Abstract

We propose a numerical method for analyzing extensively the evolution of the coherence functions of nonstationary optical pulses in dispersive, instantaneous nonlinear Kerr media. Our approach deals with the individual propagation of samples from a properly selected ensemble that reproduces the coherence properties of the input pulsed light. In contrast to the usual strategy assuming Gaussian statistics, our numerical algorithm allows us to model the propagation of arbitrary partially coherent pulses in media with strong and instantaneous nonlinearities.

© 2010 Optical Society of America

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    [CrossRef]
  5. D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
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    [CrossRef]
  8. Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
    [CrossRef]
  9. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially coherent planewave-type pulses,” Opt. Lett. 30, 2973–2975 (2005).
    [CrossRef] [PubMed]
  10. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255, 12-22 (2005).
    [CrossRef]
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    [CrossRef]
  13. A. M. Fattakhov and A. S. Chirkin, “Influence of noise on the propagation of light pulses in optical fibers,” Sov. J. Quantum Electron. 13, 1326–1330 (1983).
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    [CrossRef]
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2010 (1)

2009 (1)

2008 (1)

V. Semenov,M. Lisak, D. Anderson, T. Hansson, L. Helczynski-Wolf, and U. O¨ sterberg, “Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous Kerr media,” J. Phys. A: Math. Theor. 41, 335207 (2008).
[CrossRef]

2007 (4)

2006 (2)

G. Gbur, “Simulating fields of arbitrary spatial and temporal coherence,” Opt. Express 14, 7567–7578 (2006).
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

2005 (2)

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255, 12-22 (2005).
[CrossRef]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially coherent planewave-type pulses,” Opt. Lett. 30, 2973–2975 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (2)

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

2001 (1)

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

1998 (1)

1997 (1)

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[CrossRef]

1996 (2)

V. P. Kandidov “Monte Carlo method in nonlinear statistical optics,” Phys. USP 39, 1243–1272 (1996).
[CrossRef]

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1996).
[CrossRef]

1991 (1)

1990 (1)

1988 (1)

V. A. Aleskevich, V. A. Vysloukh, G. D. Kozhoridze, A. N. Matveev, and S. T. Terzieva, “Nonlinear propagation of a partly coherent pulse in a fiber waveguide and the role of higher-order dispersion,” Sov. J. Quantum Electron. 18, 207–211 (1988).
[CrossRef]

1983 (1)

A. M. Fattakhov and A. S. Chirkin, “Influence of noise on the propagation of light pulses in optical fibers,” Sov. J. Quantum Electron. 13, 1326–1330 (1983).
[CrossRef]

1982 (1)

Agrawal, G. P.

Aleskevich, V. A.

V. A. Aleskevich, V. A. Vysloukh, G. D. Kozhoridze, A. N. Matveev, and S. T. Terzieva, “Nonlinear propagation of a partly coherent pulse in a fiber waveguide and the role of higher-order dispersion,” Sov. J. Quantum Electron. 18, 207–211 (1988).
[CrossRef]

Anderson, D.

V. Semenov,M. Lisak, D. Anderson, T. Hansson, L. Helczynski-Wolf, and U. O¨ sterberg, “Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous Kerr media,” J. Phys. A: Math. Theor. 41, 335207 (2008).
[CrossRef]

Andres, P.

Chirkin, A. S.

A. M. Fattakhov and A. S. Chirkin, “Influence of noise on the propagation of light pulses in optical fibers,” Sov. J. Quantum Electron. 13, 1326–1330 (1983).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Corwin, K. L.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Davis, B. J.

Diddams, S. A.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Dorrer, C.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Fattakhov, A. M.

A. M. Fattakhov and A. S. Chirkin, “Influence of noise on the propagation of light pulses in optical fibers,” Sov. J. Quantum Electron. 13, 1326–1330 (1983).
[CrossRef]

Friberg, A. T.

Garnier, J.

Gbur, G.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Gouedard, C.

Gross, B.

Hansson, T.

V. Semenov,M. Lisak, D. Anderson, T. Hansson, L. Helczynski-Wolf, and U. O¨ sterberg, “Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous Kerr media,” J. Phys. A: Math. Theor. 41, 335207 (2008).
[CrossRef]

Helczynski-Wolf, L.

V. Semenov,M. Lisak, D. Anderson, T. Hansson, L. Helczynski-Wolf, and U. O¨ sterberg, “Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous Kerr media,” J. Phys. A: Math. Theor. 41, 335207 (2008).
[CrossRef]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Kandidov, V. P.

