Abstract

The higher-order coherence functions of optical fields involving phase fluctuations have not yet been studied extensively. We discuss here an experiment which can in principle yield the fourth-order coherence function. This experiment is a study of the correlation function of the irradiance fluctuations obtained in some interference experiments. Two kinds of delay are then used: optical delays in the interferometer and electrical delays with suitable variations in the correlator.

The results of the experiment are calculated theoretically with a particular statistical model of laser light. The possibility of using a spectrum analyzer instead of a correlator is discussed.

© 1968 Optical Society of America

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References

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  1. B. Picinbono and E. Boileau, J. Opt. Soc. Am. 58, 784 (1968).
    [Crossref]
  2. R. Hanbury Brown, J. Phys. Radium 20, 898 (1959).
    [Crossref]
  3. C. Freed and H. A. Haus, IEEE Quantum Electronics QE-2, 190 (1966).
    [Crossref]
  4. J. A. Armstrong and A. W. Smith, Phys. Rev. 140, 1A, 155 (1965).
    [Crossref]
  5. J. A. Armstrong and A. W. Smith, in Progress in Optics 6, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1967).
  6. A. E. Siegman, B. Daino, and K. R. Manes, IEEE Quantum Electronics QE-3, 180 (1967).
    [Crossref]
  7. J. A. Armstrong, J. Opt. Soc. Am. 56, 1024 (1966).
    [Crossref]
  8. D. Dialetis and E. Wolf, Nuovo Cimento 47, 113 (1967).
  9. H. Risken, Z. Physik 186, 85 (1965); H. Risken, C. Schmid, and W. Weidlich, 194, 337 (1966).
    [Crossref]
  10. W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966); Phys. Rev. 145, 286 (1966); F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966); F. Arecchi, E. Gatti, and A. Sona, Phys. Letters 20, 27 (1966).
    [Crossref]
  11. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).
  12. A. Blaquière, Ann. Radioélec.8, 36, 153 (1963); Analyse des systemes non linéaires (Presses Universitaires de France, Paris, 1966), p. 375.
  13. B. Picinbono and E. Boileau, Compt. Rend. 261, 5028 (1966).
  14. C. Freed and H. A. Haus, Phys. Rev. 141, 287 (1966).
    [Crossref]
  15. K. Shimoda and A. Javan, J. Appl. Phys. 36, 718 (1965).
    [Crossref]
  16. M. S. Lipsett and P. H. Lee, Appl. Opt. 5, 223 (1966).
    [Crossref]
  17. W. E. Ahearn and J. W. Crowe, IEEE Quantum Electronics QE-2, 597 (1966).
    [Crossref]
  18. A. Blanc–Lapierre and R. Fortet, Théorie des fonctions aléatoires (Masson, Paris, 1953).

1968 (1)

1967 (2)

A. E. Siegman, B. Daino, and K. R. Manes, IEEE Quantum Electronics QE-3, 180 (1967).
[Crossref]

D. Dialetis and E. Wolf, Nuovo Cimento 47, 113 (1967).

1966 (7)

J. A. Armstrong, J. Opt. Soc. Am. 56, 1024 (1966).
[Crossref]

C. Freed and H. A. Haus, IEEE Quantum Electronics QE-2, 190 (1966).
[Crossref]

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966); Phys. Rev. 145, 286 (1966); F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966); F. Arecchi, E. Gatti, and A. Sona, Phys. Letters 20, 27 (1966).
[Crossref]

B. Picinbono and E. Boileau, Compt. Rend. 261, 5028 (1966).

C. Freed and H. A. Haus, Phys. Rev. 141, 287 (1966).
[Crossref]

M. S. Lipsett and P. H. Lee, Appl. Opt. 5, 223 (1966).
[Crossref]

W. E. Ahearn and J. W. Crowe, IEEE Quantum Electronics QE-2, 597 (1966).
[Crossref]

1965 (3)

K. Shimoda and A. Javan, J. Appl. Phys. 36, 718 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, 1A, 155 (1965).
[Crossref]

H. Risken, Z. Physik 186, 85 (1965); H. Risken, C. Schmid, and W. Weidlich, 194, 337 (1966).
[Crossref]

1959 (1)

R. Hanbury Brown, J. Phys. Radium 20, 898 (1959).
[Crossref]

Ahearn, W. E.

