Abstract

We have studied the effect of geometry deformation on the mechanical frequencies and quality factors for different modes in the Whispering Gallery Mode (WGM) microresonators, that is unavoidable in the practical fabrication. The subsidence of the sphere and a more general condition with fewer symmetries and complex deformation of eccentricity, subsidence, and offset are first modeled in this paper, which could tune the mechanical frequency in a much wider spectral range than the pillar-diameter-induced perturbation. we also show that the mechanical quality factors for the non-whispering-gallery mechanical mode could be increased in the order of 4 magnitudes at a specific subsidence, and form a mechanical bound state in the continuum (BIC) which is induced by the symmetry breaking and reveals new mechanisms to confine radiation. A much broader BIC window width with higher mechanical quality factor could be achieved, which is of great importance in both fundamental research and scientific applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
  2. 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] [PubMed]
  3. H. Rokhsari, T. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005).
    [Crossref] [PubMed]
  4. R. Ma, A. Schliesser, P. Del’haye, A. Dabirian, G. Anetsberger, and T. J. Kippenberg, “Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres,” Opt. Lett. 32(15), 2200–2202 (2007).
    [Crossref] [PubMed]
  5. Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
    [Crossref]
  6. M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009).
    [Crossref] [PubMed]
  7. X. Jiang, M. Wang, M. C. Kuzyk, T. Oo, G.-L. Long, and H. Wang, “Chip-based silica microspheres for cavity optomechanics,” Opt. Express 23(21), 27260–27265 (2015).
    [Crossref] [PubMed]
  8. A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
    [Crossref]
  9. G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
    [Crossref]
  10. X. Jiang, Q. Lin, J. Rosenberg, K. Vahala, and O. Painter, “High-Q double-disk microcavities for cavity optomechanics,” Opt. Express 17(23), 20911–20919 (2009).
    [Crossref] [PubMed]
  11. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
    [Crossref] [PubMed]
  12. V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
    [Crossref] [PubMed]
  13. V. Fiore, C. Dong, M. C. Kuzyk, and H. Wang, “Optomechanical light storage in a silica microresonator,” Phys. Rev. A 87(2), 023812 (2013).
    [Crossref]
  14. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
    [Crossref] [PubMed]
  15. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013).
    [Crossref]
  16. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012).
    [Crossref] [PubMed]
  17. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
    [Crossref]
  18. L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. 527(1-2), 1–14 (2015).
    [Crossref]
  19. X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).
  20. Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).
  21. X.-F. Liu, F. Lei, M. Gao, X. Yang, G.-Q. Qin, and G.-L. Long, “Fabrication of a microtoroidal resonator with picometer precise resonant wavelength,” Opt. Lett. 41(15), 3603–3606 (2016).
    [Crossref] [PubMed]
  22. Z.-H. Zhou, C.-L. Zou, Y. Chen, Z. Shen, G.-C. Guo, and C.-H. Dong, “Broadband tuning of the optical and mechanical modes in hollow bottle-like microresonators,” Opt. Express 25(4), 4046–4053 (2017).
    [Crossref] [PubMed]
  23. H. Lamb, “On the vibrations of an elastic sphere,” Proc. Lond. Math. Soc. 13, 189–212 (1882).
  24. N. Nishiguchi and T. Sakuma, “Vibrational spectrum and specific heat of fine particles,” Solid State Commun. 38(11), 1073–1077 (1981).
    [Crossref]
  25. M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
    [Crossref] [PubMed]
  26. A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
    [Crossref]
  27. Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
    [Crossref]
  28. K. Williams and R. Muller, “Etch rates for micromachining processing,” J. Microelectromech. Syst. 5(4), 256–269 (1996).
    [Crossref]
  29. M. Ahumada, P. A. Orellana, and J. C. Retamal, “Bound states in the continuum in whispering gallery resonators,” Phys. Rev. A (Coll. Park) 98(2), 023827 (2018).
    [Crossref]
  30. F. H. Stillinger and D. R. Herrick, “Bound states in the continuum,” Phys. Rev. A 11(2), 446–454 (1975).
    [Crossref]
  31. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
    [Crossref]
  32. X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
    [Crossref] [PubMed]
  33. Y.-S. Park and H. Wang, “Radiation pressure driven mechanical oscillation in deformed silica microspheres via free-space evanescent excitation,” Opt. Express 15(25), 16471–16477 (2007).
    [Crossref] [PubMed]

2018 (2)

Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).

