Abstract

We theoretically study the frequency stability of an opto-mechanical oscillator based on resonant interaction of one mechanical, and two optical modes of the same optical microcavity. A generalized expression for the phase noise of the oscillator is derived using Langevin formalism and compared to the phase noise of existing electronic oscillators.

© 2012 OSA

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  1. T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005).
    [CrossRef] [PubMed]
  2. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006).
    [CrossRef]
  3. H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. Vahala, “Brownian noise in radiation-pressure-driven micromechnical oscillators,” Appl. Phys. Lett. 89, 261109 (2006).
    [CrossRef]
  4. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
    [CrossRef] [PubMed]
  5. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale” Science 321, 1172–1176 (2008).
    [CrossRef] [PubMed]
  6. M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).
  7. M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010).
    [CrossRef]
  8. J. Zehnpfennig, G. Bahl, M. Tomes, and T. Carmon, “Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres,” Opt. Express 19, 14240–14248 (2011).
    [CrossRef] [PubMed]
  9. K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A 78, 023832 (2008).
    [CrossRef]
  10. D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54, 329–330 (1966).
    [CrossRef]
  11. S. Tallur, S. Sridaran, S. A. Bhave, and T. Carmon, “Phase noise modeling of opto-mechanical oscillators,” Proc. of 2010 IEEE Int. Freq. Control Symp. (2010), Vol.  1, pp. 268–272.
    [CrossRef]
  12. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
    [CrossRef]
  13. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
    [CrossRef]
  14. I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
    [CrossRef] [PubMed]
  15. M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102, 113601 (2009).
    [CrossRef] [PubMed]
  16. J. Li, H. Lee, T. Chen, O. Painter, and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic systems,” arXiv.org>physics>arXiv:1201.4212v1 (2012).
  17. S. P. Smith, F. Zarinetchi, and S. Ezekiel, “Narrow linewidth stimulated Brillouin fiber laser and applications,” Opt. Lett. 16, 393–395 (1991).
    [CrossRef] [PubMed]
  18. A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: Theoretical analysis,” Phys. Rev. A 62, 023803 (2000).
    [CrossRef]
  19. A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B 18, 556–567 (2001).
    [CrossRef]
  20. J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
    [CrossRef]
  21. G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Characterization of surface acoustic wave optomechanical oscillators,” Proc. 2011 IEEE International Frequency Control Symposium (FCS) (2011), Vol. 1, pp. 1–4.
    [CrossRef]
  22. G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
    [CrossRef] [PubMed]
  23. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface acoustic wave opto-mechanical oscillator and frequency comb generator,” Opt. Lett. 36, 3338–3340 (2011).
    [CrossRef] [PubMed]
  24. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
    [CrossRef]
  25. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface-acoustic wave opto-mechanical oscillator,” Proc. 2010 IEEE International Frequency Control Symposium (FCS) (2010), Vol. 1, pp. 183–188.
    [CrossRef]

2012 (1)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
[CrossRef]

2011 (3)

2010 (2)

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010).
[CrossRef]

S. Tallur, S. Sridaran, S. A. Bhave, and T. Carmon, “Phase noise modeling of opto-mechanical oscillators,” Proc. of 2010 IEEE Int. Freq. Control Symp. (2010), Vol.  1, pp. 268–272.
[CrossRef]

2009 (3)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
[CrossRef]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef] [PubMed]

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

2008 (2)

K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A 78, 023832 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (3)

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006).
[CrossRef]

H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. Vahala, “Brownian noise in radiation-pressure-driven micromechnical oscillators,” Appl. Phys. Lett. 89, 261109 (2006).
[CrossRef]

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

2005 (1)

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

2002 (1)

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

2001 (1)

2000 (1)

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: Theoretical analysis,” Phys. Rev. A 62, 023803 (2000).
[CrossRef]

1991 (1)

1966 (1)

D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54, 329–330 (1966).
[CrossRef]

1911 (1)

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).

Bahl, G.

J. Zehnpfennig, G. Bahl, M. Tomes, and T. Carmon, “Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres,” Opt. Express 19, 14240–14248 (2011).
[CrossRef] [PubMed]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[CrossRef] [PubMed]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Characterization of surface acoustic wave optomechanical oscillators,” Proc. 2011 IEEE International Frequency Control Symposium (FCS) (2011), Vol. 1, pp. 1–4.
[CrossRef]

Bhave, S. A.

