Abstract

A stochastic theory of nonstationary light describing the random emission of elementary pulses is presented. The emission is governed by a nonhomogeneous Poisson point process determined by a time-varying emission rate. The model describes, in the appropriate limits, stationary, cyclostationary, locally stationary, and pulsed radiation, and reduces to a Gaussian theory in the limit of dense emission rate. The first- and second-order coherence theories are solved after the computation of second- and fourth-order correlation functions by use of the characteristic function. The ergodicity of second-order correlations under various types of detectors is explored and a number of observables, including optical spectrum, amplitude, and intensity correlations, are analyzed.

© 2013 Optical Society of America

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References

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  1. J. W. Goodman, Statistical Optics (Wiley, 1985).
  2. W. A. Gardner, A. Napolitano, and L. Paura, “Cyclo-stationarity: half a century of research,” Signal Process. 86, 639–697 (2006).
  3. R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inform. Theory 3, 182–187 (1956).
  4. B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
    [CrossRef]
  5. E. Ip and J. M. Kahn, “Power spectra of return-to-zero optical signals,” J. Lightwave Technol. 24, 1610–1618 (2006).
    [CrossRef]
  6. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006).
    [CrossRef]
  7. J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
    [CrossRef]
  8. F. Gori and C. Palma, “Partially-coherent sources which give rise to highly directional beams,” Opt. Commun. 27, 185–188 (1978).
    [CrossRef]
  9. M. Korhonen, A. T. Friberg, J. Turunen, and G. Genty, “Elementary field representation of supercontinuum,” J. Opt. Soc. Am. B 30, 21–26 (2013).
    [CrossRef]
  10. R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. 43, 913–949 (1980).
    [CrossRef]
  11. S. H. Chen and P. Tartaglia, “Light scattering from N non-interacting particles,” Opt. Commun. 6, 119–124 (1972).
    [CrossRef]
  12. M. C. Teich and B. E. A. Saleh, “Branching processes in quantum electronics,” IEEE J. Selected Topics Quantum. Electron. 6, 1450–1457 (2000).
    [CrossRef]
  13. B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27, 360–374 (1983).
    [CrossRef]
  14. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  16. W. A. Gardner, Introduction to Random Processes (Macmillan, 1986).
  17. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).
  18. S. O. Rice, “Mathematical analysis of random noise,” Bell Systems Tech. J. 23, 282-332 (1944) and 24, 46–156 (1945). Reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003), Sect. 2.5.
  19. H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand, 1965), Sect. 2.3.
  20. J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977).
    [CrossRef]
  21. M. Nazarathy, W. V. Sorin, D. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).
    [CrossRef]
  22. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394–396 (2004).
    [CrossRef]

2013 (1)

2011 (1)

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

2007 (1)

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

2006 (3)

2004 (1)

2000 (1)

M. C. Teich and B. E. A. Saleh, “Branching processes in quantum electronics,” IEEE J. Selected Topics Quantum. Electron. 6, 1450–1457 (2000).
[CrossRef]

1989 (1)

M. Nazarathy, W. V. Sorin, D. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).
[CrossRef]

1983 (1)

B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27, 360–374 (1983).
[CrossRef]

1980 (1)

R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. 43, 913–949 (1980).
[CrossRef]

1978 (1)

F. Gori and C. Palma, “Partially-coherent sources which give rise to highly directional beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

1977 (1)

1972 (1)

S. H. Chen and P. Tartaglia, “Light scattering from N non-interacting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

1956 (1)

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inform. Theory 3, 182–187 (1956).

Agrawal, G. P.

Baney, D. M.

M. Nazarathy, W. V. Sorin, D. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).
[CrossRef]

Chen, S. H.

S. H. Chen and P. Tartaglia, “Light scattering from N non-interacting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

Davis, B. J.

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

Eberly, J. H.

Friberg, A. T.

Gardner, W. A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclo-stationarity: half a century of research,” Signal Process. 86, 639–697 (2006).

W. A. Gardner, Introduction to Random Processes (Macmillan, 1986).

Genty, G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gori, F.

F. Gori and C. Palma, “Partially-coherent sources which give rise to highly directional beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Ip, E.

Kahn, J. M.

Korhonen, M.

Loudon, R.

