Abstract

We present a theoretical analysis and the results of measurements of thermorefractive noise in microcavities. These measurements may be considered direct observations of fundamental fluctuations of temperature in solid media. Our experimentally measured noise spectra are in agreement with our theoretical model.

© 2004 Optical Society of America

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References

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  1. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
    [CrossRef] [PubMed]
  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]
  3. V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational wave antennae,” Phys. Lett. A 271, 303–307 (2000).
    [CrossRef]
  4. Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses,” Phys. Rev. D 62, 122002 (2000).
    [CrossRef]
  5. G. Cagnoli and P. A. Willems, “Effects of nonlinear thermoelastic damping in highly stressed fibers,” Phys. Rev. B 65, 174111 (2002).
    [CrossRef]
  6. W. H. Glenn, “Noise in interferometric optical systems: An optical Nyquist theorem,” IEEE J. Quantum Electron. 25, 1218–1224 (1989).
    [CrossRef]
  7. K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28, 53–54 (1992).
    [CrossRef]
  8. K. H. Wanser, A. D. Kersey, and A. Dandridge, “Measurement of fundamental thermal phase fluctuations in optical fiber,” in International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 255–258.
  9. 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]
  10. M. L. Gorodetsky, V. S. Ilchenko, and A. A. Savchenkov, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [CrossRef] [PubMed]
  11. M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).
  12. M. L. Gorodetsky, “Thermodynamical fluctuations in optical microspheres,” in Laser Resonators IV, A. Kudryashov and H. Paxton, eds., Proc. SPIE 4270, 147–153 (2001).
    [CrossRef]
  13. M. L. Gorodetsky, S. P. Vyatchanin, and P. A. Willems, unpublished analysis.
  14. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
    [CrossRef]
  15. M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
    [CrossRef]

2002 (1)

G. Cagnoli and P. A. Willems, “Effects of nonlinear thermoelastic damping in highly stressed fibers,” Phys. Rev. B 65, 174111 (2002).
[CrossRef]

2001 (1)

M. L. Gorodetsky, “Thermodynamical fluctuations in optical microspheres,” in Laser Resonators IV, A. Kudryashov and H. Paxton, eds., Proc. SPIE 4270, 147–153 (2001).
[CrossRef]

2000 (3)

M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational wave antennae,” Phys. Lett. A 271, 303–307 (2000).
[CrossRef]

Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses,” Phys. Rev. D 62, 122002 (2000).
[CrossRef]

1999 (1)

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]

1996 (1)

1994 (1)

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

1992 (3)

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28, 53–54 (1992).
[CrossRef]

1989 (2)

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]

W. H. Glenn, “Noise in interferometric optical systems: An optical Nyquist theorem,” IEEE J. Quantum Electron. 25, 1218–1224 (1989).
[CrossRef]

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational wave antennae,” Phys. Lett. A 271, 303–307 (2000).
[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]

Cagnoli, G.

G. Cagnoli and P. A. Willems, “Effects of nonlinear thermoelastic damping in highly stressed fibers,” Phys. Rev. B 65, 174111 (2002).
[CrossRef]

Drever, R. W. P.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Glenn, W. H.

W. H. Glenn, “Noise in interferometric optical systems: An optical Nyquist theorem,” IEEE J. Quantum Electron. 25, 1218–1224 (1989).
[CrossRef]

Gorodetsky, M. L.

M. L. Gorodetsky, “Thermodynamical fluctuations in optical microspheres,” in Laser Resonators IV, A. Kudryashov and H. Paxton, eds., Proc. SPIE 4270, 147–153 (2001).
[CrossRef]

M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational wave antennae,” Phys. Lett. A 271, 303–307 (2000).
[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]

M. L. Gorodetsky, V. S. Ilchenko, and A. A. Savchenkov, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

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]

Gursel, Y.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Ilchenko, V. S.

M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17, 1051–1057 (2000).
[CrossRef]

M. L. Gorodetsky, V. S. Ilchenko, and A. A. Savchenkov, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

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]

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Liu, Y. T.

Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses,” Phys. Rev. D 62, 122002 (2000).
[CrossRef]

Pryamikov, A. D.

Raab, F. J.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Savchenkov, A. A.

Shoemaker, D.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Sievers, L.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Srero, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Thorne, K. S.

Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses,” Phys. Rev. D 62, 122002 (2000).
[CrossRef]

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Vyatchanin, S. P.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational wave antennae,” Phys. Lett. A 271, 303–307 (2000).
[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]

Wanser, K. H.

