Abstract

We discuss thermodynamic as well as quantum limitations of the stability of resonance frequencies of solid-state whispering-gallery-mode resonators. We show that the relative frequency stability of a millimeter scale resonator can reach one part per 1012 per 1 s integration time.

© 2007 Optical Society of America

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  1. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
    [CrossRef]
  2. Ch. Salomon, D. Hils, and J. L. Hall, "Laser stabilization at the millihertz level," J. Opt. Soc. Am. B 5, 1576-1587 (1988).
    [CrossRef]
  3. T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of 2-diode laser-pumped Nd:YAG lasers locked to a Fabry-Perot interferometer," IEEE J. Quantum Electron. 28, 1106-1117 (1992).
    [CrossRef]
  4. J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
    [CrossRef]
  5. S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
    [CrossRef]
  6. B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, 3799-3802 (1999).
    [CrossRef]
  7. M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, "Simple and compact 1-Hz laser system via an improved mounting configuration of a reference cavity," Opt. Lett. 30, 1815-1817 (2005).
    [CrossRef] [PubMed]
  8. T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
    [CrossRef]
  9. K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93, 250602 (2004).
    [CrossRef]
  10. V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
    [CrossRef]
  11. X. S. Yao and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. B 13, 1725-1735 (1996).
    [CrossRef]
  12. X. S. Yao and L. Maleki, "Dual microwave and optical oscillator," Opt. Lett. 22, 1867-1869 (1997).
    [CrossRef]
  13. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
    [CrossRef] [PubMed]
  14. A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
    [CrossRef]
  15. T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, "Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005).
    [CrossRef] [PubMed]
  16. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
    [CrossRef] [PubMed]
  17. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities," IEEE J. Sel. Top. Quantum Electron. 12, 96-107 (2006).
    [CrossRef]
  18. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804(R) (2004).
    [CrossRef]
  19. S. Logunov and S. Kuchinsky, "Experimental and theoretical study of bulk light scattering in CaF2 monocrystals," J. Appl. Phys. 98, 053501 (2005).
    [CrossRef]
  20. M. L. Gorodetsky and I. S. Grudinin, "Fundamental thermal fluctuations in microspheres," J. Opt. Soc. Am. B 21, 697-705 (2004).
    [CrossRef]
  21. A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
    [CrossRef] [PubMed]
  22. L. D. Landau and E. M. Lifshitz, Course Of Theoretical Physics Vol. 5: Statistical Physics (Part 1) (Elsevier, 1980).
  23. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
    [CrossRef]
  24. V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
    [CrossRef]
  25. V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (U. Chicago Press, 1985).
  26. K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
    [CrossRef]
  27. V. B. Braginsky and S. P. Vyatchanin, "Frequency fluctuations of nonlinear origin in self-sustained optical oscillators," Phys. Lett. A 279, 154-162 (2001).
    [CrossRef]
  28. S. Chang, A. G. Mann, A. N. Luiten, and D. G. Blair, "Measurements of radiation pressure effect in cryogenic sapphire dielectric resonators," Phys. Rev. Lett. 79, 2141-2144 (1997).
    [CrossRef]
  29. C. Audoin, "Enrico Fermi," Course LXVIII, in Proceedings of the International School of Physics, A.F.Milone and P.Giacomo, eds. (International School of Physics, 1980), p. 169.

2006 (3)

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
[CrossRef]

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities," IEEE J. Sel. Top. Quantum Electron. 12, 96-107 (2006).
[CrossRef]

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

2005 (5)

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

S. Logunov and S. Kuchinsky, "Experimental and theoretical study of bulk light scattering in CaF2 monocrystals," J. Appl. Phys. 98, 053501 (2005).
[CrossRef]

M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, "Simple and compact 1-Hz laser system via an improved mounting configuration of a reference cavity," Opt. Lett. 30, 1815-1817 (2005).
[CrossRef] [PubMed]

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

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

2004 (5)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

M. L. Gorodetsky and I. S. Grudinin, "Fundamental thermal fluctuations in microspheres," J. Opt. Soc. Am. B 21, 697-705 (2004).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

2001 (1)

V. B. Braginsky and S. P. Vyatchanin, "Frequency fluctuations of nonlinear origin in self-sustained optical oscillators," Phys. Lett. A 279, 154-162 (2001).
[CrossRef]

1999 (2)

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, 3799-3802 (1999).
[CrossRef]