V. P. Kandidov “Monte Carlo method in nonlinear statistical optics,” Phys. USP 39, 1243–1272 (1996).
[CrossRef]

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Kozhoridze, G. D.

V. A. Aleskevich, V. A. Vysloukh, G. D. Kozhoridze, A. N. Matveev, and S. T. Terzieva, “Nonlinear propagation of a partly coherent pulse in a fiber waveguide and the role of higher-order dispersion,” Sov. J. Quantum Electron. 18, 207–211 (1988).
[CrossRef]

Lajunen, H.

Lancis, J.

Lin, Q.

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

Lisak, M.

V. Semenov,M. Lisak, D. Anderson, T. Hansson, L. Helczynski-Wolf, and U. O¨ sterberg, “Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous Kerr media,” J. Phys. A: Math. Theor. 41, 335207 (2008).
[CrossRef]

Liu, Y.

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1996).
[CrossRef]

Manassah, J. T.

Matveev, A. N.

V. A. Aleskevich, V. A. Vysloukh, G. D. Kozhoridze, A. N. Matveev, and S. T. Terzieva, “Nonlinear propagation of a partly coherent pulse in a fiber waveguide and the role of higher-order dispersion,” Sov. J. Quantum Electron. 18, 207–211 (1988).
[CrossRef]

Migus, A.

Mitchell, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[CrossRef]

Newbury, N. R.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Park, S.-G.

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1996).
[CrossRef]

Paschotta, R.

Picozzi, A.

Ponomarenko, S. A.

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Segev, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[CrossRef]

Semenov, V.

V. Semenov,M. Lisak, D. Anderson, T. Hansson, L. Helczynski-Wolf, and U. O¨ sterberg, “Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous Kerr media,” J. Phys. A: Math. Theor. 41, 335207 (2008).
[CrossRef]

Silvestre, E.

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

Starikov, A.

Tervo, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, and 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]

Terzieva, S. T.

V. A. Aleskevich, V. A. Vysloukh, G. D. Kozhoridze, A. N. Matveev, and S. T. Terzieva, “Nonlinear propagation of a partly coherent pulse in a fiber waveguide and the role of higher-order dispersion,” Sov. J. Quantum Electron. 18, 207–211 (1988).
[CrossRef]

Torres-Company, V.

Turunen, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255, 12-22 (2005).
[CrossRef]

Vahimaa, P.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, and 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]

Videau, L.

Vysloukh, V. A.

V. A. Aleskevich, V. A. Vysloukh, G. D. Kozhoridze, A. N. Matveev, and S. T. Terzieva, “Nonlinear propagation of a partly coherent pulse in a fiber waveguide and the role of higher-order dispersion,” Sov. J. Quantum Electron. 18, 207–211 (1988).
[CrossRef]

Walmsley, I. A.

Wang, L. G.

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

Washburn, B. R.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Weber, K.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Weiner, A. M.

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1996).
[CrossRef]

Windeler, R. S.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

Wolf, E.

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255, 12-22 (2005).
[CrossRef]

Zhu, S. Y.

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Phys. B (1)

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, B. R. Washburn, K. Weber, and R. S. Windeler, “Fundamental amplitude noise limitations to supercontinuum spectra generated in a microstructured fiber,” Appl. Phys. B 77, 269–277 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Liu, S.-G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (2)

J. Phys. A: Math. Theor. (1)

V. Semenov,M. Lisak, D. Anderson, T. Hansson, L. Helczynski-Wolf, and U. O¨ sterberg, “Mathematical basis for analysis of partially coherent wave propagation in nonlinear, non-instantaneous Kerr media,” J. Phys. A: Math. Theor. 41, 335207 (2008).
[CrossRef]

Nature (2)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef] [PubMed]

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[CrossRef]

Opt. Commun. (2)

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255, 12-22 (2005).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. E (1)

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Phys. USP (1)

V. P. Kandidov “Monte Carlo method in nonlinear statistical optics,” Phys. USP 39, 1243–1272 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Sov. J. Quantum Electron. (2)

V. A. Aleskevich, V. A. Vysloukh, G. D. Kozhoridze, A. N. Matveev, and S. T. Terzieva, “Nonlinear propagation of a partly coherent pulse in a fiber waveguide and the role of higher-order dispersion,” Sov. J. Quantum Electron. 18, 207–211 (1988).
[CrossRef]

A. M. Fattakhov and A. S. Chirkin, “Influence of noise on the propagation of light pulses in optical fibers,” Sov. J. Quantum Electron. 13, 1326–1330 (1983).
[CrossRef]

Other (5)

S. B. Cavalcanti, “Theory of incoherent self-phase modulation of non-stationary pulses,” N. J. Phys. 4, 19.1–19.11 (2002).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge Univ. Press, Cambridge, UK, 1995).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).

