Abstract

Silicon-on-insulator (SOI) provides an important material system for photonic-electronic integration. Further integration of mechanical devices in SOI is hampered by incompatible under-etch or release steps required to prevent leakage of mechanical energy into the substrate. The purpose of this work is to demonstrate co-integration of nanomechanical structures on the surface of SOI. Silicon fin waveguides are used to generate co-localized and interacting optical and mechanical resonances. Radiation pressure interaction from laser driving is demonstrated. Our work enables co-integration of electronic, photonic, and phononic degrees of freedom on a single platform.

© 2017 Optical Society of America

The ability to confine and guide photons with low loss in thin films of silicon on silicon oxide has made silicon-on-insulator (SOI) a leading platform for photonic circuits [1]. Integrating mechanical devices into the same platform would greatly enhance the capabilities of the silicon photonics toolbox. Unfortunately, due to silicon’s mechanical properties, mechanical waves, or phonons, are not guided in the device layer; hence, oxide release processes are needed to suspend the silicon device and confine mechanical motion [24]. This greatly limits the integration of electronic, photonic, and phononic elements and prevents the generation of silicon phononic circuits that place mechanical and optical waves on the same footing in the SOI platform. Moreover, releasing the silicon devices limits their ability to dissipate heat [5], greatly complicating cryogenic optomechanical experiments [6]. In this work, we demonstrate a SOI nanomechanical device that uses fins to confine motion to the device layer without requiring a release process. Incorporating these fins into the first SOI optomechanical device, we use laser light to transduce their motion. The large optomechanical coupling in these structures enables us to observe thermal Brownian motion of the fin, as well as the optomechanical spring effect [7]. Our demonstration opens a route to new silicon optomechanical mass and force sensors, acousto-optic modulators, optical and microwave filters, and hybrid electronic-photonic-phononic structures that take advantage of the coupled dynamics in a semiconductor to obtain novel functionality.

In this work we demonstrate the first SOI optomechanical devices by fashioning optical and mechanical resonators from fin waveguides. We begin by describing the physics of the fin mechanical waveguides and resonators. We then outline how optical resonators can be patterned into the same device layer and describe the optomechanical experiment. Finally, we present measurements of the thermal mechanical Brownian motion in a structure with an engineered multimode spectrum and deduce the optomechanical coupling rates from the optical spring effect.

At optical frequencies, the index contrast between silicon (nSi=3.48) and silicon oxide (nOx=1.44) enables optical waveguiding in SOI via total internal reflection. In contrast to optical waves, mechanical waves, or phonons, propagate more quickly in silicon than in silicon oxide. This makes it impossible to obtain total internal reflection of mechanical waves in silicon clad with silicon oxide. One approach to obtain confinement that has been used in piezoelectric bulk acoustic wave devices [8], and more recently in silicon resonant body transistors [9], is to pattern a mechanical reflector directly into the substrate to prevent the mechanical waves from leaking away. A different approach is to sufficiently reduce the mechanical wave propagation phase velocity in the silicon layer so that the leakage is eliminated. Despite inconvenient material parameters, phonons can be confined in this way by leveraging the role that geometry and free boundaries play in their dispersion. A key observation is that tall, thin devices are more compliant than short, wide ones. We find that fin waveguides, such as that shown in Fig. 1, can guide phonons on SOI. Similar mechanical waveguides on the surface of crystals were studied several decades ago [10,11], and more recently, we have shown theoretically that fins on SOI can guide both photons and phonons and mediate large interactions between them [12].

 figure: Fig. 1.

Fig. 1. (a) Dispersion diagram for guided waves in silicon fins with dimensions h=340nm and w=100nm on an oxide substrate. (b) Mechanical frequencies for the E-B model of a fin (solid line) and a fin on glass (dashed line) are compared. E-B model starts to break down for small values of h/w, and there is a deviation from linearity in the full solution. Additionally, the fins on the oxide generally have lower frequency due to the “softer” clamping boundary condition. Simulated radiation limited quality factors (dot-dashed line) for the fins are overlaid with Qm measurements (stars). Corresponding solutions are marked with a red circle on (a) and (b).

