Abstract

Periodically structured materials can sustain both optical and mechanical excitations which are tailored by the geometry. Here we analyze the properties of dispersively coupled planar photonic and phononic crystals: optomechanical crystals. In particular, the properties of co-resonant optical and mechanical cavities in quasi-1D (patterned nanobeam) and quasi-2D (patterned membrane) geometries are studied. It is shown that the mechanical Q and optomechanical coupling in these structures can vary by many orders of magnitude with modest changes in geometry. An intuitive picture is developed based upon a perturbation theory for shifting material boundaries that allows the optomechanical properties to be designed and optimized. Several designs are presented with mechanical frequency ~1–10 GHz, optical Q-factor Qo>107, motional masses m eff≈100 femtograms, optomechanical coupling length L OM<5 µm, and clampinig losses that are exponentially suppressed with increasing number of phononic crystal periods (radiation-limited mechanical Q-factor Qm>107 for total device size less than 30 µm).

© 2009 Optical Society of America

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2009 (8)

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nature Physics 3(4), 201–205 (2009).

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat Phys 5(4), 276–280 (2009).
[CrossRef]

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009).
[CrossRef]

M. Tomes and T. Carmon, “Photonic Micro-Electromechanical Systems Vibrating at X-band (11-GHz) Rates,” Phys. Rev. Lett. 102(11), 113601 (pages 4) (2009).
[CrossRef]

R. H. O. III and I. El-Kady, “Microfabricated phononic crystal devices and applications,” Measurement Scie. Technol. 20(1), 012,002 (13pp) (2009).

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lončar, “Coupled photonic crystal nanobeam cavities,” Appl. Phys. Lett. 95(3), 031,102–3 (2009).

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (pages 3) (2009).
[CrossRef]

2008 (7)

A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one-dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 084–089 (2008).

M.W. McCutcheon and M. Lončar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16(23), 136–145 (2008).

M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q Nanocavity with 1D Photonic Gap,” Opt. Express 16(15), 905–102 (2008).

Y. Yi, “Geometric effects on thermoelastic damping in MEMS resonators,” J. Sound Vibration 309(3–5), 588–599 (2008).
[CrossRef]

R. H. O. III, I. F. El-Kady, M. F. Su, M. R. Tuck, and J. G. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sensors Act. A: Physical 145–146, 87–93 (2008).

K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A (Atomic, Molecular, and Optical Physics) 78(2), 023832 (pages 4) (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[CrossRef]

2007 (5)

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics,” Opt. Express 15(25), 172–205 (2007).

Y.-C. Wen, L.-C. Chou, H.-H. Lin, V. Gusev, K.-H. Lin, and C.-K. Sun, “Efficient generation of coherent acoustic phonons in (111) InGaAs/GaAs multiple quantum wells through piezoelectric effects,” Appl. Phys. Lett. 90(17), 172102 (pages 3) (2007).
[CrossRef]

N. D. Lanzillotti-Kimura, A. Fainstein, A. Huynh, B. Perrin, B. Jusserand, A. Miard, and A. Lemaitre, “Coherent Generation of Acoustic Phonons in an Optical Microcavity,” Phys. Rev. Lett. 99(21), 217405 (pages 4) (2007).
[CrossRef]

T. Carmon and K. J. Vahala, “Optomechanical Modal Spectroscopy of Optoexcited Vibrations of a Micron-Scale on-Chip Sphere at Greater than 1 GHz,” Phys. Rev. Lett. 98(123901) (2007).
[CrossRef] [PubMed]

T. Gorishnyy, J.-H. Jang, C. Koh, and E. L. Thomas, “Direct observation of a hypersonic band gap in two-dimensional single crystalline phononic structures,” Appl. Phys. Lett. 91(12), 121915 (pages 3) (2007).
[CrossRef]

2006 (8)

