Abstract

Optomechanical cavities have proven to be an exceptional tool to explore fundamental and applied aspects of the interaction between mechanical and optical waves. Here we demonstrate a novel optomechanical cavity based on a disk with a radial mechanical bandgap. This design confines light and mechanical waves through distinct physical mechanisms which allows for independent control of the mechanical and optical properties. Simulations foresee an optomechanical coupling rate g0 reaching 2π × 100 kHz for mechanical frequencies around 5 GHz as well as anchor loss suppression of 60 dB. Our device design is not limited by unique material properties and could be easily adapted to allow for large optomechanical coupling and high mechanical quality factors with other promising materials. Finally, our devices were fabricated in a commercial silicon photonics facility, demonstrating g0/2π = 23 kHz for mechanical modes with frequencies around 2 GHz and mechanical Q-factors as high as 2300 at room temperature, also showing that our approach can be easily scalable and useful as a new platform for multimode optomechanics.

© 2017 Optical Society of America

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References

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2016 (3)

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

K. C. Balram, M. I. Davanço, J. D. Song, and K. Srinivasan, “Coherent coupling between radiofrequency, optical and acoustic waves in piezo-optomechanical circuits,” Nature Photon. 10, 346–352 (2016).
[Crossref]

R. Van Laer, R. Baets, and D. Van Thourhout, “Unifying Brillouin scattering and cavity optomechanics,” Phys. Rev. A 93, 053828 (2016).
[Crossref]

2015 (6)

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nature Photon. 9, 199–203 (2015).
[Crossref]

M. Schmidt, V. Peano, and F. Marquardt, “Optomechanical Dirac physics,” New J. Phys. 17, 023025 (2015).
[Crossref]

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological Phases of Sound and Light,” Phy. Rev. X 5, 031011 (2015).

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
[Crossref]

K. E. Grutter, M. I. Davanço, and K. Srinivasan, “Slot-mode optomechanical crystals: a versatile platform for multimode optomechanics,” Optica 2, 994–1001 (2015).
[Crossref]

2014 (7)

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2014).
[Crossref]

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4, 6842 (2014).
[Crossref] [PubMed]

S. Forstner, E. Sheridan, J. Knittel, C. L. Humphreys, G. A. Brawley, H. Rubinsztein-Dunlop, and W. P. Bowen, “Ultrasensitive Optomechanical Magnetometry,” Adv. Mater. 26, 6348–6353 (2014).
[Crossref] [PubMed]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-Dimensional Phononic-Photonic Band Gap Optomechanical Crystal Cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref] [PubMed]

J.-M. Yi, A. Cuche, E. Devaux, C. Genet, and T. W. Ebbesen, “Beaming Visible Light with a Plasmonic Aperture Antenna,” ACS Photonics 1, 365–370 (2014).
[Crossref] [PubMed]

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, F. M. Guilherme, G. S. Wiederhecker, N. C. Frateschi, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179–20186 (2014).
[Crossref]

2013 (3)

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111, 213902 (2013).
[Crossref] [PubMed]

Y. Chen, “Macroscopic quantum mechanics: theory and experimental concepts of optomechanics,” J. Phys. B: At. Mol. Opt. Phys. 46, 104001 (2013).
[Crossref]

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J 48, 2071–2102 (2013).
[Crossref]

2012 (9)

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012).
[Crossref]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nature Commun. 3, 1196 (2012).
[Crossref]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101, 081115 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity Optomechanical Magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref] [PubMed]

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12, 2299–2305 (2012).
[Crossref] [PubMed]

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nature Photon. 6, 768–772 (2012).
[Crossref]

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of Micromechanical Oscillators Using Light,” Phys. Rev. Lett. 109, 233906 (2012).
[Crossref]

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Lončar, “All optical reconfiguration of optomechanical filters,” Nature Commun. 3, 846 (2012).
[Crossref]

W. C. Jiang, X. Lu, J. Zhang, and Q. Lin, “High-frequency silicon optomechanical oscillator with an ultralow threshold,” Opt. Express 20, 15991–15996 (2012).
[Crossref] [PubMed]

