Abstract

The concept of spectral representation of nonstationary wave fields is considered, and as an example the far-zone diffraction from a slit is discussed. In particular it is shown that the red shift of the spectrum of the diffraction pattern for the off-axis points of observation is the same for stationary polychromatic and nonstationary light sources with the same spectral composition and that under certain conditions a slit splits the Gaussian-shaped pulse into two separate pulses.

© 1995 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  2. L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961).
    [Crossref]
  3. L. Mandel and E. Wolf, J. Opt. Soc. Am. 66, 529 (1976).
    [Crossref]
  4. E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
    [Crossref] [PubMed]
  5. E. Wolf, Opt. Commun. 62, 12 (1987).
    [Crossref]
  6. E. Wolf, Nature (London) 326, 363 (1987).
    [Crossref]
  7. E. Wolf, Phys. Rev. Lett. 58, 2646 (1987).
    [Crossref] [PubMed]
  8. M. F. Bocko, D. H. Douglas, and R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
    [Crossref] [PubMed]
  9. G. M. Morris and D. Faklis, Opt. Commun. 62, 5 (1987).
    [Crossref]
  10. B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
    [Crossref]
  11. D. Faklis and G. M. Morris, Opt. Lett. 13, 4 (1988).
    [Crossref] [PubMed]
  12. F. Gori, G. Guattari, C. Palma, and C. Padovani, Opt. Commun. 67, 1 (1988).
    [Crossref]
  13. G. Indebetouw, J. Mod. Opt. 36, 251 (1989).
    [Crossref]
  14. H. C. Kandpal, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 73, 169 (1989).
    [Crossref]
  15. T. Foley, Opt. Commun. 75, 347 (1990).
    [Crossref]
  16. T. Foley, J. Opt. Soc. Am. A 8, 1099 (1991).
    [Crossref]
  17. D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
    [Crossref]
  18. D. F. V. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
    [Crossref]
  19. M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
    [Crossref]
  20. H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
    [Crossref]
  21. G. S. Agarwal and D. F. V. James, J. Mod. Opt. 40, 1431 (1993).
    [Crossref]
  22. M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341 (1995).
    [Crossref]
  23. R. Barakat, J. Opt. Soc. Am. A 10, 180 (1993).
    [Crossref]
  24. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).
  25. Yu. V. Prokhorov and Yu. A. Rosanov, Probability Theory (Springer-Verlag, Berlin, 1969).
    [Crossref]
  26. O. M. Loéve, Probability Theory (Van Nostrand, New York, 1955).
  27. G. Korn and T. Korn, Mathematical Handbook (McGraw-Hill, New York, 1968).
  28. R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
    [Crossref]
  29. M. Kálal, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 686 (1993).
  30. Zs. Bor and A. P. Kovács, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 524 (1993).
  31. F. Gori, M. Santarsiero, and G. Guattari, J. Opt. Soc. Am. A 10, 673 (1993).
    [Crossref]

1995 (1)

1993 (6)

R. Barakat, J. Opt. Soc. Am. A 10, 180 (1993).
[Crossref]

M. Kálal, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 686 (1993).

Zs. Bor and A. P. Kovács, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 524 (1993).

F. Gori, M. Santarsiero, and G. Guattari, J. Opt. Soc. Am. A 10, 673 (1993).
[Crossref]

H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
[Crossref]

G. S. Agarwal and D. F. V. James, J. Mod. Opt. 40, 1431 (1993).
[Crossref]

1992 (1)

M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
[Crossref]

1991 (3)

T. Foley, J. Opt. Soc. Am. A 8, 1099 (1991).
[Crossref]

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[Crossref]

D. F. V. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[Crossref]

1990 (1)

T. Foley, Opt. Commun. 75, 347 (1990).
[Crossref]

1989 (2)

G. Indebetouw, J. Mod. Opt. 36, 251 (1989).
[Crossref]

H. C. Kandpal, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 73, 169 (1989).
[Crossref]

1988 (2)

