Abstract

To improve the detection limit of optical sensors, it is of paramount importance to understand light–matter interaction processes at a fundamental level. At room temperature, the ultimate detection limit is governed amongst others by fundamental thermodynamic fluctuations. Their effect on the properties of light that propagates in amorphous materials is not well understood. Here, we unveil and model for the first time a dominating high-frequency (terahertz-range) noise contribution in the phase of laser light having propagated in optical waveguides, leading to Raman-like sidebands in the optical output spectrum. A salient feature of our approach is to consider a mean relaxation time of the spontaneous random heat flux in the medium, which leads to a spatial correlation of the thermo-refractive noise. The resulting phase noise can be several orders of magnitude larger than what was predicted by earlier models. Our model allows us to explain the origin and specificities of the background that is observed in the Raman optical spectra of silicon nitride waveguides and silica optical fibers. Not only do these findings add a previously unknown dimension to the fundamental knowledge about noise in light–matter interaction, they also need to be taken into account in any optical system in which dynamic fluctuations at the picosecond (ps) or sub-ps level play a role.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

A. Dhakal, P. Wuytens, A. Raza, N. Le Thomas, and R. Baets, “Silicon nitride background in nanophotonic waveguide enhanced Raman spectroscopy,” Materials 10, 140 (2017).
[Crossref]

2016 (2)

S. A. Holmstrom, T. H. Stievater, D. A. Kozak, M. W. Pruessner, N. Tyndall, W. S. Rabinovich, R. Andrew McGill, and J. B. Khurgin, “Trace gas Raman spectroscopy using functionalized waveguides,” Optica 3, 891–896 (2016).
[Crossref]

J. Dong, J. Huang, T. Li, and L. Liu, “Observation of fundamental thermal noise in optical fibers down to infrasonic frequencies,” Appl. Phys. Lett. 108, 021108 (2016).
[Crossref]

2014 (1)

2013 (1)

A. R. Zanatta and I. B. Gallo, “The thermo optic coefficient of amorphous SiN films in the near-infrared and visible regions and its experimental determination,” Appl. Phys. Express 6, 042402 (2013).
[Crossref]

2012 (2)

S. Foster, “Low-frequency thermal noise in optical fiber cavities,” Phys. Rev. A 86, 043801 (2012).
[Crossref]

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48, 720–727 (2012).
[Crossref]

2010 (1)

2007 (1)

S. Foster, A. Tikhomirov, and M. Milnes, “Fundamental thermal noise in distributed feedback fiber lasers,” IEEE J. Quantum Electron. 43, 378–384 (2007).
[Crossref]

2004 (1)

2003 (1)

P.-H. Chavanis, “Generalized thermodynamics and Fokker-Planck equations: applications to stellar dynamics and two-dimensional turbulence,” Phys. Rev. E 68, 036108 (2003).
[Crossref]

2002 (1)

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

1996 (2)

R. J. Hill, “Corrections to Taylor’s frozen turbulence approximation,” Atmos. Res. 40, 153–175 (1996).
[Crossref]

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Thermodynamic phase noise in fibre interferometers,” Opt. Quantum Electron. 28, 43–49 (1996).
[Crossref]

1992 (1)

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

1989 (1)

D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys. 61, 41–73 (1989).
[Crossref]

1988 (1)

D. Jou and J. Casas-Vázquez, “Extended irreversible thermodynamics of heat conduction,” Eur. J. Phys. 9, 329–333 (1988).
[Crossref]

1980 (1)

D. Jou, J. M. Rubi, and J. Casas-Vazquez, “Hydrodynamical fluctuations in extended irreversible thermodynamics,” Physica 101A, 588–598 (1980).
[Crossref]

1970 (1)

R. Shuker and R. W. Gammon, “Raman-scattering selection-rules breaking and the density of states in amorphous materials,” Phys. Rev. Lett. 25, 222–225 (1970).
[Crossref]

1957 (1)

S. M. Rytov, “Correlation theory of thermal fluctuations in an isotropic medium,” J. Exp. Theor. Phys. 33, 166–178 (1957).

1951 (1)

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[Crossref]

1946 (1)

J. S. Hey, S. J. Parsons, and J. W. Phillips, “Fluctuations in cosmic radiation at radiofrequencies,” Nature 158, 247–249 (1946).
[Crossref]

1938 (1)

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London Ser. A 164, 476–490 (1938).
[Crossref]

1931 (1)

L. Onsager, “Reciprocal relations in irreversible processes. I.,” Phys. Rev. 37, 405–426 (1931).
[Crossref]

Andrew McGill, R.

Annovazzi-Lodi, V.