2011 (2)

Y. C. Jun, K. C. Y. Huang, and M. L. Brongersma, “Plasmonic beaming and active control over fluorescent emission,” Nature Commun. 2, 283 (2011).
[Crossref]

J. Chan, T. P. MayerAlegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

2010 (3)

M. L. Gorodetsky, A. Schliesser, G. Anetsberger, S. Deléglise, and T. J. Kippenberg, “Determination of the vacuum optomechanical coupling rate using frequency noise calibration,” Opt. Express 18, 23236–23246 (2010).
[Crossref] [PubMed]

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?” J. Microelectromech. Syst. 19, 229–238 (2010).
[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE J. Sel. Top. Quantum Phys. 16, 316 (2010).
[Crossref]

2009 (2)

2005 (1)

2003 (2)

J. S. A. Yariv, “Annular Bragg defect mode resonators,” JOSA B 20, 2285–2291 (2003).
[Crossref]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

2002 (2)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming Light from a Subwavelength Aperture,” Science 297, 820–822 (2002).
[Crossref] [PubMed]

S. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[Crossref]

1998 (1)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Houdre, and U. Oesterle, “High-finesse disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 73, 1314–1316 (1998).
[Crossref]

1978 (1)

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber*,” JOSA 68, 1196–1201 (1978).
[Crossref]

Alegre, T. P.

R. da SilvaBenevides, G. de Oliveira Luiz, F. G. Santos, G. Wiederhecker, and T. P. Alegre, in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTh2A.99.

Anant, V.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

Anetsberger, G.

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

Aspelmeyer, M.

R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature 530, 313–316 (2016).
[Crossref] [PubMed]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

J. Chan, T. P. MayerAlegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

Auld, B. A.

B. A. Auld, Acoustic Fields and Waves in Solids, (Krieger, 1990).

Baets, R.

R. Van Laer, R. Baets, and D. Van Thourhout, “Unifying Brillouin scattering and cavity optomechanics,” Phys. Rev. A 93, 053828 (2016).
[Crossref]

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nature Photon. 9, 199–203 (2015).
[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE J. Sel. Top. Quantum Phys. 16, 316 (2010).
[Crossref]

Bagheri, M.

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111, 213902 (2013).
[Crossref] [PubMed]

Balram, K. C.

K. C. Balram, M. I. Davanço, J. D. Song, and K. Srinivasan, “Coherent coupling between radiofrequency, optical and acoustic waves in piezo-optomechanical circuits,” Nature Photon. 10, 346–352 (2016).
[Crossref]

Barea, L. A. M.

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, F. M. Guilherme, G. S. Wiederhecker, N. C. Frateschi, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179–20186 (2014).
[Crossref]

Barnard, A.

M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of Micromechanical Oscillators Using Light,” Phys. Rev. Lett. 109, 233906 (2012).
[Crossref]

Bazin, A.

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
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Zhang, M.

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ACS Photonics (1)

J.-M. Yi, A. Cuche, E. Devaux, C. Genet, and T. W. Ebbesen, “Beaming Visible Light with a Plasmonic Aperture Antenna,” ACS Photonics 1, 365–370 (2014).
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Figures (12)

Fig. 1
Fig. 1

Optomechanical ring cavity. a) Schematic of a bullseye optomechanical cavity (green) supported by a central pedestal (blue). The inset illustrates the mechanical motion localized at the disk’s edge. b) Finite-element simulation (FEM) of the electric field intensity for the fundamental transverse-electric (TE) mode of a floating silicon ring resonator, which is defined by the external radius R and the ring width wring. Inset: darker (lighter) colors represent larger (lower) electric field intensity. c) Mechanical dispersion of a silicon ring as a function of wring for R = 8 µm; only modes with g0/2π > 1 kHz are shown. The marker size represents the optomechanical coupling, g0, between each mechanical mode to the TE fundamental optical mode shown in (b). Red dots represent the first order mechanical dilatational mode, whose frequency can be also approximated by a Fabry-Perot type expression vL/2wring (dashed line), where vL = 8433 m/s is the velocity of longitudinal waves in bulk silicon. Insets show the cross-section of the a few mechanical modes for wring = 1.5 µm. Red (blue) color accounts for larger (smaller) displacement field. d) Optical dispersion of TE modes as function of the ring width with azimuthal number fixed to 83 and R = 8 µm. The optical resonance frequency is quite independent of wring for larger width; for the fundamental TE mode, such frequency is constant down to wring = 1 µm. For smaller ring width, the fundamental TE mode’s frequency varies only a few percent with wring whereas the mechanical frequency changes by a factor close to 2.