D. Faklis and G. M. Morris, Opt. Lett. 13, 4 (1988).
[Crossref] [PubMed]

F. Gori, G. Guattari, C. Palma, and C. Padovani, Opt. Commun. 67, 1 (1988).
[Crossref]

1987 (6)

E. Wolf, Opt. Commun. 62, 12 (1987).
[Crossref]

E. Wolf, Nature (London) 326, 363 (1987).
[Crossref]

E. Wolf, Phys. Rev. Lett. 58, 2646 (1987).
[Crossref] [PubMed]

M. F. Bocko, D. H. Douglas, and R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

G. M. Morris and D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[Crossref]

1986 (1)

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[Crossref] [PubMed]

1982 (1)

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[Crossref]

1976 (1)

1961 (1)

Agarwal, G. S.

G. S. Agarwal and D. F. V. James, J. Mod. Opt. 40, 1431 (1993).
[Crossref]

Barakat, R.

Bertolotti, M.

Bocko, M. F.

M. F. Bocko, D. H. Douglas, and R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

Bor, Zs.

Zs. Bor and A. P. Kovács, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 524 (1993).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Cairns, B.

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[Crossref]

Douglas, D. H.

M. F. Bocko, D. H. Douglas, and R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

Faklis, D.

D. Faklis and G. M. Morris, Opt. Lett. 13, 4 (1988).
[Crossref] [PubMed]

G. M. Morris and D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

Ferrari, A.

Foley, T.

Gase, R.

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[Crossref]

Gori, F.

F. Gori, M. Santarsiero, and G. Guattari, J. Opt. Soc. Am. A 10, 673 (1993).
[Crossref]

M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, Opt. Commun. 67, 1 (1988).
[Crossref]

Guattari, G.

F. Gori, M. Santarsiero, and G. Guattari, J. Opt. Soc. Am. A 10, 673 (1993).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, Opt. Commun. 67, 1 (1988).
[Crossref]

Indebetouw, G.

G. Indebetouw, J. Mod. Opt. 36, 251 (1989).
[Crossref]

James, D. F. V.

G. S. Agarwal and D. F. V. James, J. Mod. Opt. 40, 1431 (1993).
[Crossref]

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[Crossref]

D. F. V. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[Crossref]

Joshi, K. C.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
[Crossref]

H. C. Kandpal, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 73, 169 (1989).
[Crossref]

Kálal, M.

M. Kálal, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 686 (1993).

Kandpal, H. C.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
[Crossref]

H. C. Kandpal, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 73, 169 (1989).
[Crossref]

Knox, R. S.

M. F. Bocko, D. H. Douglas, and R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

Korn, G.

G. Korn and T. Korn, Mathematical Handbook (McGraw-Hill, New York, 1968).

Korn, T.

G. Korn and T. Korn, Mathematical Handbook (McGraw-Hill, New York, 1968).

Kovács, A. P.

Zs. Bor and A. P. Kovács, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 524 (1993).

Loéve, O. M.

O. M. Loéve, Probability Theory (Van Nostrand, New York, 1955).

Mandel, L.

Mehta, D. S.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
[Crossref]

Morris, G. M.

D. Faklis and G. M. Morris, Opt. Lett. 13, 4 (1988).
[Crossref] [PubMed]

G. M. Morris and D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, Opt. Commun. 67, 1 (1988).
[Crossref]

Palma, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, Opt. Commun. 67, 1 (1988).
[Crossref]

Prokhorov, Yu. V.

Yu. V. Prokhorov and Yu. A. Rosanov, Probability Theory (Springer-Verlag, Berlin, 1969).
[Crossref]

Rosanov, Yu. A.

Yu. V. Prokhorov and Yu. A. Rosanov, Probability Theory (Springer-Verlag, Berlin, 1969).
[Crossref]

Santarsiero, M.

Saxena, K.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
[Crossref]

Schubert, M.

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[Crossref]

Sereda, L.

