Abstract

A coherent-mode representation for spatially and spectrally partially coherent pulses is derived both in the space–frequency domain and in the space–time domain. It is shown that both the cross-spectral density and the mutual coherence function of partially coherent pulses can be expressed as a sum of spatially and spectrally and temporally completely coherent modes. The concept of the effective degree of coherence for nonstationary fields is introduced. As an application of the theory, the propagation of Gaussian Schell-model pulsed beams in the space–frequency domain is considered and their coherent-mode representation is presented.

© 2005 Optical Society of America

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
    [CrossRef]
  2. B. Cairns, E. Wolf, “The instantaneous cross-spectral density of non-stationary wavefields,” Opt. Commun. 62, 215–218 (1986).
    [CrossRef]
  3. M. Bertolotti, A. Ferrari, L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
    [CrossRef]
  4. L. Sereda, M. Bertolotti, A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. A 15, 695–705 (1998).
    [CrossRef]
  5. M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
    [CrossRef]
  6. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
    [CrossRef]
  7. S. A. Ponomarenko, G. P. Agrawal, E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394–396 (2004).
    [CrossRef] [PubMed]
  8. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
    [CrossRef] [PubMed]
  9. H. Lajunen, J. Tervo, P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
    [CrossRef]
  10. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” submitted to Opt. Commun..
  11. M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A 1, 711–715 (1984).
    [CrossRef]
  12. P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A, Pure Appl. Opt. 6, S41–S44 (2004).
    [CrossRef]
  13. I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
    [CrossRef]
  14. L. Wang, Q. Lin, H. Chen, S. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056612 (2003).
    [CrossRef]
  15. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).
  16. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  17. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  18. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [CrossRef]
  19. G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
    [CrossRef]
  20. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
    [CrossRef]
  21. J. Tervo, T. Setälä, A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
    [CrossRef]
  22. L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
    [CrossRef]
  23. T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef]
  24. F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, 1978), p. 245.
  25. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  26. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  27. R. W. Ziolkowski, J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  28. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  29. Z. Dačić, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
    [CrossRef]
  30. Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123, 5–10 (1996).
    [CrossRef]
  31. G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
    [CrossRef]
  32. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  33. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  34. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  35. A. E. Siegman, Lasers (University Science, 1986).

2004 (5)

2003 (3)

2002 (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

1998 (2)

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

L. Sereda, M. Bertolotti, A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. A 15, 695–705 (1998).
[CrossRef]

1997 (1)

M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

1996 (1)

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123, 5–10 (1996).
[CrossRef]

1995 (1)

1993 (1)

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

1992 (1)

1988 (1)

1986 (4)

B. Cairns, E. Wolf, “The instantaneous cross-spectral density of non-stationary wavefields,” Opt. Commun. 62, 215–218 (1986).
[CrossRef]

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
[CrossRef]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[CrossRef]

1984 (1)

1982 (3)

1981 (2)

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

1980 (2)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

Agrawal, G. P.

S. A. Ponomarenko, G. P. Agrawal, E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394–396 (2004).
[CrossRef] [PubMed]

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

Bastiaans, M. J.

Bertolotti, M.

Borghi, R.

Cairns, B.

B. Cairns, E. Wolf, “The instantaneous cross-spectral density of non-stationary wavefields,” Opt. Commun. 62, 215–218 (1986).
[CrossRef]

Chen, H.

L. Wang, Q. Lin, H. Chen, S. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056612 (2003).
[CrossRef]

Christov, I. P.

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
[CrossRef]

Dacic, Z.

Ferrari, A.

Friberg, A. T.

T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef]

J. Tervo, T. Setälä, A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gori, F.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Guattari, G.

Judkins, J. B.

Lajunen, H.

Lin, Q.

L. Wang, Q. Lin, H. Chen, S. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056612 (2003).
[CrossRef]

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123, 5–10 (1996).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Piquero, G.

Ponomarenko, S. A.

Riesz, F.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, 1978), p. 245.

Santarsiero, M.

Sereda, L.

Setälä, T.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Simon, R.

Starikov, A.

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Sz.-Nagy, B.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, 1978), p. 245.

Tervo, J.

Turunen, J.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” submitted to Opt. Commun..

Vahimaa, P.

H. Lajunen, J. Tervo, P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[CrossRef]

P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A, Pure Appl. Opt. 6, S41–S44 (2004).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” submitted to Opt. Commun..