1998 (1)

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

1997 (3)

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

X. S. Yao and L. Maleki, "Dual microwave and optical oscillator," Opt. Lett. 22, 1867-1869 (1997).
[CrossRef]

S. Chang, A. G. Mann, A. N. Luiten, and D. G. Blair, "Measurements of radiation pressure effect in cryogenic sapphire dielectric resonators," Phys. Rev. Lett. 79, 2141-2144 (1997).
[CrossRef]

1996 (1)

1992 (2)

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of 2-diode laser-pumped Nd:YAG lasers locked to a Fabry-Perot interferometer," IEEE J. Quantum Electron. 28, 1106-1117 (1992).
[CrossRef]

J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
[CrossRef]

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

1988 (1)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

Audoin, C.

C. Audoin, "Enrico Fermi," Course LXVIII, in Proceedings of the International School of Physics, A.F.Milone and P.Giacomo, eds. (International School of Physics, 1980), p. 169.

Bergquist, J. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, 3799-3802 (1999).
[CrossRef]

Blair, D. G.

S. Chang, A. G. Mann, A. N. Luiten, and D. G. Blair, "Measurements of radiation pressure effect in cryogenic sapphire dielectric resonators," Phys. Rev. Lett. 79, 2141-2144 (1997).
[CrossRef]

Braginsky, V. B.

V. B. Braginsky and S. P. Vyatchanin, "Frequency fluctuations of nonlinear origin in self-sustained optical oscillators," Phys. Lett. A 279, 154-162 (2001).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (U. Chicago Press, 1985).

Byer, R. L.

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of 2-diode laser-pumped Nd:YAG lasers locked to a Fabry-Perot interferometer," IEEE J. Quantum Electron. 28, 1106-1117 (1992).
[CrossRef]

Camp, J.

K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Carmon, T.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities," IEEE J. Sel. Top. Quantum Electron. 12, 96-107 (2006).
[CrossRef]

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

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Chang, S.

S. Chang, A. G. Mann, A. N. Luiten, and D. G. Blair, "Measurements of radiation pressure effect in cryogenic sapphire dielectric resonators," Phys. Rev. Lett. 79, 2141-2144 (1997).
[CrossRef]

Cruz, F. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, 3799-3802 (1999).
[CrossRef]

Day, T.

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of 2-diode laser-pumped Nd:YAG lasers locked to a Fabry-Perot interferometer," IEEE J. Quantum Electron. 28, 1106-1117 (1992).
[CrossRef]

Dirscherl, J.

J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
[CrossRef]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

Goda, K.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

Gorodetsky, M. L.

M. L. Gorodetsky and I. S. Grudinin, "Fundamental thermal fluctuations in microspheres," J. Opt. Soc. Am. B 21, 697-705 (2004).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

Grudinin, I. S.

Gustafson, E. K.

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of 2-diode laser-pumped Nd:YAG lasers locked to a Fabry-Perot interferometer," IEEE J. Quantum Electron. 28, 1106-1117 (1992).
[CrossRef]

Hall, J. L.

Hils, D.

Hollberg, L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

Ilchenko, V. S.

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

Itano, W. M.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, 3799-3802 (1999).
[CrossRef]

Kemery, A.

K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Kippenberg, T. J.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities," IEEE J. Sel. Top. Quantum Electron. 12, 96-107 (2006).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

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

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

Kuchinsky, S.

S. Logunov and S. Kuchinsky, "Experimental and theoretical study of bulk light scattering in CaF2 monocrystals," J. Appl. Phys. 98, 053501 (2005).
[CrossRef]

Lam, P. K.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Course Of Theoretical Physics Vol. 5: Statistical Physics (Part 1) (Elsevier, 1980).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Course Of Theoretical Physics Vol. 5: Statistical Physics (Part 1) (Elsevier, 1980).

Logunov, S.

S. Logunov and S. Kuchinsky, "Experimental and theoretical study of bulk light scattering in CaF2 monocrystals," J. Appl. Phys. 98, 053501 (2005).
[CrossRef]

Luiten, A. N.

S. Chang, A. G. Mann, A. N. Luiten, and D. G. Blair, "Measurements of radiation pressure effect in cryogenic sapphire dielectric resonators," Phys. Rev. Lett. 79, 2141-2144 (1997).
[CrossRef]

Ma, L.-S.

Maleki, L.