S. Tallur, S. Sridaran, S. A. Bhave, and T. Carmon, “Phase noise modeling of opto-mechanical oscillators,” Proc. of 2010 IEEE Int. Freq. Control Symp. (2010), Vol.  1, pp. 268–272.
[CrossRef]

Blake, M.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

Carmon, T.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[CrossRef] [PubMed]

J. Zehnpfennig, G. Bahl, M. Tomes, and T. Carmon, “Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres,” Opt. Express 19, 14240–14248 (2011).
[CrossRef] [PubMed]

S. Tallur, S. Sridaran, S. A. Bhave, and T. Carmon, “Phase noise modeling of opto-mechanical oscillators,” Proc. of 2010 IEEE Int. Freq. Control Symp. (2010), Vol.  1, pp. 268–272.
[CrossRef]

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

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

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Characterization of surface acoustic wave optomechanical oscillators,” Proc. 2011 IEEE International Frequency Control Symposium (FCS) (2011), Vol. 1, pp. 1–4.
[CrossRef]

Chen, T.

J. Li, H. Lee, T. Chen, O. Painter, and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic systems,” arXiv.org>physics>arXiv:1201.4212v1 (2012).

Debut, A.

A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B 18, 556–567 (2001).
[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: Theoretical analysis,” Phys. Rev. A 62, 023803 (2000).
[CrossRef]

Ezekiel, S.

Geng, J.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

Grudinin, I. S.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef] [PubMed]

Hajimiri, A.

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006).
[CrossRef]

H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. Vahala, “Brownian noise in radiation-pressure-driven micromechnical oscillators,” Appl. Phys. Lett. 89, 261109 (2006).
[CrossRef]

Hossein-Zadeh, M.

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010).
[CrossRef]

H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. Vahala, “Brownian noise in radiation-pressure-driven micromechnical oscillators,” Appl. Phys. Lett. 89, 261109 (2006).
[CrossRef]

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006).
[CrossRef]

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface acoustic wave opto-mechanical oscillator and frequency comb generator,” Opt. Lett. 36, 3338–3340 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface-acoustic wave opto-mechanical oscillator,” Proc. 2010 IEEE International Frequency Control Symposium (FCS) (2010), Vol. 1, pp. 183–188.
[CrossRef]

Jiang, S.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

Kippenberg, T.

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

Kippenberg, T. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[CrossRef] [PubMed]

Lee, H.

J. Li, H. Lee, T. Chen, O. Painter, and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic systems,” arXiv.org>physics>arXiv:1201.4212v1 (2012).

Leeson, D. B.

D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54, 329–330 (1966).
[CrossRef]

Li, J.

J. Li, H. Lee, T. Chen, O. Painter, and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic systems,” arXiv.org>physics>arXiv:1201.4212v1 (2012).

Maleki, L.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface acoustic wave opto-mechanical oscillator and frequency comb generator,” Opt. Lett. 36, 3338–3340 (2011).
[CrossRef] [PubMed]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface-acoustic wave opto-mechanical oscillator,” Proc. 2010 IEEE International Frequency Control Symposium (FCS) (2010), Vol. 1, pp. 183–188.
[CrossRef]

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface acoustic wave opto-mechanical oscillator and frequency comb generator,” Opt. Lett. 36, 3338–3340 (2011).
[CrossRef] [PubMed]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface-acoustic wave opto-mechanical oscillator,” Proc. 2010 IEEE International Frequency Control Symposium (FCS) (2010), Vol. 1, pp. 183–188.
[CrossRef]

Painter, O.

J. Li, H. Lee, T. Chen, O. Painter, and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic systems,” arXiv.org>physics>arXiv:1201.4212v1 (2012).

Randoux, S.

A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B 18, 556–567 (2001).
[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: Theoretical analysis,” Phys. Rev. A 62, 023803 (2000).
[CrossRef]

Rokhsari, H.

