Abstract

We consider the propagation of a partially coherent spatial beam in both self-focusing and self-defocusing nonlinear media. Using a Gaussian–Schell model, we derive an equation governing the width of highly incoherent beams as they propagate in both types of media and confirm its validity by using numerical simulations. Experiments performed in a biased photorefractive crystal match the predicted scaling.

© 2009 Optical Society of America

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  1. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
    [CrossRef]
  2. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997)
    [CrossRef]
  3. D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, Opt. Lett. 22, 1080 (1997).
    [CrossRef] [PubMed]
  4. V. Shkunov and D. Anderson, Phys. Rev. Lett. 81, 2682 (1998).
    [CrossRef]
  5. M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
    [CrossRef] [PubMed]
  6. V. P. Nayyar, J. Opt. Soc. Am. B 14, 2248 (1997).
    [CrossRef]
  7. D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
    [CrossRef]
  8. M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
    [CrossRef] [PubMed]
  9. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
    [CrossRef] [PubMed]
  10. D. V. Dylov and J. W. Flesicher, Phys. Rev. Lett. 100, 103903 (2008).
    [CrossRef] [PubMed]
  11. N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
    [CrossRef]
  12. C. Sun, C. Barsi, and J. W. Fleischer, Opt. Express 16, 20676 (2008).
    [CrossRef] [PubMed]
  13. C. Weilnau and C. Denz, J. Opt. A 5, S529 (2003).
    [CrossRef]

2008 (2)

2003 (1)

C. Weilnau and C. Denz, J. Opt. A 5, S529 (2003).
[CrossRef]

2001 (1)

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
[CrossRef]

2000 (2)

M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
[CrossRef] [PubMed]

1998 (2)

V. Shkunov and D. Anderson, Phys. Rev. Lett. 81, 2682 (1998).
[CrossRef]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

1997 (4)

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997)
[CrossRef]

V. P. Nayyar, J. Opt. Soc. Am. B 14, 2248 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, Opt. Lett. 22, 1080 (1997).
[CrossRef] [PubMed]

1996 (1)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Akhmediev, N.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

Anderson, D.

V. Shkunov and D. Anderson, Phys. Rev. Lett. 81, 2682 (1998).
[CrossRef]

Barsi, C.

Chen, Z.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Christodoulides, D. N.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997)
[CrossRef]

D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, Opt. Lett. 22, 1080 (1997).
[CrossRef] [PubMed]

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, Opt. Lett. 22, 1080 (1997).
[CrossRef] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997)
[CrossRef]

Denz, C.

C. Weilnau and C. Denz, J. Opt. A 5, S529 (2003).
[CrossRef]

Dylov, D. V.

D. V. Dylov and J. W. Flesicher, Phys. Rev. Lett. 100, 103903 (2008).
[CrossRef] [PubMed]

Eugenieva, E.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
[CrossRef] [PubMed]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Fleischer, J. W.

Flesicher, J. W.

D. V. Dylov and J. W. Flesicher, Phys. Rev. Lett. 100, 103903 (2008).
[CrossRef] [PubMed]

Joseph, R. I.

Kip, D.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
[CrossRef] [PubMed]

Krolikowski, W.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

Mitchell, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997)
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Nayyar, V. P.

Segev, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
[CrossRef]

M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997)
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Shkunov, V.

V. Shkunov and D. Anderson, Phys. Rev. Lett. 81, 2682 (1998).
[CrossRef]

Snyder, A. W.

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

Soljacic, M.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

Sun, C.

Vishwanath, A.

M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

Weilnau, C.

C. Weilnau and C. Denz, J. Opt. A 5, S529 (2003).
[CrossRef]

J. Opt. A (1)

C. Weilnau and C. Denz, J. Opt. A 5, S529 (2003).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, Phys. Rev. E 63, 035601 (2001).
[CrossRef]

Phys. Rev. Lett. (7)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

D. V. Dylov and J. W. Flesicher, Phys. Rev. Lett. 100, 103903 (2008).
[CrossRef] [PubMed]

N. Akhmediev, W. Krolikowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632 (1998).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997)
[CrossRef]

V. Shkunov and D. Anderson, Phys. Rev. Lett. 81, 2682 (1998).
[CrossRef]

M. Soljacic, M. Segev, T. H. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef] [PubMed]

Science (1)

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Simulation results for small correlation lengths. (a) Plot of FWHM versus Δ n at fixed propagation distance z. Input beam width a 0 = 25 μ m and correlation length l c = 1 , 1.5 , 2 μ m . (b) Plot of simulation results in (a) using scaling from Eq. (4).

Fig. 2
Fig. 2

Simulation results for l c = 6 , 10 , 16 μ m with a 0 = 25 μ m . (a) Plot of FWHM versus Δ n . (b) Plot of simulation results in (a) using scaling from Eq. (4). The dashed line represents the highly incoherent limit in Fig. 1b.

Fig. 3
Fig. 3

Experimental results for a 0 = 25 μ m , l c = 10 μ m for focusing and defocusing nonlinearities. (a) Input beam. (b)–(f) Output at (b) 0 V, (c) +120 V, (d) −40 V, (e) 240 V, and (f) −80 V. Note the ellipticity of the beam due to anisotropy in the crystal. (g) Cross sections of beam between −120 and +320 V.

Fig. 4
Fig. 4

Experimental results of partially coherent self-focusing and self-defocusing. (a) Plot of FWHM versus Δ n for correlation lengths l c = 6 , 10 , 21 μ m . (b) Plot of experimental results in (a) using scaling from Eq. (4). The dashed line represents the highly incoherent limit in Fig. 1b.

Equations (5)

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( [ 2 i k 0 z + 2 ( r p ) ] + γ ( Γ 11 Γ 22 ) ) Γ 12 = 0
Γ 12 ( z ) = I 0 exp [ r 2 a ( z ) 2 p 2 l ( z ) 2 + i p r ψ ( z ) ] exp [ S ( z ) ] ,
1 a ( z ) d a d z = 1 l ( z ) d l d z = d S d z = ψ k 0 ,
d ψ d z = 4 k 0 l 2 a 2 ψ 2 k 0 2 γ I 0 k 0 a 2 e S ( z ) .
a ( z ) 2 a 0 2 = 1 + z 2 L D 2 ( 1 Δ n Δ n cr ) ,

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