Abstract

We investigate spatial localization in a quadratic nonlinear medium in the presence of randomness. By means of numerical simulations and theoretical analyses we show that, in the down conversion regime, the transverse random modulation of the nonlinear susceptibility generates localizations of the fundamental wave that grow exponentially in propagation. The localization length is optically controlled by the pump intensity that determines the amplification rate. The results also apply to cubic nonlinearities.

© 2013 Optical Society of America

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  1. J. D. Bodyfelt, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 104, 164102 (2010).
    [CrossRef]
  2. Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Phys. Rev. Lett. 64, 1693 (1990).
    [CrossRef]
  3. I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
    [CrossRef]
  4. S. Fishman, Y. Krivolapov, and A. Soffer, Nonlinearity 25, R53 (2012).
    [CrossRef]
  5. G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
    [CrossRef]
  6. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  7. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
    [CrossRef]
  8. M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).
    [CrossRef]
  9. T. Sperling, W. Buehrer, C. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2012).
    [CrossRef]
  10. V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
    [CrossRef]
  11. K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 103, 210402 (2009).
    [CrossRef]
  12. C. Conti, Phys. Rev. A 86, 061801 (2012).
    [CrossRef]
  13. V. Folli and C. Conti, Opt. Lett. 37, 332 (2012).
    [CrossRef]
  14. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
    [CrossRef]
  15. N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
    [CrossRef]
  16. V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
    [CrossRef]
  17. S. Flach, D. Krimer, and C. Skokos, Phys. Rev. Lett. 102, 024104 (2009).
  18. L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
    [CrossRef]
  19. C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 85, 2502 (2000).
    [CrossRef]
  20. C. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  21. K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, Appl. Phys. Lett. 98, 161113 (2011).
    [CrossRef]
  22. Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
    [CrossRef]
  23. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).
  24. G. Modugno, Rep. Prog. Phys. 73, 102401 (2010).
    [CrossRef]
  25. C. Conti, S. Trillo, P. D. Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, J. Opt. Soc. Am. B 19, 852 (2002).
    [CrossRef]
  26. S. Stivala, A. C. Busacca, A. Pasquazi, R. L. Oliveri, R. Morandotti, and G. Assanto, Opt. Lett. 35, 363 (2010).
    [CrossRef]
  27. I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disorder Systems (Wiley, 1988).

2013 (1)

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).
[CrossRef]

2012 (6)

T. Sperling, W. Buehrer, C. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2012).
[CrossRef]

S. Fishman, Y. Krivolapov, and A. Soffer, Nonlinearity 25, R53 (2012).
[CrossRef]

C. Conti, Phys. Rev. A 86, 061801 (2012).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 37, 332 (2012).
[CrossRef]

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).

2011 (3)

K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, Appl. Phys. Lett. 98, 161113 (2011).
[CrossRef]

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
[CrossRef]

2010 (5)

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

J. D. Bodyfelt, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 104, 164102 (2010).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

G. Modugno, Rep. Prog. Phys. 73, 102401 (2010).
[CrossRef]

S. Stivala, A. C. Busacca, A. Pasquazi, R. L. Oliveri, R. Morandotti, and G. Assanto, Opt. Lett. 35, 363 (2010).
[CrossRef]

2009 (3)

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 103, 210402 (2009).
[CrossRef]

S. Flach, D. Krimer, and C. Skokos, Phys. Rev. Lett. 102, 024104 (2009).

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

2008 (2)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

2002 (1)

2000 (1)

C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 85, 2502 (2000).
[CrossRef]

1997 (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

1994 (1)

1990 (1)

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Phys. Rev. Lett. 64, 1693 (1990).
[CrossRef]

1958 (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Aegerter, C.

T. Sperling, W. Buehrer, C. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2012).
[CrossRef]

Anderson, P. W.

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Angelani, L.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

Arie, A.

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

Assanto, G.

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Bliokh, K. Y.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Bliokh, Y. P.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Bodyfelt, J. D.

J. D. Bodyfelt, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 104, 164102 (2010).
[CrossRef]

Bramati, A.

Buehrer, W.

T. Sperling, W. Buehrer, C. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2012).
[CrossRef]

Busacca, A. C.

Chinaglia, W.

Christodoulides, D. N.