Wang, L.

L. Wang, Q. Lin, H. Chen, S. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056612 (2003).
[CrossRef]

Wang, Z.

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123, 5–10 (1996).
[CrossRef]

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

Wolf, E.

S. A. Ponomarenko, G. P. Agrawal, E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. 29, 394–396 (2004).
[CrossRef] [PubMed]

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

Z. Dačić, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[CrossRef]

B. Cairns, E. Wolf, “The instantaneous cross-spectral density of non-stationary wavefields,” Opt. Commun. 62, 215–218 (1986).
[CrossRef]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

Wyrowski, F.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” submitted to Opt. Commun..

Xu, Z.

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123, 5–10 (1996).
[CrossRef]

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123, 5–10 (1996).
[CrossRef]

Zhu, S.

L. Wang, Q. Lin, H. Chen, S. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056612 (2003).
[CrossRef]

Ziolkowski, R. W.

J. Mod. Opt. (1)

G. S. Agarwal, E. Wolf, “Higher-order coherence functions in the space–frequency domain,” J. Mod. Opt. 40, 1489–1496 (1993).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

P. Vahimaa, J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A, Pure Appl. Opt. 6, S41–S44 (2004).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

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

Opt. Acta (1)

I. P. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–77 (1986).
[CrossRef]

Opt. Commun. (9)

B. Cairns, E. Wolf, “The instantaneous cross-spectral density of non-stationary wavefields,” Opt. Commun. 62, 215–218 (1986).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123, 5–10 (1996).
[CrossRef]

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E (1)

L. Wang, Q. Lin, H. Chen, S. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056612 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic process to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

Other (6)

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” submitted to Opt. Commun..

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, 1978), p. 245.

A. E. Siegman, Lasers (University Science, 1986).

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Equations (87)

