Abstract

An exact solution describing the self-similar dynamics of partially coherent light beams in nonlinear and noninstantaneous Kerr media is presented and analyzed. The description is based on the Wigner formalism for analyzing the propagation of partially coherent light. The solution for the Wigner distribution corresponds to a transverse beam intensity profile of a parabolic form, and the effects of the partial coherence on the beam dynamics are analyzed. The presence of partial coherence in the parabolic beam is shown to increase the diffraction effect, thus weakening the nonlinear self-focusing and increasing the defocusing rate. In the case of an almost coherent beam and a strongly nonlinear situation in a defocusing medium, the new solution is shown to reduce to a previously given parabolic similarity solution for coherent high intensity beam–pulse propagation.

© 2008 Optical Society of America

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  1. G. A. Pasmanik, “Self-action of incoherent light beams,” Sov. Phys. JETP 39, 234-238 (1974).
  2. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490-493 (1996).
    [CrossRef] [PubMed]
  3. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990-4993 (1997).
    [CrossRef]
  4. D. N. Christodoulides and T. H. Coskun, “Theory of incoherent self-focusing in biased photo-refractive media,” Phys. Rev. Lett. 78, 646-649 (1997).
    [CrossRef]
  5. A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422-1424 (1998).
    [CrossRef]
  6. V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683-2686 (1998).
    [CrossRef]
  7. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
    [CrossRef]
  8. D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation to inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
    [CrossRef]
  9. M. Lisak, L. Helczynski, and D. Anderson, “Relation between the Wigner formalism and the three approaches describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321-323 (2003).
    [CrossRef]
  10. A. W. Snyder and M. Mitchell, “Mighty morphing spatial solitons and bullets,” Opt. Lett. 22, 16-18 (1997).
    [CrossRef] [PubMed]
  11. D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, “Incoherent spatial solitons in saturable nonlinear media,” Opt. Lett. 22, 1080-1082 (1997).
    [CrossRef] [PubMed]
  12. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking free pulses in nonlinear-optical fibres,” J. Opt. Soc. Am. B 10, 1185-1190 (1993).
    [CrossRef]
  13. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifier,” Opt. Lett. 25, 1753-1755 (2000).
    [CrossRef]
  14. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
    [CrossRef] [PubMed]
  15. M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).
  16. D. Dragoman, “Wigner distribution function in nonlinear optics,” Appl. Opt. 35, 4142-4146 (1998).
    [CrossRef]
  17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press1995).

2003 (2)

M. Lisak, L. Helczynski, and D. Anderson, “Relation between the Wigner formalism and the three approaches describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321-323 (2003).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[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 to inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

2000 (1)

1998 (3)

D. Dragoman, “Wigner distribution function in nonlinear optics,” Appl. Opt. 35, 4142-4146 (1998).
[CrossRef]

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422-1424 (1998).
[CrossRef]

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

1997 (4)

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

D. N. Christodoulides and T. H. Coskun, “Theory of incoherent self-focusing in biased photo-refractive media,” Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

A. W. Snyder and M. Mitchell, “Mighty morphing spatial solitons and bullets,” Opt. Lett. 22, 16-18 (1997).
[CrossRef] [PubMed]

D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, “Incoherent spatial solitons in saturable nonlinear media,” Opt. Lett. 22, 1080-1082 (1997).
[CrossRef] [PubMed]

1996 (1)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

1993 (1)

1974 (1)

G. A. Pasmanik, “Self-action of incoherent light beams,” Sov. Phys. JETP 39, 234-238 (1974).

Anderson, D.

M. Lisak, L. Helczynski, and D. Anderson, “Relation between the Wigner formalism and the three approaches describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321-323 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking free pulses in nonlinear-optical fibres,” J. Opt. Soc. Am. B 10, 1185-1190 (1993).
[CrossRef]

Anderson, D. Z.

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

Chen, Z.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

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 to inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

D. N. Christodoulides and T. H. Coskun, “Theory of incoherent self-focusing in biased photo-refractive media,” Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, “Incoherent spatial solitons in saturable nonlinear media,” Opt. Lett. 22, 1080-1082 (1997).
[CrossRef] [PubMed]

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 to inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

D. N. Christodoulides and T. H. Coskun, “Theory of incoherent self-focusing in biased photo-refractive media,” Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, “Incoherent spatial solitons in saturable nonlinear media,” Opt. Lett. 22, 1080-1082 (1997).
[CrossRef] [PubMed]

Desaix, M.