P. Velha, J. C. Rodier, P. Lalanne, J. D. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultracompact silicon-on-insulator ridge-waveguide mirrors with high reflectance,” Appl. Phys. Lett. 89, 171121 (2006).
[CrossRef]

M. Maldovan and E. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B 83(4), 595–600 (2006).
[CrossRef]

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88(25), 251907 (pages 3) (2006).
[CrossRef]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J. M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-Sensitivity Optical Monitoring of a Micromechanical Resonator with a Quantum-Limited Optomechanical Sensor,” Phys. Rev. Lett. 97, 133601 (2006).
[CrossRef] [PubMed]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef] [PubMed]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–73 (2006).
[CrossRef] [PubMed]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

2005 (4)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials 4, 207–210 (2005).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef]

D. Bindel and S. Govindjee, “Elastic PMLs for resonator anchor loss simulation,” Tech report UCB/SEMM-2005/01. Submitted to IJNME 64(6), 789–818 (2005).
[CrossRef]

C. Sauvan, P. Lalanne, and J. P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B 71, 165118 (2005).
[CrossRef]

2003 (3)

O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like equation for photon states of locally perturbed photonic crystals,” Phys. Rev. B 68, 035,214 (2003).

A. Duwel, J. Gorman, M. Weinstein, J. Borenstein, and P. Ward, “Experimental study of thermoelastic damping in MEMS gyros,” Sensors Act. A: Physical 103(1–2), 70–75 (2003).
[CrossRef]

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, and P. A. Deymier, “Transmission and dispersion relations of perfect and defect-containing waveguide structures in phononic band gap materials,” Phys. Rev. B 68(2), 024,302 (2003).

2002 (6)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[CrossRef]

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002).

V. Braginsky and S. P. Vyachanin, “Low quantum noise tranquilizer for Fabry Perot interferometer,” Phys. Lett. A 293(5–6), 228–234 (2002).
[CrossRef]

M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of Acoustical Vibrations in a Semiconductor Planar Phonon Cavity,” Phys. Rev. Lett. 89(22), 227,402 (2002).

B. H. Houston, D. M. Photiadis, M. H. Marcus, J. A. Bucaro, X. Liu, and J. F. Vignola, “Thermoelastic loss in microscale oscillators,” Appl. Phys. Lett. 80(7), 1300–1302 (2002).
[CrossRef]

W. Fon, K. C. Schwab, J. M. Worlock, and M. L. Roukes, “Phonon scattering mechanisms in suspended nanostructures from 4 to 40 K,” Phys. Rev. B 66(4) (2002).

2001 (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett. A 287(5–6), 331–338 (2001).
[CrossRef]

2000 (1)

R. Lifshitz and M. L. Roukes, “Thermoelastic damping in micro- and nanomechanical systems,” Phys. Rev. B 61(8) (2000).

1999 (3)

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: Towards observation of quantum limits,” Phys. Rev. A 59(2), 1038–1044 (1999).
[CrossRef]

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1824 (1999).
[CrossRef] [PubMed]

1998 (3)

J. V. Sánchez-Pérez, D. Caballero, R. Mártinez-Sala, C. Rubio, J. Sánchez-Dehesa, F. Meseguer, J. Llinares, and F. Gálvez, “Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders,” Phys. Rev. Lett. 80(24), 5325–5328 (1998).
[CrossRef]

W. M. Robertson and J. F. R. III, “Measurement of acoustic stop bands in two-dimensional periodic scattering arrays,” J. Acoustical Soc. Am. 104(2), 694–699 (1998).
[CrossRef]

F. R. Montero de Espinosa, E. Jiménez, and M. Torres, “Ultrasonic Band Gap in a Periodic Two-Dimensional Composite,” Phys. Rev. Lett. 80(6), 1208–1211 (1998).
[CrossRef]

1997 (1)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-Bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

1993 (1)

M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. 71(13), 2022–2025 (1993).
[CrossRef]

1985 (1)

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B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials 4, 207–210 (2005).
[CrossRef]

Sorel, M.

