Abstract

We demonstrate the nonlinear recovery of diffused images in a self-focusing photorefractive medium. The method is based on the convolution property of nonlinearity, in which related modes reinforce each other as they propagate. The resulting mode coupling enables energy transfer from the scattered light to the underlying signal. The dynamics is well described by a model in which the signal seeds a modulation instability in the diffused background.

© 2011 Optical Society of America

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  1. D. V. Dylov and J. W. Fleischer, Nature Photon. 4, 323(2010).
    [Crossref]
  2. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851(1993).
    [Crossref]
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    [Crossref]
  4. K. Kondo, Y. Ichioka, and T. Suzuki, Appl. Opt. 16, 2554(1977).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, Science 290, 495 (2000).
    [Crossref] [PubMed]
  8. D. V. Dylov and J. W. Fleischer, Phys. Rev. Lett. 100, 103903 (2008).
    [Crossref] [PubMed]
  9. D. V. Dylov and J. W. Fleischer, Phys. Rev. A 78, 061804(2008).
    [Crossref]
  10. J. Klinger, H. Martin, and Z. Chen, Opt. Lett. 26, 271 (2001).
    [Crossref]
  11. V. V. Shkunov and D. Anderson, Phys. Rev. Lett. 81, 2683(1998).
    [Crossref]
  12. O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, Phys. Rev. E 73, 015601 (2006).
    [Crossref]
  13. J. Parravicini, P. Minzioni, V. Degiorgio, and E. DelRe, Opt. Lett. 34, 3908 (2009).
    [Crossref] [PubMed]
  14. D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
    [Crossref]
  15. D. V. Dylov and J. W. Fleischer, Opt. Lett. 35, 2149 (2010).
    [Crossref] [PubMed]
  16. E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, Nature Photon. 5, 39 (2011).
    [Crossref]
  17. L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
    [Crossref]
  18. Z. G. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, Proc. Natl. Acad. Sci. USA 99, 5223 (2002).
    [Crossref]
  19. M. A. Oleary, D. A. Boas, B. Chance, and A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
    [Crossref]

2011 (1)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, Nature Photon. 5, 39 (2011).
[Crossref]

2010 (2)

D. V. Dylov and J. W. Fleischer, Opt. Lett. 35, 2149 (2010).
[Crossref] [PubMed]

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

2009 (2)

2008 (2)

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]

2006 (1)

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

2004 (1)

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

2002 (1)

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)

2000 (2)

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]

1998 (2)

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[Crossref]

V. V. Shkunov and D. 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]

1993 (1)

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851(1993).
[Crossref]

1992 (1)

M. A. Oleary, D. A. Boas, B. Chance, and A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

1977 (1)

Agranat, A. J.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, Nature Photon. 5, 39 (2011).
[Crossref]

Anderson, D.

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

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

Boas, D. A.

M. A. Oleary, D. A. Boas, B. Chance, and A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Buljan, H.

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

Chance, B.

M. A. Oleary, D. A. Boas, B. Chance, and A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Chen, Z.

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]

Christodoulides, D. N.

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

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]

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]

Conti, C.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, Nature Photon. 5, 39 (2011).
[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.

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

Cross, M. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851(1993).
[Crossref]

Degiorgio, V.

DelRe, E.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, Nature Photon. 5, 39 (2011).
[Crossref]

J. Parravicini, P. Minzioni, V. Degiorgio, and E. DelRe, Opt. Lett. 34, 3908 (2009).
[Crossref] [PubMed]

Dylov, D. V.

D. V. Dylov and J. W. Fleischer, Opt. Lett. 35, 2149 (2010).
[Crossref] [PubMed]

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

C. Sun, D. V. Dylov, and J. W. Fleischer, Opt. Lett. 34, 3003(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]

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, Nature Photon. 4, 323(2010).
[Crossref]

D. V. Dylov and J. W. Fleischer, Opt. Lett. 35, 2149 (2010).
[Crossref] [PubMed]

C. Sun, D. V. Dylov, and J. W. Fleischer, Opt. Lett. 34, 3003(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]

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

Gammaitoni, L.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[Crossref]

Hanggi, P.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[Crossref]

Helczynski-Wolf, L.

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

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851(1993).
[Crossref]

Ichioka, Y.

Jung, P.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[Crossref]

Kip, D.

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

Klinger, J.

Kondo, K.

Lisak, M.

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

Marchesoni, F.

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[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]

J. Klinger, H. Martin, and Z. Chen, Opt. Lett. 26, 271 (2001).
[Crossref]

Minzioni, P.

Mitchell, M.

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

Oleary, M. A.

M. A. Oleary, D. A. Boas, B. Chance, and A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Parravicini, J.

Schwartz, T.

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

Sears, S. M.

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]

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]

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. Anderson, Phys. Rev. Lett. 81, 2683(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. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[Crossref] [PubMed]

Spinozzi, E.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, Nature Photon. 5, 39 (2011).
[Crossref]

Sun, C.

Suzuki, T.

Vishwanath, A.

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

Yodh, A. G.

M. A. Oleary, D. A. Boas, B. Chance, and A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

Appl. Opt. (1)

Nature Photon. (2)

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

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, Nature Photon. 5, 39 (2011).
[Crossref]

Opt. Lett. (4)

Phys. Rev. A (1)

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

Phys. Rev. E (2)

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

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

Phys. Rev. Lett. (5)

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

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]

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

M. A. Oleary, D. A. Boas, B. Chance, and A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[Crossref]

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]

Rev. Mod. Phys. (2)

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851(1993).
[Crossref]

L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
[Crossref]

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

Schematic of the experimental setup. (a) Coherent image of a resolution chart is diffused and sent onto a nonlinear SBN : 60 photorefractive crystal. Output from the crystal is imaged onto a CCD camera. (b) Resolution chart input, without the diffuser in place and (c) diffused chart input.

Fig. 2
Fig. 2

Numerical simulation of seeded instability with increasing nonlinearity. (a) Signal, (b) diffused linear output intensity, and (c) nonlinear output for Δ n / n 0 = 6 × 10 4 . (d) Output intensity as a function of increasing nonlinearity. (e) Cross-section of bars in (b) and (c), showing intensity modulations apparent in the nonlinear case.

Fig. 3
Fig. 3

Experimental output with increasing nonlinearity. (a) Undiffused image of the resolution chart. (b) Output after linear propagation. (c) Nonlinear output for weak nonlinearity ( δ n = 1.7 × 10 4 ). (d)–(f) Output for stronger nonlinearity, above the threshold for incoherent modulation instability [(d)  δ n = 1.7 × 10 4 , (e)  δ n = 2.0 × 10 4 , and (f)  δ n = 2.3 × 10 4 ]. Numbers below the frames are the corresponding input– output correlation coefficients, C = I chart · I out NL .

Fig. 4
Fig. 4

Single versus averaged multishot reconstruction. (a) Section of original resolution chart, (b) diffused output. (c),(d) Nonlinear output for δ n = 2.3 × 10 4 (c) for single shot and (d) averaged over 10 separate runs. Frames are labeled with the corresponding input–output correlation coefficients.

Equations (2)

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i ψ j z + 1 2 k 2 ψ j x 2 + k n 0 Δ n ( I ) ψ j = 0 ,
g ˜ ( α ˜ ) | α ˜ | = 1 2 δ n α ˜ 2 Δ k k ,

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