Dragoman, D.

Dudley, J. M.

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 to inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Fedele, R.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

Hall, B.

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifier,” Opt. Lett. 25, 1753-1755 (2000).
[CrossRef]

Hasegawa, A.

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

Helczynski, L.

M. Lisak, L. Helczynski, and D. Anderson, “Relation between the Wigner formalism and the three approaches describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321-323 (2003).
[CrossRef]

Joseph, R. I.

Karlsson, M.

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifier,” Opt. Lett. 25, 1753-1755 (2000).
[CrossRef]

Lisak, M.

M. Lisak, L. Helczynski, and D. Anderson, “Relation between the Wigner formalism and the three approaches describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321-323 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking free pulses in nonlinear-optical fibres,” J. Opt. Soc. Am. B 10, 1185-1190 (1993).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press1995).

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422-1424 (1998).
[CrossRef]

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 to inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

A. W. Snyder and M. Mitchell, “Mighty morphing spatial solitons and bullets,” Opt. Lett. 22, 16-18 (1997).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

Pasmanik, G. A.

G. A. Pasmanik, “Self-action of incoherent light beams,” Sov. Phys. JETP 39, 234-238 (1974).

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifier,” Opt. Lett. 25, 1753-1755 (2000).
[CrossRef]

Quiroga-Teixeiro, M. L.

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 to inertial nonlinear media,” Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

Semenov, V.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

Shkunov, V. V.

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

Shukla, P. K.

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422-1424 (1998).
[CrossRef]

A. W. Snyder and M. Mitchell, “Mighty morphing spatial solitons and bullets,” Opt. Lett. 22, 16-18 (1997).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press1995).

Appl. Opt. (1)

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

Opt. Commun. (1)

M. Lisak, L. Helczynski, and D. Anderson, “Relation between the Wigner formalism and the three approaches describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220, 321-323 (2003).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (2)

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002).
[CrossRef]

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

Phys. Rev. Lett. (6)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

D. N. Christodoulides and T. H. Coskun, “Theory of incoherent self-focusing in biased photo-refractive media,” Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

A. W. Snyder and D. J. Mitchell, “Big incoherent solitons,” Phys. Rev. Lett. 80, 1422-1424 (1998).
[CrossRef]

V. V. Shkunov and D. Z. Anderson, “Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media,” Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

Phys. Scr., T (1)

M. Lisak, B. Hall, D. Anderson, R. Fedele, V. Semenov, P. K. Shukla, and A. Hasegawa, “Nonlinear dynamics of partially incoherent optical waves based on the Wigner transform method,” Phys. Scr., T 98, 12-17 (2002).

Sov. Phys. JETP (1)

G. A. Pasmanik, “Self-action of incoherent light beams,” Sov. Phys. JETP 39, 234-238 (1974).

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press1995).

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Equations (47)