R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. 43, 913–949 (1980).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Napolitano, A.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclo-stationarity: half a century of research,” Signal Process. 86, 639–697 (2006).

Nazarathy, M.

M. Nazarathy, W. V. Sorin, D. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).
[CrossRef]

Newton, S. A.

M. Nazarathy, W. V. Sorin, D. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).
[CrossRef]

Palma, C.

F. Gori and C. Palma, “Partially-coherent sources which give rise to highly directional beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).

Paura, L.

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclo-stationarity: half a century of research,” Signal Process. 86, 639–697 (2006).

Ponomarenko, S. A.

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Systems Tech. J. 23, 282-332 (1944) and 24, 46–156 (1945). Reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003), Sect. 2.5.

Rowe, H. E.

H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand, 1965), Sect. 2.3.

Saleh, B. E. A.

M. C. Teich and B. E. A. Saleh, “Branching processes in quantum electronics,” IEEE J. Selected Topics Quantum. Electron. 6, 1450–1457 (2000).
[CrossRef]

B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27, 360–374 (1983).
[CrossRef]

Silverman, R. A.

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inform. Theory 3, 182–187 (1956).

Sorin, W. V.

M. Nazarathy, W. V. Sorin, D. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).
[CrossRef]

Stoler, D.

B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27, 360–374 (1983).
[CrossRef]

Tartaglia, P.

S. H. Chen and P. Tartaglia, “Light scattering from N non-interacting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

Teich, M. C.

M. C. Teich and B. E. A. Saleh, “Branching processes in quantum electronics,” IEEE J. Selected Topics Quantum. Electron. 6, 1450–1457 (2000).
[CrossRef]

B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27, 360–374 (1983).
[CrossRef]

Turunen, J.

Vahimaa, P.

Wódkiewicz, K.

Wolf, E.

IEEE J. Selected Topics Quantum. Electron. (1)

M. C. Teich and B. E. A. Saleh, “Branching processes in quantum electronics,” IEEE J. Selected Topics Quantum. Electron. 6, 1450–1457 (2000).
[CrossRef]

IRE Trans. Inform. Theory (1)

R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inform. Theory 3, 182–187 (1956).

J. Lightwave Technol. (2)

E. Ip and J. M. Kahn, “Power spectra of return-to-zero optical signals,” J. Lightwave Technol. 24, 1610–1618 (2006).
[CrossRef]

M. Nazarathy, W. V. Sorin, D. M. Baney, and S. A. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).
[CrossRef]

J. Mod. Opt. (1)

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

S. H. Chen and P. Tartaglia, “Light scattering from N non-interacting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

F. Gori and C. Palma, “Partially-coherent sources which give rise to highly directional beams,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

B. E. A. Saleh, D. Stoler, and M. C. Teich, “Coherence and photon statistics for optical fields generated by Poisson random emissions,” Phys. Rev. A 27, 360–374 (1983).
[CrossRef]

Rep. Prog. Phys. (1)

R. Loudon, “Non-classical effects in the statistical properties of light,” Rep. Prog. Phys. 43, 913–949 (1980).
[CrossRef]

Signal Process. (1)

W. A. Gardner, A. Napolitano, and L. Paura, “Cyclo-stationarity: half a century of research,” Signal Process. 86, 639–697 (2006).

Other (7)

J. W. Goodman, Statistical Optics (Wiley, 1985).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

W. A. Gardner, Introduction to Random Processes (Macmillan, 1986).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

S. O. Rice, “Mathematical analysis of random noise,” Bell Systems Tech. J. 23, 282-332 (1944) and 24, 46–156 (1945). Reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes (Dover, 2003), Sect. 2.5.

H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand, 1965), Sect. 2.3.

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Figures (3)

Fig. 1.
Fig. 1.

Representation of the optical wave of elementary-field light randomly triggered by Poisson impulses.

Fig. 2.
Fig. 2.

Schematics of the Fourier transforms Λ ( ν ) and E 0 ( ν ) of the emission rate λ ( t ) and the elementary pulse e 0 ( t ) . B , bandwidth of the emission rate; ν o , optical central frequency; Δ ν , optical spectral width; T s , T i , T f , integration times of slow, intermediate and fast detectors, respectively.

