Abstract

Bound states in the continuum (BICs), an emerging type of long-lived resonances different from the cavity-based ones, have been explored in several classical systems, including photonic crystals and surface acoustic waves. Here, we reveal symmetry-protected mechanical BICs in the structure of slab-on-substrate optomechanical crystals. Using a group theory approach, we identified all the mechanical BICs at the Γ point in optomechanical crystals with C4v and C6v symmetries as examples, and analyzed their coupling with the co-localized optical BICs and guided resonances due to both moving boundary and photo-elastic effects. We verified the theoretical analysis with numerical simulations of specific optomechanical crystals which support substantial optomechanical interactions between the mechanical BICs and optical resonances. Due to the unique features of high-Q, large-size mechanical BICs and substrate-enabled thermal dissipation, this architecture of slab-on-substrate optomechanical crystals might be useful for exploring macroscopic quantum mechanical physics and enabling new applications such as high-throughput sensing and free-space beam steering.

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

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References

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  3. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 16048 (2016).
    [Crossref]
  4. J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).
  5. J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  9. S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
    [Crossref]
  10. A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).
  11. T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
    [Crossref]
  12. A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
    [Crossref]
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    [Crossref] [PubMed]
  14. A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (2012).
    [Crossref]
  15. K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
    [Crossref]
  16. R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
    [Crossref] [PubMed]
  17. J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
    [Crossref] [PubMed]
  18. S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).
  19. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).
  20. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
    [Crossref]
  21. 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]
  22. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
    [Crossref]
  23. S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
    [Crossref]

2018 (4)

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

2017 (1)

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

2016 (2)

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

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

2015 (2)

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

2014 (1)

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

2013 (1)

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

2012 (3)

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photons. 6, 768 (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]

2009 (1)

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

2006 (1)

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

2002 (2)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
[Crossref]

2001 (1)

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

1969 (1)

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

1929 (1)

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Aspelmeyer, M.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

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

Azzam, S. I.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Bahari, B.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Becker, T.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Blasius, T. D.

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

Boltasseva, A.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Camacho, R. M.

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

Cerjan, A.

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

Chan, J.

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]

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

Chua, S.-L.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Cohen, J. D.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Davydov, S. Y.

S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
[Crossref]

Eichenfield, M.

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

Englert, B.-G.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Every, A.

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

Fainman, Y.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Fan, S.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

Fang, K.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

Farnell, G.

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

Gröblacher, S.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

Gu, Q.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Hill, J. T.

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]

Hong, S.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

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, 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

Inui, T.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

Joannopoulos, J. D.

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

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

Johnson, S. G.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

Kanté, B.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Kildishev, A. V.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Kippenberg, T. J.

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

Kodigala, A.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Krause, A. G.

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

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Lepetit, T.

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

Lim, T.

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

Lin, Q.

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

Löschnauer, C.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Luan, X.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

MacCabe, G. S.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

Marinkovic, I.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Marquardt, F.

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

Marsili, F.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Matheny, M. H.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

Maznev, A.

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

Meenehan, S.

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]

Meenehan, S. M.

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

Minkov, M.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

Ochiai, T.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

Painter, O.

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[Crossref]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

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]

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

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

Qiu, W.

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Rechtsman, M. C.

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

Riedinger, R.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

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]

Sakoda, K.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Shalaev, V. M.

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Shapira, O.

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Shaw, M. D.

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

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, 16048 (2016).
[Crossref]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

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, 16048 (2016).
[Crossref]

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

Vahala, K. J.

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

Varcoe, B. T.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

von Neumann, J.

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Wallucks, A.

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

Walther, H.

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Wigner, E.

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Williamson, I. A.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

Winger, M.

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

Xiao, M.

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

Zhen, B.