A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one-dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 084–089 (2008).

Srinivasan, K.

O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like equation for photon states of locally perturbed photonic crystals,” Phys. Rev. B 68, 035,214 (2003).

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002).

Steinmeyer, G.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-Bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Steinsland, E.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: Towards observation of quantum limits,” Phys. Rev. A 59(2), 1038–1044 (1999).
[CrossRef]

Strigin, S. E.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett. A 287(5–6), 331–338 (2001).
[CrossRef]

Su, M. F.

R. H. O. III, I. F. El-Kady, M. F. Su, M. R. Tuck, and J. G. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sensors Act. A: Physical 145–146, 87–93 (2008).

Sun, C.-K.

Y.-C. Wen, L.-C. Chou, H.-H. Lin, V. Gusev, K.-H. Lin, and C.-K. Sun, “Efficient generation of coherent acoustic phonons in (111) InGaAs/GaAs multiple quantum wells through piezoelectric effects,” Appl. Phys. Lett. 90(17), 172102 (pages 3) (2007).
[CrossRef]

Taniyama, H.

M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q Nanocavity with 1D Photonic Gap,” Opt. Express 16(15), 905–102 (2008).

Thierry-Mieg, V.

M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of Acoustical Vibrations in a Semiconductor Planar Phonon Cavity,” Phys. Rev. Lett. 89(22), 227,402 (2002).

Thoen, E. R.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-Bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Thomas, E.

M. Maldovan and E. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B 83(4), 595–600 (2006).
[CrossRef]

Thomas, E. L.

T. Gorishnyy, J.-H. Jang, C. Koh, and E. L. Thomas, “Direct observation of a hypersonic band gap in two-dimensional single crystalline phononic structures,” Appl. Phys. Lett. 91(12), 121915 (pages 3) (2007).
[CrossRef]

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88(25), 251907 (pages 3) (2006).
[CrossRef]

Thorne, K. S.

V. B. Braginskiĭ, F. Y. Khalili, and K. S. Thorne, Quantum measurement (Cambridge University Press, Cambridge, 1992).
[CrossRef]

Tittonen, I.

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: Towards observation of quantum limits,” Phys. Rev. A 59(2), 1038–1044 (1999).
[CrossRef]

Tomes, M.

M. Tomes and T. Carmon, “Photonic Micro-Electromechanical Systems Vibrating at X-band (11-GHz) Rates,” Phys. Rev. Lett. 102(11), 113601 (pages 4) (2009).
[CrossRef]

Torres, M.

F. R. Montero de Espinosa, E. Jiménez, and M. Torres, “Ultrasonic Band Gap in a Periodic Two-Dimensional Composite,” Phys. Rev. Lett. 80(6), 1208–1211 (1998).
[CrossRef]

Trigo, M.

M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of Acoustical Vibrations in a Semiconductor Planar Phonon Cavity,” Phys. Rev. Lett. 89(22), 227,402 (2002).

Tuck, M. R.

R. H. O. III, I. F. El-Kady, M. F. Su, M. R. Tuck, and J. G. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sensors Act. A: Physical 145–146, 87–93 (2008).

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef]

K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A (Atomic, Molecular, and Optical Physics) 78(2), 023832 (pages 4) (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics,” Opt. Express 15(25), 172–205 (2007).

T. Carmon and K. J. Vahala, “Optomechanical Modal Spectroscopy of Optoexcited Vibrations of a Micron-Scale on-Chip Sphere at Greater than 1 GHz,” Phys. Rev. Lett. 98(123901) (2007).
[CrossRef] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef]

M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical Crystals,” Nature, DOI:10.1038/nature08524 (2009).
[PubMed]

Vasseur, J. O.

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, and P. A. Deymier, “Transmission and dispersion relations of perfect and defect-containing waveguide structures in phononic band gap materials,” Phys. Rev. B 68(2), 024,302 (2003).