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006).
[CrossRef]

H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. Vahala, “Brownian noise in radiation-pressure-driven micromechnical oscillators,” Appl. Phys. Lett. 89, 261109 (2006).
[CrossRef]

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

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface acoustic wave opto-mechanical oscillator and frequency comb generator,” Opt. Lett. 36, 3338–3340 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface-acoustic wave opto-mechanical oscillator,” Proc. 2010 IEEE International Frequency Control Symposium (FCS) (2010), Vol. 1, pp. 183–188.
[CrossRef]

Seidel, D.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface acoustic wave opto-mechanical oscillator and frequency comb generator,” Opt. Lett. 36, 3338–3340 (2011).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface-acoustic wave opto-mechanical oscillator,” Proc. 2010 IEEE International Frequency Control Symposium (FCS) (2010), Vol. 1, pp. 183–188.
[CrossRef]

Seidel, D. J.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
[CrossRef]

Smith, S. P.

Sridaran, S.

S. Tallur, S. Sridaran, S. A. Bhave, and T. Carmon, “Phase noise modeling of opto-mechanical oscillators,” Proc. of 2010 IEEE Int. Freq. Control Symp. (2010), Vol.  1, pp. 268–272.
[CrossRef]

Staines, S.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

Tallur, S.

S. Tallur, S. Sridaran, S. A. Bhave, and T. Carmon, “Phase noise modeling of opto-mechanical oscillators,” Proc. of 2010 IEEE Int. Freq. Control Symp. (2010), Vol.  1, pp. 268–272.
[CrossRef]

Tomes, M.

J. Zehnpfennig, G. Bahl, M. Tomes, and T. Carmon, “Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres,” Opt. Express 19, 14240–14248 (2011).
[CrossRef] [PubMed]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[CrossRef] [PubMed]

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

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Characterization of surface acoustic wave optomechanical oscillators,” Proc. 2011 IEEE International Frequency Control Symposium (FCS) (2011), Vol. 1, pp. 1–4.
[CrossRef]

Vahala, K.

H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. Vahala, “Brownian noise in radiation-pressure-driven micromechnical oscillators,” Appl. Phys. Lett. 89, 261109 (2006).
[CrossRef]

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

J. Li, H. Lee, T. Chen, O. Painter, and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic systems,” arXiv.org>physics>arXiv:1201.4212v1 (2012).

Vahala, K. J.

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010).
[CrossRef]

K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A 78, 023832 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[CrossRef] [PubMed]

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006).
[CrossRef]

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).

Wang, Z.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

Yang, L.

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

Zarinetchi, F.

Zehnpfennig, J.

J. Zehnpfennig, G. Bahl, M. Tomes, and T. Carmon, “Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres,” Opt. Express 19, 14240–14248 (2011).
[CrossRef] [PubMed]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[CrossRef] [PubMed]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Characterization of surface acoustic wave optomechanical oscillators,” Proc. 2011 IEEE International Frequency Control Symposium (FCS) (2011), Vol. 1, pp. 1–4.
[CrossRef]

Zemmouri, J.

A. Debut, S. Randoux, and J. Zemmouri, “Experimental and theoretical study of linewidth narrowing in Brillouin fiber ring lasers,” J. Opt. Soc. Am. B 18, 556–567 (2001).
[CrossRef]

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: Theoretical analysis,” Phys. Rev. A 62, 023803 (2000).
[CrossRef]

Zong, J.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. Vahala, “Brownian noise in radiation-pressure-driven micromechnical oscillators,” Appl. Phys. Lett. 89, 261109 (2006).
[CrossRef]

M. Hossein-Zadeh and K. J. Vahala, “Observation of injection locking in an optomechanical RF oscillator,” Appl. Phys. Lett. 93, 191115 (2008).

IEEE J. Sel. Top. Quantum Electron. (1)

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[CrossRef]

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

Nat. Commun. (1)

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (4)

A. Debut, S. Randoux, and J. Zemmouri, “Linewidth narrowing in Brillouin lasers: Theoretical analysis,” Phys. Rev. A 62, 023803 (2000).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A 78, 023832 (2008).
[CrossRef]

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

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

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103, 257403 (2009).
[CrossRef]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef] [PubMed]

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

Proc. IEEE (1)

D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54, 329–330 (1966).
[CrossRef]

Proc. of 2010 IEEE Int. Freq. Control Symp. (1)