Fig. 2
Fig. 2

Bullseye optical modes. a) FEM simulation of the optical dispersion of quasi-TE (azimuthal number fixed to 83) bullseye modes (inset). For the fundamental mode, it is clearly seen that optical resonance frequencies are not affected by wring down to nearly 1 µm, and shift just by a few percent down to wring = 0.5 µm for a typical mechanical grating. The marker’s color label the mode profile in (b) with the same bounding box color for wring = 1 µm. b) Optical modes living in the outer ring may evanescently couple to modes within adjacent inner rings depending on the inner ring radius — by keeping the total radius R constant, increasing wring decreases the inner ring’s radius and therefore increases its corresponding mode frequency — thus deviating from the desired ring-like modes and resulting in the avoided crossings shown in (a). However, for typical mechanical gratings, the fundamental quasi-TE mode does not couple to any other mode down to wring = 0.5 µm. (Simulation parameters: a = 1 µm, w = 200 nm, t = 70 nm, R = 8 µm.)

Fig. 3
Fig. 3

Mechanical grating design. a) Each geometrical parameter of the bullseye disk controls distinct physical properties: R defines the optical frequency and free-spectral range; wring defines the mechanical frequency while the grating parameters a, w and t determine the phononic bandgap. b) Linear phononic crystal approximation for the circular grating shown in (a). c) The blue (red) lines are the mechanical bands of the linear crystal for x-polarized or z-polarized (y-polarized) modes for a = 1 µm, w = 200 nm and t = 70 nm. The insets show the mechanical deformation for the modes at the band-edge (X-point) of selected edge states. Red (blue) colors account for larger (smaller) displacement field. d) Bandgap maps for the linear crystals for x and z-polarized waves. The colored areas correspond to regions within bandgaps for different lattice period a. The regions delimited by solid lines correspond to bandgaps between the two modes around 3 GHz and 5 GHz shown in (c). Left (Right) bandgap maps as a function of w/a (t/a) for fixed t = 70 nm (w/a = 0.25). All dimensions are compatible to Si-photonics foundry based processes.

Fig. 4
Fig. 4

Optomechanical bullseye disk. a) FEM simulations for the mechanical dispersion of a bullseye disk (solid dots) as a function of the periodicity a for wring = 1.1 µm (dots closer to 4 Ghz) and 2.0 µm (around 2 GHz), w/a = 0.25 and t = 70 nm. The gray tone shades are proportional to the mechanical density of states (DOS) of the corresponding linear crystal, where darker regions are related to higher DOS within the mechanical grating. Only the two largest g0 modes are shown. Marker sizes are proportional to the total optomechanical coupling rate. b) Type-I and type-II displacement profile of the confined mechanical modes for wring = 1.1 (2.0) µm and a = 1.0 (1.4) µm. Type-I is the first order dilatational mode at 4.2 (2.4) GHz; type-II is a high order flapping mode at 3.8 (2.2) GHz. c) Simulated mechanical radiation loss quality factor for the type-I mode as a function of the number of grating periods (wring = 1 µm and a = 1.0 µm). The solid line is a fit for the simulated points with an exponential decay curve. The insets show the normalized mechanical displacement in log scale for zero (no clamping loss suppression) and six grating periods (≈ 60 dB of isolation from the clamping region) along with the displacement profile near the ring region.