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Thermodynamic phase noise in fibre interferometers,” Opt. Quantum Electron. 28, 43–49 (1996).
[Crossref]

Baets, R.

A. Dhakal, P. Wuytens, A. Raza, N. Le Thomas, and R. Baets, “Silicon nitride background in nanophotonic waveguide enhanced Raman spectroscopy,” Materials 10, 140 (2017).
[Crossref]

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. Le Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39, 4025–4028 (2014).
[Crossref]

Bartolo, R. E.

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48, 720–727 (2012).
[Crossref]

Berne, B. J.

B. J. Berne and R. Pecora, Dynamic Light Scattering (Dover, 2000).

Callen, H. B.

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[Crossref]

Carlson, S.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Casas-Vazquez, J.

D. Jou, J. M. Rubi, and J. Casas-Vazquez, “Hydrodynamical fluctuations in extended irreversible thermodynamics,” Physica 101A, 588–598 (1980).
[Crossref]

Casas-Vázquez, J.

D. Jou and J. Casas-Vázquez, “Extended irreversible thermodynamics of heat conduction,” Eur. J. Phys. 9, 329–333 (1988).
[Crossref]

D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics, 4th ed. (Springer, 2010).

Chavanis, P.-H.

P.-H. Chavanis, “Generalized thermodynamics and Fokker-Planck equations: applications to stellar dynamics and two-dimensional turbulence,” Phys. Rev. E 68, 036108 (2003).
[Crossref]

Dandridge, A.

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48, 720–727 (2012).
[Crossref]

Dhakal, A.

A. Dhakal, P. Wuytens, A. Raza, N. Le Thomas, and R. Baets, “Silicon nitride background in nanophotonic waveguide enhanced Raman spectroscopy,” Materials 10, 140 (2017).
[Crossref]

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. Le Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39, 4025–4028 (2014).
[Crossref]

Donati, S.

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Thermodynamic phase noise in fibre interferometers,” Opt. Quantum Electron. 28, 43–49 (1996).
[Crossref]

Dong, J.

J. Dong, J. Huang, T. Li, and L. Liu, “Observation of fundamental thermal noise in optical fibers down to infrasonic frequencies,” Appl. Phys. Lett. 108, 021108 (2016).
[Crossref]

Dooryhee, E.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Fitch, A.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Foster, S.

S. Foster, “Low-frequency thermal noise in optical fiber cavities,” Phys. Rev. A 86, 043801 (2012).
[Crossref]

S. Foster, A. Tikhomirov, and M. Milnes, “Fundamental thermal noise in distributed feedback fiber lasers,” IEEE J. Quantum Electron. 43, 378–384 (2007).
[Crossref]

Frost, D. J.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Gallo, I. B.

A. R. Zanatta and I. B. Gallo, “The thermo optic coefficient of amorphous SiN films in the near-infrared and visible regions and its experimental determination,” Appl. Phys. Express 6, 042402 (2013).
[Crossref]

Gammon, R. W.

R. Shuker and R. W. Gammon, “Raman-scattering selection-rules breaking and the density of states in amorphous materials,” Phys. Rev. Lett. 25, 222–225 (1970).
[Crossref]

Gelfand, I. M.

I. M. Gelfand and G. E. Shilov, Generalized Functions (Academic, 1964).

Gerward, L.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Gorodetsky, M. L.

Grudinin, I. S.

Hey, J. S.

J. S. Hey, S. J. Parsons, and J. W. Phillips, “Fluctuations in cosmic radiation at radiofrequencies,” Nature 158, 247–249 (1946).
[Crossref]

Hill, R. J.

R. J. Hill, “Corrections to Taylor’s frozen turbulence approximation,” Atmos. Res. 40, 153–175 (1996).
[Crossref]

Holmstrom, S. A.

Houdré, R.

Huang, J.

J. Dong, J. Huang, T. Li, and L. Liu, “Observation of fundamental thermal noise in optical fibers down to infrasonic frequencies,” Appl. Phys. Lett. 108, 021108 (2016).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2, Section 17.

Jiang, J. Z.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Joseph, D. D.

D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys. 61, 41–73 (1989).
[Crossref]

Jou, D.

D. Jou and J. Casas-Vázquez, “Extended irreversible thermodynamics of heat conduction,” Eur. J. Phys. 9, 329–333 (1988).
[Crossref]

D. Jou, J. M. Rubi, and J. Casas-Vazquez, “Hydrodynamical fluctuations in extended irreversible thermodynamics,” Physica 101A, 588–598 (1980).
[Crossref]

D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics, 4th ed. (Springer, 2010).

Khurgin, J. B.

