Abstract

Optomechanical oscillators (OMOs) combine the co-existing high quality factor mechanical and optical resonances in an integrated device to realize low phase noise RF oscillations. While several attempts to model the phase noise in such oscillators have been reported, the close-to-carrier phase noise models in literature do not account for $1/f^3$ (pink noise) and higher order slopes in the phase noise spectra. Here we present a phase noise model, corroborated with an experimental characterization of phase noise of two monolithic integrated silicon OMOs, accounting for contributions of thermomechanical and adsorption-desorption (AD) noise to the phase noise. The model shows good agreement with experimental data and provides further insights into the mechanisms underlying the noise processes contributing to different slopes in the phase noise spectra in OMOs.

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

Full Article  |  PDF Article
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References

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    [Crossref]
  2. Z.-C. Zhang, Y.-P. Wang, Y.-F. Yu, and Z.-M. Zhang, “Quantum squeezing in a modulated optomechanical system,” Opt. Express 26(9), 11915–11927 (2018).
    [Crossref]
  3. X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 46(3), 035502 (2013).
    [Crossref]
  4. H. J. Eerkens, F. M. Buters, M. J. Weaver, B. Pepper, G. Welker, K. Heeck, P. Sonin, S. de Man, and D. Bouwmeester, “Optical side-band cooling of a low frequency optomechanical system,” Opt. Express 23(6), 8014–8020 (2015).
    [Crossref]
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  6. S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express 19(24), 24522–24529 (2011).
    [Crossref]
  7. Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11(4), 1753–1759 (2011).
    [Crossref]
  8. Y. K. Yong and J. R. Vig, “Resonator surface contamination-a cause of frequency fluctuations?” IEEE Trans. UFFC 36(4), 452–458 (1989).
    [Crossref]
  9. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” in Proc. IEEE Int. Freq. Ctrl. Symp. 2006, (2006), pp. 405–408.
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    [Crossref]
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    [Crossref]
  12. S. Tallur and S. A. Bhave, “Electromechanically induced ghz rate optical frequency modulation in silicon,” IEEE Photonics J. 4(5), 1474–1483 (2012).
    [Crossref]
  13. S. Tallur and S. A. Bhave, “Partial gap transduced mems optoacoustic oscillator beyond gigahertz,” J. Microelectromech. Syst. 24(2), 422–430 (2015).
    [Crossref]
  14. D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54(2), 329–330 (1966).
    [Crossref]
  15. I. Langmuir, “The adsorption of gases on plane surfaces of glass mica and platinum.,” J. Am. Chem. Soc. 40(9), 1361–1403 (1918).
    [Crossref]
  16. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, 1986).
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    [Crossref]
  18. J. Kritzenberger, H. C. Gaede, J. S. Shore, A. Pines, and A. T. Bell, “129xe nmr study of tio2 (anatase)-supported v2o5 catalysts,” J. Phys. Chem. 98(40), 10173–10179 (1994).
    [Crossref]
  19. F. Rittner, D. Paschek, and B. Boddenberg, “Simulation studies of the adsorption of xenon on the (110) face of rutile,” Langmuir 11(8), 3097–3102 (1995).
    [Crossref]
  20. S. Tallur and S. A. Bhave, “Non-linear dynamics in opto-mechanical oscillators,” in Proc. IEEE Transducers 2015, (2015), pp. 993–996.
  21. Y. Marcus, “The sizes of molecules-revisited,” J. Phys. Org. Chem. 16(7), 398–408 (2003).
    [Crossref]
  22. T. Raeker, “Physical adsorption: Forces and phenomena (bruch, l.w.; cole, milton w.; zaremba, eugene),” J. Chem. Educ. 75(12), 1557 (1998).
    [Crossref]
  23. P. Thomas, J. Gray, X. Zhu, and C. Fong, “Surface diffusion of xe on nb (1 1 0),” Chem. Phys. Lett. 381(3-4), 376–380 (2003).
    [Crossref]
  24. G. Ambati, J. Bell, and J. Sharp, “In-plane vibrations of annular rings,” J. Sound Vib. 47(3), 415–432 (1976).
    [Crossref]

2018 (1)

2015 (2)

2014 (2)

K. Y. Fong, M. Poot, X. Han, and H. X. Tang, “Phase noise of self-sustained optomechanical oscillators,” Phys. Rev. A 90(2), 023825 (2014).
[Crossref]

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

2013 (1)

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 46(3), 035502 (2013).
[Crossref]

2012 (1)

S. Tallur and S. A. Bhave, “Electromechanically induced ghz rate optical frequency modulation in silicon,” IEEE Photonics J. 4(5), 1474–1483 (2012).
[Crossref]