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U ̃ ( r , ω ) = 1 2 π U ( r , t ) exp ( i ω t ) d t ,
U ( r , t ) = 0 U ̃ ( r , ω ) exp ( i ω t ) d ω ,
D U ( r , t ) 2 d t d 3 r < ,
D 0 U ̃ ( r , ω ) 2 d ω d 3 r < .
Γ ( r 1 , r 2 , t 1 , t 2 ) = U * ( r 1 , t 1 ) U ( r 2 , t 2 ) = 1 N n = 1 N U n * ( r 1 , t 1 ) U n ( r 2 , t 2 ) ,
W ( r 1 , r 2 , ω 1 , ω 2 ) = U ̃ * ( r 1 , ω 1 ) U ̃ ( r 2 , ω 2 ) = 1 N n = 1 N U ̃ n * ( r 1 , ω 1 ) U ̃ n ( r 2 , ω 2 ) .
W ( r 1 , r 2 , ω 1 , ω 2 ) = 1 ( 2 π ) 2 Γ ( r 1 , r 2 , t 1 , t 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 ,
Γ ( r 1 , r 2 , t 1 , t 2 ) = W ( r 1 , r 2 , ω 1 , ω 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d ω 1 d ω 2 ,
γ ( r 1 , r 2 , t 1 , t 2 ) = Γ ( r 1 , r 2 , t 1 , t 2 ) [ I ( r 1 , t 1 ) ] 1 2 [ I ( r 2 , t 2 ) ] 1 2 ,
μ ( r 1 , r 2 , ω 1 , ω 2 ) = W ( r 1 , r 2 , ω 1 , ω 2 ) [ S ( r 1 , ω 1 ) ] 1 2 [ S ( r 2 , ω 2 ) ] 1 2 ,
k 2 Γ ( r 1 , r 2 , t 1 , t 2 ) = 1 c 2 2 t k 2 Γ ( r 1 , r 2 , t 1 , t 2 ) ,
k 2 W ( r 1 , r 2 , ω 1 , ω 2 ) + ( ω k c ) 2 W ( r 1 , r 2 , ω 1 , ω 2 ) = 0
Γ ( r 2 , r 1 , t 2 , t 1 ) = Γ * ( r 1 , r 2 , t 1 , t 2 ) ,
W ( r 2 , r 1 , ω 2 , ω 1 ) = W * ( r 1 , r 2 , ω 1 , ω 2 ) ,
D 0 g ( r , ω ) U ̃ ( r , ω ) d ω d 3 r
0 D 0 g ( r , ω ) U ̃ ( r , ω ) d ω d 3 r 2 = D 0 g * ( r 1 , ω 1 ) g ( r 2 , ω 2 ) U ̃ * ( r 1 , ω 1 ) U ̃ ( r 2 , ω 2 ) d ω 1 d ω 2 d 3 r 1 d 3 r 2 = D 0 g * ( r 1 , ω 1 ) g ( r 2 , ω 2 ) U ̃ * ( r 1 , ω 1 ) U ̃ ( r 2 , ω 2 ) d ω 1 d ω 2 d 3 r 1 d 3 r 2 .
D 0 g * ( r 1 , ω 1 ) g ( r 2 , ω 2 ) W ( r 1 , r 2 , ω 1 , ω 2 ) d ω 1 d ω 2 d 3 r 1 d 3 r 2 0 ,
D f * ( r 1 , t 1 ) f ( r 2 , t 2 ) Γ ( r 1 , r 2 , t 1 , t 2 ) d t 1 d t 2 d 3 r 1 d 3 r 2
0 ,
Γ ( r 1 , r 2 , t 1 , t 2 ) 2 I ( r 1 , t 1 ) I ( r 2 , t 2 ) ,
W ( r 1 , r 2 , ω 1 , ω 2 ) 2 S ( r 1 , ω 1 ) S ( r 2 , ω 2 ) ,
D 0 W ( r 1 , r 2 , ω 1 , ω 2 ) 2 d ω 1 d ω 2 d 3 r 1 d 3 r 2
[ D 0 S ( r , ω ) d ω d 3 r ] 2
= D 0 U ̃ ( r , ω ) 2 d ω d 3 r 2 < ,
D Γ ( r 1 , r 2 , t 1 , t 2 ) 2 d t 1 d t 2 d 3 r 1 d 3 r 2 < .
W ( r 1 , r 2 , ω 1 , ω 2 ) = n λ n ϕ n * ( r 1 , ω 1 ) ϕ n ( r 2 , ω 2 ) ,
D W ( r 1 , r 2 , ω 1 , ω 2 ) ϕ n ( r 1 , ω 1 ) d ω 1 d 3 r 1 = λ n ϕ n ( r 2 , ω 2 ) ,
D ϕ n * ( r , ω ) ϕ m ( r , ω ) d ω d 3 r = δ n m ,
W n ( r 1 , r 2 , ω 1 , ω 2 ) = λ n ϕ n * ( r 1 , ω 1 ) ϕ n ( r 2 , ω 2 ) .
2 ϕ n ( r , ω ) + ( ω c ) 2 ϕ n ( r , ω ) = 0 .