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A simple approximation for the fundamental resonance frequency of a silicon fin with width w, etched into a silicon device layer of thickness h is obtained from Euler–Bernoulli (E-B) beam theory [13]:

ΩEρwh2.
The Young’s modulus, E, and density, ρ, of silicon are taken to be 165 GPa and 2330kg/m3, respectively, and we assume an isotropic medium. This motion has no longitudinal variation and represents a K=0 point for the dispersion of phonons as shown in Fig. 1(a). For larger K-vectors, the dispersion of this band can cause it to have a phase velocity in the longitudinal direction that is smaller than any other mechanical wave of the system, which leads to lossless propagation of waves in silicon on oxide [12]. This is represented by the crossing of the band from the hatched region to the unhatched, “protected” region in Fig. 1(a).

In this work, we focus our attention on the unprotected region in this diagram, near K=0 (Γ-point) of the mechanical dispersion. Γ-point mechanical modes are convenient for optomechanical design since there are no sign changes in the contribution to the optomechanical coupling due to variation of the phase of the mechanical motion along the propagation direction. We note that X-point mechanical modes can also show large optomechanical coupling [3] in certain designs, though the design process is more involved for such modes. Considering that our devices are limited by fabrication to aspect ratios of h/w<10, and the quality factor due to phonon leakage from a cantilever is known to be approximately QmC(h/w)3 where C is a factor on the order of unity [14], we expect the mechanical quality factors of our structures to be dominated by the radiation of acoustic waves into the substrate. The effect of this mechanical radiation into surface and bulk acoustic waves can be modeled numerically and is plotted for different geometric parameters in Fig. 1(b). Finite-element method simulations in COMSOL [15] of the equations of elasticity were performed to compute the dependence of mechanical frequency and Qm on fin width for an h=340nm SOI. As shown in Fig. 1(b), increasing the width of the beam causes an increase in the frequency, in qualitative agreement with the approximate E-B theory, as well as a reduction in the mechanical quality factor.

The fin geometry enables transverse confinement of motion in SOI waveguides [12]. It is important to also longitudinally confine the mechanical motion for many devices of interest. To do so, we break the translational symmetry by smoothly varying the fin’s width to make a curved fin shown in Fig. 2(a). This smoothly modulates the cutoff frequency (the K=0 mode) of the fin waveguide, causing it to support, in the thinned part, modes with frequencies below the cutoff frequency at the thicker edges of the fin. The mechanical mode profiles of the first few symmetric modes of such a structure are shown in Fig. 2(b) with exaggerated displacements. Since the modulation is smooth, we expect Qm of these localized resonances to be approximately the same as the Qm of the guided waves. This is borne out in measurements described below and summarized in Fig. 1(b).

 figure: Fig. 2.

Fig. 2. (a) SEM of the fabricated structure composed of two fins surrounding a wide beam and forming a photonic crystal cavity. The unit cell of the photonic crystal is shown with the geometric parameters defined. (b) Modulating the width of the fins leads to localized mechanical resonances with modes labeled f0 to f3 (these are only the even modes; odd modes are optically dark). (c) TE optical bands of a symmetric fin cavity unit cell have a 17-THz bandgap. Same variation that leads to trapped phonons (b) causes optical resonances to be trapped in the central region. Transverse component of the electric field for one such mode (TE00) is overlaid in the SEM in (a, d). Measurement scheme: cleaved fibers are aligned to TE grating couplers. Optical transmission spectra are recorded on an output channel and intensity fluctuations induced by the mechanics are read out upon reflection. (e) Optical transmission spectrum for the fin cavity reported here (black) is plotted for comparison alongside the transmission of a through waveguide (red, higher ηgc than measured for the cavity—see Supplement 1). In the inset, a narrower scan of the TE00 mode is shown. Detected RF power spectral density for the laser tuned to the slope of the cavity resonance transmission for optical modes TE00 and TE01 are shown in (f) and (g), respectively.