K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28, 53–54 (1992).
[CrossRef]

Weiss, R.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Whitcomb, S. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Willems, P. A.

G. Cagnoli and P. A. Willems, “Effects of nonlinear thermoelastic damping in highly stressed fibers,” Phys. Rev. B 65, 174111 (2002).
[CrossRef]

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Electron. Lett. (1)

K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28, 53–54 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. H. Glenn, “Noise in interferometric optical systems: An optical Nyquist theorem,” IEEE J. Quantum Electron. 25, 1218–1224 (1989).
[CrossRef]

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

Laser Phys. (1)

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Opt. Commun. (1)

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

Opt. Lett. (1)

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, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermo-refractive noise in gravitational wave antennae,” Phys. Lett. A 271, 303–307 (2000).
[CrossRef]

Phys. Rev. B (1)

G. Cagnoli and P. A. Willems, “Effects of nonlinear thermoelastic damping in highly stressed fibers,” Phys. Rev. B 65, 174111 (2002).
[CrossRef]

Phys. Rev. D (1)

Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses,” Phys. Rev. D 62, 122002 (2000).
[CrossRef]

Proc. SPIE (1)

M. L. Gorodetsky, “Thermodynamical fluctuations in optical microspheres,” in Laser Resonators IV, A. Kudryashov and H. Paxton, eds., Proc. SPIE 4270, 147–153 (2001).
[CrossRef]

Science (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Srero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, “The project LIGO,” Science 256, 325–335 (1992).
[CrossRef] [PubMed]

Other (2)

K. H. Wanser, A. D. Kersey, and A. Dandridge, “Measurement of fundamental thermal phase fluctuations in optical fiber,” in International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 255–258.

M. L. Gorodetsky, S. P. Vyatchanin, and P. A. Willems, unpublished analysis.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the experimental setup for measurement of noise spectra in microspheres: ADC, analog-to-digital converter.

Fig. 2
Fig. 2

Resonator of 570-µm diameter with l-m13 modes (magnification 88×) visible as a result of surface scattering on residual inhomogeneities. At the right is a reflection of the microsphere in the coupling prism.

Fig. 3
Fig. 3

Thermorefractive noise in a microsphere (138±8) µm in diameter; l-m=4. Lower curve, setup and laser noise, measured in a large, 894-µm sphere. The straight line is a theoretical estimate.

Fig. 4
Fig. 4

Thermorefractive noise in four microspheres for six different modes. Dotted lines, theoretical curves obtained for recognized modes’ parameters. Larger noise corresponds to smaller sizes of microspheres and smaller l-m values.

Equations (54)