Fig. 5
Fig. 5

Probing mechanical modes. a) Scanning electron microscope (false color) images of typical bullseye cavities from side b) and top. c) Experimental setup. DAQ: Data Acquisition System (records optical spectra); ESA: Electrical Spectrum Analyzer (records mechanical spectra); φ: electro-optical phase modulator; MZ: Mach-Zehnder interferometer (wavelength calibration); FPC: Fiber Polarization Controller. d) Typical (intentionally undercoupled) optical resonance used to address the bullseye’s mechanical modes (Qopt ≈ 4 × 104) and exhibiting forward-backward splitting due to surface roughness back-scattering. The extrinsic coupling is tuned close to critical coupling in order to interrogate the mechanical modes. e) The transmitted noise spectrum is composed by the external phase modulator tone (pink) and several peaks due to the device’s thermal motion. The higher frequency family of peaks (red) is attributed to type-I modes whereas the lower frequency one (blue) has to do with the type-II mechanical modes. The spectrum noise-floor is above the detector shot noise (not shown) while the the detector dark noise was typically 10 dB below the shot-noise. f) Fitting the noise spectrum to the typical thermal mechanical response allows us to determine the mechanical mode frequency, quality factor and, along with knowledge of the external phase modulation, g0. For a = 1400 nm, the highest peak belonging to the type-I family shows g0/2π = 23.1 ± 0.2 kHz and Qm = 2360 ± 40.

Fig. 6
Fig. 6

Bullseye experimental demonstration. a) Mechanical spectra as a function of a for fixed wring = 2 µm, w/a = 0.2815, t = 70 nm and silicon layer of 220 nm. The vertical axis is normalized in order to account only for the optomechanical response (see Appendix C). The dashed lines account for the simulated edges for shear-horizontal (upper) and longitudinal (lower) waves, which dictate the isolation between type-II and type-I modes, respectively, and the pedestal. b) Simulated mechanical modes for an axisymmetric structure with the same parameters as those in (a). Larger (smaller) marker size accounts for larger (smaller) optomechanical coupling rate to the first order TE mode shown in Fig. 3(a). The red (blue) color is a guide to the mechanical modes of type-I (type-II). The gray tone shades are proportional to the mechanical DOS for the corresponding linear crystal, where darker (lighter) regions are related to higher (smaller) density of states (DOS) within the mechanical grating. c) Calculated mechanical mode displacement profile for selected type-I modes. Red (blue) colors represent larger (zero) mechanical displacement.

Fig. 7
Fig. 7

The linear grating approximation. a) Linear and b) circular gratings used to illustrate the validity of the linear crystal approximation in predicting bandgaps. Material properties: ρ1 = 2329 kg/m3, ρ2 = 0.001ρ1, c11 = 165.6 GPa, c44 = 79.5 GPa, v L , j = c 11 / ρ j, v S , j = c 44 / ρ j (j = 1, 2); geometric parameters: a = 1000 nm, w/a = 0.2815. c) Transmission spectra for linear and circular gratings (for varying r0) calculated through the transfer-matrix method. Such spectra clearly show that the linear approximation works better in the short-wavelength limit, where ωr0/vL ≫ 1.

Fig. 8
Fig. 8

Pump-probe measurements. a) An amplitude modulated pump laser (red) locked to the half-linewidth of an optical resonance harmonically drives the mechanical mode through radiation pressure force; a probe laser (blue) tuned to an optical resonance with equal transverse profile but a few free spectral ranges away, reads the driven motion through the optomechanical interaction. b) The signal gain compared to the thermal driven motion strongly depends on the modulation amplitude and input power of the pump laser. The dashed line shows the unity gain, above which the pump-probe scheme is actually advantageous. c) In order to avoid contamination of the probe signal by the pump, the beams are launched in opposite directions and isolated from each other through optical circulators. A bandpass filter tuned to the probe band further improves the degree of isolation. d) A Fano lineshape arises from interference between the driven mechanical motion and the broadband Kerr effect. Such lineshape may be fitted to yield the mechanical mode’s frequency and quality factor. e) The pump-probe measurement agrees to the thermally driven one; although having a similar signal-to-noise ratio, the pump-probe spectrum has a much high extinction.