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

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

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]

J. Acoust. Soc. Am. (1)

T. Lim and G. Farnell, “Character of pseudo surface waves on anisotropic crystals,” J. Acoust. Soc. Am. 45, 845–851 (1969).
[Crossref]

Nat. Photons. (2)

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

K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photons. 10, 489 (2016).
[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, 16048 (2016).
[Crossref]

Nature (5)

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499, 188 (2013).
[Crossref] [PubMed]

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541, 196 (2017).
[Crossref] [PubMed]

R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556, 473 (2018).
[Crossref] [PubMed]

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature 520, 522 (2015).
[Crossref] [PubMed]

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

Phys. Rev. B (3)

A. Maznev and A. Every, “Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures,” Phys. Rev. B 97, 014108 (2018).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Phys. Rev. Lett. (3)

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012).
[Crossref] [PubMed]

M. Minkov, I. A. Williamson, M. Xiao, and S. Fan, “Zero-index bound states in the continuum,” Phys. Rev. Lett. 121, 263901 (2018).
[Crossref]

S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, “Formation of bound states in the continuum in hybrid plasmonic-photonic systems,” Phys. Rev. Lett. 121, 253901 (2018).
[Crossref]

Phys. Rev. X (1)

S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili, M. D. Shaw, and O. Painter, “Pulsed excitation dynamics of an optomechanical crystal resonator near its quantum ground state of motion,” Phys. Rev. X 5, 041002 (2015).

Phys. Z (1)

J. von Neumann and E. Wigner, “Über merkwürdige diskrete eigenwerte,” Phys. Z 30, 465–467 (1929).

Rep. Prog. Phys. (1)

H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325 (2006).
[Crossref]

Rev. Mod. Phys. (1)

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

Semiconductors (1)

S. Y. Davydov, “Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN,” Semiconductors 36, 41–44 (2002).
[Crossref]

Other (2)

A. Cerjan, C. W. Hsu, and M. C. Rechtsman, “Bound states in the continuum through environmental design,” arXiv e-prints arXiv:1901.07126 (2019).

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and its applications in physics(Springer Science & Business Media, 2012).

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Figures (6)

Fig. 1
Fig. 1 Schematic of the structure of a slab-on-substrate optomechancial crystal.
Fig. 2
Fig. 2 (a) Top view of the unit cell of an optomechanical crystal with C4v symmetry. a = 1   μm. Red and white areas are AlN and air, respectively. (b) The phononic bandstructure. The red dots indicate three mechanical BICs at the Γ point and the green dot indicates a pair of degenerate mechanical guided resonances. The gray shaded region indicates the region below substrate sound line ( ω = c T k ). (c) The photonic bandstructure. The red dots indicate two optical BICs and the green dot indicates a pair of degenerate optical guided resonances. The gray shaded region indicates the region below substrate light line ( ω = c n k ).
Fig. 3
Fig. 3 Mechanical and optical modes at the Γ point. (a-c) Total displacement ( | Q |) (left) and z-component of the displacement (Qz) at the interface of the slab and substrate (right) of the three mechanical BICs with frequency 2.53 GHz (a), 2.77 GHz (b) and 2.93 GHz (c). (d-f) | E | 2 (left) and Ez (right) of the two optical BICs with frequency 182 THz (d) and 194 THz (e), and one of the degenerate guided resonance with frequency 190 THz (f).
Fig. 4
Fig. 4 Quality factor (Q) of the five mechanical modes (red and green dots in Fig. 2(b)) as the Bloch wavevector is scanned near the Γ point. The BICs have significantly higher quality factor than the guided resonances.
Fig. 5
Fig. 5 (a) Top view of the unit cell of an optomechanical crystal with C6v symmetry. a = 1   μm. Red and white areas are AlN and air, respectively. (b) The phononic bandstructure. The red dots indicate three mechanical BICs at the Γ point. The gray shaded region indicates the region below substrate sound line ( ω = c T k ). (c) The photonic bandstructure. The red dots indicate two pairs of degenerate optical BICs and two non-degenerate BICs. The green dot indicates a pair of degenerate optical guided resonances. The gray shaded region indicates the region below substrate light line ( ω = c n k ).
Fig. 6
Fig. 6 (a-d) Mechanical modes at Γ point. Total displacement (left) and z component of the displacement at the interface between the slab and substrate (right) of the four mechanical BICs with frequency 2.69 GHz (a), 3.90 GHz (b) and 3.95 GHz (2) (c, d). (e-l) Optical modes at Γ point. | E | 2 (left) and Ez (right) of the six optical BICs with frequency 198 THz (e), 202 THz (2) (f, g), 209 THz (h), 222 THz (2) (i, j) and guided resonances with frequency 226 THz (2) (k, l).