Kozak, D. A.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 3rd ed. (Pergamon, 1966), Section 132.

L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 2nd ed. (Pergamon, 1970), Section 121, p. 372.

L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 2nd ed. (Pergamon, 1980), Section 114.

Le Thomas, N.

Lebon, G.

D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics, 4th ed. (Springer, 2010).

Li, T.

J. Dong, J. Huang, T. Li, and L. Liu, “Observation of fundamental thermal noise in optical fibers down to infrasonic frequencies,” Appl. Phys. Lett. 108, 021108 (2016).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 2nd ed. (Pergamon, 1980), Section 114.

L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 2nd ed. (Pergamon, 1970), Section 121, p. 372.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 3rd ed. (Pergamon, 1966), Section 132.

Lindelov, H.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Liu, L.

J. Dong, J. Huang, T. Li, and L. Liu, “Observation of fundamental thermal noise in optical fibers down to infrasonic frequencies,” Appl. Phys. Lett. 108, 021108 (2016).
[Crossref]

Merlo, S.

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Thermodynamic phase noise in fibre interferometers,” Opt. Quantum Electron. 28, 43–49 (1996).
[Crossref]

Mezouar, M.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Milnes, M.

S. Foster, A. Tikhomirov, and M. Milnes, “Fundamental thermal noise in distributed feedback fiber lasers,” IEEE J. Quantum Electron. 43, 378–384 (2007).
[Crossref]

Mori-Sanchez, P.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Onsager, L.

L. Onsager, “Reciprocal relations in irreversible processes. I.,” Phys. Rev. 37, 405–426 (1931).
[Crossref]

Parsons, S. J.

J. S. Hey, S. J. Parsons, and J. W. Phillips, “Fluctuations in cosmic radiation at radiofrequencies,” Nature 158, 247–249 (1946).
[Crossref]

Pecora, R.

B. J. Berne and R. Pecora, Dynamic Light Scattering (Dover, 2000).

Peyskens, F.

Phillips, J. W.

J. S. Hey, S. J. Parsons, and J. W. Phillips, “Fluctuations in cosmic radiation at radiofrequencies,” Nature 158, 247–249 (1946).
[Crossref]

Pottier, N.

N. Pottier, Nonequilibrium Statistical Physics, 1st ed. (Oxford University, 2010).

Preziosi, L.

D. D. Joseph and L. Preziosi, “Heat waves,” Rev. Mod. Phys. 61, 41–73 (1989).
[Crossref]

Prigogine, I.

I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, 3rd ed. (Interscience, 1961).

Pruessner, M. W.

Rabinovich, W. S.

Raza, A.

A. Dhakal, P. Wuytens, A. Raza, N. Le Thomas, and R. Baets, “Silicon nitride background in nanophotonic waveguide enhanced Raman spectroscopy,” Materials 10, 140 (2017).
[Crossref]

Recio, J. M.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Rubi, J. M.

D. Jou, J. M. Rubi, and J. Casas-Vazquez, “Hydrodynamical fluctuations in extended irreversible thermodynamics,” Physica 101A, 588–598 (1980).
[Crossref]

Rytov, S. M.

S. M. Rytov, “Correlation theory of thermal fluctuations in an isotropic medium,” J. Exp. Theor. Phys. 33, 166–178 (1957).

Shilov, G. E.

I. M. Gelfand and G. E. Shilov, Generalized Functions (Academic, 1964).

Shuker, R.

R. Shuker and R. W. Gammon, “Raman-scattering selection-rules breaking and the density of states in amorphous materials,” Phys. Rev. Lett. 25, 222–225 (1970).
[Crossref]

Ståhl, K.

J. Z. Jiang, H. Lindelov, L. Gerward, K. Ståhl, J. M. Recio, P. Mori-Sanchez, S. Carlson, M. Mezouar, E. Dooryhee, A. Fitch, and D. J. Frost, “Compressibility and thermal expansion of cubic silicon nitride,” Phys. Rev. B 65, 161202 (2002).
[Crossref]

Stievater, T. H.

Subramanian, A. Z.

Taylor, G. I.

G. I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. London Ser. A 164, 476–490 (1938).
[Crossref]

Tikhomirov, A.

S. Foster, A. Tikhomirov, and M. Milnes, “Fundamental thermal noise in distributed feedback fiber lasers,” IEEE J. Quantum Electron. 43, 378–384 (2007).
[Crossref]

Tveten, A. B.

R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48, 720–727 (2012).
[Crossref]

Tyndall, N.

Wanser, K. H.

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

Welton, T. A.

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[Crossref]

Wuytens, P.