W. E. Ahearn and J. W. Crowe, IEEE Quantum Electronics QE-2, 597 (1966).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, J. Opt. Soc. Am. 56, 1024 (1966).
[Crossref]

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, 1A, 155 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, in Progress in Optics 6, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1967).

Blanc–Lapierre, A.

A. Blanc–Lapierre and R. Fortet, Théorie des fonctions aléatoires (Masson, Paris, 1953).

Blaquière, A.

A. Blaquière, Ann. Radioélec.8, 36, 153 (1963); Analyse des systemes non linéaires (Presses Universitaires de France, Paris, 1966), p. 375.

Boileau, E.

B. Picinbono and E. Boileau, J. Opt. Soc. Am. 58, 784 (1968).
[Crossref]

B. Picinbono and E. Boileau, Compt. Rend. 261, 5028 (1966).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).

Crowe, J. W.

W. E. Ahearn and J. W. Crowe, IEEE Quantum Electronics QE-2, 597 (1966).
[Crossref]

Daino, B.

A. E. Siegman, B. Daino, and K. R. Manes, IEEE Quantum Electronics QE-3, 180 (1967).
[Crossref]

Dialetis, D.

D. Dialetis and E. Wolf, Nuovo Cimento 47, 113 (1967).

Fortet, R.

A. Blanc–Lapierre and R. Fortet, Théorie des fonctions aléatoires (Masson, Paris, 1953).

Freed, C.

C. Freed and H. A. Haus, IEEE Quantum Electronics QE-2, 190 (1966).
[Crossref]

C. Freed and H. A. Haus, Phys. Rev. 141, 287 (1966).
[Crossref]

Hanbury Brown, R.

R. Hanbury Brown, J. Phys. Radium 20, 898 (1959).
[Crossref]

Haus, H. A.

C. Freed and H. A. Haus, IEEE Quantum Electronics QE-2, 190 (1966).
[Crossref]

C. Freed and H. A. Haus, Phys. Rev. 141, 287 (1966).
[Crossref]

Javan, A.

K. Shimoda and A. Javan, J. Appl. Phys. 36, 718 (1965).
[Crossref]

Lee, P. H.

M. S. Lipsett and P. H. Lee, Appl. Opt. 5, 223 (1966).
[Crossref]

Lipsett, M. S.

M. S. Lipsett and P. H. Lee, Appl. Opt. 5, 223 (1966).
[Crossref]

Manes, K. R.

A. E. Siegman, B. Daino, and K. R. Manes, IEEE Quantum Electronics QE-3, 180 (1967).
[Crossref]

Martienssen, W.

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966); Phys. Rev. 145, 286 (1966); F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966); F. Arecchi, E. Gatti, and A. Sona, Phys. Letters 20, 27 (1966).
[Crossref]

Picinbono, B.

B. Picinbono and E. Boileau, J. Opt. Soc. Am. 58, 784 (1968).
[Crossref]

B. Picinbono and E. Boileau, Compt. Rend. 261, 5028 (1966).

Risken, H.

H. Risken, Z. Physik 186, 85 (1965); H. Risken, C. Schmid, and W. Weidlich, 194, 337 (1966).
[Crossref]

Shimoda, K.

K. Shimoda and A. Javan, J. Appl. Phys. 36, 718 (1965).
[Crossref]

Siegman, A. E.

A. E. Siegman, B. Daino, and K. R. Manes, IEEE Quantum Electronics QE-3, 180 (1967).
[Crossref]

Smith, A. W.

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, 1A, 155 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, in Progress in Optics 6, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1967).

Spiller, E.

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966); Phys. Rev. 145, 286 (1966); F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966); F. Arecchi, E. Gatti, and A. Sona, Phys. Letters 20, 27 (1966).
[Crossref]

Wolf, E.