Velha, P.

P. Velha, J. C. Rodier, P. Lalanne, J. D. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultracompact silicon-on-insulator ridge-waveguide mirrors with high reflectance,” Appl. Phys. Lett. 89, 171121 (2006).
[CrossRef]

Vignes, E.

P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2(11), 1830–1840 (1985).
[CrossRef]

A. Dorsel, J. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical Bistability and Mirror Confinement Induced by Radiation Pressure,” Phys. Rev. Lett. 51(17), 1550–1553 (1983).
[CrossRef]

Vignola, J. F.

B. H. Houston, D. M. Photiadis, M. H. Marcus, J. A. Bucaro, X. Liu, and J. F. Vignola, “Thermoelastic loss in microscale oscillators,” Appl. Phys. Lett. 80(7), 1300–1302 (2002).
[CrossRef]

Villeneuve, P. R.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-Bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Vyachanin, S. P.

V. Braginsky and S. P. Vyachanin, “Low quantum noise tranquilizer for Fabry Perot interferometer,” Phys. Lett. A 293(5–6), 228–234 (2002).
[CrossRef]

Vyatchanin, S. P.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett. A 287(5–6), 331–338 (2001).
[CrossRef]

Walther, H.

A. Dorsel, J. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical Bistability and Mirror Confinement Induced by Radiation Pressure,” Phys. Rev. Lett. 51(17), 1550–1553 (1983).
[CrossRef]

Ward, P.

A. Duwel, J. Gorman, M. Weinstein, J. Borenstein, and P. Ward, “Experimental study of thermoelastic damping in MEMS gyros,” Sensors Act. A: Physical 103(1–2), 70–75 (2003).
[CrossRef]

Weinstein, M.

A. Duwel, J. Gorman, M. Weinstein, J. Borenstein, and P. Ward, “Experimental study of thermoelastic damping in MEMS gyros,” Sensors Act. A: Physical 103(1–2), 70–75 (2003).
[CrossRef]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[CrossRef]

Wen, Y.-C.

Y.-C. Wen, L.-C. Chou, H.-H. Lin, V. Gusev, K.-H. Lin, and C.-K. Sun, “Efficient generation of coherent acoustic phonons in (111) InGaAs/GaAs multiple quantum wells through piezoelectric effects,” Appl. Phys. Lett. 90(17), 172102 (pages 3) (2007).
[CrossRef]

Worlock, J. M.

W. Fon, K. C. Schwab, J. M. Worlock, and M. L. Roukes, “Phonon scattering mechanisms in suspended nanostructures from 4 to 40 K,” Phys. Rev. B 66(4) (2002).

Wright, E. M.

P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2(11), 1830–1840 (1985).
[CrossRef]

Yariv, A.

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1824 (1999).
[CrossRef] [PubMed]

Yi, Y.

Y. Yi, “Geometric effects on thermoelastic damping in MEMS resonators,” J. Sound Vibration 309(3–5), 588–599 (2008).
[CrossRef]

Zeilinger, A.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef] [PubMed]

Appl. Phys. B (1)

M. Maldovan and E. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B 83(4), 595–600 (2006).
[CrossRef]

Appl. Phys. Lett. (7)

P. Velha, J. C. Rodier, P. Lalanne, J. D. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultracompact silicon-on-insulator ridge-waveguide mirrors with high reflectance,” Appl. Phys. Lett. 89, 171121 (2006).
[CrossRef]

T. Gorishnyy, J.-H. Jang, C. Koh, and E. L. Thomas, “Direct observation of a hypersonic band gap in two-dimensional single crystalline phononic structures,” Appl. Phys. Lett. 91(12), 121915 (pages 3) (2007).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lončar, “Coupled photonic crystal nanobeam cavities,” Appl. Phys. Lett. 95(3), 031,102–3 (2009).