2011 (2)

S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express 19(24), 24522–24529 (2011).
[Crossref]

Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11(4), 1753–1759 (2011).
[Crossref]

2007 (1)

2003 (2)

Y. Marcus, “The sizes of molecules-revisited,” J. Phys. Org. Chem. 16(7), 398–408 (2003).
[Crossref]

P. Thomas, J. Gray, X. Zhu, and C. Fong, “Surface diffusion of xe on nb (1 1 0),” Chem. Phys. Lett. 381(3-4), 376–380 (2003).
[Crossref]

1998 (1)

T. Raeker, “Physical adsorption: Forces and phenomena (bruch, l.w.; cole, milton w.; zaremba, eugene),” J. Chem. Educ. 75(12), 1557 (1998).
[Crossref]

1995 (1)

F. Rittner, D. Paschek, and B. Boddenberg, “Simulation studies of the adsorption of xenon on the (110) face of rutile,” Langmuir 11(8), 3097–3102 (1995).
[Crossref]

1994 (1)

J. Kritzenberger, H. C. Gaede, J. S. Shore, A. Pines, and A. T. Bell, “129xe nmr study of tio2 (anatase)-supported v2o5 catalysts,” J. Phys. Chem. 98(40), 10173–10179 (1994).
[Crossref]

1989 (1)

Y. K. Yong and J. R. Vig, “Resonator surface contamination-a cause of frequency fluctuations?” IEEE Trans. UFFC 36(4), 452–458 (1989).
[Crossref]

1976 (1)

G. Ambati, J. Bell, and J. Sharp, “In-plane vibrations of annular rings,” J. Sound Vib. 47(3), 415–432 (1976).
[Crossref]

1966 (1)

D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54(2), 329–330 (1966).
[Crossref]

1930 (1)

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55(0), 117–258 (1930).
[Crossref]

1918 (1)

I. Langmuir, “The adsorption of gases on plane surfaces of glass mica and platinum.,” J. Am. Chem. Soc. 40(9), 1361–1403 (1918).
[Crossref]

Ambati, G.

G. Ambati, J. Bell, and J. Sharp, “In-plane vibrations of annular rings,” J. Sound Vib. 47(3), 415–432 (1976).
[Crossref]

Bell, A. T.

J. Kritzenberger, H. C. Gaede, J. S. Shore, A. Pines, and A. T. Bell, “129xe nmr study of tio2 (anatase)-supported v2o5 catalysts,” J. Phys. Chem. 98(40), 10173–10179 (1994).
[Crossref]

Bell, J.

G. Ambati, J. Bell, and J. Sharp, “In-plane vibrations of annular rings,” J. Sound Vib. 47(3), 415–432 (1976).
[Crossref]

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, 1986).

Bhave, S. A.

S. Tallur and S. A. Bhave, “Partial gap transduced mems optoacoustic oscillator beyond gigahertz,” J. Microelectromech. Syst. 24(2), 422–430 (2015).
[Crossref]

S. Tallur and S. A. Bhave, “Electromechanically induced ghz rate optical frequency modulation in silicon,” IEEE Photonics J. 4(5), 1474–1483 (2012).
[Crossref]

S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express 19(24), 24522–24529 (2011).
[Crossref]

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

S. Tallur and S. A. Bhave, “Non-linear dynamics in opto-mechanical oscillators,” in Proc. IEEE Transducers 2015, (2015), pp. 993–996.

Boddenberg, B.

F. Rittner, D. Paschek, and B. Boddenberg, “Simulation studies of the adsorption of xenon on the (110) face of rutile,” Langmuir 11(8), 3097–3102 (1995).
[Crossref]

Bouwmeester, D.

Buters, F. M.

Callegari, C.

Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11(4), 1753–1759 (2011).
[Crossref]

Carmon, T.

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

de Man, S.

Eerkens, H. J.

Feng, X. L.

Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11(4), 1753–1759 (2011).
[Crossref]

Fong, C.

P. Thomas, J. Gray, X. Zhu, and C. Fong, “Surface diffusion of xe on nb (1 1 0),” Chem. Phys. Lett. 381(3-4), 376–380 (2003).
[Crossref]

Fong, K. Y.

K. Y. Fong, M. Poot, X. Han, and H. X. Tang, “Phase noise of self-sustained optomechanical oscillators,” Phys. Rev. A 90(2), 023825 (2014).
[Crossref]

Gaede, H. C.