Γ ( r 1 , r 2 , t 1 , t 2 ) = n λ n ξ n * ( r 1 , t 1 ) ξ n ( r 2 , t 2 ) ,
ξ n ( r , t ) = ϕ n ( r , ω ) exp ( i ω t ) d ω ,
D ξ n * ( r , t ) ξ m ( r , t ) d t d 3 r = 2 π δ n m
Γ n ( r 1 , r 2 , t 1 , t 2 ) = λ n ξ n * ( r 1 , t 1 ) ξ n ( r 2 , t 2 )
U ̃ ( r , ω ) = n a n ϕ n ( r , ω ) ,
W ( r 1 , r 2 , ω 1 , ω 2 ) 2 = n m λ n λ m ϕ n * ( r 1 , ω 1 ) ϕ n ( r 2 , ω 2 ) ϕ m ( r 1 , ω 1 ) ϕ m * ( r 2 , ω 2 )
D D W ( r 1 , r 2 , ω 1 , ω 2 ) 2 d ω 1 d ω 2 d 3 r 1 d 3 r 2 = n λ n 2
S ( r , ω ) = W ( r , r , ω , ω ) = n λ n ϕ n ( r , ω ) 2 .
D S ( r , ω ) d ω d 3 r = n λ n .
W ( r 1 , r 2 , ω 1 , ω 2 ) 2 = S ( r 1 , ω 1 ) S ( r 2 , ω 2 ) μ ( r 1 , r 2 , ω 1 , ω 2 ) 2 ,
μ ¯ 2 = D S ( r 1 , ω 1 ) S ( r 2 , ω 2 ) μ ( r 1 , r 2 , ω 1 , ω 2 ) 2 d ω 1 d ω 2 d 3 r 1 d 3 r 2 S ( r 1 , ω 1 ) S ( r 2 , ω 2 ) d ω 1 d ω 2 = n λ n 2 ( n λ n ) 2
γ ¯ 2
= D I ( r 1 , t 1 ) I ( r 2 , t 2 ) γ ( r 1 , r 2 , t 1 , t 2 ) 2 d t 1 d t 2 d 3 r 1 d 3 r 2 I ( r 1 , t 1 ) I ( r 2 , t 2 ) d t 1 d t 2
γ ¯ 2 = n λ n 2 ( n λ n ) 2 .
γ ¯ μ ¯ ,
Γ ( x 1 , y 1 , 0 , x 2 , y 2 , 0 , t 1 , t 2 ) = Γ 0 exp [ R ( x 1 , y 1 , x 2 , y 2 ) T ( t 1 , t 2 ) ] ,
R ( x 1 , y 1 , x 2 , y 2 ) = x 1 2 + x 2 2 + y 1 2 + y 2 2 w 0 2 + ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ 0 2 ,
T ( t 1 , t 2 ) = t 1 2 + t 2 2 2 T 2 + ( t 1 t 2 ) 2 2 T c 2 + i ω 0 ( t 1 t 2 ) ,
W ( x 1 , y 1 , 0 , x 2 , y 2 , 0 , ω 1 , ω 2 ) = W 0 exp [ R ( x 1 , y 1 , x 2 , y 2 ) F ( ω 1 , ω 2 ) ] ,
F ( ω 1 , ω 2 ) = ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 2 Ω 2 + ( ω 1 ω 2 ) 2 2 Ω c 2 .
Ω 2 = 1 T 2 + 2 T c 2 ,
Ω c = T c Ω T ,
W ( r 1 , r 2 , ω 1 , ω 2 ) = ω 1 ω 2 4 z 1 z 2 π 2 c 2 exp [ i ( ω 2 z 2 ω 1 z 1 ) c ] × W ( x 1 , y 1 , 0 , x 2 , y 2 , 0 , ω 1 , ω 2 ) exp [ i ω 2 ( x 2 x 2 ) 2 2 c z 2 i ω 1 ( x 1 x 1 ) 2 2 c z 1 ] exp [ i ω 2 ( y 2 y 2 ) 2 2 c z 2 i ω 1 ( y 1 y 1 ) 2 2 c z 1 ] d x 1 d x 2 d y 1 d y 2 .
W ( r 1 , r 2 , ω 1 , ω 2 ) = W ( r 1 , r 2 , ω 1 , ω 2 ) exp { i arg [ W ( r 1 , r 2 , ω 1 , ω 2 ) ] } .
g j = ω j 2 c z j ,
W ( r 1 , r 2 , ω 1 , ω 2 ) = W 0 g 1 g 2 W ( g 1 , g 2 ) exp [ F ( ω 1 , ω 2 ) ] exp { 1 W 2 ( g 1 , g 2 ) [ g 1 2 g 2 2 R ( x 1 , y 1 , x 2 , y 2 ) + 1 w 0 2 w c 2 X ( r 1 , r 2 , ω 1 , ω 2 ) ] } ,
1 w c 2 = 1 w 0 2 + 1 σ 0 2
X ( r 1 , r 2 , ω 1 , ω 2 ) = g 1 2 ( x 1 2 + y 1 2 ) + g 2 2 ( x 2 2 + y 2 2 ) w 0 2 + ( g 1 x 1 g 2 x 2 ) 2 + ( g 1 y 1 g 2 y 2 ) 2 2 σ 0 2