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (pages 3) (2009).
[CrossRef]

B. H. Houston, D. M. Photiadis, M. H. Marcus, J. A. Bucaro, X. Liu, and J. F. Vignola, “Thermoelastic loss in microscale oscillators,” Appl. Phys. Lett. 80(7), 1300–1302 (2002).
[CrossRef]

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88(25), 251907 (pages 3) (2006).
[CrossRef]

Y.-C. Wen, L.-C. Chou, H.-H. Lin, V. Gusev, K.-H. Lin, and C.-K. Sun, “Efficient generation of coherent acoustic phonons in (111) InGaAs/GaAs multiple quantum wells through piezoelectric effects,” Appl. Phys. Lett. 90(17), 172102 (pages 3) (2007).
[CrossRef]

Eur. Phys. J. D (1)

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

J. Acoustical Soc. Am. (1)

W. M. Robertson and J. F. R. III, “Measurement of acoustic stop bands in two-dimensional periodic scattering arrays,” J. Acoustical Soc. Am. 104(2), 694–699 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2(11), 1830–1840 (1985).
[CrossRef]

J. Sound Vibration (1)

Y. Yi, “Geometric effects on thermoelastic damping in MEMS resonators,” J. Sound Vibration 309(3–5), 588–599 (2008).
[CrossRef]

Measurement Scie. Technol. (1)

R. H. O. III and I. El-Kady, “Microfabricated phononic crystal devices and applications,” Measurement Scie. Technol. 20(1), 012,002 (13pp) (2009).

Nat Phys (1)

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat Phys 5(4), 276–280 (2009).
[CrossRef]

Nature (5)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-Bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[CrossRef] [PubMed]

O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–73 (2006).
[CrossRef] [PubMed]

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature 444, 75–78 (2006).
[CrossRef] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[CrossRef]

Nature Materials (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials 4, 207–210 (2005).
[CrossRef]

Nature Physics (1)

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nature Physics 3(4), 201–205 (2009).

Opt. Express (6)

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics,” Opt. Express 15(25), 172–205 (2007).

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009).
[CrossRef]

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10(15), 670–684 (2002).

A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one-dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 084–089 (2008).

M.W. McCutcheon and M. Lončar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express 16(23), 136–145 (2008).

M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q Nanocavity with 1D Photonic Gap,” Opt. Express 16(15), 905–102 (2008).

Phys. Lett. A (2)

V. Braginsky and S. P. Vyachanin, “Low quantum noise tranquilizer for Fabry Perot interferometer,” Phys. Lett. A 293(5–6), 228–234 (2002).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett. A 287(5–6), 331–338 (2001).
[CrossRef]

Phys. Rev. (1)

T. A. Read, “The Internal Friction of Single Metal Crystals,” Phys. Rev. 58(4) (1940).

Phys. Rev. A (1)

I. Tittonen, G. Breitenbach, T. Kalkbrenner, T. Müller, R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F. de Rooij, “Interferometric measurements of the position of a macroscopic body: Towards observation of quantum limits,” Phys. Rev. A 59(2), 1038–1044 (1999).
[CrossRef]

Phys. Rev. A (Atomic, Molecular, and Optical Physics) (1)

K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A (Atomic, Molecular, and Optical Physics) 78(2), 023832 (pages 4) (2008).
[CrossRef]

Phys. Rev. B (5)

A. Khelif, B. Djafari-Rouhani, J. O. Vasseur, and P. A. Deymier, “Transmission and dispersion relations of perfect and defect-containing waveguide structures in phononic band gap materials,” Phys. Rev. B 68(2), 024,302 (2003).

R. Lifshitz and M. L. Roukes, “Thermoelastic damping in micro- and nanomechanical systems,” Phys. Rev. B 61(8) (2000).

C. Sauvan, P. Lalanne, and J. P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B 71, 165118 (2005).
[CrossRef]

W. Fon, K. C. Schwab, J. M. Worlock, and M. L. Roukes, “Phonon scattering mechanisms in suspended nanostructures from 4 to 40 K,” Phys. Rev. B 66(4) (2002).