M. Ahumada, P. A. Orellana, and J. C. Retamal, “Bound states in the continuum in whispering gallery resonators,” Phys. Rev. A (Coll. Park) 98(2), 023827 (2018).
[Crossref]

2017 (2)

Z.-H. Zhou, C.-L. Zou, Y. Chen, Z. Shen, G.-C. Guo, and C.-H. Dong, “Broadband tuning of the optical and mechanical modes in hollow bottle-like microresonators,” Opt. Express 25(4), 4046–4053 (2017).
[Crossref] [PubMed]

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).

2016 (5)

X.-F. Liu, F. Lei, M. Gao, X. Yang, G.-Q. Qin, and G.-L. Long, “Fabrication of a microtoroidal resonator with picometer precise resonant wavelength,” Opt. Lett. 41(15), 3603–3606 (2016).
[Crossref] [PubMed]

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

2015 (2)

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. 527(1-2), 1–14 (2015).
[Crossref]

X. Jiang, M. Wang, M. C. Kuzyk, T. Oo, G.-L. Long, and H. Wang, “Chip-based silica microspheres for cavity optomechanics,” Opt. Express 23(21), 27260–27265 (2015).
[Crossref] [PubMed]

2013 (2)

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013).
[Crossref]

V. Fiore, C. Dong, M. C. Kuzyk, and H. Wang, “Optomechanical light storage in a silica microresonator,” Phys. Rev. A 87(2), 023812 (2013).
[Crossref]

2012 (2)

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref] [PubMed]

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012).
[Crossref] [PubMed]

2011 (1)

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

2010 (1)

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[Crossref]

2009 (4)

X. Jiang, Q. Lin, J. Rosenberg, K. Vahala, and O. Painter, “High-Q double-disk microcavities for cavity optomechanics,” Opt. Express 17(23), 20911–20919 (2009).
[Crossref] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
[Crossref]

M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009).
[Crossref] [PubMed]

2008 (2)

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
[Crossref]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
[Crossref]

2007 (2)

2005 (3)

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[Crossref] [PubMed]

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] [PubMed]

H. Rokhsari, T. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005).
[Crossref] [PubMed]

2003 (1)

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
[Crossref] [PubMed]

1996 (1)

K. Williams and R. Muller, “Etch rates for micromachining processing,” J. Microelectromech. Syst. 5(4), 256–269 (1996).
[Crossref]

1981 (1)

N. Nishiguchi and T. Sakuma, “Vibrational spectrum and specific heat of fine particles,” Solid State Commun. 38(11), 1073–1077 (1981).
[Crossref]

1975 (1)

F. H. Stillinger and D. R. Herrick, “Bound states in the continuum,” Phys. Rev. A 11(2), 446–454 (1975).
[Crossref]

1882 (1)

H. Lamb, “On the vibrations of an elastic sphere,” Proc. Lond. Math. Soc. 13, 189–212 (1882).

Ahumada, M.

M. Ahumada, P. A. Orellana, and J. C. Retamal, “Bound states in the continuum in whispering gallery resonators,” Phys. Rev. A (Coll. Park) 98(2), 023827 (2018).
[Crossref]

Anetsberger, G.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
[Crossref]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
[Crossref]

R. Ma, A. Schliesser, P. Del’haye, A. Dabirian, G. Anetsberger, and T. J. Kippenberg, “Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres,” Opt. Lett. 32(15), 2200–2202 (2007).
[Crossref] [PubMed]

Arcizet, O.