D. Dialetis and E. Wolf, Nuovo Cimento 47, 113 (1967).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).

Appl. Opt. (1)

M. S. Lipsett and P. H. Lee, Appl. Opt. 5, 223 (1966).
[Crossref]

Compt. Rend. (1)

B. Picinbono and E. Boileau, Compt. Rend. 261, 5028 (1966).

IEEE Quantum Electronics (3)

W. E. Ahearn and J. W. Crowe, IEEE Quantum Electronics QE-2, 597 (1966).
[Crossref]

C. Freed and H. A. Haus, IEEE Quantum Electronics QE-2, 190 (1966).
[Crossref]

A. E. Siegman, B. Daino, and K. R. Manes, IEEE Quantum Electronics QE-3, 180 (1967).
[Crossref]

J. Appl. Phys. (1)

K. Shimoda and A. Javan, J. Appl. Phys. 36, 718 (1965).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. Radium (1)

R. Hanbury Brown, J. Phys. Radium 20, 898 (1959).
[Crossref]

Nuovo Cimento (1)

D. Dialetis and E. Wolf, Nuovo Cimento 47, 113 (1967).

Phys. Rev. (2)

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, 1A, 155 (1965).
[Crossref]

C. Freed and H. A. Haus, Phys. Rev. 141, 287 (1966).
[Crossref]

Phys. Rev. Letters (1)

W. Martienssen and E. Spiller, Phys. Rev. Letters 16, 531 (1966); Phys. Rev. 145, 286 (1966); F. T. Arecchi, A. Berné, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966); F. Arecchi, E. Gatti, and A. Sona, Phys. Letters 20, 27 (1966).
[Crossref]

Z. Physik (1)

H. Risken, Z. Physik 186, 85 (1965); H. Risken, C. Schmid, and W. Weidlich, 194, 337 (1966).
[Crossref]

Other (4)

J. A. Armstrong and A. W. Smith, in Progress in Optics 6, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1967).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).

A. Blaquière, Ann. Radioélec.8, 36, 153 (1963); Analyse des systemes non linéaires (Presses Universitaires de France, Paris, 1966), p. 375.

A. Blanc–Lapierre and R. Fortet, Théorie des fonctions aléatoires (Masson, Paris, 1953).

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Figures (6)

Fig. 1
Fig. 1

Principles of experiment. In M and M′ we observe with two photodetectors interference from the sources S1 and S2. The signals are multiplied and averaged to obtain the correlation function. By varying the delays τ, we can obtain the fourth-order coherence functions.

Fig. 2
Fig. 2

Experiment with a Michelson interferometer. τ is the optical delay and θ is performed on the electric signal.

Fig. 3
Fig. 3

(a) Variation of the irradiance of the interference field; MI is a bright fringe and MIII a dark fringe. (b) and (c): F(τ,θ) and G1(τ,θ) are defined by

Fig. 4
Fig. 4

Variation of the parameters involved in Fig. 3 as θ is varied (with ττφ and ττb).

Fig. 5
Fig. 5

Case ττb, the curves of P and F(τ,θ) are not much different from those of Fig. 4, but G1(τ,θ) is different.

Fig. 6
Fig. 6

Dependence on the statistical model. The same curves as in Fig. 4(b), but with another statistical model of the light. τ is arbitrary, fixed.

Equations (53)

