Abstract

We examine the nonlinear coupling and modulation instability of a coherent beam with one that is partially spatially incoherent. Using a mutual coherence approach, we derive the growth rate for perturbations and show that the presence of any amount of coherent component eliminates the nonlinear threshold for instability. The fraction of coherent light is shown to determine the gain and characteristic period of the resulting patterns. Theoretical considerations are confirmed by numerical simulation and by experimental observations in a self-focusing photorefractive crystal.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  2. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
    [CrossRef] [PubMed]
  3. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
    [CrossRef] [PubMed]
  4. O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
    [CrossRef]
  5. A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
    [CrossRef] [PubMed]
  6. A. Sauter, S. Pitois, G. Millot, and A. Picozzi, Opt. Lett. 30, 2143 (2005).
    [CrossRef] [PubMed]
  7. V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
    [CrossRef]
  8. D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
    [CrossRef]
  9. D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc. Am. B 19, 502 (2002).
    [CrossRef]
  10. D. V. Dylov and J. W. Fleischer, Opt. Lett. 34, 2673 (2009).
    [CrossRef] [PubMed]
  11. W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
    [CrossRef]
  12. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
    [CrossRef]
  13. Z. Chen, J. Klinger, and D. N. Christodoulides, Phys. Rev. E 66, 066601 (2002).
    [CrossRef]
  14. T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
    [CrossRef] [PubMed]
  15. T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, Opt. Lett. 25, 826 (2000).
    [CrossRef]
  16. Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
    [CrossRef]
  17. D. V. Dylov and J. W. Fleischer, Phys. Rev. Lett. 100, 103903 (2008).
    [CrossRef] [PubMed]
  18. D. V. Dylov and J. W. Fleischer, Phys. Rev. A 78, 061804 (2008).
    [CrossRef]
  19. A. Picozzi, Opt. Express 15, 9063 (2007).
    [CrossRef] [PubMed]
  20. D. V. Dylov and J. W. Fleischer, Nat. Photon. 4, 323 (2010).
    [CrossRef]

2010 (1)

D. V. Dylov and J. W. Fleischer, Nat. Photon. 4, 323 (2010).
[CrossRef]

2009 (1)

2008 (3)

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef] [PubMed]

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

D. V. Dylov and J. W. Fleischer, Phys. Rev. A 78, 061804 (2008).
[CrossRef]

2007 (1)

2006 (1)

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
[CrossRef]

2005 (1)

2004 (1)

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

2002 (3)

D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc. Am. B 19, 502 (2002).
[CrossRef]

Z. Chen, J. Klinger, and D. N. Christodoulides, Phys. Rev. E 66, 066601 (2002).
[CrossRef]

Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
[CrossRef]

2001 (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

2000 (4)

M. Soljacic, M. Segev, T. 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]

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, Opt. Lett. 25, 826 (2000).
[CrossRef]

1998 (1)

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

1997 (1)

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

1964 (1)

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Anderson, D.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

Anderson, D. Z.

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

Buljan, H.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Chen, Z.

Z. Chen, J. Klinger, and D. N. Christodoulides, Phys. Rev. E 66, 066601 (2002).
[CrossRef]

Chen, Z. G.

Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
[CrossRef]

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

Christodoulides, D. N.

Z. Chen, J. Klinger, and D. N. Christodoulides, Phys. Rev. E 66, 066601 (2002).
[CrossRef]

Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc. Am. B 19, 502 (2002).
[CrossRef]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, Opt. Lett. 25, 826 (2000).
[CrossRef]

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

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

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

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

Cohen, O.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Coskun, T.

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

Coskun, T. H.

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, Opt. Lett. 25, 826 (2000).
[CrossRef]

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

Dylov, D. V.

D. V. Dylov and J. W. Fleischer, Nat. Photon. 4, 323 (2010).
[CrossRef]

D. V. Dylov and J. W. Fleischer, Opt. Lett. 34, 2673 (2009).
[CrossRef] [PubMed]

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

D. V. Dylov and J. W. Fleischer, Phys. Rev. A 78, 061804 (2008).
[CrossRef]

Eugenieva, E.

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

Fleischer, J. W.