Vaishya, J. C.

H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
[Crossref]

H. C. Kandpal, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 73, 169 (1989).
[Crossref]

Wolf, E.

D. F. V. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[Crossref]

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[Crossref]

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[Crossref]

E. Wolf, Opt. Commun. 62, 12 (1987).
[Crossref]

E. Wolf, Nature (London) 326, 363 (1987).
[Crossref]

E. Wolf, Phys. Rev. Lett. 58, 2646 (1987).
[Crossref] [PubMed]

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[Crossref] [PubMed]

L. Mandel and E. Wolf, J. Opt. Soc. Am. 66, 529 (1976).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Yaglom, A. M.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).

J. Mod. Opt. (2)

G. Indebetouw, J. Mod. Opt. 36, 251 (1989).
[Crossref]

G. S. Agarwal and D. F. V. James, J. Mod. Opt. 40, 1431 (1993).
[Crossref]

J. Opt. Soc. Am. (2)

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

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

Nature (London) (1)

E. Wolf, Nature (London) 326, 363 (1987).
[Crossref]

Opt. Acta (1)

R. Gase and M. Schubert, Opt. Acta 29, 1331 (1982).
[Crossref]

Opt. Commun. (8)

G. M. Morris and D. Faklis, Opt. Commun. 62, 5 (1987).
[Crossref]

B. Cairns and E. Wolf, Opt. Commun. 62, 215 (1987).
[Crossref]

E. Wolf, Opt. Commun. 62, 12 (1987).
[Crossref]

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[Crossref]

H. C. Kandpal, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 73, 169 (1989).
[Crossref]

T. Foley, Opt. Commun. 75, 347 (1990).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, Opt. Commun. 67, 1 (1988).
[Crossref]

H. C. Kandpal, K. Saxena, D. S. Mehta, J. C. Vaishya, and K. C. Joshi, Opt. Commun. 99, 157 (1993).
[Crossref]

Opt. Lett. (1)

Phys. Lett. A (2)

D. F. V. James and E. Wolf, Phys. Lett. A 157, 6 (1991).
[Crossref]

M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
[Crossref]

Phys. Rev. Lett. (3)

E. Wolf, Phys. Rev. Lett. 58, 2646 (1987).
[Crossref] [PubMed]

M. F. Bocko, D. H. Douglas, and R. S. Knox, Phys. Rev. Lett. 58, 2649 (1987).
[Crossref] [PubMed]

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[Crossref] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

M. Kálal, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 686 (1993).

Zs. Bor and A. P. Kovács, Proc. Soc. Photo-Opt. Instrum. Eng. 1983, 524 (1993).

Other (5)

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).

Yu. V. Prokhorov and Yu. A. Rosanov, Probability Theory (Springer-Verlag, Berlin, 1969).
[Crossref]

O. M. Loéve, Probability Theory (Van Nostrand, New York, 1955).

G. Korn and T. Korn, Mathematical Handbook (McGraw-Hill, New York, 1968).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the diffraction of light from an extended source.

Fig. 2
Fig. 2

Geometry of the diffraction of a plane wave from a slit.

Fig. 3
Fig. 3

Spectra at different diffraction angles, normalized to the maximum value of the spectral density at the center of the diffraction pattern. Δω/ω0 = 0.45, β = 2π(a/λ0)sin Θ.

Fig. 4
Fig. 4

Spectra at different diffraction angles, normalized to their own maximum values. Δω/ω0 = 0.45, β = 2π(a/λ0)sin Θ.

Fig. 5
Fig. 5

Spectra at different diffraction angles, normalized to the maximum value of the spectral density at the center of the diffraction pattern. Δω/ω0 = 0.1, β = 2π(a/λ0)sin Θ.

Fig. 6
Fig. 6

Spatial distribution of the intensity of the diffraction pattern of a two-component source for different values of α = ω01/ω02. The intensity is normalized to its maximum value at the center.