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J ( t ) = Z ( t - τ 1 ) + Z ( t - τ 2 ) 2
J ( t ) = Z ( t - τ 3 ) + Z ( t - τ 4 ) 2 .
E [ Z ( t 1 ) Z * ( t 2 ) ] = Γ ( t 1 - t 2 ) ,
E [ Z ( t 1 ) Z ( t 2 ) Z * ( t 3 ) Z * ( t 4 ) ] = Γ ( t 1 , t 2 , t 3 , t 4 ) ,
I ( t ) = Z ( t ) 2             and             I = E [ I ( t ) ] ,
E [ J ( t ) J ( t ) ] = Γ ( - τ 1 , - τ 3 , - τ 1 , - τ 3 ) + Γ ( - τ 1 , - τ 4 , - τ 1 , - τ 4 ) + Γ ( - τ 2 , - τ 3 , - τ 2 , - τ 3 ) + Γ ( - τ 2 , - τ 4 , - τ 2 , - τ 4 ) + 2 Re [ Γ ( - τ 1 , - τ 3 , - τ 1 , - τ 4 ) + Γ ( - τ 2 , - τ 3 , - τ 2 , - τ 4 ) + Γ ( - τ 1 , - τ 3 , - τ 2 , - τ 3 ) + Γ ( - τ 1 , - τ 4 , - τ 2 , - τ 4 ) + Γ ( - τ 1 , - τ 3 , - τ 2 , - τ 4 ) + Γ ( - τ 1 , - τ 4 , - τ 2 , - τ 3 ) ] .
Γ ( - τ i , - τ j , - τ i , - τ j ) = Γ ( t - τ i , t - τ j , t - τ i , t - τ j ) = E [ I ( t - τ i ) I ( t - τ j ) ] ,
Γ ( - τ i , - τ j , - τ i , - τ k )
E [ I ( t - τ i ) Z ( t - τ j ) Z * ( t - τ k ) ] .
Γ ( t 1 , t 2 ) = exp [ i ω 0 ( t 1 - t 2 ) ] γ ( t 1 - t 2 ) Γ ( t 1 , t 2 , t 3 , t 4 ) = exp [ i ω 0 ( t 1 + t 2 - t 3 - t 4 ) ] × γ ( t 1 - t 2 , t 1 - t 3 , t 1 - t 4 ) .
τ = τ 1 - τ 2 ;             τ = τ 3 - τ 4 ;             θ = τ 1 - τ 3 ,
E [ J ( t ) ] = 2 [ I + γ ( τ ) cos ω 0 τ ] E [ J ( t ) ] = 2 [ I + γ ( τ ) cos ω 0 τ ]
E [ J ( t ) J ( t ) ] = γ ( θ , 0 , θ ) + γ ( θ + τ , 0 , θ + τ ) + γ ( θ - τ , 0 , θ - τ ) + γ ( θ + τ - τ , 0 , θ + τ - τ ) + 2 [ γ ( θ , 0 , θ + τ ) + γ ( θ - τ , 0 , θ + τ - τ ) ] cos ω 0 τ + 2 [ γ ( θ , τ , θ ) + γ ( θ + τ , τ , θ + τ ) ] cos ω 0 τ + 2 [ γ ( θ , τ , θ + τ ) cos ω 0 ( τ + τ ) + γ ( θ + τ , τ , θ ) cos ω 0 ( τ - τ ) ] .
E [ Δ J Δ J ] = E [ J J ] - E [ J ] E [ J ] = S ( τ , τ , θ ) S ( τ , τ , θ ) = γ ( θ , 0 , θ ) + γ ( θ + τ , 0 , θ + τ ) + γ ( θ - τ , 0 , θ - τ ) + γ ( θ + τ - τ , 0 , θ + τ - τ ) - 4 I 2 + 2 [ γ ( θ , 0 , θ + τ ) + γ ( θ - τ , 0 , θ + τ - τ ) - 2 I γ ( τ ) ] cos ω 0 τ + 2 [ γ ( θ , τ , θ ) + γ ( θ + τ , τ , θ + τ ) - 2 I γ ( τ ) ] cos ω 0 τ + 2 [ γ ( θ , τ , θ + τ ) cos ω 0 ( τ + τ ) + γ ( θ + τ , τ , θ ) cos ω 0 ( τ - τ ) ] - 4 γ ( τ ) γ ( τ ) cos ω 0 τ cos ω 0 τ .
S ( τ , τ , θ ) = B + C cos ω 0 x + D cos 2 ω 0 x ,