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To optically read out the motion, the curved fins are incorporated into a photonic crystal cavity. Figure 2(a) is a scanning electron micrograph (SEM) of a curved fin cavity consisting of a photonic crystal waveguide with adjacent curved fins. The unit cell of such a photonic crystal is characterized by a lattice constant a, optical waveguide width wWG, circular hole radius r, fin width w, and a gap g. These geometric parameters are outlined in Fig. 2(a). Photonic waveguides that are symmetric with respect to transverse reflections have electric fields that are antisymmetric (TE) or symmetric (TM) under a transverse reflection. Figure 2(a) shows the TE optical bands for this unit cell. We focus on TE bands, since these waves exhibit a larger bandgap, though we have also made similar structures with TM guided waves. The TE resonances have a 17-THz bandgap that varies with w. Conveniently, the same perturbation that confines mechanical modes, that is, reducing the width of the fins, increases the frequency of the fundamental TE band as shown Fig. 2(c), pushing the photonic crystal band edge into the bandgap and leading to confined optical resonances.

We measure transmission of a laser light through a photonic crystal with the curved fin defect. Laser light from an external cavity diode laser (Santec TSL-550-A) is swept over a range of wavelengths (1520–1570 nm), and the resulting transmitted field is detected on a photodiode. Devices with curved fins give rise to the TE00, TE01, and TE02 resonances, which appear as peaks in the transmission spectrum in Fig. 2(e). Fits to this spectrum yield (Q,κe/2π,κi/2π) of (25k, 2.8 GHz, 2.6 GHz) and (6.5k, 12 GHz, 5.1 GHz) for the first two resonances. Radiative losses of the TE00 mode (κ/2π=0.4GHz and κTM/2π=0.5GHz) computed from finite element method (FEM) solutions of the optical resonances in COMSOL [15] cannot account for the measured intrinsic losses κi. Separate measurements of optical Q on ring and disk resonances show far higher intrinsic Q>106. The excess losses are likely due to intermodal scattering enhanced by roughness inside the slot. The total transmitted power from one fiber to the other is 1.5% of the input power on resonance. The efficiency of the grating couplers ηgc=17±1% is measured on reflection (further description in Supplement 1).

Due to the very small amount of mechanical energy in the substrate, the two curved fins of symmetric fin cavities have essentially degenerate mechanical spectra that are only different due to inhomogeneity or disorder. In order to remove this degeneracy so that we can more clearly resolve the modes, the two fins are fabricated with different widths. By making the center of the fins 65- and 90-nm wide, and having the widths increasing parabolically in both directions by 30 nm over 7.5 μm, we cause a shift of 145 MHz between the fundamental frequencies of the two fins, in close agreement with the splitting we expect from simulations shown in Fig. 1(b) (156 MHz). This perturbation, however, has consequences for the optical spectrum, as it breaks the transverse symmetry plane and induces scattering between the quasi-TE and quasi-TM modes. For our cavity, simulations show an asymmetry-induced loss rate, as reported above, of κTM=2π×518MHz from the nearly TE mode into propagating TM waves (details are found in Supplement 1).

We expect improvements in the design of the photonic crystals to give us access to the sideband resolved regime, as well as to higher order and possibly symmetry protected mechanical modes in these structures. Nonetheless, the design presented has a number of very desirable features. The mechanical and optical design problems are largely decoupled. The fins can be designed first to engineer a mechanical response of interest. The photonic waveguide then offers independent degrees of freedom for engineering the optical modes. For the cavity measured and discussed below, a, w, and r are not varied along the cavity.