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

X. S. Yao and L. Maleki, "Dual microwave and optical oscillator," Opt. Lett. 22, 1867-1869 (1997).
[CrossRef]

X. S. Yao and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. B 13, 1725-1735 (1996).
[CrossRef]

Mann, A. G.

S. Chang, A. G. Mann, A. N. Luiten, and D. G. Blair, "Measurements of radiation pressure effect in cryogenic sapphire dielectric resonators," Phys. Rev. Lett. 79, 2141-2144 (1997).
[CrossRef]

Matsko, A. B.

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

Mavalvala, N.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

McClelland, D. E.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

McKenzie, K.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

Mikhailov, E. E.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

Mitrofanov, V. P.

V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (U. Chicago Press, 1985).

Mlynek, J.

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

Mohageg, M.

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

Nazarova, T.

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
[CrossRef]

Neizert, B.

J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
[CrossRef]

Notcutt, M.

Numata, K.

K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

Panov, V. I.

V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (U. Chicago Press, 1985).

Riehle, F.

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
[CrossRef]

Rokhsari, H.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities," IEEE J. Sel. Top. Quantum Electron. 12, 96-107 (2006).
[CrossRef]

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

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Ruoso, G.

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

Salomon, Ch.

Savchenkov, A. A.

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

Scherer, A.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

Schiller, S.

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

Seel, S.

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

Spillane, S. M.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Sterr, U.

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
[CrossRef]

Storz, R.

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

Strekalov, D.

A. A. Savchenkov, I. S. Grudinin, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Morphology-dependent photonic circuit elements," Opt. Lett. 31, 1313-1315 (2006).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

Vahala, K. J.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities," IEEE J. Sel. Top. Quantum Electron. 12, 96-107 (2006).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

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

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Vassiliev, V. V.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Velichansky, V. L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Vyatchanin, S. P.

V. B. Braginsky and S. P. Vyatchanin, "Frequency fluctuations of nonlinear origin in self-sustained optical oscillators," Phys. Lett. A 279, 154-162 (2001).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

Walther, H.

J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

Wegener, T.

J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
[CrossRef]

Yang, L.

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

Yao, X. S.

Yarovitsky, A. V.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Ye, J.

Young, B. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, 3799-3802 (1999).
[CrossRef]

Appl. Phys. B (2)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83, 531-536 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of 2-diode laser-pumped Nd:YAG lasers locked to a Fabry-Perot interferometer," IEEE J. Quantum Electron. 28, 1106-1117 (1992).
[CrossRef]

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

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities," IEEE J. Sel. Top. Quantum Electron. 12, 96-107 (2006).
[CrossRef]

J. Appl. Phys. (1)

S. Logunov and S. Kuchinsky, "Experimental and theoretical study of bulk light scattering in CaF2 monocrystals," J. Appl. Phys. 98, 053501 (2005).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

J. Dirscherl, B. Neizert, T. Wegener, and H. Walther, "A dye laser spectrometer for high resolution spectroscopy," Opt. Commun. 91, 131-139 (1992).
[CrossRef]

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (3)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393-397 (1989).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

V. B. Braginsky and S. P. Vyatchanin, "Frequency fluctuations of nonlinear origin in self-sustained optical oscillators," Phys. Lett. A 279, 154-162 (2001).
[CrossRef]

Phys. Rev. A (2)

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005).
[CrossRef]

A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, "Kilohertz optical resonances in dielectric crystal cavities," Phys. Rev. A 70, 051804(R) (2004).
[CrossRef]

Phys. Rev. Lett. (8)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, "Low threshold optical oscillations in a whispering gallery mode CaF2 resonator," Phys. Rev. Lett. 93, 243905 (2004).
[CrossRef]

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

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, "Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1Hz level," Phys. Rev. Lett. 78, 4741-4744 (1997).
[CrossRef]

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82, 3799-3802 (1999).
[CrossRef]

K. Numata, A. Kemery, and J. Camp, "Thermal-noise limit in the frequency stabilization of lasers with rigid cavities," Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

S. Chang, A. G. Mann, A. N. Luiten, and D. G. Blair, "Measurements of radiation pressure effect in cryogenic sapphire dielectric resonators," Phys. Rev. Lett. 79, 2141-2144 (1997).
[CrossRef]

Other (3)

C. Audoin, "Enrico Fermi," Course LXVIII, in Proceedings of the International School of Physics, A.F.Milone and P.Giacomo, eds. (International School of Physics, 1980), p. 169.