Fig. 3.
Fig. 3.

Scheme of the intensity interferometer.

Equations (68)

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W e ( ν 1 , ν 2 ) = E * ( ν 1 ) E ( ν 2 ) = E 0 * ( ν 1 ) G ( ν 2 ν 1 ) E 0 ( ν 2 ) .
e ( t ) = k e 0 ( t τ k ) = e 0 ( t ) k δ ( t τ k ) ,
λ ( t ) = λ ¯ + n = 1 N ( λ n e j 2 π f n t + cc ) + Δ λ ( t ) ,
Φ ( s ) = exp ( q = 2 , 4 s q e ( t q ) + q = 1 , 3 s q e * ( t q ) ) ,
Φ ( s ) = exp d u λ ( u ) [ exp β ( u , s ) 1 ]
β ( u , s ) = q = 2 , 4 s q e 0 ( t q u ) + q = 1 , 3 s q e 0 * ( t q u ) .
e ( t 2 ) = Φ ( s ) s 2 | s = 0 = d u λ ( u ) e 0 ( t 2 u ) = λ ( t 2 ) e 0 ( t 2 ) .
Γ e ( t 1 , t 2 ) = e * ( t 1 ) e ( t 2 ) = d u λ ( u ) e 0 * ( t 1 u ) e 0 ( t 2 u ) .
Γ e ( t , t + τ ) = λ ¯ Γ 0 ( τ ) + n = 1 N [ λ n A 0 ( f n , τ ) e j 2 π f n t + λ n * A 0 ( f n , τ ) e j 2 π f n t ] + Δ Γ e ( t , t + τ ) ,
Γ 0 ( τ ) = d u e 0 * ( u ) e 0 ( u + τ )
A 0 ( f , τ ) = d u e 0 * ( u ) e 0 ( u + τ ) e j 2 π f u
Δ Γ e ( t , t + τ ) = d u Δ λ ( t u ) e 0 * ( u ) e 0 ( u + τ )
W e ( ν 1 , ν 2 ) = E * ( ν 1 ) E ( ν 2 ) = E 0 * ( ν 1 ) Λ ( ν 2 ν 1 ) E 0 ( ν 2 ) .
W e ( ν 1 , ν 2 ) = λ ¯ | E 0 ( ν 1 ) | 2 δ ( ν 2 ν 1 ) + n [ λ n E 0 * ( ν 1 ) E 0 ( ν 2 ) δ ( ν 2 ν 1 f n ) + λ n * E 0 ( ν 1 ) E 0 * ( ν 2 ) δ ( ν 2 ν 1 + f n ) ] + E 0 * ( ν 1 ) Δ Λ ( ν 2 ν 1 ) E 0 ( ν 2 ) .
e ( t 2 ) e ( t 4 ) = d u λ ( u ) e 0 ( t 2 u ) e 0 ( t 4 u ) FT Λ ( ν 2 + ν 4 ) E 0 ( ν 2 ) E 0 ( ν 4 ) ,
f ( t , x , y , ) ¯ = lim L 1 2 L L L d t f ( t , x , y ) .
γ ¯ ( τ ) = Γ e ( t , t + τ ) ¯ Γ e ( t , t ) ¯ = e * ( t ) e ( t + τ ) ¯ i ( t ) ¯ ,
t ¯ c = d τ | γ ¯ ( τ ) | 2 .
Γ e ( t , t + τ ) ¯ = λ ¯ Γ 0 ( τ ) .
Γ 0 ( 0 ) = d u | e 0 ( u ) | 2 = d u p ( u ) ε p ,
t ¯ c = 1 ε p 2 | Γ 0 ( τ ) | 2 d τ .
Γ e ( t 1 , t 2 , t 3 , t 4 ) = e * ( t 1 ) e ( t 2 ) e * ( t 3 ) e ( t 4 ) = Γ e ( t 1 , t 2 ) Γ e ( t 3 , t 4 ) + Γ e ( t 1 , t 4 ) Γ e ( t 3 , t 2 ) + d η λ ( η ) e 0 * ( t 1 η ) e 0 ( t 2 η ) e 0 * ( t 3 η ) e 0 ( t 4 η ) .
lim κ Φ ( s ) = exp ( d u λ ( u ) β ( u , s ) 2 / 2 ) = exp [ k = 1 , 3 n = 2 , 4 s k s n Γ e ( t k , t n ) ] ,