D = 2 [ γ ( θ , τ 0 , θ + τ 0 ) - γ ( τ 0 ) γ ( τ 0 ) ] .
Z ( t ) = A [ 1 + B ( t ) ] exp i [ ω 0 t + Φ ( t ) ] ,
γ ( τ ) = A 2 γ b ( τ ) γ φ ( τ ) , γ ( τ 1 , τ 2 , τ 3 ) = A 4 γ b ( τ 1 , τ 2 , τ 3 ) γ φ ( τ 1 , τ 2 , τ 3 ) ,
( 1 / A 4 ) E [ Δ J ( t ) Δ J ( t + θ ) ] = 2 γ ˜ b ( θ , θ , 0 ) + γ ˜ b ( θ + τ , θ + τ , 0 ) + γ ˜ b ( θ - τ , θ - τ , 0 ) + 2 γ φ ( τ ) cos ω 0 τ [ γ ˜ b ( θ , 0 , θ + τ ) + γ ˜ b ( θ - τ , 0 , θ ) + γ ˜ b ( θ , τ , θ ) + γ ˜ b ( θ + τ , τ , θ + τ ) ] + 2 γ b ( θ , τ , θ + τ ) × [ γ φ ( θ , τ , θ + τ ) cos 2 ω 0 τ + γ φ ( θ + τ , τ , θ ) ] - 2 γ b 2 ( τ ) γ φ 2 ( τ ) [ cos 2 ω 0 τ + 1 ] .
E [ B ( t ) B ( t - τ ) ] = σ b 2 C b ( τ ) , E [ F ( t ) F ( t - τ ) ] = C F ( τ ) .
γ b ( τ 1 , τ 2 , τ 3 ) = 1 + σ b 2 [ i C b ( τ i ) + i > j C b ( τ i - τ j ) ] + σ b 4 i j k C b ( τ i ) C b ( τ j - τ k ) ,
γ φ ( τ ) = exp - 0 τ ( τ - τ ) C F ( τ ) d τ ,
γ φ ( τ 1 , τ 2 , τ 3 ) = γ φ ( τ 2 ) γ φ ( τ 3 ) γ φ ( τ 2 - τ 1 ) γ φ ( τ 3 - τ 1 ) γ φ ( τ 1 ) γ φ ( τ 3 - τ 2 )
( 1 / A 4 ) E [ Δ J ( t ) Δ J ( t + θ ) ] = F ( τ , θ ) + σ b 2 G ( τ , θ ) + σ b 4 H ( τ , θ ) .
P = γ φ ( θ + τ ) γ φ ( θ - τ ) / γ φ 2 ( θ ) ,
F ( τ , θ ) = 2 γ φ 2 ( τ ) [ ( 1 - P ) / P ] ( 1 - P cos 2 ω 0 τ ) ,
G ( τ , θ ) = 2 C b ( τ ) F ( τ , θ ) + 2 [ 2 C b ( θ ) + C b ( θ + τ ) + C b ( θ - τ ) ] × [ 2 + 4 γ φ ( τ ) cos ω 0 τ + γ φ 2 ( τ ) ( P cos 2 ω 0 τ + 1 / P ) ] ,
H ( τ , θ ) = C b 2 ( τ ) F ( τ , θ ) + 2 { 2 C b 2 ( θ ) + C b 2 ( θ + τ ) + C b 2 ( θ - τ ) + 4 C b ( θ ) [ C b ( θ + τ ) + C b ( θ - τ ) ] γ φ ( τ ) cos ω 0 τ + [ C b 2 ( θ ) + C b ( θ + τ ) C b ( θ - τ ) ] γ φ 2 ( τ ) × ( P cos 2 ω 0 τ + 1 / P ) } .
( 1 / A 4 ) E [ Δ J ( t ) Δ J ( t + θ ) ] = 32 σ b 2 C b ( θ ) [ 2 + σ b 2 C b ( θ ) ] .
F ( τ , 0 ) = 2 [ 1 - γ φ 2 ( τ ) ] [ 1 - γ φ 2 ( τ ) cos 2 ω 0 τ ] .
1 A 4 E [ Δ J ( t ) Δ J ( t + θ ) ] ( 1 + σ b 2 ) 2 F ( τ , θ ) + 4 σ b 2 C b ( θ ) [ 2 + σ b 2 C b ( θ ) ] × [ 2 + 4 γ φ ( τ ) cos ω 0 τ + γ φ 2 ( τ ) ( P cos 2 ω 0 τ + 1 P ) ] .
1 A 4 E [ Δ J ( t ) Δ J ( t + θ ) ] F ( τ , θ ) + 8 σ b 2 C b ( θ ) [ 2 + 4 γ φ ( τ ) cos ω 0 τ + γ φ ( τ ) ( P cos 2 ω 0 τ + 1 P ) ] = F ( τ , θ ) + σ b 2 G 1 ( τ , θ ) .