O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like equation for photon states of locally perturbed photonic crystals,” Phys. Rev. B 68, 035,214 (2003).

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[CrossRef]

Phys. Rev. Lett. (11)

A. Dorsel, J. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical Bistability and Mirror Confinement Induced by Radiation Pressure,” Phys. Rev. Lett. 51(17), 1550–1553 (1983).
[CrossRef]

J. V. Sánchez-Pérez, D. Caballero, R. Mártinez-Sala, C. Rubio, J. Sánchez-Dehesa, F. Meseguer, J. Llinares, and F. Gálvez, “Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders,” Phys. Rev. Lett. 80(24), 5325–5328 (1998).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef]

N. D. Lanzillotti-Kimura, A. Fainstein, A. Huynh, B. Perrin, B. Jusserand, A. Miard, and A. Lemaitre, “Coherent Generation of Acoustic Phonons in an Optical Microcavity,” Phys. Rev. Lett. 99(21), 217405 (pages 4) (2007).
[CrossRef]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

T. Carmon and K. J. Vahala, “Optomechanical Modal Spectroscopy of Optoexcited Vibrations of a Micron-Scale on-Chip Sphere at Greater than 1 GHz,” Phys. Rev. Lett. 98(123901) (2007).
[CrossRef] [PubMed]

M. Tomes and T. Carmon, “Photonic Micro-Electromechanical Systems Vibrating at X-band (11-GHz) Rates,” Phys. Rev. Lett. 102(11), 113601 (pages 4) (2009).
[CrossRef]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J. M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “High-Sensitivity Optical Monitoring of a Micromechanical Resonator with a Quantum-Limited Optomechanical Sensor,” Phys. Rev. Lett. 97, 133601 (2006).
[CrossRef] [PubMed]

M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of Acoustical Vibrations in a Semiconductor Planar Phonon Cavity,” Phys. Rev. Lett. 89(22), 227,402 (2002).

M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. 71(13), 2022–2025 (1993).
[CrossRef]

F. R. Montero de Espinosa, E. Jiménez, and M. Torres, “Ultrasonic Band Gap in a Periodic Two-Dimensional Composite,” Phys. Rev. Lett. 80(6), 1208–1211 (1998).
[CrossRef]

Phys. Zeit. Sowjet. (1)

L. Landau and G. Rumer, “On the absorption of sound in solids,” Phys. Zeit. Sowjet. 11(18) (1937).

Science (2)

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1824 (1999).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321(5893), 1172–1176 (2008).
[CrossRef]

Sensors Act. A: Physical (2)

R. H. O. III, I. F. El-Kady, M. F. Su, M. R. Tuck, and J. G. Fleming, “Microfabricated VHF acoustic crystals and waveguides,” Sensors Act. A: Physical 145–146, 87–93 (2008).

A. Duwel, J. Gorman, M. Weinstein, J. Borenstein, and P. Ward, “Experimental study of thermoelastic damping in MEMS gyros,” Sensors Act. A: Physical 103(1–2), 70–75 (2003).
[CrossRef]

Tech report UCB/SEMM-2005/01. Submitted to IJNME (1)

D. Bindel and S. Govindjee, “Elastic PMLs for resonator anchor loss simulation,” Tech report UCB/SEMM-2005/01. Submitted to IJNME 64(6), 789–818 (2005).
[CrossRef]

Other (8)

S. Mohammadi, A. A. Eftekhar, and A. Adibi, “Large Simultaneous Band Gaps for Photonic and Phononic Crystal Slabs,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, p. CFY1 (Optical Society of America, 2008).
[PubMed]

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H. Kolsky, Stress waves in solids (Dover Publications, Inc., 1963).