Fig. 9
Fig. 9

Effect of anisotropy on mechanical modes. a) A three-dimensional FEM simulation shows the dilatational mode and 3 extra mechanical modes for an isotropic approximation (η = 0), all of which couple to the ring-like optical mode at similar optomechanical coupling rates. These 4 modes are hybridized as silicon’s anisotropy is taken into account (η > 0, such that the exact anisotropy is approached as η approaches 1). The measured splitting of nearly 10 MHz, however, may be better understood as a combination between anisotropy and fluctuations in wring (see Fig. 10). b) Numerical calculations for the dilatational mode shows that the total optomechanical coupling rate g0 (black) diminishes towards the measured values as anisotropy is taken into account following the trend of the dominant photoelastic contribution (red); the smaller moving boundary contribution (green) barely changes.

Fig. 10
Fig. 10

Effect of eccentricity on mechanical modes. a) Simulated mechanical dispersion as a function of b/a for a silicon ring (inset) with anisotropic stiffness. A splitting of ∼ 10 MHz arises as b exceeds a by 0.5%. The eccentricity-induced mode hybridization is such that the former type A mode gives origin to two branches with similar mode profiles, hence similar optomechanical couplings. b) A similar splitting behavior also shows up for bullseye simulations (performed for half-structure with symmetric boundary conditions). Because the real fluctuations are random, such mechanism may be responsible for the multiple peaks observed within the RF spectra.

Fig. 11
Fig. 11

Scaling the bullseye disk fabrication. a) An array of bullseye cavities is shown with constant wring = 2.0 µm and varying a and w. The central row is the optomechanically characterized one (see Fig. 5). SEM images of the lower row devices were fitted (red curves) as shown in the inset in order to check the geometrical parameters. b) The ring-width wring matches the nominal value within the fitting error, as well as the c) groove width w. The fitting uncertainty is dominated by systematic error, ±2 pixels, intrinsic to our fitting algorithm. d) w/a also matches the nominal values within error bars. The higher error bars for smaller w/a values is due to the relative importance of the ± 2 pixels of systematic error compared to both w and a.

Fig. 12
Fig. 12

Mechanical modes for wring = 1.5 µm. Pump-probe spectra as function of a for bullseye disks with wring = 1.5 µm and w/a = 0.28; the mechanical mode frequencies along with extinction and quality factor approximately matches the linear crystal DOS (gray scale), like observed for wring = 2.0 µm.

Equations (25)