S. Tallur, S. Sridaran, S. A. Bhave, and T. Carmon, “Phase noise modeling of opto-mechanical oscillators,” Proc. of 2010 IEEE Int. Freq. Control Symp. (2010), Vol.  1, pp. 268–272.
[CrossRef]

Proc. SPIE (1)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. J. Seidel, and L. Maleki, “Surface acoustic wave frequency comb,” Proc. SPIE 8236, 82361M (2012).
[CrossRef]

Science (1)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale” Science 321, 1172–1176 (2008).
[CrossRef] [PubMed]

Other (3)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Surface-acoustic wave opto-mechanical oscillator,” Proc. 2010 IEEE International Frequency Control Symposium (FCS) (2010), Vol. 1, pp. 183–188.
[CrossRef]

J. Li, H. Lee, T. Chen, O. Painter, and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic systems,” arXiv.org>physics>arXiv:1201.4212v1 (2012).

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Characterization of surface acoustic wave optomechanical oscillators,” Proc. 2011 IEEE International Frequency Control Symposium (FCS) (2011), Vol. 1, pp. 1–4.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

(a) Phase noise of the 100 MHz opto-mechanical oscillator (1) versus offset frequency f = ω/2π characterized with parameters defined in the text. Curve (2) defines the contribution from the phase noise of the pumping laser. Curve (3) stands for the thermal and white shot noise defined in Eq. (67). Curve (4) shows phase diffusion of the opto-mechanical oscillator. Curve (5) stands for phase noise of a commercially available 100 MHz ovenized quartz oscillator. (b) Phase noise of the 1 GHz opto-mechanical oscillator (1) characterized with parameters defined in the text. Curve (2) defines the contribution from the phase noise of the pumping laser. Curve (3) stands for the thermal and white shot noise defined in Eq. (67). Curve (4) shows phase diffusion of the opto-mechanical oscillator. Curve (5) stands for phase noise of a commercially available 100 MHz ovenized quartz oscillator corrected by 20 dB to reflect the frequency difference.

Equations (69)