V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation (U. Chicago Press, 1985).

L. D. Landau and E. M. Lifshitz, Course Of Theoretical Physics Vol. 5: Statistical Physics (Part 1) (Elsevier, 1980).

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Figures (8)

Fig. 1
Fig. 1

Spectral power density of the frequency noise S δ ω ω ( Ω ) = α n 2 S u ¯ ( Ω ) resulting from the thermorefractive fluctuations in a low-contrast WGM resonator.

Fig. 2
Fig. 2

Spectral density of the frequency noise S δ ω ω ( Ω ) = α n 2 S u ¯ ( Ω ) . The solid (black) curve is derived from the exact solution given by Eq. (B19), found by the summation of contributions of modes with m = 0 (dashed blue curve) and m = 2 (dotted green curve).

Fig. 3
Fig. 3

Spectral density of the frequency noise S δ ω ω ( Ω ) = S Δ R R ( Ω ) due to thermoelastic noise.

Fig. 4
Fig. 4

Thermodynamic uncertainty of the absolute frequency of a WGM versus mode volume calculated for a spherical resonator.

Fig. 5
Fig. 5

Spectral density of the photothermal fluctuations of a WGM frequency ( S δ ω ω 1 2 = α n S u ¯ 1 2 ) calculated for a fluorite resonator with R = 0.3 cm and L = 0.01 cm interrogated with coherent 1.55 μ m light of 1 mW power assuming that the light is absorbed in the resonator. The solid (dashed) curve stands for the simulation (analytical calculations).

Fig. 6
Fig. 6

Spectral density of the ponderomotive backaction fluctuations of frequency [ S δ ω ω 1 2 = ( 1 + K ϵ 2 ) S Δ R R 1 2 ] calculated for a fluorite resonator with R = 0.3 cm and L = 0.01 cm interrogated with 1 mW coherent light.

Fig. 7
Fig. 7

Spectral density of the frequency noise S δ ω ω 1 2 ( Ω ) resulting from the self-phase-modulation effect in the CaF 2 resonator. The noise is calculated for a fluorite resonator with R = 0.3 cm and L = 0.01 cm interrogated with 1 mW coherent light.

Fig. 8
Fig. 8

Spectral density of the frequency noise S δ ω ω ( Ω ) = α n 2 S u ¯ ( Ω ) . The solid (black) curve is derived from the exact solution in Eq. (B19). The dashed (red) curve is the analytical approximation given by Eq. (10).

Tables (1)

Tables Icon

Table 1 Parameters of the Calcium Fluoride, Sapphire, and Fused Silica

Equations (102)