J. Kritzenberger, H. C. Gaede, J. S. Shore, A. Pines, and A. T. Bell, “129xe nmr study of tio2 (anatase)-supported v2o5 catalysts,” J. Phys. Chem. 98(40), 10173–10179 (1994).
[Crossref]

Gray, J.

P. Thomas, J. Gray, X. Zhu, and C. Fong, “Surface diffusion of xe on nb (1 1 0),” Chem. Phys. Lett. 381(3-4), 376–380 (2003).
[Crossref]

Gu, T.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Hajimiri, A.

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” in Proc. IEEE Int. Freq. Ctrl. Symp. 2006, (2006), pp. 405–408.

Han, X.

K. Y. Fong, M. Poot, X. Han, and H. X. Tang, “Phase noise of self-sustained optomechanical oscillators,” Phys. Rev. A 90(2), 023825 (2014).
[Crossref]

Hati, A.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Heeck, K.

Hossein-Zadeh, M.

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” in Proc. IEEE Int. Freq. Ctrl. Symp. 2006, (2006), pp. 405–408.

Howe, D. A.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Hsieh, P.-C.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Huang, S.-W.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Huang, Y.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Kippenberg, T.

Kritzenberger, J.

J. Kritzenberger, H. C. Gaede, J. S. Shore, A. Pines, and A. T. Bell, “129xe nmr study of tio2 (anatase)-supported v2o5 catalysts,” J. Phys. Chem. 98(40), 10173–10179 (1994).
[Crossref]

Kwong, D.-L.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Langmuir, I.

I. Langmuir, “The adsorption of gases on plane surfaces of glass mica and platinum.,” J. Am. Chem. Soc. 40(9), 1361–1403 (1918).
[Crossref]

Leeson, D. B.

D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54(2), 329–330 (1966).
[Crossref]

Li, Y.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Li, Y.-J.

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 46(3), 035502 (2013).
[Crossref]

Lo, G.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Luan, X.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Marcus, Y.

Y. Marcus, “The sizes of molecules-revisited,” J. Phys. Org. Chem. 16(7), 398–408 (2003).
[Crossref]

McMillan, J. F.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Paschek, D.

F. Rittner, D. Paschek, and B. Boddenberg, “Simulation studies of the adsorption of xenon on the (110) face of rutile,” Langmuir 11(8), 3097–3102 (1995).
[Crossref]

Pepper, B.

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley-Interscience, 1986).

Pines, A.

J. Kritzenberger, H. C. Gaede, J. S. Shore, A. Pines, and A. T. Bell, “129xe nmr study of tio2 (anatase)-supported v2o5 catalysts,” J. Phys. Chem. 98(40), 10173–10179 (1994).
[Crossref]

Poot, M.

K. Y. Fong, M. Poot, X. Han, and H. X. Tang, “Phase noise of self-sustained optomechanical oscillators,” Phys. Rev. A 90(2), 023825 (2014).
[Crossref]

Raeker, T.

T. Raeker, “Physical adsorption: Forces and phenomena (bruch, l.w.; cole, milton w.; zaremba, eugene),” J. Chem. Educ. 75(12), 1557 (1998).
[Crossref]

Rittner, F.

F. Rittner, D. Paschek, and B. Boddenberg, “Simulation studies of the adsorption of xenon on the (110) face of rutile,” Langmuir 11(8), 3097–3102 (1995).
[Crossref]

Rokhsari, H.

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” in Proc. IEEE Int. Freq. Ctrl. Symp. 2006, (2006), pp. 405–408.

Roukes, M. L.

Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11(4), 1753–1759 (2011).
[Crossref]

Sharp, J.

G. Ambati, J. Bell, and J. Sharp, “In-plane vibrations of annular rings,” J. Sound Vib. 47(3), 415–432 (1976).
[Crossref]

Shore, J. S.

J. Kritzenberger, H. C. Gaede, J. S. Shore, A. Pines, and A. T. Bell, “129xe nmr study of tio2 (anatase)-supported v2o5 catalysts,” J. Phys. Chem. 98(40), 10173–10179 (1994).
[Crossref]

Sonin, P.

Sridaran, S.

S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express 19(24), 24522–24529 (2011).
[Crossref]

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

Tallur, S.

S. Tallur and S. A. Bhave, “Partial gap transduced mems optoacoustic oscillator beyond gigahertz,” J. Microelectromech. Syst. 24(2), 422–430 (2015).
[Crossref]

S. Tallur and S. A. Bhave, “Electromechanically induced ghz rate optical frequency modulation in silicon,” IEEE Photonics J. 4(5), 1474–1483 (2012).
[Crossref]

S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express 19(24), 24522–24529 (2011).
[Crossref]

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

S. Tallur and S. A. Bhave, “Non-linear dynamics in opto-mechanical oscillators,” in Proc. IEEE Transducers 2015, (2015), pp. 993–996.