W e ( ν 1 , ν 2 ) Λ ( ν 1 ν 2 ) | E 0 ( ν 2 ) | 2
Γ e ( t , t + τ ) λ ( t ) Γ 0 ( τ ) λ ( t + τ ) Γ 0 ( τ )
d η λ ( η ) e 0 * ( t 1 η ) e 0 ( t 2 η ) e 0 * ( t 3 η ) e 0 ( t 4 η ) = d η λ ( t 1 η ) e 0 * ( η ) × e 0 ( t 2 t 1 + η ) e 0 * ( t 3 t 1 + η ) e 0 ( t 4 t 1 + η ) ,
λ ( t ) d η e 0 * ( η ) e 0 ( η + u ) e 0 * ( η + w ) e 0 ( η + v ) = λ ( t ) Γ 0 ( u , w , v ) ,
Γ ^ T ( t , τ ) = 1 T t T t d η e * ( η ) e ( η + τ )
lim T | Γ ^ T ( t , τ ) Γ ( τ ) | 2 = 0 .
1 / T s f n B 1 / T i Δ ν 1 / t ¯ c 1 / T f .
Γ ^ T i ( t , τ ) = 1 T i t T i t d η e * ( η ) e ( η + τ ) = 1 T i t T i t d η λ ( η ) Γ 0 ( τ ) = λ ( t ) Γ 0 ( τ ) = Γ e ( t , t + τ ) .
| Γ ^ T i ( t , τ ) Γ e ( t , t + τ ) | 2 Γ e ( t , t ) Γ e ( t + τ , t + τ ) t ¯ c T i + 1 λ ( t ) T i .
Γ ^ T s ( t , τ ) = 1 T s t T s t d η λ ( η ) Γ 0 ( τ ) = λ ¯ Γ 0 ( τ ) = Γ e ( t , t + τ ) ¯ ,
i ( t ) = d u λ ( t u ) | e 0 ( u ) | 2 = λ ( t ) p ( t ) .
i ( t ) ¯ = d u λ ( t u ) ¯ p ( u ) = λ ¯ d u p ( u ) = λ ¯ ε p .
G ( ν ) = d τ Γ e ( t , t + τ ) ¯ e j 2 π ν τ .
G ^ δ ν , T s ( ν , t ) = 1 T s t T s t d u | h ν , δ ν ( u ) e ( u ) | 2 .
| h ν , δ ν ( t ) e ( t ) | 2 ¯ = d u d v h ν , δ ν * ( u ) h ν , δ ν ( v ) Γ e ( t u , t v ) ¯ = 0 d η | H ν , Δ ν ( η ) | 2 G ( η ) .
G ( ν ) = λ ¯ | E 0 ( ν ) | 2 .
1 T s t T s t d t | e ( t ) + e ( t + τ ) | 2 .
| e ( t ) + e ( t + τ ) | 2 ¯ = 2 i ( t ) ¯ + 2 Re Γ e ( t , t + τ ) ¯ = 2 λ ¯ ε p + 2 λ ¯ Re Γ 0 ( τ ) ,
Γ e ( t , t + τ ) = α = N N λ α A 0 ( f α , τ ) e j 2 π f α t + Δ Γ e ( t , t + τ ) ,
Δ i ( t ) Δ i ( t + τ ) ¯ = i ( t ) i ( t + τ ) ¯ i ( t ) i ( t + τ ) ¯ = α | λ α A 0 ( f α , τ ) | 2 + λ ¯ Γ p ( τ ) ,
λ ( t u ) λ ( t v ) ¯ = α = N N | λ α | 2 e j 2 π f α ( v u ) ,
Γ p ( τ ) = d u p ( u ) p ( u + τ ) .
Δ i ( t ) Δ i ( t + τ ) ¯ = λ 2 ¯ | Γ 0 ( τ ) | 2 + λ ¯ Γ p ( τ ) .
Δ i ( t ) Δ i ( t + τ ) ¯ = | Γ e ( t , t + τ ) | 2 ¯ + i ( t ) ¯ ξ ( τ ) ,
Δ i ( t ) 2 ¯ = i ( t ) 2 ¯ + i ( t ) ¯ ξ ( 0 ) .
i ( t ) i ( t + τ ) ¯ = α = N N | λ α P ( f α ) | 2 e j 2 π f α τ ,