D. V. Dylov and J. W. Fleischer, Nat. Photon. 4, 323 (2010).
[CrossRef]

D. V. Dylov and J. W. Fleischer, Opt. Lett. 34, 2673 (2009).
[CrossRef] [PubMed]

D. V. Dylov and J. W. Fleischer, Phys. Rev. A 78, 061804 (2008).
[CrossRef]

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

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Grandpierre, A. G.

Helczynski-Wolf, L.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

Kim, Y. R.

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

Kip, D.

D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc. Am. B 19, 502 (2002).
[CrossRef]

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

Klinger, J.

Z. Chen, J. Klinger, and D. N. Christodoulides, Phys. Rev. E 66, 066601 (2002).
[CrossRef]

Lisak, M.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

Martienssen, W.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Martin, H.

Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
[CrossRef]

Millot, G.

Mitchell, M.

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

Picozzi, A.

Pitois, S.

Sauter, A.

Schwartz, T.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Sears, S. M.

D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc. Am. B 19, 502 (2002).
[CrossRef]

Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
[CrossRef]

Segev, M.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc. Am. B 19, 502 (2002).
[CrossRef]

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, Opt. Lett. 25, 826 (2000).
[CrossRef]

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

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

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

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

Semenov, V.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

Shkunov, V. V.

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

Soljacic, M.

D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, J. Opt. Soc. Am. B 19, 502 (2002).
[CrossRef]

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, T. 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]

Spiller, E.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Vishwanath, A.

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

Am. J. Phys. (1)

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

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

Nat. Photon. (1)

D. V. Dylov and J. W. Fleischer, Nat. Photon. 4, 323 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

D. V. Dylov and J. W. Fleischer, Phys. Rev. A 78, 061804 (2008).
[CrossRef]

Phys. Rev. E (3)

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
[CrossRef]

Z. Chen, J. Klinger, and D. N. Christodoulides, Phys. Rev. E 66, 066601 (2002).
[CrossRef]

Phys. Rev. Lett. (6)

T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. G. Chen, M. Soljacic, and M. Segev, Phys. Rev. Lett. 84, 2374 (2000).
[CrossRef] [PubMed]

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

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef] [PubMed]

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

V. V. Shkunov and D. Z. Anderson, Phys. Rev. Lett. 81, 2683 (1998).
[CrossRef]

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

Proc. Natl. Acad. Sci. USA (1)

Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
[CrossRef]

Science (1)

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

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Growth rate of coherent–incoherent modulation instability. Shown are growth rates for (a) equal intensities [exact solution using Eq. (4)] and (b) varied intensity ratios [numerical]. In (a), the solid line is the 50/50 mixture, the dots are the purely coherent case, and the dashes are the purely incoherent case.

Fig. 2
Fig. 2

Experimental results of the coupling between a coherent plane wave and an incoherent distribution with uniform intensity. The ratio of the intensities is shown on the upper right of each frame: coherent/incoherent. The output pictures after 8 mm propagation at fixed nonlinearity ( δ n = 1.74 × 10 4 , i.e., below the threshold for the purely incoherent case) show that any amount of coherent component makes the mixture unstable. Total intensity is fixed at 1.25 W / cm 2 .

Fig. 3
Fig. 3

Characteristic spatial period of the instability. (a) Typical cross sections of the intensity for different fractions of coherent component in the mixture (total intensity 1.25 W / cm 2 ). (b) Measurements of the period of modulation instability (average characteristic feature size) as a function of coherency for three different values of total intensity.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

i B z + 1 2 k ( 2 x 1 2 2 x 2 2 ) B + k n 0 γ ( I 1 I 2 ) B = 0 ,
1 = k 2 γ n 0 + d k x B ^ 0 ( k x + α / 2 ) B ^ 0 ( k x α / 2 ) i g k + α k x ,
B ^ 0 ( k x ) = I 0 2 Δ k π ( k x 2 + Δ k 2 ) + 1 2 m 0 [ J m δ ( k x + k m ) J m δ ( k x k m ) ] ,
g ˜ = | α ˜ | 2 θ 0 + | α ˜ | 2 δ n α ˜ 2 + θ 0 2 + [ δ n 2 + 4 θ 0 2 ( δ n α ˜ 2 ) ] 1 / 2 ,

Metrics