Tang, H. X.

K. Y. Fong, M. Poot, X. Han, and H. X. Tang, “Phase noise of self-sustained optomechanical oscillators,” Phys. Rev. A 90(2), 023825 (2014).
[Crossref]

Thomas, P.

P. Thomas, J. Gray, X. Zhu, and C. Fong, “Surface diffusion of xe on nb (1 1 0),” Chem. Phys. Lett. 381(3-4), 376–380 (2003).
[Crossref]

Vahala, K.

T. Kippenberg and K. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[Crossref]

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” in Proc. IEEE Int. Freq. Ctrl. Symp. 2006, (2006), pp. 405–408.

Vig, J. R.

Y. K. Yong and J. R. Vig, “Resonator surface contamination-a cause of frequency fluctuations?” IEEE Trans. UFFC 36(4), 452–458 (1989).
[Crossref]

Wang, D.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Wang, Y.-P.

Weaver, M. J.

Welker, G.

Wen, G.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Wiener, N.

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55(0), 117–258 (1930).
[Crossref]

Wong, C. W.

X. Luan, Y. Huang, Y. Li, J. F. McMillan, J. Zheng, S.-W. Huang, P.-C. Hsieh, T. Gu, D. Wang, A. Hati, D. A. Howe, G. Wen, M. Yu, G. Lo, D.-L. Kwong, and C. W. Wong, “An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset,” Sci. Rep. 4(1), 6842 (2014).
[Crossref]

Xu, X.-W.

X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 46(3), 035502 (2013).
[Crossref]

Yang, Y. T.

Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11(4), 1753–1759 (2011).
[Crossref]

Yong, Y. K.

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Figures (4)

Fig. 1.
Fig. 1. Scanning Electron Micrographs (SEM) of the two coupled ring silicon OMOs studied in this work. The rings differ in their lateral dimensions, as tabulated in Table 1. Panels (a) and (b) show the smaller (OMO$_1$) and larger (OMO$_2$) devices, respectively.
Fig. 2.
Fig. 2. Block diagram of experimental setup used for characterizing the OMO phase noise.
Fig. 3.
Fig. 3. Phase noise spectra for the radial mode for OMO$_1$ (oscillation frequency 176 MHz) at (a) room temperature and pressure (with external electrical feedback using DC bias $V_{dc} = +17 V$; RF signal power, $P_{RF} = -18 dBm$; laser wavelength, $\lambda _0 = 1595.30 nm$), and (b) at low temperature in vacuum ($P_{RF} = -13.7 dBm$; $\lambda _0 = 1595.30 nm$). The inset in panel (b) shows the mode shape of the radial breathing mode simulated in COMSOL Multiphysics FEM software.
Fig. 4.
Fig. 4. Phase noise spectra of OMO$_2$ for (a) radial breathing mode at oscillation frequency $70 MHz$ ($P_{RF} = -24.62 dBm$; $\lambda _0 = 1590 nm$), and (b) wineglass mode at frequency $58 MHz$ ($P_{RF} = -9 dBm$; $\lambda _0 = 1590 nm$). Insets show the corresponding mode shapes simulated in COMSOL Multiphysics FEM software.

Tables (1)

Tables Icon

Table 1. Parameters used for the phase noise models. The sticking coefficient ( s ) is assumed to be equal to 1 for all cases.

Equations (5)

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S ϕ , T N ( f ) = { 10 log 10 [ A 2 2 π Δ Ω Δ Ω 2 + ( 2 π f ) 2 ] , f < Δ Ω 0 2 π 10 log 10 [ A 2 2 π Δ Ω Δ Ω 2 + Δ Ω 0 2 ] , f Δ Ω 0 2 π
R ( t ) = i = 1 N ξ i 2 E [ b i ( t + τ ) b i ( t ) ] = σ 2 n = 0 ( 1 ) n P ( A n ) = σ 2 e λ | t |
σ 2 = p ( 1 p ) i = 1 N ξ i 2 = p ( 1 p ) ( Δ f ) 2 N
S f ( f ) = F [ σ 2 e | t | / τ r ] = 2 σ 2 τ r 1 + ( 2 π f τ r ) 2
S ϕ , A D N ( f ) = 2 r a r d / N ( r a + r d ) 3 + ( 2 π f ) 2 ( r a + r d ) f 0 2 f 2 ( ρ A ρ A e f f ) 2

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