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A ˙ = Γ A A i g C B + F A ,
B ˙ = Γ B B i g C A + F B ,
C ˙ = Γ C C i g B A + F C .
Γ A = i ( ω a ω 0 ) + γ + γ c a , Γ B = i ( ω b ω ) + γ + γ c b , Γ C = i ( ω c ω M ) + γ M ,
g = ω 0 K ε h ¯ 2 m * L 2 ω c ,
F A = F A + F c A + F r A , F A = e i ϕ F A 2 P γ c a h ¯ ω 0 ,
F c A ( t ) F c A ( t ) = 2 γ c a δ ( t t ) ,
F r A ( t ) F r A ( t ) = 2 γ δ ( t t ) ,
F B = F B + F c B + F r B , F B = 0 ,
F c B ( t ) F c B ( t ) = 2 γ c b δ ( t t ) ,
F r B ( t ) F r B ( t ) = 2 γ δ ( t t ) ,
F C = 0 ,
F C ( t ) F C ( t ) = 2 γ M ( n ¯ t h + 1 ) δ ( t t )
A = | A | e i ϕ A ,
B = | B | e i ϕ B ,
C = | C | e i ϕ C ,
| A ˙ | = ( γ + γ c a ) | A | g | C | | B | sin ϕ + | F A | cos ( ϕ F A ϕ A ) + F A r ,
| B ˙ | = ( γ + γ c b ) | B | + g | C | | A | sin ϕ + F B r ,
| C ˙ | = γ M | C | + g | B | | A | sin ϕ + F C r ;
ϕ ˙ A = ( ω a ω 0 ) g | C | | B | | A | cos ϕ + | F A | | A | sin ( ϕ F A ϕ A ) + F A i | A | ,
ϕ ˙ B = ( ω b ω ) g | C | | A | | B | cos ϕ + F B i | B | ,
ϕ ˙ C = ( ω c ω M ) g | B | | A | | C | cos ϕ + F C i | C | ;
ϕ = ϕ A ϕ B ϕ C ,
F A r = 1 2 ( F c A e i ϕ A + F c A e i ϕ A ) + 1 2 ( F r A e i ϕ A + F r A e i ϕ A ) ,
F B r = 1 2 ( F c B e i ϕ B + F c B e i ϕ B ) + 1 2 ( F r B e i ϕ B + F r B e i ϕ B ) ,
F C r = 1 2 ( F C e i ϕ C + F C e i ϕ C ) ,
F A i = 1 2 i ( F c A e i ϕ A F c A e i ϕ A ) + 1 2 i ( F r A e i ϕ A F r A e i ϕ A ) ,
F B i = 1 2 i ( F c B e i ϕ B F c B e i ϕ B ) + 1 2 i ( F r B e i ϕ B F r B e i ϕ B ) ,
F C i = 1 2 i ( F C e i ϕ C F C e i ϕ C ) .
ϕ ˙ = ( ω a ω b ω c ) + g cos ϕ ( | B | | A | | C | + | C | | A | | B | | C | | B | | A | ) + | F A | | A | sin ( ϕ F A ϕ A ) + F A i | A | F B i | B | F C i | C | ,
ψ ˙ = ( ω a ω b ω c ) g sin ψ ( | B | | A | | C | + | C | | A | | B | | C | | B | | A | ) + | F A | | A | sin ( ϕ F A ϕ A ) + F A i | A | F B i | B | F C i | C | .
( ω a ω b ω c ) + g sin ψ ( | B | | A | | C | + | C | | A | | B | | C | | B | | A | ) = | F A | | A | sin ( ϕ F A ϕ A ) ,
δ ϕ ˙ + g δ ϕ cos ψ ( | B | | A | | C | + | C | | A | | B | | C | | B | | A | ) = δ [ g sin ψ ( | B | | A | | C | + | C | | A | | B | | C | | B | | A | ) + | F A | | A | sin ( ϕ F A ϕ A ) ] + F A i | A | F B i | B | F C i | C | ,
| B | 2 | C | 2 = γ M γ + γ c b ,
| A | 2 = γ M γ + γ c b | Γ B | 2 g 2
ω b ω ω c ω M = γ + γ c b γ M ,
e i ψ = e i ϕ Γ B .
| B | | A | | C | + | C | | A | | B | | C | | B | | A | = g | Γ B | | B | 2 + ( γ M γ + γ c b + γ + γ c b γ M ) γ M γ + γ c b | Γ B | g ,
g ( | B | | A | | C | + | C | | A | | B | | C | | B | | A | ) γ M + γ + γ c b .
δ [ g sin ψ ( | B | | A | | C | + | C | | A | | B | | C | | B | | A | ) + | F A | | A | sin ( ϕ F A ϕ A ) ] ( δ ϕ F A δ ϕ A ) | F A | | A | ,
δ ϕ ˙ + ( γ M + γ + γ c b ) δ ϕ = ( δ ϕ F A δ ϕ A ) | F A | | A | + F A i | A | F B i | B | F C i | C | ,
δ ϕ ˙ A = ( δ ϕ F A δ ϕ A ) | F A | | A | + F A i | A | .
| F A | | A | = γ + γ c a .
δ ϕ ˙ A + ( γ + γ c a ) δ ϕ A = F A i | A | ,