Fig. 7
Fig. 7

Spatial distribution of the intensity of the diffraction pattern for different sources. The intensity is normalized to its maximum value at the center.

Fig. 8
Fig. 8

Time dependence of the intensity of the diffraction pattern at different diffraction angles for a pulsed source with γ = 1/ω0τ = 0.32. The intensity is normalized to its maximum value at the center, and β = 2π(a/λ0)sin Θ.

Fig. 9
Fig. 9

Time dependence of the intensity of the diffraction pattern for a pulsed source with γ = 1/ω0τ = 0.32 for two values of the diffraction angle β = 2π(a/λ0)sin Θ. Intensities are normalized to their own maximum values.

Fig. 10
Fig. 10

Time dependence of the intensity of the diffraction pattern at the diffraction angle β = 2π(a/λ0)sin Θ = 3.75 for two values of γ = 1/ω0τ. Intensities are normalized to their own maximum values.

Equations (65)

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Q ( ρ , t ) = - exp ( i ω t ) d U Q ( ρ , ω ) ,             ρ D ,
d U Q ( ρ , ω ) = U Q ( ρ , ω + d ω ) - U Q ( ρ , ω )
d U Q ( ρ , ω 1 ) d U Q * ( ρ , ω 2 ) = 0             if ω 1 ω 2 ,
S Q ( ρ 1 , ρ 2 ; ω 1 , ω 2 ) = U Q ( ρ 1 , ω 1 ) U Q * ( ρ 2 , ω 2 ) ,
S Q ( ρ ; ω 1 , ω 2 ) = U Q ( ρ , ω 1 ) U Q * ( ρ , ω 2 ) = S Q ( ρ , ρ ; ω 1 , ω 2 )
d S Q ( ρ ; ω , ω ) = d U Q ( ρ , ω ) 2 = d I Q ( ρ ; ω ) ,
I Q ( ρ ) = - d I Q ( ρ ; ω ) .
I Q ( ρ , t ) = Q ( ρ , t ) 2 = - - exp [ i ( ω 1 - ω 2 ) t ] × d S Q ( ρ ; ω 1 , ω 2 ) .
V ( r , t ) = - exp ( i ω t ) d U F ( r , ω ) ,             r R 3 ,
d U F ( r , ω ) = ρ D exp ( - i ω v s ) s d U Q ( ρ , ω ) ,
U Q ( ρ , ω ) = U ( ρ - ρ 0 ) U Q ( ω ) ,
d U F ( r , ω ) = exp ( - i ω v s ) s d U Q ( ω ) ,
V ( r , t ) = - exp [ i ω ( t - s v ) ] s d U Q ( ω ) .
U Q ( ρ , ω ) = U ( ρ - ρ 1 ) U Q 1 ( ω ) + U ( ρ - ρ 2 ) U Q 2 ( ω ) ,
d U F ( r , ω ) = exp ( - i ω v s 1 ) s 1 d U Q 1 ( ω ) + exp ( - i ω v s 2 ) s 2 d U Q 2 ( ω ) ,
V ( r , t ) = - exp [ i ω ( t - s 1 v ) ] s 1 d U Q 1 ( ω ) + - exp [ i ω ( t - s 2 v ) ] s 2 d U Q 2 ( ω ) .