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).
[CrossRef]

Conti, C.

C. Conti, Phys. Rev. A 86, 061801 (2012).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 37, 332 (2012).
[CrossRef]

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

C. Conti, S. Trillo, P. D. Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, J. Opt. Soc. Am. B 19, 852 (2002).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 85, 2502 (2000).
[CrossRef]

D’Errico, C.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Delande, D.

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 103, 210402 (2009).
[CrossRef]

DelRe, E.

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

Fallani, L.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Fattori, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Fishman, S.

S. Fishman, Y. Krivolapov, and A. Soffer, Nonlinearity 25, R53 (2012).
[CrossRef]

Flach, S.

S. Flach, D. Krimer, and C. Skokos, Phys. Rev. Lett. 102, 024104 (2009).

Folli, V.

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 37, 332 (2012).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef]

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

Fort, C.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Freilikher, V.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Gallo, K.

K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, Appl. Phys. Lett. 98, 161113 (2011).
[CrossRef]

Garanovich, I. L.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).

Gentilini, S.

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

Ghofraniha, N.

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

Gredeskul, S. A.

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Phys. Rev. Lett. 64, 1693 (1990).
[CrossRef]

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disorder Systems (Wiley, 1988).

Inguscio, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

Kilius, J.

Kivshar, Y. S.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Phys. Rev. Lett. 64, 1693 (1990).
[CrossRef]

Kong, Y.

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

Kottos, T.

J. D. Bodyfelt, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 104, 164102 (2010).
[CrossRef]

Krimer, D.

S. Flach, D. Krimer, and C. Skokos, Phys. Rev. Lett. 102, 024104 (2009).

Krivolapov, Y.

S. Fishman, Y. Krivolapov, and A. Soffer, Nonlinearity 25, R53 (2012).
[CrossRef]

Krolikowski, W.

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Laurell, F.

K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, Appl. Phys. Lett. 98, 161113 (2011).
[CrossRef]

Leuzzi, L.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

Levenius, M.

K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, Appl. Phys. Lett. 98, 161113 (2011).
[CrossRef]

Lifshits, I. M.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disorder Systems (Wiley, 1988).

Longhi, S.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).

Maret, G.

T. Sperling, W. Buehrer, C. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2012).
[CrossRef]

Menyuk, C.

Minardi, S.

Modugno, G.

G. Modugno, Rep. Prog. Phys. 73, 102401 (2010).
[CrossRef]

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Modugno, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Morandotti, R.

Müller, C. A.

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 103, 210402 (2009).
[CrossRef]

Oliveri, R. L.

Pasiskevicius, V.

K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, Appl. Phys. Lett. 98, 161113 (2011).
[CrossRef]

Pasquazi, A.

Pastur, L. A.

I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disorder Systems (Wiley, 1988).

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Roati, G.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Roppo, V.

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

Ruocco, G.

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

Sacha, K.

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 103, 210402 (2009).
[CrossRef]

Sánchez, A.

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Phys. Rev. Lett. 64, 1693 (1990).
[CrossRef]

Schiek, R.

Segev, M.

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Shapiro, B.

J. D. Bodyfelt, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 104, 164102 (2010).
[CrossRef]

Sheng, Y.

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

Shiloh, R.

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

Silberberg, Y.

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).
[CrossRef]

Skokos, C.

S. Flach, D. Krimer, and C. Skokos, Phys. Rev. Lett. 102, 024104 (2009).

Soffer, A.

S. Fishman, Y. Krivolapov, and A. Soffer, Nonlinearity 25, R53 (2012).
[CrossRef]

Sperling, T.

T. Sperling, W. Buehrer, C. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2012).
[CrossRef]

Stivala, S.

Sukhorukov, A. A.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

C. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Trapani, P. D.

Trillo, S.

Valiulis, G.

Vázquez, L.

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Phys. Rev. Lett. 64, 1693 (1990).
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

Wang, W.

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Zaccanti, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Zakrzewski, J.

