Abstract

We derive a general law for the M2 factor of any beam generated by a partially coherent Schell-model source. The fourth power of M differs from its minimum value, attained in the coherent limit, by a term proportional to the second derivative of the modulus of the spectral degree of coherence evaluated for zero argument. Examples are given for cases of practical interest.

© 1999 Optical Society of America

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References

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  1. P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds., Proceedings of the Workshop on Laser Beam Characterization (Sociedad Española de Optica, Madrid, 1993); H. Weber, N. Reng, J. Lüdtke, P. M. Mejı́as, eds., Laser Beam Characterization (Festkörper-Laser-Institute GmbH, Berlin, 1994); M. Morin, A. Giesen, eds., Third International Workshop on Laser Beam and Optics Characterization, Proc. SPIE2870, (1996); A. Giesen, M. Morin, eds., Proceedings of the Fourth International Workshop on Laser Beam and Optics Characterization (Institut für Strahlwerkzeuge, Munich, 1997).
  2. A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  3. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  4. A. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  5. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 7–17 (1978).
    [CrossRef]
  6. E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef]
  7. R. Martı́nez-Herrero, P. M. Mejı́as, “Relation among planar sources that generate the same radiant intensity at the output of a general optical system,” J. Opt. Soc. Am. 72, 765–769 (1982).
    [CrossRef]
  8. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  9. F. Gori, M. Santarsiero, A. Sona, “The change of width for partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
    [CrossRef]
  10. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  11. R. Gase, “The multimode laser radiation as a Gaussian Schell model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
    [CrossRef]
  12. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  13. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  14. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [CrossRef]
  15. F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
    [CrossRef]
  16. R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
    [CrossRef]
  17. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  18. F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [CrossRef]
  19. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  20. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of a general partially coherent beam propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  21. B. Eppich, “Die Charakterisierung von Strahlungsfeldern mit der Wigner-Verteilung und deren Messung,” Ph.D. dissertation (Fachbereich Physik, Technisch Universität Berlin, 1998).
  22. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  23. F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1990).
    [CrossRef]
  24. F. Gori, “Collett–Wolf sources and multimode laser,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  25. J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
    [CrossRef]
  26. A. T. Friberg, J. Turunen, “Spatially partially coherent Fabry–Perot modes,” J. Opt. Soc. Am. A 11, 227–235 (1994).
    [CrossRef]
  27. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  28. 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]
  29. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  30. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  31. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
    [CrossRef]
  32. E. Wolf, “New theory of partial coherence in the space-frequency domain. I. Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  33. R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  34. A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).
  35. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980).

1998 (2)

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

1997 (1)

1996 (1)

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

1994 (3)

1993 (1)

1991 (6)

R. Gase, “The multimode laser radiation as a Gaussian Schell model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[CrossRef] [PubMed]

A. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of a general partially coherent beam propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
[CrossRef]

1990 (1)

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1990).
[CrossRef]

1988 (1)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

1987 (2)

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1986 (1)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1982 (4)

1980 (1)

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

1978 (2)

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bastiaans, M. J.

Bélanger, P. A.

Borghi, R.

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Brichkov, Yu. A.

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Collett, E.

Eppich, B.

B. Eppich, “Die Charakterisierung von Strahlungsfeldern mit der Wigner-Verteilung und deren Messung,” Ph.D. dissertation (Fachbereich Physik, Technisch Universität Berlin, 1998).

Friberg, A. T.

Gase, R.

R. Gase, “The multimode laser radiation as a Gaussian Schell model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1990).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

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

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980).

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Mandel, L.

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

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

Marti´nez-Herrero, R.

Meji´as, P. M.

Mukunda, N.

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980).

Santarsiero, M.

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-top beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1990).
[CrossRef]

Serna, J.

Siegman, A.

A. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Sona, A.

F. Gori, M. Santarsiero, A. Sona, “The change of width for partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1990).
[CrossRef]

Starikov, A.

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Sudol, R. J.

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

Tervonen, E.

Turunen, J.

Vasara, A.

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[CrossRef]

Wolf, E.

IEEE J. Quantum Electron. (1)

A. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

J. Mod. Opt. (3)

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[CrossRef]

R. Gase, “The multimode laser radiation as a Gaussian Schell model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (6)

Opt. Commun. (9)

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

F. Gori, M. Santarsiero, A. Sona, “The change of width for partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

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

F. Gori, M. Santarsiero, A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1990).
[CrossRef]

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

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Opt. Lett. (4)

Other (8)

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

P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds., Proceedings of the Workshop on Laser Beam Characterization (Sociedad Española de Optica, Madrid, 1993); H. Weber, N. Reng, J. Lüdtke, P. M. Mejı́as, eds., Laser Beam Characterization (Festkörper-Laser-Institute GmbH, Berlin, 1994); M. Morin, A. Giesen, eds., Third International Workshop on Laser Beam and Optics Characterization, Proc. SPIE2870, (1996); A. Giesen, M. Morin, eds., Proceedings of the Fourth International Workshop on Laser Beam and Optics Characterization (Institut für Strahlwerkzeuge, Munich, 1997).