d U Q 1 ( ω ) d U Q 2 * ( ω ) 0 ,
d I F ( r ; ω ) = d U F ( r , ω ) 2 .
d I F ( r ; ω ) = d U F ( r ; ω ) 2 = D D exp [ - i ω v ( s 1 - s 2 ) ] s 1 s 2 × d S Q ( ρ 1 , ρ 2 ; ω , ω ) ,
I F ( r ; t ) = V ( r ; t ) 2 = - - exp [ i ( ω 1 - ω 2 ) t ] × d S F ( r ; ω 1 , ω 2 ) ,
S F ( r ; ω 1 , ω 2 ) = U F ( r ; ω 1 ) U F * ( r ; ω 2 )
d S F ( r ; ω 1 , ω 2 ) = D D exp [ - i v ( ω 1 s 1 - ω 2 s 2 ) ] s 1 s 2 × d S Q ( ρ 1 , ρ 2 ; ω 1 , ω 2 ) ,
S F ( r 1 , r 2 ; ω 1 , ω 2 ) = U F ( r 1 ; ω 1 ) U F * ( r 2 ; ω 2 ) ,
d S F ( r 1 , r 2 ; ω 1 , ω 2 ) = D D exp [ - i v ( ω 1 s 1 - ω 2 s 2 ) ] s 1 s 2 × d S Q ( ρ 1 , ρ 2 ; ω 1 , ω 2 ) ,
Γ F ( r 1 , r 2 ; t 1 , t 2 ) = - - exp [ i ( ω 1 t 1 - ω 2 t 2 ) ] × d S F ( r 1 , r 2 ; ω 1 , ω 2 ) ,
J F ( r 1 , r 2 ; t ) = Γ F ( r 1 , r 2 ; t , t ) = - - exp [ i ( ω 1 - ω 2 ) t ] × d S F ( r 1 , r 2 ; ω 1 , ω 2 ) ,
μ F ( r 1 , r 2 ; t ) = J F ( r 1 , r 2 ; t ) [ I F ( r 1 ; t ) I F ( r 2 ; t ) ] 1 / 2 ,
d I F ( r , ω ) = d U F ( r , ω ) 2 = d I Q ( ω ) ( a R 0 ) 2 [ sin ( ω v a sin Θ ) ω v a sin Θ ] 2 ,
I F ( r ) = - d I F ( r , ω ) = ( a R 0 ) 2 - [ sin ( ω v a sin Θ ) ω v a sin Θ ] 2 d I Q ( ω ) .
d I Q ( ω ) = I 0 d U ( ω - ω 0 ) ,
I F ( r ) = I 0 ( a R 0 ) 2 [ sin ( ω 0 v a sin Θ ) ω 0 v a sin Θ ] 2 .
I Q ( ω ) = I 2 π Δ ω 2 exp - ( ω - ω 0 Δ ω ) 2 ,
I F ( r , ω ) = I 2 π Δ ω 2 exp - ( ω - ω 0 Δ ω ) 2 ( a R 0 ) 2 × [ sin ( ω v a sin Θ ) ω v a sin Θ ] 2 .
d I Q ( ω ) = I 1 d U ( ω - ω 01 ) + I 2 d U ( ω - ω 02 ) ,
I F ( r ) = I 1 ( a R 0 ) 2 [ sin ( ω 01 v a sin Θ ) ω 01 v a sin Θ ] 2 + I 2 ( a R 0 ) 2 [ sin ( ω 02 v a sin Θ ) ω 02 v a sin Θ ] 2 .
d I Q ( ω ) = I 1 d U ( ω - ω 01 ) + I 0 d U ( ω - ω 0 ) + I 2 d U ( ω - ω 02 ) ,
Q ( t ) = η exp [ - 1 2 ( t τ ) 2 + i ω 0 t ] .
U Q ( ω ) = d U Q ( ω ) d ω = η τ 2 π exp { - ½ [ ( ω - ω 0 ) τ ] 2 } ,
U F ( r , ω ) = d U F ( r , ω ) d ω = η τ 2 π exp { - ½ [ ( ω - ω 0 ) τ ] 2 } a R 0 × exp ( - i ω v R 0 ) sin ( ω v a sin Θ ) ω v a sin Θ .