The mechanical motion of the fin resonance, described by modal displacements xk, causes fluctuations in the nth optical resonance frequency ωopt,n. Here, index n refers to either of the TE00 or TE01 modes. The coupling is described perturbatively by the relation ωopt,n(x0,x1,)=ωopt,n(0)+kgOM,nkxk. The coupling rate for each mode pair gOM,nk contains boundary (>99% of the contribution) and photoelastic contributions. Expressions for these contributions can be found in Ref. [16] in terms of the mode profiles. The resulting optomechanical interaction Hamiltonian is given by

Hint=k,ng0,nk(b^k+b^k)a^na^n,
where b^k and a^n are the annihilation operators for the kth mechanical mode and the nth optical mode, respectively, and g0,nk is the respective single-photon optomechanical coupling rate.

Optomechanical coupling to thermally excited mechanical degrees of freedom causes the intensity of light reflected off the cavity to fluctuate. We detect these fluctuations by first amplifying the light coming back from the cavity in reflection using an erbium-doped fiber amplifier (EDFA, Fiberprime EDFA-C-26G-S11) and sending the amplifier output to a photodetector (Optilab PD-40-M). The resulting photocurrent is sent to a spectrum analyzer (Rohde & Schwarz FSW26). Representative detected RF spectra of the thermal Brownian motion of the fins detected on the TE00 and TE01 resonances are shown in Figs. 2(f) and 2(g), respectively. For each spectrum, an RF spectral density SVV(ω) is taken at a given detuning. In addition, a far off-resonant (nm) spectrum is recorded SVV,BG(ω). The plotted signal is SVV(ω)/SVV,BG(ω)1, so the noise level is simply 1 and the y-axis can be interpreted as the signal-to-noise ratio. The noise level is dominated by amplified spontaneous emission from the EDFA.

To understand the mechanical and optomechanical response of the system and compare with theory, we extract the mechanical quality factors Qm,k and the optomechanical coupling rates g0,nk for each mechanical mode k from the measured RF spectra. The mechanical Qs are easily inferred by fitting the RF spectra, and they are plotted in Fig. 1(b) for the two fins. To extract the g0,nk, we take advantage of the dependence of radiation pressure back-action effects on the coupling rate and the intracavity photon number, which can be independently calibrated to good precision. This approach has the advantage of not requiring precise knowledge of the gain of the optical and electronic amplifiers and detectors. Laser light modifies the dynamics of the mechanical resonator and can either stiffen (Δ>0) or soften (Δ<0) the mechanical mode, depending on the detuning Δ=ωLωopt,0. This change in the mechanical frequency, denoted as dΩk, is known as the optical spring effect [7] and can be expressed as

dΩk=2(g0,nk|α|)2Δ(ΔΩk)2+κt24,
where κt=2κe+κi is the total optical loss rate, and |α|2 is the intracavity photon number in the nth optical mode. For a fixed optical input power, we record and fit mechanical spectra to obtain the mechanical frequency for a range of detunings, Δ. The detuning itself is verified by measuring the cavity transmission to cancel out thermal shifts of the cavity frequency. The resulting shift in the mechanical frequency and fits to Eq. (3) are shown in Fig. 3(a). This procedure, repeated at different incident optical powers, is used to find the coupling rate of the TE00 and f0, TE00 and f1, and TE01 and f1 modes resulting in coupling rates, g0,nk/2π, of 290±10, 95±3, and 150±5kHz, respectively. The systematic errors are larger than statistical errors and are primarily due to the uncertainty in the photon flux incident on the cavity. Since the spring effect is proportional to g0,nk2|α|2, uncertainty in the intracavity photon number is directly propagated to the coupling rate estimate. For modes with smaller optomechanical coupling, the spring effect is an unreliable means of determining the coupling. We compare contribution to the area under the power spectral density from each of these modes to that of a mode where g0,nk has been determined using the spring effect. Assuming all modes for a single spectrum are at the same temperature, the power contributed by each peak scales as g0,nk2/Ωk2, allowing precise determination of g0,nk, even for modes with small optomechanical coupling.

 figure: Fig. 3.