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
[Crossref]

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
[Crossref]

Barbour, R.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

Cao, C.

Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).

Carmon, T.

M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009).
[Crossref] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[Crossref] [PubMed]

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] [PubMed]

H. Rokhsari, T. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005).
[Crossref] [PubMed]

Chen, H.

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Chen, Y.

Z.-H. Zhou, C.-L. Zou, Y. Chen, Z. Shen, G.-C. Guo, and C.-H. Dong, “Broadband tuning of the optical and mechanical modes in hollow bottle-like microresonators,” Opt. Express 25(4), 4046–4053 (2017).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

Dabirian, A.

Del’haye, P.

Dong, C.

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013).
[Crossref]

V. Fiore, C. Dong, M. C. Kuzyk, and H. Wang, “Optomechanical light storage in a silica microresonator,” Phys. Rev. A 87(2), 023812 (2013).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref] [PubMed]

Dong, C.-H.

Z.-H. Zhou, C.-L. Zou, Y. Chen, Z. Shen, G.-C. Guo, and C.-H. Dong, “Broadband tuning of the optical and mechanical modes in hollow bottle-like microresonators,” Opt. Express 25(4), 4046–4053 (2017).
[Crossref] [PubMed]

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Fiore, V.

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013).
[Crossref]

V. Fiore, C. Dong, M. C. Kuzyk, and H. Wang, “Optomechanical light storage in a silica microresonator,” Phys. Rev. A 87(2), 023812 (2013).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref] [PubMed]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

Gao, M.

Gao, X.

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

Gao, Y.-P.

Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Guo, G.-C.

Z.-H. Zhou, C.-L. Zou, Y. Chen, Z. Shen, G.-C. Guo, and C.-H. Dong, “Broadband tuning of the optical and mechanical modes in hollow bottle-like microresonators,” Opt. Express 25(4), 4046–4053 (2017).
[Crossref] [PubMed]

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Hafezi, M.

Herrick, D. R.

F. H. Stillinger and D. R. Herrick, “Bound states in the continuum,” Phys. Rev. A 11(2), 446–454 (1975).
[Crossref]

Hsu, C. W.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

Jiang, X.

Joannopoulos, J. D.

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Kippenberg, T.

Kippenberg, T. J.

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[Crossref]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
[Crossref]

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
[Crossref]

R. Ma, A. Schliesser, P. Del’haye, A. Dabirian, G. Anetsberger, and T. J. Kippenberg, “Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres,” Opt. Lett. 32(15), 2200–2202 (2007).
[Crossref] [PubMed]

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] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[Crossref] [PubMed]

Kuok, M. H.

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
[Crossref] [PubMed]

Kuzyk, M. C.

X. Jiang, M. Wang, M. C. Kuzyk, T. Oo, G.-L. Long, and H. Wang, “Chip-based silica microspheres for cavity optomechanics,” Opt. Express 23(21), 27260–27265 (2015).
[Crossref] [PubMed]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013).
[Crossref]

V. Fiore, C. Dong, M. C. Kuzyk, and H. Wang, “Optomechanical light storage in a silica microresonator,” Phys. Rev. A 87(2), 023812 (2013).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref] [PubMed]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

Lamb, H.

H. Lamb, “On the vibrations of an elastic sphere,” Proc. Lond. Math. Soc. 13, 189–212 (1882).

Lei, F.

Lim, H. S.

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
[Crossref] [PubMed]

Lin, Q.

Lin, X.

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

Lipson, M.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Liu, N. N.

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
[Crossref] [PubMed]

Liu, X.-F.

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).

X.-F. Liu, F. Lei, M. Gao, X. Yang, G.-Q. Qin, and G.-L. Long, “Fabrication of a microtoroidal resonator with picometer precise resonant wavelength,” Opt. Lett. 41(15), 3603–3606 (2016).
[Crossref] [PubMed]

Long, G.-L.

Ma, R.

Muller, R.

