼E18ρ

�2π

L

�4 3

8; ðG15Þ

where ρ is the density of the nanobeam material. Note thatwhile α2 vanishes in a straight nanobeam, it is nonzero indeflected nanobeams (jxj > 0).Assuming that the solution to Eq. (G1) may be written

as a combination of harmonic functions, the method ofsuccessive approximations [83–85] shows that to first orderthe fundamental frequency of oscillation is

ω ¼ ωmðxÞ þv2

ωmðxÞ�3

8α3 − 5

12ω2mðxÞ

α22

�ðG16Þ

¼ ωmðxÞ þ v2α; ðG17Þ

where v is the amplitude of oscillation and α is the effectivenonlinear frequency shift coefficient. The first term inbrackets is the well-known Duffing frequency modifica-tion, whereas the second term results from nonlinearitiesinduced by static deflection of the nanobeam.

6. Parameter estimation and comparisonwith experiment

To compare the model described above with experimen-tally measured γ0mðhÞ, as shown in Fig. 5(c), a combinationof known simulated and fit parameters are input intoEq. (G11), as summarized in Table I and described below.For a given coupler operating condition, the photo-

thermal force and the resulting γ0ðhÞ described byEq. (G11) scales linearly with ζF . Neither F or ζ areknown a priori. However, using finite element simulationsto determine F , as well as other parameters such as κt andR, allows ζ to become the sole fitting parameter whencomparing experimental and predicted values of γ0ðhÞ.To determine F from finite element simulations, it is

necessary to determine whether the nanobeam is in aprebuckled or postbuckled state. Insight into the bucklingconfiguration of the nanobeam is provided by the observednonlinear softening at the onset of self-oscillations, whichis directly proportional to v2 [Fig. 5(d)]. From these dataand Eq. (G17), the nonlinear coefficient α given in Table Ican be measured. Together with Eqs. (G14) and (G15),the corresponding nanobeam deflection amplitude of jxj ¼98 nm is inferred, in good agreement with x ¼ −122 nmpredicted from finite element simulations of the nanobeamin its postbuckled configuration (Fi=A ∼ 37 MPa), asshown in Fig. 12(b). When the nanobeam is in thispostbuckled configuration, finite element simulationsshown in Fig. 12(c) indicate that F ¼ −26 pN=μW(F=k ¼ −0.36 nm=μW). Negative F and x indicate thatthe photothermal force and nanobeam deflection, respec-tively, are in the down direction.

Inputing F and other finite element simulated parame-ters summarized in Table I into Eq. (G11), good agreementbetween predicted and observed γ0mðhÞ is found forζ ∼ 0.12 cm−1. The corresponding optical absorptionrate can be described by quality factor Qo ¼ 2πng=ζλ∼6.6 × 105, where the group index ng ∼ 2.0 of the nano-beam is predicted from numerical simulation. This absorp-tion rate is smaller than combined absorption and radiationloss rates in other single-crystal diamond nanophotonicstructures [47].Given the value of ζ obtained from fitting γ0ðhÞ, ω0

mðhÞpredicted from Eq. (G10) can be compared with measure-ments. This is shown in Fig. 13, in which the predictedω0mðhÞ is generated with dωm=dFg as a fitting parameter,

and other parameters set as in the model for γ0ðhÞ and listedin Table I. Static and dynamic thermal effects are found tobe significantly smaller than the maximum experimentallyobserved shift to ωm. Rather, the monotonic decrease inω0m, which becomes significant for h < 200 nm, follows an

h dependence consistent with static tuning by the attractiveoptical gradient force FgðhÞ pulling up on a down-bucklednanobeam. This is in contrast to the static thermal tuningdescribed by Ct, which is proportional to the power coupledinto the nanobeam, and is expected to decrease in magni-tude with decreasing h for h < 200 nm.There is good agreement between the model and

experimental observation in Fig. 13; however, the fit valuefor dωm=dFg is larger than expected from finite elementsimulations [Fig. 12(d)] and predictions of FgðhÞ (Fig. 11).As shown in Fig. 12(d), dωm=dFg is found in simulationsto be highly variable, spanning over 3 orders of magnitude,depending on compressive stress and resulting bucklingconfiguration. It is possible that imperfect nanobeam andclamping-point shape result in an enhanced sensitivity[84,86]. For example, including the triangular “arched”clamping points to approximate the fabricated structure, asshown in Fig. 12(a), enhances dωm=dFg by a factor of ∼2compared to the case of ideal clamping points. Other effectsnot included here include reflection of light from the end ofthe waveguide resulting in enhanced optical interactions,dynamic and static displacement of the fiber taper dueto optical forces [87], low-frequency fiber vibrations andpossible parametric driving for small h, breakdown of

FIG. 13. Predicted ω0mðhÞ generated using the model in

Eq. (G10) and the parameters in Table I.


041051-18

the coupler two-mode representation and the influence ofhigher-order modes for small h, and short-range effectssuch as the Casmir force [88].In conclusion, the analysis we have presented here serves

to illustrate the influence of compressive loading andbuckling on the photothermal response of nanobeamsand to show that significant photothermal forces are presentat relatively low optical absorption levels. Note thatimperfect nanobeams have significantly altered prebuck-ling and postbuckling behavior when subject to ancompressive load [89], and taking into account thesenonidealities is necessary to more accurately predict thenanobeam behavior.

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