A 0 ( f , 0 ) = d u p ( u ) e 2 π j f u = P ( f ) .
S ( f ) = d τ i ( t ) i ( t + τ ) ¯ e j 2 π f τ .
S ( f ) = | P ( f ) | 2 α | λ α | 2 δ ( f f α ) + α | λ α | 2 d τ | A 0 ( f α , τ ) | 2 e j 2 π f τ + λ ¯ P ( f ) P ( f ) .
S ( f ) = | P ( f ) | 2 α | λ α | 2 δ ( f f α ) + λ 2 ¯ | E 0 ( f ) | 2 | E 0 ( f ) | 2 + λ ¯ P ( f ) P ( f ) ,
S ( f ) = i ( t ) 2 δ ( f ) + G ( f ) G ( f ) + λ ¯ P ( f ) P ( f ) ,
λ ( t ) = n = δ ( t n T 0 ) = 1 T 0 n = e j 2 π n t / T 0 .
e ( t ) = n = e 0 ( t n T 0 ) e j θ n
Γ ( t 1 , t 2 ) = n e 0 * ( t 1 n T 0 ) e 0 ( t 2 n T 0 ) ,
λ ( t ) = n = Δ λ ( t n T 0 ) ,
| Γ ^ T i ( t , τ ) Γ e ( t , t + τ ) | 2 = | Γ ^ T i ( t , τ ) | 2 | Γ e ( t , t + τ ) | 2 = | Γ ^ T i ( t , τ ) | 2 λ ( t ) 2 | Γ 0 ( τ ) | 2 .
1 T i 2 t T i t d η d η [ λ ( η ) λ ( η + τ ) | Γ 0 ( η η ) | 2 + λ ( η ) Γ 0 ( τ , τ + η η , η η ) ] .
t T i t d η d η | Γ 0 ( η η ) | 2 = 2 0 T i d u ( T i u ) | Γ 0 ( u ) | 2 T i d u | Γ 0 ( u ) | 2 = T i ε p 2 t ¯ c ,
| Γ 0 ( u , u + v , v ) | [ d η | e 0 * ( η ) e 0 ( η + v ) | 2 d η | e 0 * ( η + u + v ) e 0 ( η + u ) | 2 ] 1 / 2 = d η | e 0 * ( η ) e 0 ( η + v ) | 2 = d η p ( η ) p ( η + v ) = Γ p ( v ) .
t T i t d η d η | Γ 0 ( τ , τ + η η , η η ) | t T i t d η d η Γ p ( η η ) 2 0 T i d u ( T i u ) Γ p ( u ) 2 T i 0 d u Γ p ( u ) = T i d u d τ p ( τ ) p ( τ + u ) = T i ε p 2 ,
t k T i t k t p T i t p d η d η λ ( η ) λ ( η + τ ) | Γ 0 ( η η ) | 2 λ ( t k ) λ ( t p + τ ) t k T i t k t p T i t p d η d η | Γ 0 ( η η ) | 2 .
1 T s 2 k = 1 M λ ( t k ) λ ( t k + τ ) = 1 T s T i 1 T s k = 1 M λ ( t k ) λ ( t k + τ ) T i = 1 T s T i t T s t d u T s λ ( u ) λ ( u + τ ) Γ λ ( t , t + τ ) ¯ T s T i ,
1 T s 2 k = 1 M λ ( t k ) = 1 T s T i 1 M k = 1 M λ ( t k ) = λ ¯ T s T i .
| Γ ^ T s ( t , τ ) Γ e ( t , t + τ ) ¯ | 2 Γ e ( t , t ) ¯ 2 t ¯ c T s Γ λ ( t , t + τ ) ¯ λ ¯ 2 + 1 λ ¯ T s
1 T s 2 t T s t d η d η λ ( η ) λ ( η + τ ) | Γ 0 ( η η ) | 2 λ B 2 T s 2 t T s t d η d η | Γ 0 ( η η ) | 2 λ B 2 T s ε p 2 t ¯ c ,

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