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δ ω ω = Δ R R + Δ n n ,
Δ n n ( Δ T ) m = α n ( Δ T ) m ,
( Δ T ) m 2 = k B T 2 C p V m ρ ,
V m = 3.4 π 3 2 ( λ 2 π n ) 7 6 R 11 6 .
u t D Δ u = F ( r , t ) ,
u ¯ ( t ) = u ( r , t ) Ψ ( r ) 2 d r ,
S u ¯ ( Ω ) = u ¯ * ( t ) u ¯ ( t + τ ) e i Ω τ d τ ,
( Δ T ) m 2 = u ¯ ( 0 ) 2 = S u ¯ ( Ω ) d Ω 2 π .
F ( r , t ) F ( r , t ) 16 π k B T 2 D L ρ C V m δ ( r r ) δ ( t t ) ;
S u ¯ ( Ω ) = k B T 2 ρ C V m R 2 12 D [ 1 + ( R 2 D Ω 9 3 ) 3 2 + 1 6 ( R 2 D Ω 8 ν 1 3 ) 2 ] 1 .
( Δ V r ) 2 V r 2 = k B T β T V r ,
Δ R R Δ V = Δ V r 3 V r .
( Δ T ) r 2 = k B T 2 C p V r ρ ,
Δ R R Δ T = α l ( Δ T ) r ,
( Δ R ) 2 R 2 = k B T β T 9 V r + α l 2 k B T 2 C p V r ρ .
( Δ R ) t + ( i Ω 0 + Γ 0 ) Δ R = F R ( t ) ,
Ω 0 = π v s R ,
Γ 0 Ω 0 2 κ T α l 2 ρ 9 C 2 .
F R * ( t ) F R ( t ) = Γ 0 k B T β T R 2 9 V r δ ( t t ) ,
S Δ R R = k B T β T 9 V r Γ 0 ( Ω Ω 0 ) 2 + Γ 0 2 .
u t D Δ u = F P ( r , t ) ,
F P ( r , t ) F P ( r , t ) = ω ν , q , m ρ 2 C 2 P a b s Ψ ( r ) 2 δ ( r r ) δ ( t t ) ,
S u ¯ ( Ω ) ω ν , q , m ρ 2 C 2 P a b s V r 2 π 2 R 3 2 64 L 3 2 Ω 2 + π 6 D 2 16 L 3 R .
δ ω ω = [ 1 + 1 2 K ε ] Δ R R ,
2 ( Δ R ) t 2 + Γ 0 ( Δ R ) t + Ω 0 2 ( Δ R ) = 2 π n δ P ( t ) m * c ,
S Δ R R = ( 2 π n m * c R ) 2 ω ν , q , m P τ 0 × 2 γ R γ R 2 + Ω 2 1 ( Ω 0 2 Ω 2 ) 2 + Γ 0 2 Ω 2 ,
Δ R 2 ( t ) 1 2 R 2 π n m * c R [ ω ν , q , m P τ 0 Ω 0 4 ( 1 + γ R Γ 0 ) ] 1 2 .
δ n n = n 2 δ P ( t ) A n ,
S δ ω ω = n 2 2 ω ν , q , m P A 2 n 2 τ 0 2 γ R γ R 2 + Ω 2 ,
δ ω ω = n 2 ω ν , q , m P A n τ 0 .
σ 2 ( τ ) = 2 π 0 S δ ω ω ( Ω ) sin 4 ( Ω τ 2 ) ( Ω τ 2 ) 2 d Ω ,
u t D Δ u = 0 ,
u n = 0 ,
u ( t ) = p , l u ̃ p , l , k m ( t ) J l ( k l , p r ) e ± i l ϕ e i k m z d k m 2 π ,
Ψ ( r ) = sin ( π z L ) 2 π J ν ( k ν , q r ) J ν + 1 ( k ν , q R ) e ± i ν ϕ R L ,
u ( t ) = p Φ p , k m ( r ) u ̃ p , k m ( t ) d k m 2 π ,
Φ p , k m Φ p , k m * d r = 2 π δ ( k m k m ) δ p , p .
Φ p , k m ( r ) = J 0 ( k p r ) J 0 ( k p R ) exp ( i k m z ) R π ,
u ̃ p , k m ( t ) t + D k p , m 2 u ̃ p , k m ( t ) = F p , k m ( t ) ,
F p , k m ( t ) F p , k m * ( t ) = 4 π 2 D R k B T 2 C p V m ρ δ ( t t ) δ ( k m k m ) δ p , p .
Φ p , k m ( r ) Ψ ( r ) 2 d r 8 π 2 sin ( k m L 2 ) k m L ( k m 2 L 2 4 π 2 ) exp ( i k m L 2 ) π R ,
S u ¯ ( Ω ) = 1 π R 2 p ν 2 3 d k m 2 π d k m 2 π [ 8 π 2 sin ( k m L 2 ) k m L ( k m 2 L 2 4 π 2 ) ] 2 u ̃ p , k m * ( t ) u ̃ p , k m ( t + τ ) e i Ω τ d τ .