δ ϕ ˙ B + δ ϕ ˙ C + ( γ M + γ + γ c b ) ( δ ϕ B + δ ϕ C ) = ( γ M + γ + γ c b ) δ ϕ A + F B i | B | + F C i | C | .
δ ϕ ˙ B γ M γ + γ c b δ ϕ ˙ C γ + γ c b γ M = γ M γ + γ c b F B i | B | γ + γ c b γ M F C i | C | .
F B i = f B i ( ω ) e i ω t d ω 2 π ,
F C i = f C i ( ω ) e i ω t d ω 2 π ,
F B i ( t ) F B i ( t ) = 1 2 ( γ + γ c b ) δ ( t t ) ,
F C i ( t ) F C i ( t ) = γ M ( n ¯ t h + 1 2 ) δ ( t t ) ,
f B i ( ω ) f B i ( ω ) = π ( γ + γ c b ) δ ( ω + ω ) ,
f C i ( ω ) f C i ( ω ) = 2 π γ M ( n ¯ t h + 1 2 ) δ ( ω + ω ) .
δ ϕ A ( ω ) = f A i | A | 1 i ω + γ + γ c a ,
δ ϕ A ( ω ) δ ϕ B ( ω ) = i ω + γ M i ω + γ M + γ + γ c b δ ϕ A ( ω ) i ω + γ M i ω ( i ω + γ M + γ + γ c b ) f B i | B | + γ + γ c b i ω ( i ω + γ M + γ + γ c b ) f C i | C | ,
δ ϕ C ( ω ) = γ M i ω + γ M + γ + γ c b δ ϕ A ( ω ) γ M i ω ( i ω + γ M + γ + γ c b ) f B i | B | + i ω + γ + γ c b i ω ( i ω + γ M + γ + γ c b ) f C i | C | .
δ ϕ C ( t ) δ ϕ C ( t τ ) = c ( ω ) e i ω t d ω 2 π .
δ ϕ C ( t ) = δ ϕ C ( ω ) e i ω t d ω 2 π ,
c = γ M 2 ω 2 + ( γ M + γ + γ c b ) 2 γ c a 2 ω 2 + ( γ + γ c a ) 2 i n + γ M 2 ω 2 [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ + γ c b 2 | B | 2 + ω 2 + ( γ + γ c b ) 2 ω 2 [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ M | C | 2 ( n ¯ t h + 1 2 ) ,
a b = ω 2 + γ M 2 ω 2 + ( γ M + γ + γ c b ) 2 γ c a 2 ω 2 + ( γ + γ c a ) 2 in + ω 2 + γ M 2 ω 2 [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ + γ c b 2 | B | 2 + ( γ + γ c b ) 2 ω 2 [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ M | C | 2 ( n ¯ t h + 1 2 ) .
b = ( γ + γ c b ) 2 ω 2 + ( γ M + γ + γ c b ) 2 γ c a 2 ω 2 + ( γ + γ c a ) 2 in + ω 2 + γ M 2 ω 2 [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ + γ c b 2 | B | 2 + ( γ + γ c b ) 2 ω 2 [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ M | C | 2 ( n ¯ t h + 1 2 ) .
| A | 2 = 2 γ c a ( γ + γ c a ) 2 P h ¯ ω 0 , | B | 2 = P Bout 2 γ c b h ¯ ω 0 ,
a b = ω 2 + γ M 2 ω 2 + ( γ M + γ + γ c b ) 2 γ c a 2 ω 2 + ( γ + γ c a ) 2 in + ω 2 + 2 γ M 2 ( n ¯ t h + 1 ) [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ c b ( γ + γ c b ) ω 2 h ¯ ω 0 P Bout ,
c = γ M 2 ω 2 + ( γ M + γ + γ c b ) 2 γ c a 2 ω 2 + ( γ + γ c a ) 2 i n + γ M 2 { n ¯ t h + 1 + ω 2 / [ 2 ( γ + γ c b ) 2 ] } [ ω 2 + ( γ M + γ + γ c b ) 2 ] 2 γ c b ( γ + γ c b ) ω 2 h ¯ ω 0 P Bout ,
b = ( γ + γ c b ) 2 ω 2 + ( γ M + γ + γ c b ) 2 γ c a 2 ω 2 + ( γ + γ c a ) 2 i n + ω 2 + 2 γ M 2 ( n ¯ t h + 1 ) [ ω 2 + ( γ M + γ + γ c b ) 2 ] γ c b ( γ + γ c b ) ω 2 h ¯ ω 0 P Bout .
σ 2 ( τ ) = 0 4 ω 2 ω 0 2 sin 4 ( ω τ / 2 ) ( ω τ / 2 ) 2 d ω 2 π .
Δ ν a b | γ M γ γ c a 2 ( γ + γ c a ) 2 Δ ν pump + γ c b ( γ + γ c b ) ( n ¯ t h + 1 ) h ¯ ω 0 π P Bout ,
Δ ν a b | γ M γ γ M 2 ( γ + γ c b ) 2 γ c a 2 ( γ + γ c a ) 2 Δ ν pump + γ c b γ + γ c b γ M 2 ( n ¯ t h + 1 ) h ¯ ω 0 π P Bout .
P D = 2 q R ρ P D + k B T P R F
ω 0 ω = ω M = γ + γ c b γ + γ c b + γ M ω c + γ M γ + γ c b + γ M ( ω 0 ω b ) .

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