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i ψ z + β 2 2 ψ x 2 + κ ψ 2 ψ = 0 ,
f ( x , p , z ) = 1 2 π + ψ * ( x + y 2 , z ) ψ ( x y 2 , z ) e i p y d y .
ψ 2 = + f ( x , p , z ) d p .
f z + p f x + 2 s ψ 2 sin ( 1 2 x p ) f = 0 ,
f z + p f x + s ψ 2 x f p = 0 .
f ( x , p , z ) = 1 π 2 L ( z ) Λ I 0 ( z ) ( 1 x 2 L 2 ( z ) ) L 2 ( z ) Λ 2 ( p L z ( z ) L ( z ) x ) 2
I ( x , z ) = + f ( x , p , z ) d p = I 0 ( z ) ( 1 x 2 L 2 ( z ) ) = 3 W 4 L ( z ) ( 1 x 2 L 2 ( z ) ) ,
W = L + L I ( x , z ) d x = 4 L ( z ) I 0 ( z ) 3 .
L z z = 3 W Λ 2 L 3 ( 1 s L Λ ) .
f ( x , p , 0 ) = 1 π 2 L I 0 Λ 1 x 2 L 2 p 2 p 0 2 ,
f ( x , p ) f G ( x , p ) = 1 π 2 L I 0 Λ exp ( x 2 2 L 2 p 2 2 p 0 2 ) .
f ( x , p ) = A 2 π a 1 + 2 a 2 θ 0 2 exp ( x 2 a 2 p 2 a 2 1 + 2 a 2 θ 0 2 ) ,
1 2 L 2 = 1 a 2 , 1 2 p 0 2 = L 4 I 0 Λ = a 2 1 + 2 a 2 θ 0 2 ,
1 π 2 L I 0 Λ = A 2 π a 1 + 2 a 2 θ 0 2 .
L c = 2 L 6 W Λ 1 ,
1 2 ( L z ) 2 + π ( L ) = 0 .
π ( L ) = 3 W Λ 4 L 2 s 3 W 2 L C 0 ,
C 0 = 3 W Λ 4 L 0 2 s 3 W 2 L 0 .
R ( i ( z ) ) a i ( z ) b 2 a 3 2 ln ( 2 a + b i ( z ) + 2 a R ( i ( z ) ) ( 2 a + b ) i ( z ) ) = γ z ,
a = 1 2 s L 0 Λ , b = 2 s L 0 Λ , γ 2 = 2 I 0 ( 0 ) Λ L 0 3 .
1 i ( z ) i ( z ) + ln ( 1 + 1 i ( z ) i ( z ) ) = 4 I 0 ( 0 ) L 0 2 z .
a R ( i ( z ) ) + b 2 [ arcsin 2 a + b i ( z ) i ( z ) b 2 4 a arcsin 2 a + b b 2 4 a ] = ( a ) 3 2 2 I 0 ( 0 ) Λ L 0 3 z .
x i p j ¯ + + x i p j f ( x , p , z ) d x d p + + f ( x , p , z ) d x d p ,
+ + f ( x , p , z ) d x d p = + ψ ( z , x ) 2 d x = constant
d d z x 2 ¯ = 2 x p ¯ ,
d d z x p ¯ = p 2 ¯ s 2 N ,
d d z p 2 ¯ = s d N d z ,
N = + ( ψ ( z , x ) 2 ) 2 d x + ψ ( z , x ) 2 d x .
p 2 s N = constant .
E = + ψ ( z , x ) 2 d x = constant ,
H = + ( ψ ( z , x ) x 2 s ψ 4 ) d x = constant .
k 2 ¯ + k 2 ψ ̃ ( z , k ) 2 d k + ψ ̃ ( z , k ) 2 d k = + ψ ( z , x ) x 2 d x + ψ ( z , x ) 2 d x ,
p 2 ¯ = p 0 2 ¯ + s + ( ψ 2 ) 2 d x + ( ψ 0 2 ) 2 d x + ψ 0 2 d x ,
p 2 ¯ p 0 2 ¯ = s + ( ψ 0 2 ) 2 d x + ψ 0 2 d x ,
p 2 ¯ = p 0 2 15 + s 4 5 [ I 0 ( z ) I 0 ( 0 ) ] ,
p 2 ¯ p 0 2 ¯ = s 4 I 0 ( 0 ) 5 ,
p 2 ¯ p 0 2 ¯ s 9 W 2 20 L 0 4 ( Λ s L 0 ) z 2 .
p 2 ¯ p 0 2 ¯ = 3 W 5 ( 1 L 0 1 L ( z ) ) .
p 2 ¯ p 0 2 ¯ = 6 W 5 L 0 ( 1 L 0 Λ ) .
ψ ( x ) = A exp ( x 2 2 a 2 ) G ( θ ) exp ( i θ x ) d θ ,
G ( θ ) = 0 , G ( θ 1 ) G ( θ 2 ) = J ( θ 1 ) δ ( θ 1 θ 2 ) .
J ( θ ) = 1 2 π θ 0 2 exp ( θ 2 θ 0 2 ) .
K ( x 1 , x 2 ) = ψ ( x 1 ) ψ * ( x 2 ) = K 0 ( x 1 , x 2 ) exp [ θ 0 2 2 ( x 1 x 2 ) 2 ] ,
K 0 ( x 1 , x 2 ) = A 2 exp ( x 1 2 + x 2 2 2 a 2 ) = A 2 exp [ ( x 1 + x 2 ) 2 4 a 2 ( x 1 x 2 ) 2 4 a 2 ] .
I ( x ) = K ( x , x ) = K 0 ( x , x ) = A 2 exp ( x 2 a 2 ) ,
f ( x , p ) = 1 2 π K ( x + ξ 2 , x ξ 2 ) exp ( i p ξ ) d ξ = A 2 π a 1 + 2 a 2 θ 0 2 exp ( x 2 a 2 p 2 a 2 1 + 2 a 2 θ 0 2 ) ,
K ( x 1 , x 2 ) = I ( x 1 ) I ( x 2 ) μ ( x 1 x 2 ) ,

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