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

B. Eppich, “Die Charakterisierung von Strahlungsfeldern mit der Wigner-Verteilung und deren Messung,” Ph.D. dissertation (Fachbereich Physik, Technisch Universität Berlin, 1998).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

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Figures (3)

Fig. 1
Fig. 1

Pictorial representation of the propagated beam for (a) the partially coherent source and (b) the associated coherent source.

Fig. 2
Fig. 2

Degree of coherence, as a function of the normalized variable u/σ0, for a GSM source (dashed curve) and a JSM source (solid curve) with the same global coherence parameter η=2.

Fig. 3
Fig. 3

Far-field intensity profiles for a GSM source (dashed curves) and a JSM source (solid curves) with the same global coherence parameter, as functions of the normalized spatial frequency σ0p. Values of η are (a) 2, (b) 1, and (c) 0.5.

Equations (57)

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M2=4πσζσ,
p=θ/λ,
σζ2=1N(x-x¯ζ)2Iζ(x)dx,
σ2=1N(p-p¯)2I(p)dp,
N=Iζ(x)dx=I(p)dp,
x¯ζ=1NxIζ(x)dx,
p¯=1NpI(p)dp
σζ2=σ02-λ2σ2ζ2.
ζ=J2πλσ2,
J=1NIm pW˜0(-p1, p2)p1p2=pp1=pdp,
M2=(16π2σ02σ2-J2)1/2.
σ2=14π2N2W0(x1, x2)x1x2x2=xx1=xdx,
W0(x1, x2)=T*(x1)T(x2)g(x1-x2),
g(u)=α(u)exp[iψ(u)],
α(u)=α(-u),ψ(u)=-ψ(-u);
J=Im T*(x)T(x)dx,
2W0x1x2x2=xx1=x=|T(x)|2g(0)-|T(x)|2g(0)+{T*(x)T(x)-[T(x)]*T(x)}g(0).
g(0)=iψ(0),g(0)=α(0)-ψ2(0).
σ2=14π2N|T(x)|2dx-14π2[α(0)-ψ2(0)]+i ψ(0)2π2NT*(x)T(x)dx,
T*(x)T(x)dx=-[T(x)]*T(x)dx,
σ2=1Np2|T˜(p)|2dp-14π2[α(0)-ψ2(0)]-ψ(0)2πNp|T˜(p)|2dp,
I(c)(p)=|T˜(p)|2.
σ2=σ(c)2-α(0)4π2+p¯(c)-12πψ(0)2.
p¯=1NpW˜0(-p, p)dp.
p¯=i2πg(0)-1NT*(x)T(x)dx,
p¯=p¯(c)-12πψ(0).
σ2=σ(c)2-α(0)4π2.
M2=[16π2σ02σ2-4σ02α(0)-J2]1/2.
M2=[Mc4-4σ02α(0)]1/2.
Mr2=2πσζσ,
Mr2=[Mcr4-σ022α(0, 0)]1/2,
T(x)=T0 exp-x24σ02,
g(u)=exp-u22σμ2.
M2=[1-4σ02α(0)]1/2.
α(u)=-1σμ21-u2σμ2exp-u22σμ2,
M2=1+4 σ02σμ21/2,
η=σμ/σ0.
T(r)=T0 exp-r22σ02,
g(u)=J0(β|u|),
2α=2g=1rddrr ddrJ0(βr)=-βrddr[rJ1(βr)],
ddx[xJ1(x)]=xJ0(x),
2α(0, 0)=-β2.
Mr2=(1+β2σ02)1/2.
η=2σ0β,
σμ=2/β.
W0(r1, r2)=n=-+λnϕn*(r1)ϕn(r2),
λn=πσ02T02 exp-β2σ022Inβ2σ022,
ϕn(r)=ϕ0n exp-r22σ02Jn(βr)exp(-inθ),
ϕ0n=T0/λn.
I()(p)=W˜0(p,-p).
W˜0(p1, p2)=n=-+λnϕ˜n*(-p1)ϕ˜n(p2)
I()(p)=n=-+λn|ϕ˜n(p)|2.
ϕ˜n(p)=πσ02(-i)nϕ0n exp-β2σ022exp(-2π2σ02p2)×In(2πβσ02p)exp(-inφ),
I()(p)=4π2σ04 exp(-β2σ02)exp(-4π2σ02p2)×n=-+In2(2πβσ02p)=4π2σ04 exp(-β2σ02)×exp(-4π2σ02p2)I0(4πβσ02p),
σ2=0I0(4πβσ02p)exp(-4π2σ02p2)p3dp0I0(4πβσ02p)exp(-4π2σ02p2)pdp.
σ2=14π2σ02(1+β2σ02).
Mr2=(1+β2σ02)1/2,

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