u ̃ p , k m ( t ) = u ̃ p , k m ( Ω ) e i Ω t d Ω 2 π ,
F p , k m ( t ) = F p , k m ( Ω ) e i Ω t d Ω 2 π ,
F p , k m * ( Ω ) F p , k m ( Ω ) = 8 π 3 D R k B T 2 C p V m ρ δ ( Ω Ω ) δ ( k m k m ) δ p , p .
u ̃ p , k m ( t ) = F p , k m ( Ω ) i Ω + D k p , m 2 e i Ω t d Ω 2 π .
u ̃ p , k m * ( t ) u ̃ p , k m ( t + τ ) = 4 π 2 D R k B T 2 C p V m ρ × δ ( k m k m ) δ p , p e i Ω τ Ω 2 + D 2 k p , m 4 d Ω 2 π .
S u ¯ ( Ω ) = k B T 2 C p V m ρ 2 D R p ν 2 3 d k m 2 π × [ 8 π 2 sin ( k m L 2 ) k m L ( k m 2 L 2 4 π 2 ) ] 2 1 Ω 2 + D 2 k p , m 4 .
( Δ T ) m 2 = S u ¯ ( Ω ) d Ω 2 π = k B T 2 C p V m ρ 1 R p ν 2 3 d k m 2 π [ 8 π 2 sin ( k m L 2 ) k m L ( k m 2 L 2 4 π 2 ) ] 2 1 k p , m 2 .
u t D Δ u = 0 ,
u n = 0 ,
u ( t ) = p , l , m u ̃ p , l , m ( t ) cos ( π m z L ) J l ( k l , p r ) e ± i l ϕ ,
Ψ ( r ) = sin ( π z L ) 2 π J ν ( k ν , q r ) J ν + 1 ( k ν , q R ) e ± i ν ϕ R L ,
u ( t ) = p , m Φ p , m ( r ) u ̃ p , m ( t ) ,
Φ p , m Φ p , m d r = δ m , m δ p , p .
Φ p , 0 ( r ) = J 0 ( k p r ) π J 0 ( k p R ) 1 R L ,
Φ p , 2 ( r ) = cos ( 2 π z L ) 2 π J 0 ( k p r ) J 0 ( k p R ) 1 R L ,
u ̃ p , m ( t ) t + D k p , m 2 u ̃ p , m ( t ) = F p , m ( t ) ,
F p , m * ( t ) F p , m ( t ) = 16 π k B T 2 D L ρ C V m δ ( t t ) δ m , m δ p , p .
Φ p , 0 ( r ) Ψ ( r ) 2 d r 1 π R 2 L , ν 2 3 p ,
Φ p , 2 ( r ) Ψ ( r ) 2 d r 1 π R 2 L , ν 2 3 p ,
S u ¯ ( Ω ) = 1 π R 2 L p ν 2 3 m = 0 , 2 u ̃ p , m * ( t ) u ̃ p , m ( t + τ ) e i Ω τ d τ .
u ̃ p , m ( t ) = u ̃ p , m ( Ω ) e i Ω t d Ω 2 π ,
F p , m ( t ) = F p , m ( Ω ) e i Ω t d Ω 2 π ,
F p , m * ( Ω ) F p , m ( Ω ) = 32 π 2 k B T 2 D L ρ C V m δ ( Ω Ω ) δ m , m δ p , p .
u ̃ p , m ( t ) = F p , m ( Ω ) i Ω + D k p , m 2 e i Ω t d Ω 2 π .
u ̃ p , m * ( t ) u ̃ p , m ( t + τ ) = 16 π k B T 2 D L ρ C V m δ m , m δ p , p e i Ω τ Ω 2 + D 2 k p , m 4 d Ω 2 π .
S u ¯ ( Ω ) = k B T 2 ρ C V m 16 D R 2 p ν 2 3 m = 0 , 2 1 Ω 2 + D 2 k p , m 4 .
( Δ T ) m 2 = S u ¯ ( Ω ) d Ω 2 π = k B T 2 ρ C V m 8 R 2 p ν 2 3 m = 0 , 2 1 k p , m 2 .
p ν 2 3 1 k p , 0 2 = R 2 p ν 2 3 1 β p 2 R 2 p = 1 1 β p 2 = R 2 8 ,
p ν 2 3 1 k p , 2 2 R 2 π 2 p = 1 1 p 2 + 4 R 2 π 2 L 2 R L 4 .
S u ¯ ( Ω ) Ω D R 2 = k B T 2 ρ C V m R 2 12 D ,
S u ¯ ( Ω ) Ω 4 D L 2 , D ν 4 3 R 2 = k B T 2 ρ C V m 32 D ν 2 3 R 2 Ω 2 .
Δ U + Ω 2 β S ρ U = 0 ,
U = u m , n , p ( r ) Ω 2 β S ρ ,