Tables (9)

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Table 1 Character table for the C 4 v point group

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Table 2 Character table for the C 6 v point group

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Table 3 Mechanical and optical BICs at the Γ point

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Table 4 Moving boundary effect g OM , MB, C4v

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Table 5 Moving boundary effect g OM , MB, C 6 v

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Table 6 Photo-elastic effect g OM , PE, C 4 v

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Table 7 Photo-elastic effect g OM , PE, C 6 v

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Table 8 Optomechanical couplings between the optical guided resonance 190 THz (E) and mechanical BICs in a unit cell of the C4v cross-structure optomechanical crystal

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Table 9 Optomechanical couplings between one mechanical BIC (3.95 GHz, E 2) and optical modes in a unit cell of the C 6 v cross-structure optomechanical crystal

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

ρ 2 Q t 2 = Y 2 ( 1 + ν ) 2 Q + Y 2 ( 1 + ν ) ( 1 2 ν ) Q ,
Q k ( r ) e i ω t = u k ( r ) e i ( k ρ ω t ) ,
u k ( r ) = j = 0 f j ( z ) e i G j ρ ,
2 Q T ( L ) t 2 = c T ( L ) 2 2 Q T ( L ) ,
Q k , ( r ) = e i k ρ j = 0 ( A T , j e i k T , z j z + A L , j e i k L , z j z ) e i G j ρ ,
k T ( L ) , z j = ω 2 c T ( L ) 2 | k + G j | 2 ,
Q 0 = Q T , 0 + Q L , 0 = ( u e x + v e y ) e i k T , z 0 z + w e z e i k L , z 0 z .
Q 1 , T = l A T , l e i k T , z l z e i G l ρ ,
1 c 2 2 E t 2 = 1 n 2 2 E ,
E k ( r ) = e i k ρ j = 0 A j e i k z j z + i G j ρ ,
k z j = ω 2 ( c / n ) 2 | k + G j | 2 .
g 0 = g 0 , MB + g 0 , PE 2 m eff ω m ( g OM , MB + g OM , PE ) ,
g OM , MB = ω o 2 ( Q n ^ ) ( Δ ϵ | E | 2 Δ ϵ 1 | D | 2 ) d S E * D d V ,
g OM , PE = ω o 2 ϵ 0 n 4 E i * E j p i j k l S k l d V E * D d V ,
g 0 = g ¯ 0 N 1 N n , m e i k ( n a 1 + m a 2 ) ,
g 0 = g ¯ 0 N .
G = g 0 N n p = g ¯ 0 n p ,
( Q n ^ ) f d S = α α + π ( Q n ^ ) f d S + α α π ( Q n ^ ) f d S   = θ = 0 π ( Q ( α + θ ) + σ α Q ( α θ ) ) n ^ ( α + θ ) f ( α + θ ) d S .
( Q n ^ ) f d S = 0 π ( Q n ^ ) f d S + π 2 π ( Q n ^ ) f d S   = θ = 0 π ( Q ( θ ) + C 2 Q ( θ + π ) ) n ^ ( θ ) f ( θ ) d S .
p i j k l ( θ ) = R i q ( θ ) R j r ( θ ) R k s ( θ ) R l t ( θ ) p q r s t ,
R ( θ ) = ( cos ( θ ) sin ( θ ) 0 sin ( θ ) cos ( θ ) 0 0 0 1 ) ,

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