W 2 ( g 1 , g 2 ) = u ( g 1 , g 1 ) u ( g 2 , g 2 ) + ( g 1 g 2 ) 2 4 σ 0 4 ,
u ( g 1 , g 2 ) = 1 w 0 2 w c 2 + g 1 g 2 ,
arg [ W ( r 1 , r 2 , ω 1 , ω 2 ) ] = η ( g 1 , g 2 ) + a g 1 g 2 ( g 1 g 2 ) W 2 ( g 1 , g 2 ) R ( x 1 , y 1 , x 2 , y 2 ) u ( g 1 , g 2 ) W 2 ( g 1 , g 2 ) Y ( r 1 , r 2 , ω 1 , ω 2 ) + ( ω 2 z 2 ω 1 z 1 ) c ,
a = 1 w 0 2 + 1 2 σ 0 2 .
Y ( r 1 , r 2 , ω 1 , ω 2 ) = g 1 ( x 1 2 + y 1 2 ) g 2 ( x 2 2 + y 2 2 ) w 0 2 w c 2 ,
η ( g 1 , g 2 ) = tan 1 [ a ( g 1 g 2 ) u ( g 1 , g 2 ) ]
S ( r , ω ) = W 0 w 0 2 w 2 ( z , ω ) exp [ 2 ( x 2 + y 2 ) w 2 ( z , ω ) ] exp [ ( ω ω 0 ) 2 Ω 2 ] ,
w 2 ( z , ω ) = w 0 2 [ 1 + ( 2 c z w 0 w c ω ) 2 ] .
a 1 = 1 w 0 2 , b 1 = 1 2 σ 0 2 ,
a 2 = 1 2 Ω 2 , b 2 = 1 2 Ω c 2 ,
d j = ( a j 2 + 2 a j b j ) 1 2 ,
exp [ R ( x 1 , y 1 , x 2 , y 2 ) ] = m = 0 n = 0 ϵ m n ψ m n * ( x 1 , y 1 , 0 ) ψ m n ( x 2 , y 2 , 0 ) ,
exp [ F ( ω 1 , ω 2 ) ] = k = 0 α k φ k * ( ω 1 ) φ k ( ω 2 ) .
ψ m n ( x , y , 0 ) = ( 2 d 1 2 m + n π m ! n ! ) 1 2 H m ( x 2 d 1 ) H n ( y 2 d 1 ) exp [ d 1 ( x 2 + y 2 ) ] ,
ϵ m n = π a 1 + b 1 + d 1 ( b 1 a 1 + b 1 + d 1 ) m + n .
φ k ( ω ) = 1 2 k k ! ( 2 d 2 π ) 1 4 H k [ ( ω ω 0 ) 2 d 2 ] exp [ d 2 ( ω ω 0 ) 2 ] ,
α k = ( π a 2 + b 2 + d 2 ) 1 2 ( b 2 a 2 + b 2 + d 2 ) k ,
W ( x 1 , y 1 , 0 , x 2 , y 2 , 0 , ω 1 , ω 2 ) = m = 0 n = 0 k = 0 λ m n k ϕ m n k * ( x 1 , y 1 , 0 , ω 1 ) ϕ m n k ( x 2 , y 2 , 0 , ω 2 ) ,
ϕ m n k ( x , y , 0 , ω ) = ψ m n ( x , y , 0 ) φ k ( ω ) ,
λ m n k = W 0 ϵ m n α k .
D ϕ m n k * ( x , y , 0 , ω ) ϕ m n k ( x , y , 0 , ω ) d ω d x d y
= δ m m δ n n δ k k .
ϕ m n k ( r , ω ) = ( 2 π ) 1 2 exp { i [ ω z c ( m + n + 1 ) θ ( z , ω ) ] } φ k ( ω ) w ( z , ω ) 2 m + n m ! n ! × H m [ x 2 w ( z , ω ) ] H n [ y 2 w ( z , ω ) ] exp [ i ω ( x 2 + y 2 ) 2 c R ( z , ω ) x 2 + y 2 w 2 ( z , ω ) ] .
R ( z , ω ) = z [ 1 + ( w 0 w c ω 2 c z ) 2 ] .
tan θ ( z , ω ) = w 2 ( z ) ω 2 c R ( z ) = 2 c z w c 2 ω .
W ( r 1 , r 2 , ω 1 , ω 2 ) = m , n , k = 0 λ m n k ϕ m n k * ( r 1 , ω 1 ) ϕ m n k ( r 2 , ω 2 ) ,
μ ¯ 2 = m , n , k = 0 λ m n k 2 ( m , n , k = 0 λ m n k ) 2 = ( a 1 + d 1 a 1 + 2 b 1 + d 1 ) 2 a 2 + d 2 a 2 + 2 b 2 + d 2 = 1 ( 1 + w 0 2 σ 0 2 ) ( 1 + 2 Ω 2 Ω c 2 ) 1 2 .
μ ¯ 2 = w c 2 w 0 2 Ω T ,
T 2 = 1 Ω 2 + 2 Ω c 2

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