u m , n , p ( r ) = u 0 π 2 J n + 1 2 ( Ω n , p r ( β S ρ ) 1 2 ) [ Ω n , p r ( β S ρ ) 1 2 ] 1 2 P n m ( cos θ ) e ± i m ϕ ,
J n + 1 2 ( Ω n , p R ( β S ρ ) 1 2 ) = 0 .
u 0 , 0 , p ( r ) = u p 0 sin ( Ω 0 , p r ( β S ρ ) 1 2 ) Ω 0 , p r ( β S ρ ) 1 2 ,
sin ( Ω 0 , p R ( β S ρ ) 1 2 ) = 0 .
Ω 0 , p = π p R ( β S ρ ) 1 2 .
U r ( p ) = u p 0 R 2 π 2 p 2 r [ sin ( Ω 0 , p r ( β S ρ ) 1 2 ) Ω 0 , p r ( β S ρ ) 1 2 ] .
U r r ( p ) = U r ( p ) r , U θ θ ( p ) = U ϕ ϕ ( p ) = U r ( p ) r .
E p = 1 2 β T ( U r r ( p ) + U θ θ ( p ) + U ϕ ϕ ( p ) ) 2 d V = π 2 β T 0 R ( U r r ( p ) + U θ θ ( p ) + U ϕ ϕ ( p ) ) 2 r 2 d r = u p 0 2 2 p 2 R 3 β T .
U r ( p ) 2 R = 1 π 4 p 2 k B T β T R .
( Δ R ) 2 R 2 = p U r ( p ) 2 R R 2 = 2 π k B T β T 9 V r ,
u ̃ p , k m ( t ) t + D k p , m 2 u ̃ p , k m ( t ) = F p , k m ( t ) ,
F p , k m * ( t ) F p , k m ( t ) = g 1 ( p , p , m , m ) ω ν , q , m ρ 2 C 2 P a b s δ ( t t ) ,
g 1 ( p , p , m , m ) = Ψ ( r ) 2 Φ p , k m ( r ) Φ p , k m * ( r ) d r .
u ̃ p , m ( t ) = F p , k m ( Ω ) i Ω + D k p , m 2 e i Ω t d Ω 2 π ,
u ̃ p , k m * ( t ) u ̃ p , k m ( t + τ ) = ω ν , q , m ρ 2 C 2 P a b s g 1 ( p , p , m , m ) e i Ω r ( i Ω + D k p , m 2 ) ( i Ω + D k p , m 2 ) d Ω 2 π .
S u ¯ ( Ω ) = ω ν , q , m ρ 2 C 2 P a b s p , p d k m 2 π d k m 2 π g 1 ( p , p , m , m ) g 2 * ( p , m ) g 2 ( p , m ) ( i Ω + D k p , m 2 ) ( i Ω + D k p , m 2 ) ,
g 2 ( p , m ) = Ψ ( r ) 2 Φ p , k m ( r ) d r .
S u ¯ ( Ω ) ω ν , q , m ρ 2 C 2 P a b s L 2 V r 2 p ν 2 3 π 4 L π 4 L d k m i Ω + D k p , m 2 2 .
( Δ T ) 2 = S u ¯ ( Ω ) d Ω 2 π = ω ν , q , m ρ 2 C 2 L 2 P a b s D V r 2 p , p ν 2 3 π 4 L π 4 L π 4 L π 4 L d k m d k m k p , m 2 + k p , m 2 ω ν , q , m ρ 2 C 2 π 3 32 R 2 P a b s D V r 2 .
S u ¯ ( 0 ) ω ν , q , m ρ 2 C 2 P a b s V r 2 L 3 2 R 5 2 4 D 2 .
P P p = 2 γ R γ R + γ R 0 1 τ 0 ( γ R + γ R 0 ) ,
a ̇ + ( γ R + i ω ν , q , m ) a = γ R a 0 e i ω ν , q , m t + 2 γ R f ( t ) ,
a = a 0 e i ω ν , q , m t + 2 γ R f ( ω ) γ R + i ( ω ν , q , m ω ) e i ω t d ω 2 π ,
δ P ( t ) = ω ν , q , m P τ 0 2 γ R γ R + i ω ( f ( ω + ω ν , q , m ) + f ( ω ν , q , m ω ) ) e i ω t d ω 2 π .
Δ R ( t ) = Δ R ( Ω ) e i Ω t d Ω 2 π ,
Δ R ( Ω ) = 2 π n m * c ω ν , q , m P τ 0 × 2 γ R γ R + i Ω f ( Ω + ω ν , q , m ) + f ( ω ν , q , m Ω ) Ω 0 2 Ω 2 i Γ 0 Ω .
Δ R 2 ( t ) = ( 2 π n m * c ) 2 ω ν , q , m P τ 0 × 2 γ R γ R 2 + Ω 2 d Ω 2 π ( Ω 0 2 Ω 2 ) 2 + Γ 0 2 Ω 2 .

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