A. Dhakal, P. Wuytens, A. Raza, N. Le Thomas, and R. Baets, “Silicon nitride background in nanophotonic waveguide enhanced Raman spectroscopy,” Materials 10, 140 (2017).
[Crossref]

A. Dhakal, A. Z. Subramanian, P. Wuytens, F. Peyskens, N. Le Thomas, and R. Baets, “Evanescent excitation and collection of spontaneous Raman spectra using silicon nitride nanophotonic waveguides,” Opt. Lett. 39, 4025–4028 (2014).
[Crossref]

Zanatta, A. R.

A. R. Zanatta and I. B. Gallo, “The thermo optic coefficient of amorphous SiN films in the near-infrared and visible regions and its experimental determination,” Appl. Phys. Express 6, 042402 (2013).
[Crossref]

Appl. Phys. Express (1)

A. R. Zanatta and I. B. Gallo, “The thermo optic coefficient of amorphous SiN films in the near-infrared and visible regions and its experimental determination,” Appl. Phys. Express 6, 042402 (2013).
[Crossref]

Appl. Phys. Lett. (1)

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Other (13)

Note that considering a non-vanishing correlation time τ of the random heat flux is equivalent to replacing the Fourier law with the Cattaneo law [22]. Cattaneo law is a damped version of the Fourier law where the relaxation time of the heat-flux is equal to the correlation time τ. In contrast to the Fourier law, it does not lead to infinite speed of propagation.

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Supplementary Material (1)

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» Supplement 1       Supplementary informations

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Figures (6)

Fig. 1.
Fig. 1. Positioning of the current model (blue box) against the conventional treatment of the thermo-refractive noise in waveguides. The optical spectrum S(Ω) is considered at high frequencies. See the main text for the definition of the different symbols.
Fig. 2.
Fig. 2. Theoretical optical spectra with standard thermal diffusion (blue line) and with extended correlation of the heat flux (black line). The parameters are those of a typical silicon nitride waveguide (see main text). Red dashed curve: outcome of Rytov’s theory, where quantum effects are included in addition to the standard thermal diffusion. The blue area corresponds to the frequency range relevant for Raman spectroscopy.
Fig. 3.
Fig. 3. Schematic representation of the propagating medium with the input and output optical fields. The energy distribution of the guided mode is represented by a Gaussian profile. The gray color pattern represents an arbitrary fluctuation of the temperature field δT(x,r,t).
Fig. 4.
Fig. 4. Optical spectrum of a single-frequency pump laser beam after propagating through a 1 cm long silicon nitride waveguide (SiN). It is normalized according to the maximum of the laser optical spectrum Spump(0) at the input of the waveguide and plotted versus the frequency Ω<0 relative to the frequency of the laser beam at the input of the waveguide. The spectral resolution of the spectrometer is dωspectro2π=9×1010  Hz. The inset shows a schematic of the cross section of the SiN waveguide on top of a 2 μm thick silicon oxide layer. Red curve: experimental data. Black curve: outcome of the model fit [see Eq. (10)]. Dotted curve: model fit with a characteristic time γ0 fixed to be equal to /(kBT0) and T0=298  K.
Fig. 5.
Fig. 5. Variation of the intensity S of the experimental spectral noise versus the temperature T normalized to the spectral noise S0 at T0=297.6  K. Squares with error bars: experimental data. Blue line: linear variation. Red line: quadratic variation. The error bars correspond to the alignment error of the light coupling into the waveguide.
Fig. 6.
Fig. 6. Variation of the intensity S0 of the experimental Stokes spectrum at Ω=4×1014  Hz versus the waveguide length. Dots with error bars: experimental data normalized to the value S0 at a length of 1 cm. Dashed line: linear fit.

Equations (10)

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ds=duθτρκ0θ2q.dq.
δ2s(δu,δu)=1CVT2(δu)2τκ0ρCV2T2δu·δu.
=(τκ0ρCV)1/2=τDT.
δT(r1)δT(r2)=kBT02ρCV3{2π2|r1r2|exp|r1r2|}.
δT(r1)δT(r2)=kBT02ρCV3{8ππexp|r1r2|22}.
Δφ(L;t)=k0L{+|A0m(r)|2δn(x,r;t)dS}dx.
Δφ(L;t1)Δφ(L;t2)*=2πk2Lκ|A(κ)|2ϕn(κ;t1,t2)dκ.
ϕn(κ,|t1t2|)=δn23exp(2+2σv2|t1t2|24κ2),
γ=2+2  W22σv,
S(Ω)=A02{δ(Ω)+4π2δn2Lλ0222+2W2γeγ|Ω|},

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