K. Williams and R. Muller, “Etch rates for micromachining processing,” J. Microelectromech. Syst. 5(4), 256–269 (1996).
[Crossref]

Ng, S. C.

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
[Crossref] [PubMed]

Nishiguchi, N.

N. Nishiguchi and T. Sakuma, “Vibrational spectrum and specific heat of fine particles,” Solid State Commun. 38(11), 1073–1077 (1981).
[Crossref]

Oo, T.

Orellana, P. A.

M. Ahumada, P. A. Orellana, and J. C. Retamal, “Bound states in the continuum in whispering gallery resonators,” Phys. Rev. A (Coll. Park) 98(2), 023827 (2018).
[Crossref]

Painter, O.

Park, Y.-S.

Qin, G.-Q.

Rabl, P.

Retamal, J. C.

M. Ahumada, P. A. Orellana, and J. C. Retamal, “Bound states in the continuum in whispering gallery resonators,” Phys. Rev. A (Coll. Park) 98(2), 023827 (2018).
[Crossref]

Rivière, R.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
[Crossref]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
[Crossref]

Rokhsari, H.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[Crossref] [PubMed]

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] [PubMed]

H. Rokhsari, T. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005).
[Crossref] [PubMed]

Rosenberg, J.

Sakuma, T.

N. Nishiguchi and T. Sakuma, “Vibrational spectrum and specific heat of fine particles,” Solid State Commun. 38(11), 1073–1077 (1981).
[Crossref]

Scherer, A.

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] [PubMed]

Schliesser, A.

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[Crossref]

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
[Crossref]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
[Crossref]

R. Ma, A. Schliesser, P. Del’haye, A. Dabirian, G. Anetsberger, and T. J. Kippenberg, “Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres,” Opt. Lett. 32(15), 2200–2202 (2007).
[Crossref] [PubMed]

Shen, Z.

Z.-H. Zhou, C.-L. Zou, Y. Chen, Z. Shen, G.-C. Guo, and C.-H. Dong, “Broadband tuning of the optical and mechanical modes in hollow bottle-like microresonators,” Opt. Express 25(4), 4046–4053 (2017).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

Soljacic, M.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

Stillinger, F. H.

F. H. Stillinger and D. R. Herrick, “Bound states in the continuum,” Phys. Rev. A 11(2), 446–454 (1975).
[Crossref]

Stone, A. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Sun, F.-W.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Tian, L.

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. 527(1-2), 1–14 (2015).
[Crossref]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

Tomes, M.

M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009).
[Crossref] [PubMed]

Vahala, K.

Vahala, K. J.

H. Rokhsari, T. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005).
[Crossref] [PubMed]

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] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[Crossref] [PubMed]

Wang, C.

Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).

Wang, H.

X. Jiang, M. Wang, M. C. Kuzyk, T. Oo, G.-L. Long, and H. Wang, “Chip-based silica microspheres for cavity optomechanics,” Opt. Express 23(21), 27260–27265 (2015).
[Crossref] [PubMed]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013).
[Crossref]

V. Fiore, C. Dong, M. C. Kuzyk, and H. Wang, “Optomechanical light storage in a silica microresonator,” Phys. Rev. A 87(2), 023812 (2013).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref] [PubMed]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
[Crossref]

Y.-S. Park and H. Wang, “Radiation pressure driven mechanical oscillation in deformed silica microspheres via free-space evanescent excitation,” Opt. Express 15(25), 16471–16477 (2007).
[Crossref] [PubMed]

Wang, M.

Wang, T.-J.

Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).

Wang, Z. K.

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
[Crossref] [PubMed]

Wiederhecker, G. S.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Williams, K.

K. Williams and R. Muller, “Etch rates for micromachining processing,” J. Microelectromech. Syst. 5(4), 256–269 (1996).
[Crossref]

Xiao, Y.-F.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Xiong, X.

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

Yang, L.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[Crossref] [PubMed]

Yang, X.

Yang, Y.

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

Zhang, Y.

Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).