1 A 4 E [ Δ J Δ J ] av = S a = γ ( θ , 0 , θ ) + γ ( θ + τ , 0 , θ + τ ) + γ ( θ - τ , 0 , θ - τ ) + γ ( θ + τ - τ , 0 , θ + τ - τ ) - 4 I 2 .
S b = S a + 2 [ γ ( θ + τ , τ , θ ) - γ 2 ( τ ) ] .
S c = S b + C cos ω 0 τ av + D cos 2 ω 0 τ av .
S c = S b + C g cos [ ω 0 τ av ] + D g 4 cos [ 2 ω 0 τ av ] .
γ ( τ ) cos ω 0 τ av = γ ( τ ) g cos [ ω 0 τ av ] .
F ( τ , 0 ) = 2 [ 1 - γ φ 4 ( τ ) ] 8 ( τ / τ φ ) 2 , G 1 ( τ , 0 ) 16.
S a 16 σ b 2 C b ( θ ) , S b - S a [ 2 γ φ 2 ( τ ) / P ] { 4 σ b 2 C b ( θ ) + 1 - P } , C 32 σ b 2 C b ( θ ) γ φ ( τ ) , D 2 γ φ 2 ( τ ) { 4 σ b 2 C b ( θ ) - ( 1 - P ) } .
S b - S a 1 2 S a             and             D 1 4 C 1 2 S a .
E [ J ( t ) J ( t + θ ) ] = E 2 [ J ] + E [ Δ J ( t ) Δ J ( t + θ ) ] .
E [ B ( t ) B ( t - τ 1 ) B ( t - τ 2 ) B ( t - τ 3 ) ] = E [ B ( t ) B ( t + τ 1 ) B ( t + τ 2 ) B ( t + τ 3 ) ] ,
γ ˜ b ( θ , 0 , θ + τ ) + γ ˜ b ( θ - τ , 0 , θ ) + γ ˜ b ( θ , τ , θ ) + γ ˜ b ( θ + τ , τ , θ + τ ) = 2 [ γ ˜ b ( θ , θ , τ ) + γ ˜ b ( θ + τ , θ + τ , τ ) ] .
γ ˜ b ( θ , 0 , τ ) , γ ˜ b ( θ , θ + τ , τ ) , γ ˜ φ ( θ , τ , θ + τ ) = γ φ ( θ , τ , θ + τ ) - γ φ 2 ( τ )
γ ˜ φ ( θ + τ , τ , θ ) = γ φ ( θ + τ , τ , θ ) - γ φ 2 ( τ ) .
1 A 4 g J ( ν ) = 2 g 1 ( ν , 0 ) [ 1 + cos 2 π ν τ ] + 4 γ φ ( τ ) cos ω 0 τ g ( ν , τ ) [ 1 + exp ( i 2 π ν τ ) ] + 2 g 2 ( ν , τ ) * [ h ( ν ) cos 2 ω 0 τ + k ( ν ) ] + 2 γ b 2 ( τ ) [ h ( ν ) cos 2 ω 0 τ + k ( ν ) ] + 2 γ φ 2 ( τ ) [ 1 + cos 2 ω 0 τ ] g 2 ( ν , τ ) .
1 A 4 g J ( ν ) av = 2 g 1 ( ν , 0 ) [ 1 + cos 2 π ν τ ] + 2 g 2 ( ν , τ ) * k ( ν ) + 2 γ b 2 ( τ ) k ( ν ) + 2 γ φ 2 ( τ ) g 2 ( ν , τ ) .
g 1 ( ν , 0 ) = 4 σ b 2 g b ( ν ) , g 2 ( ν , τ ) = 2 σ b 2 g b ( ν ) [ 1 + cos 2 π ν τ ] ,
k ( ν ) = [ 1 - γ φ 2 ( τ ) ] g φ ( ν ) .
1 A 4 g J ( ν ) av = 4 σ b 2 [ 2 + γ φ 2 ( τ ) ] g b ( ν ) [ 1 + cos 2 π ν τ ] + 2 γ b 2 ( τ ) [ 1 - γ φ 2 ( τ ) ] g φ ( ν ) + 4 σ b 2 [ 1 - γ φ 2 ( τ ) ] × { g b ( ν ) [ 1 + cos 2 π ν τ ] } * g φ ( ν ) .
Z ( t ) = Z ( t ) + α Z ( t - τ ) + + α n Z ( t - n τ ) + ,
G ( ν ) = 1 / [ 1 - α exp ( - 2 π i ν τ ) ] .
E [ Δ J ( t ) Δ J ( t + θ ) ] = A 4 [ F ( τ , θ ) + σ b 2 G 1 ( τ , θ ) ] .