R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 1983).

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

Supplementary Material (2)

» Media 1: AVI (1056 KB)     
» Media 2: AVI (1179 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Amplitude of 30 random realizations (red, thin lines) corresponding to 10 ps GSMPs with non-Gaussian statistics fixed by Eq. (12) and coherence times: (a) Tc = 50 ps, and (b) Tc = 10 ps. The average amplitude (black, thick line) and the envelopes (blue, dashed line) are shown for a set of 20000 random realizations. The absolute value of the corresponding mutual coherence functions are shown in the insets. As expected, each individual realization has the same energy.

Fig. 2.
Fig. 2.

Variation of the error function for random samples fixed by Eq. (12) corresponding to 10 ps GSMP input fields with coherence time equal to Tc = 10 ps and Tc = 50 ps. The curves fitted to the computed data, which follow a L −1/2 decay rate, are plotted in solid line.

Fig. 3.
Fig. 3.

Variation of the error function for random samples fixed by Eq. (12) corresponding to 10 ps GSMPs with coherence time Tc = 10 ps propagated in a nonlinear medium with soliton number: (a) N 2 = 1 (fiber 1), and (b) N2 = 2500 (fiber 2). The curves fitted to the computed data, which follow a L −1/2 decay rate, are plotted in solid line.

Fig. 4.
Fig. 4.

Spectral amplitudes of 30 random realizations (red, thin lines) fixed by Eq. (12) corresponding to 10 ps GSMPs with coherence time Tc = 10 ps propagated in fiber 2 distances: (a) z = 0, (b) z = 15, and (c) z = 30 m. The average amplitude (black, thick line) and the envelopes (blue, dashed line) are shown for a set of 20000 random realizations.

Fig. 5.
Fig. 5.

Coherence functions corresponding to 10 ps GSMPs with non-Gaussian statistics fixed by Eq. (12) and coherence time Tc = 10 ps after propagation in fiber 2 from z = 0 to z = 30 m. (Media 1).

Fig. 6.
Fig. 6.

Coherence functions corresponding to 10 ps GSMPs with non-Gaussian statistics fixed by Eq. (12) and coherence time Tc = 50 ps after propagation in fiber 2 from z = 0 to z = 30 m. (Media 2).

Equations (13)

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Γ ( t 1 , t 2 ; z ) = U * ( t 1 , z ) U ( t 2 , z ) ,
γ ( t 1 , t 2 ; z ) = Γ ( t 1 , t 2 ; z ) I ( t 1 , z ) I ( t 2 , z ) ,
W ( ω 1 , ω 2 ; z ) = U ˜ * ( ω 1 , z ) U ˜ ( ω 2 , z ) ,
W ( ω 1 , ω 2 ; z ) = 1 ( 2 π ) 2 Γ ( t 1 , t 2 ; z ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 ,
i U z β 2 2 2 U t 2 = γ U 2 U ,
i z Γ + β 2 2 ( 2 t 1 2 2 t 2 2 ) Γ = γ ( U 1 2 U 1 * U 2 U 2 2 U 1 * U 2 ) ,
Γ L ( t 1 , t 2 ) = 1 L Σ i = 1 L U i * ( t 1 ) U i ( t 2 ) .
Γ ( t 1 , t 2 ) = Σ n λ n ϕ n * ( t 1 ) ϕ n ( t 2 ) ,
Γ ( t 1 , t 2 ) ϕ n ( t 1 ) dt 1 = λ n ϕ n ( t 2 ) .
U ( t ) = Σ n a n ϕ n ( t ) .
a n * a m = λ n δ mn ,
a n = λ n exp ( i φ n ) ,
ε L 2 = Γ 0 ( t 1 , t 2 ) Γ L ( t 1 , t 2 ) 2 dt 1 dt 2 Γ 0 ( t 1 , t 2 ) 2 dt 1 dt 2 .

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