Zhang, Y.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Zhen, B.

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Zhou, Z.-H.

Zou, C.-L.

Z.-H. Zhou, C.-L. Zou, Y. Chen, Z. Shen, G.-C. Guo, and C.-H. Dong, “Broadband tuning of the optical and mechanical modes in hollow bottle-like microresonators,” Opt. Express 25(4), 4046–4053 (2017).
[Crossref] [PubMed]

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Zou, X.-B.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

Adv. At. Mol. Opt. Phys. (1)

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[Crossref]

Ann. Phys. (1)

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. 527(1-2), 1–14 (2015).
[Crossref]

J. Microelectromech. Syst. (1)

K. Williams and R. Muller, “Etch rates for micromachining processing,” J. Microelectromech. Syst. 5(4), 256–269 (1996).
[Crossref]

Nat. Photonics (2)

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
[Crossref]

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2(10), 627–633 (2008).
[Crossref]

Nat. Phys. (2)

Y.-S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5(7), 489–493 (2009).
[Crossref]

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
[Crossref]

Nat. Rev. Mater. (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Nature (1)

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

New J. Phys. (1)

Y. Chen, Z. Shen, X. Xiong, C.-H. Dong, C.-L. Zou, and G.-C. Guo, “Mechanical bound state in the continuum for optomechanical microresonators,” New J. Phys. 18(6), 063031 (2016).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. A (3)

F. H. Stillinger and D. R. Herrick, “Bound states in the continuum,” Phys. Rev. A 11(2), 446–454 (1975).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Transient optomechanically induced transparency in a silica microsphere,” Phys. Rev. A 87(5), 055802 (2013).
[Crossref]

V. Fiore, C. Dong, M. C. Kuzyk, and H. Wang, “Optomechanical light storage in a silica microresonator,” Phys. Rev. A 87(2), 023812 (2013).
[Crossref]

Phys. Rev. A (Coll. Park) (3)

M. Ahumada, P. A. Orellana, and J. C. Retamal, “Bound states in the continuum in whispering gallery resonators,” Phys. Rev. A (Coll. Park) 98(2), 023827 (2018).
[Crossref]

X.-F. Liu, T.-J. Wang, Y.-P. Gao, C. Cao, and C. Wang, “Chiral microresonator assisted by Rydberg-atom ensembles,” Phys. Rev. A (Coll. Park) 98, 023826 (2017).

Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A (Coll. Park) 96, 033824 (2018).

Phys. Rev. Lett. (5)

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, “Brillouin study of the quantization of acoustic modes in nanospheres,” Phys. Rev. Lett. 90(25 Pt 1), 255502 (2003).
[Crossref] [PubMed]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107(13), 133601 (2011).
[Crossref] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[Crossref] [PubMed]

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] [PubMed]

M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009).
[Crossref] [PubMed]

Proc. Lond. Math. Soc. (1)

H. Lamb, “On the vibrations of an elastic sphere,” Proc. Lond. Math. Soc. 13, 189–212 (1882).

Sci. Rep. (1)

X. Gao, C. W. Hsu, B. Zhen, X. Lin, J. D. Joannopoulos, M. Soljačić, and H. Chen, “Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs,” Sci. Rep. 6(1), 31908 (2016).
[Crossref] [PubMed]

Science (1)

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref] [PubMed]

Solid State Commun. (1)