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u2=κT2ρCV,
ΔE+(0+2nδn) ω2c2 E=0,
δωω0=-1n V|E02|δndr=-1n dndTu¯,
ut-DΔu=F(r, t),
BrtF=F(r, t)F(r, t)=2κT2DρCΔδ(r-r)δ(t-t),
u¯=u(r, t)|E0(r)|2dr,
u(r, t)= F(q, Ω)Dq2+iΩ exp(iΩt+iq·r) dΩdq(2π)4
BqΩF=F(q, Ω)F*(q, Ω)=(2π)4 2κT2DρCq2δ(q-q)δ(Ω-Ω).
Su¯(Ω)=4κT2DρC  q2|E0(r)|2|E0(r)|2D2q4+Ω2×exp[iq·(r-r)]drdr dq(2π)3=4κT2DρC  q2|G(q)|2D2q4+Ω2 dq(2π)3,
G(q)=|E0|2 exp(-iq·r)dr
u2=0Su¯(Ω) dΩ2π=κT2ρC |G(q)|2 dq(2π)3.
Veff-1=|G(q)|2 dq(2π)3=|E(r)|4dr.
F(r, t)=Fν(Ω)Φν(r)exp(iΩt) dΩ2π.
Su¯(Ω)=4κT2DρC ν qν2|Gν|2D2qν4+Ω2,
Sδω/ω(Ω)=Su¯dndT2=κT2lπ3/2n2R2λ*ρCΩ dndT2×1(1-b2/d2)1/2 1[1+(Ωτb)3/4]2,
δωω=-1n dndTu,
E(r, θ, ϕ)Eθ(r, θ, ϕ) nl1/4n2-1R3/2π3/4iθ exp(-l cos2 θ/2+ilϕ)×jl(knr)/jl(knR)rRexp[-γ(r-R)]r>R,
knRl+1/2+1.8558(l+1/2)1/3-nn2-1,
eˆθ(r, θ, ϕ)1π2bdR0×exp-(r-R0)22b2-r2 cos2 θ2d2+ilϕ,
knR0l+1/2+0.71(l+1/2)1/3,
dR0l-1/2,
b1kn jl(knR0)jl(knR0)1/20.84R0l-2/3.
|G(q)|=12π2bdR0 0R02π0π exp-r2 cos2 θd2-(r-R0)2b2exp{iqr[cos θ cos ϑ+sin θ sin ϑ cos(ϕ-φ)]}sin θdθdϕr2drexp-(qd cos ϑ)24exp-(qb sin θ)24×1πqR0 sin ϑ1/2.
Su¯=8κT2DρC  q2|G(q)|2a4q4+Ω2 dq(2π)32κT2DρCπ3R 00π exp-(qd cos ϑ)22×exp-(qb sin θ)22 q3dϑdqD2q4+Ω2,limx 0π exp[-x2 cos2(ϑ)]dϑπx,
Sδω/ω(Ω)κT2Dπ5/2n2ρCR 2d2-b2 dndT2×0 q2 exp(-q2b2/2)D2q4+Ω2 dq2π,
0 q2 exp(-q2b2/2)D2q4+Ω2 dq2π
24D3/2Ω 1(1+(Ωτb)3/4)2.
Sδω/ω(Ω)=dndT2 κT2lπ3/2n2R2λ*ρCΩ×1(1-b2/d2)1/2 1[1+(Ωτb)3/4]2.
u(r, θ, ϕ, t)rr=R=0,
ΦL,M,N=CL,M,NjL(qLNr)PLM(cos θ)×cos(Mϕ)sin(Mϕ),
CL,M,N2=2L+1π(1+δ0M) (L-M)!(L+M)!×ξL,N2R3[ξLN2-L(L+1)]jL2(ξLN),
GL,N=CL,0,NπbdR0 exp-(r-R0)2b2-r2 cos2 θd2jL(qLNr)×PL(cos θ)r2dr sin θdθ.
PL(cos θ)2πL 1-14LcosL+12ψ+Lπ2,
jL(z)=1z sin(z-Lπ/2),
ξLN=qLNRπ(2N+L-1)2,
jL(ξLN)(-1)N-1ξLN,L=2K.
|GL,N|22R(L, N)π2R3 exp-(L+1/2)2d22R02-ξLN2b22R02;
R(L, N)=(1-1/4L)2(1+1/2L)1-L(L+1)/ξLN2,L>0,
R(0, N)=π/4.
Su¯(Ω)8κT2π2ρDCR K=0N=1 ξLN2 exp{-[(L+1/2)2d2/2R02]-(ξLN2b2/2R02)}ξLN4+(Ω2R4/D2) R(L, N).
Su8κT2π2ρCDR 0-xx 2π2x24π4x4+Ω2τR2×exp(-π2x2b2/R2)exp[-(x+y)2/l]dydx8κT2π2ρCDR l8π 0 t2t4+Ω2τR2×exp[-t2b2/(2R2)]erf2tπldt.
B(τ)=x(t)x(t+τ)=-S±(Ω)exp(iΩτ) dΩ2π,
Xj=-T/2T/2W(t)x(t)exp(-iΩjt)dt,Ωj=2πjT.
|Xj|2=-T/2T/2x(t)x(t)exp[iωj(t-t)]dtdt=-S(ω) 4 sin2[(ωj-ω)(T/2)](ωj-ω)2 dω2π.
Ys=-T/2T/2Y0 cos(Ωt)exp(-iΩjt)dt,
Ysj2=Y02T24TS±(ωj)=|Xj|s2.
Sδf/fY0f0 T2 1Hz,
Sδf/fY0f0 T3 1Hz.
Δθ=[2(l+1/2-m)/l]1/2.
q=1:l=-0.5+tlq-2.287tlq1/3+0.1718tlq-1/3,
q=2:l=-0.5+tlq-4.617tlq1/3+0.6944tlq-1/3,
q=3:l=-0.5+tlq-6.895tlq1/3+1.518tlq-1/3,
q=4:l=-0.5+tlq-9.190tlq1/3+2.632tlq-1/3,
tlq=2πniRλlmq+pnini2-1.

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