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 103, 210402 (2009).
[CrossRef]

Appl. Phys. Lett. (2)

K. Gallo, M. Levenius, F. Laurell, and V. Pasiskevicius, Appl. Phys. Lett. 98, 161113 (2011).
[CrossRef]

Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie, and W. Krolikowski, Appl. Phys. Lett. 98, 241114 (2011).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Photonics (2)

M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat. Photonics 7, 197 (2013).
[CrossRef]

T. Sperling, W. Buehrer, C. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2012).
[CrossRef]

Nature (2)

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Nonlinearity (1)

S. Fishman, Y. Krivolapov, and A. Soffer, Nonlinearity 25, R53 (2012).
[CrossRef]

Opt. Lett. (3)

Phys. Rep. (1)

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rep. 518, 1 (2012).

Phys. Rev. (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Phys. Rev. A (2)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

C. Conti, Phys. Rev. A 86, 061801 (2012).
[CrossRef]

Phys. Rev. Lett. (9)

N. Ghofraniha, S. Gentilini, V. Folli, E. DelRe, and C. Conti, Phys. Rev. Lett. 109, 243902 (2012).
[CrossRef]

S. Flach, D. Krimer, and C. Skokos, Phys. Rev. Lett. 102, 024104 (2009).

L. Leuzzi, C. Conti, V. Folli, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 102, 083901 (2009).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 85, 2502 (2000).
[CrossRef]

J. D. Bodyfelt, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 104, 164102 (2010).
[CrossRef]

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, Phys. Rev. Lett. 64, 1693 (1990).
[CrossRef]

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Rep. Prog. Phys. (1)

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Other (1)

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Figures (5)

Fig. 1.
Fig. 1.

A1(x,z)2 after Eq. (1) for δk=0.1, A1(x,0)=0.01, d0=1 with A2(x,0)=A20r(x) calculated for A20=2 (a) and A20=7 (b). In the latter case, the localized states are strongly enhanced by the more effective disordered potential V(x)=d0A20r(x).

Fig. 2.
Fig. 2.

(a) Numerically calculated gain λR after Eq. (1) versus δk for A1(x,0)=0.01 with A20=2 (dashed line) and A20=3 (dotted line). The eigenvalues of the ground state are E01.2 and E01.7, respectively. The theoretical values of the instability region δk1=2E0 and δk2=6E0 are in a good agreement with the simulations. Data are fitted by Eq. (7) with V00=2E0, f2(δk)=(3bδk/2)(b+δk/2) (dashed lines) for: A20=2 (b) and A20=3 (c). The parameters are b=0.9±0.7 (b) and b=1.5±0.3 (c). Solid lines in (b) and (c), represent numerical data.

Fig. 3.
Fig. 3.

(a) Numerically found localization lengths for the ground state of Eq. (4) versus SH amplitude A20 (continuous line) with A10=0.01, δk=0.1, and for 10 realizations of disorder. Dashed line is the fit after Eq. (10). (b) Gain λR versus A20 (continuous line) for δk=0.1 and A10=0.01. Dashed line is the fit after Eq. (11).

Fig. 4.
Fig. 4.

(a) FF amplitude |A1(x,z)| at z=1 (continuous line) and the ground state arbitrarily scaled for A01=0.01, δk=0.1, A20=10. (b) Evolution of the peak value of the FF wave (continuous thick line) and coefficients cm(z), in Eq. (12), for m=0 (dashed line) and m>0 (continuous thin lines).

Fig. 5.
Fig. 5.

Histogram of the minimum difference between the growth rate of A1(x,z) and φk, with 50 realizations of disorder, A01=0.01 and δk=0.1. (a) A20=1, (b) A20=2, (c) A20=3, and (d) A20=4.

Equations (12)

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izA1+x2A1+d(x)A1*A2=0izA2+12x2A2+δkA2+12d(x)A12=0,
izA1+x2A1V(x)A1*δk2A1=0,
A1(x,z)=ncn(z)φn(x).
2φnx2+V(x)φn(x)=Enφn.
idcmdzcm(Em+δk2)+nVmn(cncn*)=0,
cm(z)=c^meλRz,
λR2=(2VmmEmδk/2)(Em+δk/2).
V(x)φm2(x)dx=Emφm2dx+φmxxφmdx.
Vmm=Em1lm22Em<0,
l=l˜/V02/3=l˜/(d0A20)2/3,
max(λRm)=Vmm=2Em=2E˜m(d0A20)4/3.
A1(x,z)=mcm(z)φm(x).

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