I F ( r , ω ) = U F ( r , ω ) 2 = I τ 2 2 π exp { - [ ( ω - ω 0 ) τ ] 2 } ( a R 0 ) 2 × [ sin ( ω v a sin Θ ) ω v a sin Θ ] 2 ,
I = η 2 .
s Q ( ρ 1 , ρ 2 ; ω 1 , ω 2 ) = 2 s Q ( ρ 1 , ρ 2 ; ω 1 , ω 2 ) ω 1 ω 2 = A ( ρ 1 ) A ( ρ 2 ) I τ 2 2 π exp { - ½ [ ( ω 1 - ω 0 ) 2 + ( ω 2 - ω 0 ) 2 ] τ 2 } ,
s F ( r ; ω 1 , ω 2 ) = 2 s F ( r ; ω 1 , ω 2 ) ω 1 ω 2 = I τ 2 2 π × exp { - ½ [ ( ω 1 - ω 0 ) + ( ω 2 - ω 0 ) ] 2 τ 2 } × ( a R 0 ) 2 exp ( - i ω 1 - ω 2 v R 0 ) × [ sin ( ω 1 v a sin Θ ) ω 1 v a sin Θ ] 2 × [ sin ( ω 2 v a sin Θ ) ω 2 v a sin Θ ] 2 ,
I F ( r , t ) = I ( a R 0 ) 2 C ( r , t ) 2 ,
C ( r , t ) = - exp [ i ω ( t - R 0 v ) ] τ 2 π × exp [ - ½ ( ω - ω 0 ) 2 τ 2 ] sin ( ω v a sin Θ ) ω v a sin Θ d ω .
Q ( ρ , t ) = A ( ρ ) η exp ( i ω 0 t ) ,
U Q ( ρ , ω ) = A ( ρ ) η U ( ω - ω 0 ) ,
W Q ( ρ , ω ) = A 2 ( ρ ) I U ( ω - ω 0 )
S Q ( ρ ; ω 1 , ω 2 ) = A 2 ( ρ ) I U ( ω 1 - ω 0 ) U ( ω 2 - ω 0 ) ,
d I Q ( ρ ; ω ) = d S Q ( ρ ; ω , ω ) = S Q ( ρ ; ω + d ω , ω + d ω ) - S Q ( ρ ; ω + d ω , ω ) - S Q ( ρ ; ω , + ω + d ω ) + S Q ( ρ ; ω , ω ) = { 0 if ω ω 0 A 2 ( r ) I if ω = ω 0 .
d W Q ( ρ , ω ) = W Q ( ρ , ω + d ω ) - W Q ( ρ , ω ) = { 0 if ω ω 0 A 2 ( r ) I if ω = ω 0 .
d I Q ( ρ ; ω ) = d S Q ( ρ ; ω , ω ) = d W Q ( ρ ; ω ) .
d I Q ( ρ ; ω ) = w Q ( ρ ; ω ) d ω ,
d I Q ( ρ ; ω ) = s Q ( ρ ; ω , ω ) d ω d ω ;
I Q ( ρ ; ω ) = - ω d I Q ( ρ ; ν ) = - ω - ω d S Q ( ρ ; ν 1 , ν 2 ) ,
I Q ( ρ ; ω ) = d 2 I Q ( ρ ; ω ) d ω 2 = s Q ( ρ ; ω , ω ) ,
I Q ( ρ ; ω ) = d I Q ( ρ ; ω ) d ω = w Q ( ρ ; ω ) .
U Q ( ρ , ω ) = U Q ( ω ) A ( ρ ) ,
A ( ρ ) = A ( x ) = { 1 if x [ - a , a ] 0 otherwise ,
d U F ( r , ω ) = d U Q ( ω ) - a a exp ( - i ω v s ) s d x ,
s = R 0 - Δ s ,
Δ s = x sin Θ
d U F ( r , ω ) = d U Q ( ω ) - a a exp [ - i ω v ( R 0 - x sin Θ ) ] R 0 - x sin Θ d x .
λ , a R 0
d U F ( r , ω ) = d U Q ( ω ) exp ( - i ω v R 0 ) R 0 × - a a exp ( i ω v x sin Θ ) d x = d U Q ( ω ) a R 0 exp ( - i ω v R 0 ) sin ( ω v a sin Θ ) ω v a sin Θ .

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