B. A. Auld, Acoustic Fields in Waves and Solids (Robert E. Krieger Publishing Company, 1973).

MIT Photonic Bands (MPB) is a free software package for the solution of the electromagnetic eigenmodes of periodic structures. MPB has been developed at MIT, http://ab-initio.mit.edu/wiki/index.php/MPB.

COMSOL is a multiphysics software package for performing finite-element-method (FEM) simulations. See COMSOL AB, http://www.comsol.com/. We use the COMSOL multiphysics software package to perform both optical and mechanical numerical simulations of the optomechanical crystal systems.

M. Eichenfield, J. Chan, R. Camacho, K. J. Vahala, and O. Painter, “Optomechanical Crystals,” Nature, DOI:10.1038/nature08524 (2009).
[PubMed]

V. B. Braginskiĭ, F. Y. Khalili, and K. S. Thorne, Quantum measurement (Cambridge University Press, Cambridge, 1992).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

(a) General geometry of the periodic nanobeam structure’s projection (infinite structure, no defect). (b) Optical band diagram of the nanobeam’s projection. The band from which all localized optical modes will be derived is shown in dark black, with Ey of the optical mode at the X point shown to the right of the diagram. The harmonic spatial potential created by the defect, along with the first three optical modes are shown as emanating from the X-point band-edge. (c) Mechanical band diagram of the nanobeam’s projection. The three bands that can form optomechanically coupled mechanical defect modes are colored. The bottom-most mode is from the X point of the red band; the Γ points of the green and blue bands correspond to the middle and top mechanical modes, respectively. The frequencies of the defect modes that form from the band edges are shown as short, horizontal bars.

Fig. 2.
Fig. 2.

Mechanical band diagram and corresponding normalized displacement profiles of the unit cell at the Γ (kx =0) and X (kx =π/Λ) points. In the band diagram, the mirror symmetry σ z , (across the plane defined by z=0) is indicated by color: red corresponds to even vector parity (pz =1) and blue to odd vector parity (pz =-1). Mirror symmetry σ y (across the plane defined by y=0) plane is indicated by the line shape: solid corresponds to even vector parity (py =1) and dashed to odd vector parity (py =-1). The mechanical mode profiles are all viewed from a direction normal to the z=0 plane unless labeled “yz”, in which case the viewing angle is normal to the x=0 plane. The pinch, accordion, and breathing mode bands are b, i, and j, respectively. As torsional modes can be difficult to interpret without isometric views, it is noted for the reader that the mechanical modes for band e at X, band f at Γ, and band h at X are all torsional mechanical modes.

Fig. 3.
Fig. 3.

(a) Schematic illustration of actual nanobeam optomechanical crystal with defect and clamps at substrate. (b) Localized optical modes of the nanobeam OMC. The colors of the names correspond to the illustration of the inverted potential in Fig. 1(b). Localized, optomechanically-coupled mechanical modes of the nanobeam OMC. The colors of the names correspond to the colored bands and horizontal bars showing the modal frequencies in Fig. 1(c).

Fig. 4.
Fig. 4.

(a) In-phase and (b) in-quadrature mechanical displacement field (log10(|q|2/max(|q|2)) of the fundamental breathing mode of nanobeam OMC structure with weakly absorbing “pad”, showing the propagating nature of the radiated mechanical waves in the pad region. (c) Dependence of Q m on the total length of the structure; the number of mirror holes on each side is (NT -15)/2. This shows the oscillatory Q m of the pinch and breathing modes, which are coupled to waveguide modes, and the exponentially-increasing Q m of the accordion mode. (d) Mechanical band structure of the nanobeam OMC, with arrow tails indicating the frequency and high-symmetry point of the breathing (blue) and pinch (red) modes, and arrow heads indicating the equi-frequency waveguide mode that acts as the dominant source of parasitic coupling. The effective bandgap of the accordion mode is shown in transparent green, with its frequency indicated as a horizontal green bar at the Γ point.

Fig. 5.
Fig. 5.