Fig. 3. (a) Optical mechanical spring effect is fit to determine the optomechanical coupling rate for TE00 and f0 modes. (b) Resulting g0,nk (black) between TE00 and four mechanical modes of the 65 nm fin are fit (as described in the text) and seen to agree well with simulated coupling rates (red). (c) Same as (b) but for optical mode TE01.

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The resulting estimates of the coupling rates for the two optical modes are compared to simulations of the interaction performed in COMSOL [15] and are found to be in excellent qualitative and quantitative agreement as shown in Figs. 3(b) and 3(c).

We have demonstrated the first fully release-free SOI optomechanical devices. Coupling to surface and bulk acoustic waves limits their mechanical quality factors to 100, which is already sufficient for observing optomechanical back-action. Further work is required to make resonators access the protected mechanical modes and exhibit low loss rates, despite the glass substrate (see Supplement 1). Since our devices do not require special release steps, they are completely compatible with silicon photonic foundry processes and can be produced at scale and integrated easily with on-chip electronics and photonics. Our demonstration is a step towards silicon phononic systems with the potential to combine optical, mechanical, and electronic functionality in an integrated platform.

Funding

National Science Foundation (NSF) (ECCS-1509107, ECCS-1542152); Office of Naval Research (ONR) (MURI QOMAND).

Acknowledgment

The authors thank Raphaël Van Laer for critical reading of the manuscript and valuable input. ASN is supported by the Terman and Hellman Fellowships. RNP is supported by the NSF Graduate Research Fellowships Program. YDD is supported by the Stanford UAR Major Grants program. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-1542152.

 

See Supplement 1 for supporting content.

REFERENCES

1. R. Soref, IEEE J. Sel. Top. Quantum Electron. 12, 1678 (2006). [CrossRef]  

2. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, Nature 456, 480 (2008). [CrossRef]  

3. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009). [CrossRef]  

4. B. J. Eggleton, C. G. Poulton, and R. Pant, Adv. Opt. Photon. 5, 536 (2013). [CrossRef]  

5. L.-D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, Opt. Express 17, 21108 (2009). [CrossRef]  

6. S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys. Rev. A 90, 011803 (2014). [CrossRef]  

7. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014). [CrossRef]  

8. K. Y. Hashimoto and R. F. Bulk, Acoustic Wave Filters for Communications, 1st ed. (Artech House, 2015).

9. B. Bahr, R. Marathe, and D. Weinstein, J. Microelectromech. Syst. 24, 1520 (2015). [CrossRef]  

10. E. Ash, R. De La Rue, and R. Humphryes, IEEE Trans. Microw. Theory Tech. 17, 882 (1969). [CrossRef]  

11. I. Mason, R. de la Rue, R. Schmidt, E. Ash, and P. Lagasse, Electron. Lett. 7, 395 (1971). [CrossRef]  

12. C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, APL Photon. 1, 071301 (2016). [CrossRef]  

13. A. N. Cleland, Foundations of Nanomechanics (Springer-Verlag, 2003).

14. K. Yasumura, T. Stowe, E. Chow, T. Pfafman, T. Kenny, B. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000). [CrossRef]  

15. COMSOL AB, Stockholm, COMSOL v. 5.0, https://www.comsol.com/.

16. A. H. Safavi-Naeini and O. Painter, Cavity Optomechanics, M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, eds., Quantum Science and Technology (Springer, 2014), pp. 195–231.