N. Nishiguchi and T. Sakuma, “Vibrational spectrum and specific heat of fine particles,” Solid State Commun. 38(11), 1073–1077 (1981).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Finite element simulation of mechanical vibration modes of free silica microspheres without supporting structure. (a) Eigenfrequencies of free silica microsphere for the spheroidal modes with n = 1, l = 0, 1, 2, 3, 4, 5. (b) Displacement patterns of the (n, l, m) modes with n = 1, 2, l = 0, 1, 2, 3, 4, 5 and m = 0. Here, n = 1 corresponds to the surface modes and n ≥ 2 to the inner modes. The displacement of a slice through the equator (original sphere outlined in black) and the magnitude (color coded) are shown.
Fig. 2
Fig. 2 Chip-based silica microsphere model.
Fig. 3
Fig. 3 Perturbation of the relative pillar height h on mechanical frequencies (a)(c) and quality factors (b)(d) for RBMs and non-RBMs, respectively. The relative pillar diameter d is fixed to 0.1 with a sphere diameter of 40 μm. The maximum varying range of the mechanical frequency is around 3 MHz for RBMs, while 100 kHz for non-RBMs. The mechanical quality factor of the RBMs (m = 0) is relatively low compared to the non-RBMs, about one order of magnitude lower compared with m = 1 and eight orders of magnitude lower for the acoustic whispering gallery modes with m ≥ 2.
Fig. 4
Fig. 4 Pillar-diameter-induced perturbation on mechanical frequencies (a)(c) and quality factors (b)(d) for RBMs and non-RBMs, respectively. The relative pillar height h is fixed to 1.5 with a sphere diameter of 40 μm. The maximum varying range of the mechanical frequency is around 6 MHz for RBMs, less than 3 MHz for non-RBMs with m = 1, and hundred kHz for acoustic whispering gallery modes with m ≥ 2. As for mode (1,4,1), the changing range is 2.81 MHz, which is around 1.9% of the mode frequency. The mechanical quality factor decreases in three orders of magnitude for RBMs and oscillates in a long-diameter-period for m ≥ 2.
Fig. 5
Fig. 5 Cross sectional drawings of the (a) spheroid, (b) subsided spheroid and (c) off-centered subsided spheroid in the symmetry plane. The bottom inset is a zoomed-in 3D model.
Fig. 6
Fig. 6 (a) Eigenfrequencies of a free spheroid. (b) Frequency intersection for adjacent modes. (c) Sphere-eccentricity-induced perturbation on mechanical frequency for a chip-based spheroid. The mechanical frequency of RBMs (m = 0) is monotonically increasing and the frequency changing rate decreases with increased m, reaching the minimum when |m| = l. (d) Sphere-eccentricity-induced perturbation on mechanical quality factor. The relative pillar diameter d is fixed to 0.1 with an expected sphere diameter of 40 μm.
Fig. 7
Fig. 7 (a)(b) and (c)(d) are the spheroid-subsidence-induced perturbation on mechanical frequencies and quality factors for different modes, respectively. The relative pillar diameter d and relative eccentricity ε are both fixed to 0.1 with an expected sphere diameter D of 40 μm.
Fig. 8
Fig. 8 Displacement distribution of (a) mode (1,3,1) and (b) mode (1,3,0). The relative pillar diameter d and relative eccentricity ε are both fixed to 0.1 with an expected sphere diameter D of 40 μm, and the relative subsidence is 0.11 for mechanical BIC in (1,3,1) mode.
Fig. 9
Fig. 9 Sphere-offset-induced perturbation on mechanical frequencies (a)(c) and quality factors (b)(d) for RBMs and non-RBMs, respectively. The relative pillar diameter d is fixed to 0.1 with a sphere diameter of 40 μm.
Fig. 10
Fig. 10 (a) Frequency difference δν of nondegenerate azimuthal modes with the same |m|. (b) Mechanical quality factor of nondegenerate mechanical BIC modes. Compared with the chip-based microsphere in Fig. 4(d), the off-centered subsided spheroid has a much broader BIC window width, about 11 and 4 times broader for the (1,2,1) and (1,4,1) mode, respectively. The centered subsided spheroid has a similar window width as the off-centered one, about 8 and 5 times broader than the sphere, but much higher maximum mechanical quality factors could be achieved at the BIC point.

Equations (2)

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ρ u ¨ =(λ+2μ)(u)μ×(×u),
ϕ k (r,t)= l,m A k (l,m) j l ( 2π v n,l,m r V k ) Y l m (θ,ψ) e 2πi ν n,l,m t ,

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