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g 0 x zpf = ω x = ω 2 E | ϵ / x | E E | ϵ | E ,
S II ( Ω ) = g 0 2 K ( Ω ) x zpf 2 Ω 2 S x x ( Ω ) + ϕ 0 2 2 K ( Ω ) ENBW δ ( Ω Ω m o d ) + S shot ( Ω ) ,
v L 2 ( v ( r ) ) v S 2 × ( × v ( r ) ) + ω 2 v ( r ) = 0
v rect = ( A e i ω x / v L + B e i ω x / v L ) x ^
v circ = ( A H 1 ( 1 ) ( ω r v L ) + B H 1 ( 2 ) ( ω r v L ) ) r ^ ,
( A 2 B 2 ) = M 21 ( A 1 B 1 ) ,
( A N B N ) = M N , N 1 M 32 M 21 ( A 1 B 1 ) .
M 21 = ( v L , 1 ρ 1 + v L , 2 ρ 2 2 v L , 2 ρ 2 exp { 2 i w ω v L , 1 i ( v L , 1 + v L , 2 ) w ω v L , 1 v L , 2 } v L , 2 ρ 2 v L , 1 ρ 1 2 v L , 2 ρ 2 exp { i ( v L , 1 + v L , 2 ) w ω v L , 1 v L , 2 } v L , 2 ρ 2 v L , 1 ρ 1 2 v L , 2 ρ 2 exp { 2 i w ω v L , 1 + i ( v L , 1 v L , 2 ) w ω v L , 1 v L , 2 } v L , 1 ρ 1 + v L , 2 ρ 2 2 v L , 2 ρ 2 exp { i ( v L , 1 v L , 2 ) w ω v L , 1 v L , 2 } )
M 21 = i π 4 v L , 2 2 ρ 2 ( d 1 t 12 t 21 d 2 )
d 1 = H 1 ( 1 ) ( a ω v L , 1 ) { a v L , 2 ρ 2 ω H 0 ( 2 ) ( a ω v L , 2 ) + 2 ( v S , 1 2 ρ 1 v S , 2 2 ρ 2 ) H 1 ( 2 ) ( a ω v L , 2 ) } a v L , 1 ρ 1 ω H 0 ( 1 ) ( a ω v L , 1 ) H 1 ( 2 ) ( a ω v L , 2 ) d 2 = H 1 ( 1 ) ( a ω v L , 2 ) { a v L , 1 ρ 1 ω H 0 ( 2 ) ( a ω v L , 1 ) + 2 ( v S , 2 2 ρ 2 v S , 1 2 ρ 1 ) H 1 ( 2 ) ( a ω v L , 1 ) } a v L , 2 ρ 2 ω H 0 ( 1 ) ( a ω v L , 2 ) H 1 ( 2 ) ( a ω v L , 1 ) t 12 = a v L , 2 ρ 2 ω H 0 ( 2 ) ( a ω v L , 2 ) H 1 ( 2 ) ( a ω v L , 1 ) + { 2 ( v S , 1 2 ρ 1 v S , 2 2 ρ 2 ) H 1 ( 2 ) ( a ω v L , 1 ) a v L , 1 ρ 1 ω H 0 ( 2 ) ( a ω v L , 1 ) } H 1 ( 2 ) ( a ω v L , 2 ) t 21 = a v L , 2 ρ 2 ω H 0 ( 1 ) ( a ω v L , 2 ) H 1 ( 1 ) ( a ω v L , 1 ) { 2 ( v S , 1 2 ρ 1 v S , 2 2 ρ 2 ) H 1 ( 1 ) ( a ω v L , 1 ) a v L , 1 ρ 1 ω H 0 ( 1 ) ( a ω v L , 1 ) } H 1 ( 1 ) ( a ω v L , 2 )
S x x ( thermal ) ( Ω ) = 2 γ k B T / m eff ( Ω m 2 Ω 2 ) 2 + ( γ Ω ) 2
a ˙ = ( i Δ + κ 2 ) a i G x a + κ e a in ( 1 + δ cos Ω mod t )
x ¨ + γ x ˙ + Ω m 2 x = G m eff | a | 2 ,
a = α 0 + α e i Ω mod t + α + e i Ω mod t x = x 0 + 1 2 ( x 1 e i Ω mod t + x 1 * e i Ω mod t )
x 1 = 2 G m eff α 0 * α + α 0 α + * Ω m 2 Ω mod 2 + i γ Ω mod
α 0 = κ e a in i Δ + κ / 2
α = κ e a in δ i ( Δ + Ω mod ) + κ / 2
α + = κ e a in δ i ( Δ Ω mod ) + κ / 2
Δ x 2 = ( x x ) 2 = | x 1 | 2 2 .
S x x ( driven ) ( Ω mod ) = 2 π ENBW Δ x 2 = 4 π ENBW ( G m eff ) 2 | ( α 0 * α + α 0 α + * ) | 2 ( Ω m 2 Ω mod 2 ) 2 + ( γ Ω mod ) 2
gain = 2 π ENBW | G ( α 0 * α + α 0 α + * ) | 2 γ m eff k B T
S I I ( Ω mod ) = | k NL + k OM Ω m 2 Ω mod 2 + i γ Ω mod | 2
C ( η ) = ( c 11 c 12 c 12 0 0 0 c 12 c 11 c 12 0 0 0 c 12 c 12 c 11 0 0 0 0 0 0 c 44 * ( η ) 0 0 0 0 0 0 c 44 * ( η ) 0 0 0 0 0 0 c 44 * ( η ) )
c 44 * ( η ) = η c 44 + ( 1 η ) c 11 c 12 2
Δ r = s λ ξ p ( 1 i ) Δ w ξ

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