References

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  1. R. Soref, IEEE J. Sel. Top. Quantum Electron. 12, 1678 (2006).
    [Crossref]
  2. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, Nature 456, 480 (2008).
    [Crossref]
  3. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
    [Crossref]
  4. B. J. Eggleton, C. G. Poulton, and R. Pant, Adv. Opt. Photon. 5, 536 (2013).
    [Crossref]
  5. L.-D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, Opt. Express 17, 21108 (2009).
    [Crossref]
  6. S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys. Rev. A 90, 011803 (2014).
    [Crossref]
  7. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
    [Crossref]
  8. K. Y. Hashimoto and R. F. Bulk, Acoustic Wave Filters for Communications, 1st ed. (Artech House, 2015).
  9. B. Bahr, R. Marathe, and D. Weinstein, J. Microelectromech. Syst. 24, 1520 (2015).
    [Crossref]
  10. E. Ash, R. De La Rue, and R. Humphryes, IEEE Trans. Microw. Theory Tech. 17, 882 (1969).
    [Crossref]
  11. I. Mason, R. de la Rue, R. Schmidt, E. Ash, and P. Lagasse, Electron. Lett. 7, 395 (1971).
    [Crossref]
  12. C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, APL Photon. 1, 071301 (2016).
    [Crossref]
  13. A. N. Cleland, Foundations of Nanomechanics (Springer-Verlag, 2003).
  14. K. Yasumura, T. Stowe, E. Chow, T. Pfafman, T. Kenny, B. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000).
    [Crossref]
  15. COMSOL AB, Stockholm, COMSOL v. 5.0, https://www.comsol.com/ .
  16. A. H. Safavi-Naeini and O. Painter, Cavity Optomechanics, M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, eds., Quantum Science and Technology (Springer, 2014), pp. 195–231.

2016 (1)

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, APL Photon. 1, 071301 (2016).
[Crossref]

2015 (1)

B. Bahr, R. Marathe, and D. Weinstein, J. Microelectromech. Syst. 24, 1520 (2015).
[Crossref]

2014 (2)

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys. Rev. A 90, 011803 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

2013 (1)

2009 (2)

L.-D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, Opt. Express 17, 21108 (2009).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

2008 (1)

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, Nature 456, 480 (2008).
[Crossref]

2006 (1)

R. Soref, IEEE J. Sel. Top. Quantum Electron. 12, 1678 (2006).
[Crossref]

2000 (1)

K. Yasumura, T. Stowe, E. Chow, T. Pfafman, T. Kenny, B. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000).
[Crossref]

1971 (1)

I. Mason, R. de la Rue, R. Schmidt, E. Ash, and P. Lagasse, Electron. Lett. 7, 395 (1971).
[Crossref]

1969 (1)

E. Ash, R. De La Rue, and R. Humphryes, IEEE Trans. Microw. Theory Tech. 17, 882 (1969).
[Crossref]

Ash, E.

I. Mason, R. de la Rue, R. Schmidt, E. Ash, and P. Lagasse, Electron. Lett. 7, 395 (1971).
[Crossref]

E. Ash, R. De La Rue, and R. Humphryes, IEEE Trans. Microw. Theory Tech. 17, 882 (1969).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys. Rev. A 90, 011803 (2014).
[Crossref]

Baehr-Jones, T.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, Nature 456, 480 (2008).
[Crossref]

Bahr, B.

B. Bahr, R. Marathe, and D. Weinstein, J. Microelectromech. Syst. 24, 1520 (2015).
[Crossref]

Bulk, R. F.

K. Y. Hashimoto and R. F. Bulk, Acoustic Wave Filters for Communications, 1st ed. (Artech House, 2015).

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

Chan, J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

Chow, E.

K. Yasumura, T. Stowe, E. Chow, T. Pfafman, T. Kenny, B. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000).
[Crossref]

Cleland, A. N.

A. N. Cleland, Foundations of Nanomechanics (Springer-Verlag, 2003).

Cohen, J. D.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys. Rev. A 90, 011803 (2014).
[Crossref]

de la Rue, R.

I. Mason, R. de la Rue, R. Schmidt, E. Ash, and P. Lagasse, Electron. Lett. 7, 395 (1971).
[Crossref]

E. Ash, R. De La Rue, and R. Humphryes, IEEE Trans. Microw. Theory Tech. 17, 882 (1969).
[Crossref]

Eggleton, B. J.