For the fundamental breathing mode and the fundamental optical mode in the nominal structure, (a) FEM simulation of individual unit cell contributions to the total optomechanical coupling (each point computed by integrating ζOM (Equation 8) over the respective unit cell), (b) surface plot of the optomechanical coupling density, ζOM. (c) surface plot of the normal displacement profile, Θm (Equation 10), (d) surface plot of the electromagnetic energy functional, Θo (Equation 11). In (d), there is significant optomechanical coupling density in the corner of the holes, where the crossbar meets the rail. Without the fillets, the field amplitude is concentrated in the corner and difficult to see. For this reason, the corners have been filleted to allow the optomechanical coupling density in the corners to be visualized. The fillets do not significantly affecting the optomechanical coupling (confirmed by simulation).

Fig. 6.
Fig. 6.

For the fundamental breathing mode and the fundamental optical mode, (a) the dependence of the optomechanical coupling on the rail thickness (with oscillations in the data arising from accidental degeneracies with the cantilever modes), (b) the optical and mechanical mode profiles for rail thicknesses of 100 nm, 190 nm and 400 nm circled in red, green and blue respectively in (a), (c) comparison of the mechanical mode profiles when coupled (orange) and not coupled (purple) to cantilever modes, with the corresponding effect on in L OM highlighted in (a).

Fig. 7.
Fig. 7.

For the fundamental pinch mode and the fundamental optical mode in the nominal structure, (a) FEM simulation of individual unit cell contributions to the total optomechanical coupling, (b) surface plot of the optomechanical coupling density, (c) surface plot of the normal displacement profile (Equation 10), (d) surface plot of the electromagnetic energy functional (Equation 11).

Fig. 8.
Fig. 8.

For the accordion mode and the fundamental optical mode in the nominal structure, (a) FEM simulation of individual unit cell contributions to the total optomechanical coupling, (b) surface plot of the optomechanical coupling density, (c) surface plot of the normal displacement profile (Equation 10), (d) surface plot of the electromagnetic energy functional (Equation 11).

Fig. 9.
Fig. 9.

For the accordion mode with the fundamental optical mode, (a), the effective length as a function of total beam width, (b), individual unit cell contributions to the total optomechanical coupling for a structure with a beam width of 700 nm (circled in (a)), mode frequency of 3.97 GHz and effective motional mass of 334 fg, with accompanying mechanical mode plot. The narrower mechanical mode (represented here by the deformation of the structure with color indicating relative strain) envelope results in drastically different optomechanical coupling contributions compared to Fig. 8.

Fig. 10.
Fig. 10.

(a) Fundamental optical mode of the double-heterostructrure OMC (geometry identical to that described in Ref. [35]), with λ0≈1.5 µm, Q rad≈2.7×107, and V eff=1.2 (λ0/n)3. (b) Breathing mechanical mode of the double-heterostructure OMC, with νm=9.3 GHz, and m eff=322 femtograms. (c) Optomechanical coupling integrand plotted on the double-heterostructure OMC system’s surface; the structure has an L OM=1.75 µm for the optical-mechanical mode-pair from 10(a) and 10(b).

Equations (11)

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(c:sQ(r))=p2Q(r)t2 ,
c1 =1E1vv000v1v000vv10000002(1+v)0000002(1+v)0000002(1+v) ,
σy =[100010000] σz = [100010001] ,
σx =[100010001] .
ωo (α) =ωoα=α0+(αα0)dωodαα=α0+12(αα0)2d2ωodα2α=α0+...
ωo (α)=ωoα=α0+(αα0)dωodαα=α0ωo+(αα0)gOMωo+(αα0)ωoLOM ,
dωodα =ωo2dAdhdα[ΔεE2Δ(ε1)D2]dVεE2
1LOM =14 d A (q.n̂)[Δεe2Δ(ε1)d2]
ζOM (r)14(q·n̂)[Δεe2Δ(ε1)d2].
Θm (r)q·n̂
Θo (r)Δεe2Δ(ε1)d2,

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