Eichenfield, M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
[Crossref]

Gröblacher, S.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys. Rev. A 90, 011803 (2014).
[Crossref]

Haret, L.-D.

Hashimoto, K. Y.

K. Y. Hashimoto and R. F. Bulk, Acoustic Wave Filters for Communications, 1st ed. (Artech House, 2015).

Hill, J. T.

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, APL Photon. 1, 071301 (2016).
[Crossref]

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, Phys. Rev. A 90, 011803 (2014).
[Crossref]

Hochberg, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, Nature 456, 480 (2008).
[Crossref]

Humphryes, R.

E. Ash, R. De La Rue, and R. Humphryes, IEEE Trans. Microw. Theory Tech. 17, 882 (1969).
[Crossref]

Kenny, T.

K. Yasumura, T. Stowe, E. Chow, T. Pfafman, T. Kenny, B. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 (2000).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Kuramochi, E.

Lagasse, P.

I. Mason, R. de la Rue, R. Schmidt, E. Ash, and P. Lagasse, Electron. Lett. 7, 395 (1971).
[Crossref]

Li, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, Nature 456, 480 (2008).
[Crossref]

Marathe, R.

B. Bahr, R. Marathe, and D. Weinstein, J. Microelectromech. Syst. 24, 1520 (2015).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Mason, I.

I. Mason, R. de la Rue, R. Schmidt, E. Ash, and P. Lagasse, Electron. Lett. 7, 395 (1971).
[Crossref]

Meenehan, S. M.

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Supplementary Material (1)

NameDescription
» Supplement 1       Supplement 1

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Figures (3)

Fig. 1.
Fig. 1. (a) Dispersion diagram for guided waves in silicon fins with dimensions h=340nm and w=100nm on an oxide substrate. (b) Mechanical frequencies for the E-B model of a fin (solid line) and a fin on glass (dashed line) are compared. E-B model starts to break down for small values of h/w, and there is a deviation from linearity in the full solution. Additionally, the fins on the oxide generally have lower frequency due to the “softer” clamping boundary condition. Simulated radiation limited quality factors (dot-dashed line) for the fins are overlaid with Qm measurements (stars). Corresponding solutions are marked with a red circle on (a) and (b).
Fig. 2.
Fig. 2. (a) SEM of the fabricated structure composed of two fins surrounding a wide beam and forming a photonic crystal cavity. The unit cell of the photonic crystal is shown with the geometric parameters defined. (b) Modulating the width of the fins leads to localized mechanical resonances with modes labeled f0 to f3 (these are only the even modes; odd modes are optically dark). (c) TE optical bands of a symmetric fin cavity unit cell have a 17-THz bandgap. Same variation that leads to trapped phonons (b) causes optical resonances to be trapped in the central region. Transverse component of the electric field for one such mode (TE00) is overlaid in the SEM in (a, d). Measurement scheme: cleaved fibers are aligned to TE grating couplers. Optical transmission spectra are recorded on an output channel and intensity fluctuations induced by the mechanics are read out upon reflection. (e) Optical transmission spectrum for the fin cavity reported here (black) is plotted for comparison alongside the transmission of a through waveguide (red, higher ηgc than measured for the cavity—see Supplement 1). In the inset, a narrower scan of the TE00 mode is shown. Detected RF power spectral density for the laser tuned to the slope of the cavity resonance transmission for optical modes TE00 and TE01 are shown in (f) and (g), respectively.
Fig. 3.
Fig. 3. (a) Optical mechanical spring effect is fit to determine the optomechanical coupling rate for TE00 and f0 modes. (b) Resulting g0,nk (black) between TE00 and four mechanical modes of the 65 nm fin are fit (as described in the text) and seen to agree well with simulated coupling rates (red). (c) Same as (b) but for optical mode TE01.

Equations (3)

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ΩEρwh2.
Hint=k,ng0,nk(b^k+b^k)a^na^n,
dΩk=2(g0,nk|α|)2Δ(ΔΩk)2+κt24,

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