Abstract

We report Anderson localization in two-dimensional optical waveguide arrays with disorder in waveguide separation introduced along one axis of the array, in an uncorrelated fashion for each waveguide row. We show that the anisotropic nature of such disorder induces a strong localization along both array axes. The degree of localization in the cross-axis remains weaker than that in the direction in which disorder is introduced. This effect is illustrated both theoretically and experimentally.

© 2012 Optical Society of America

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  1. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  2. P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
    [CrossRef]
  3. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).
  4. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
    [CrossRef]
  5. A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Nature 404, 850 (2000).
    [CrossRef]
  6. S. S. Abdullaev and F. K. Abdullaev, Sov. J. Radiofizika 23, 766 (1980).
  7. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
    [CrossRef]
  8. H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
    [CrossRef]
  9. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef]
  10. Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, arXiv:1203.0838 (2012).
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    [CrossRef]
  12. A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
    [CrossRef]
  13. L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, Opt. Express 19, 13636 (2011).
    [CrossRef]

2011 (1)

2010 (1)

2008 (1)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

2004 (1)

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

2000 (1)

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Nature 404, 850 (2000).
[CrossRef]

1997 (1)

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

1989 (1)

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

1985 (1)

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

1980 (1)

S. S. Abdullaev and F. K. Abdullaev, Sov. J. Radiofizika 23, 766 (1980).

1958 (1)

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

Abdullaev, F. K.

S. S. Abdullaev and F. K. Abdullaev, Sov. J. Radiofizika 23, 766 (1980).

Abdullaev, S. S.

S. S. Abdullaev and F. K. Abdullaev, Sov. J. Radiofizika 23, 766 (1980).

Abouraddy, A. F.

Anderson, P. W.

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

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Bartelt, H.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Bartolini, P.

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

Chabanov, A. A.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Nature 404, 850 (2000).
[CrossRef]

Christodoulides, D. N.

de Raedt, H.

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

de Vries, P.

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

Di Giuseppe, G.

Dreisow, F.

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, arXiv:1203.0838 (2012).

Genack, A. Z.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Nature 404, 850 (2000).
[CrossRef]

Heinrich, M.

Kartashov, Y. V.

Keil, R.

Kobelke, J.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Krivolapov, Y.

Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, arXiv:1203.0838 (2012).

Lagendijk, A.

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

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Lederer, F.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Lee, P. A.

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

Levi, L.

Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, arXiv:1203.0838 (2012).

Martin, L.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Nolte, S.

Perez-Leija, A.

Pertsch, T.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Peschel, U.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Ramakrishnan, T. V.

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

Righini, R.

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

Saleh, B. E. A.

Schuster, K.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, arXiv:1203.0838 (2012).

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Stoytchev, M.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Nature 404, 850 (2000).
[CrossRef]

Szameit, A.

Torner, L.

Tünnermann, A.

A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
[CrossRef]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Vysloukh, V. A.

Wiersma, D. S.

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

Wilkinson, M.

Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, arXiv:1203.0838 (2012).

Zeil, P.

Nature (3)

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

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, Nature 404, 850 (2000).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

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

Phys. Rev. Lett. (3)

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Rev. Mod. Phys. (1)

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

Sov. J. Radiofizika (1)

S. S. Abdullaev and F. K. Abdullaev, Sov. J. Radiofizika 23, 766 (1980).

Other (2)

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).

Y. Krivolapov, L. Levi, S. Fishman, M. Segev, and M. Wilkinson, arXiv:1203.0838 (2012).

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Figures (4)

Fig. 1.
Fig. 1.

Comparison of output intensity distribution obtained by averaging over different realizations of arrays (left) and for excitation of different waveguides (right) at Sη=0.4.

Fig. 2.
Fig. 2.

Microscopic images of disordered waveguide arrays (left), experimental (center), and theoretical (right) averaged output intensity distributions in linear case. The disorder level is Sη=0.0, 0.2, 0.4 and 0.6 from first to fourth row.

Fig. 3.
Fig. 3.

Experimental output intensity distributions for excitation of different waveguides of disordered array with Sη=0.4 resulting in the formation of weakly (left and center) and strongly localized (right) patterns.

Fig. 4.
Fig. 4.

(a) Theoretically calculated and (b) experimentally determined exponential decay rates αη, αζ versus disorder strength Sη, and (c) inverse form-factor of averaged output intensity distribution versus input amplitude at Sη=0.4.

Equations (1)

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iqξ=12(2qη2+2qζ2)q|q|2pR(η,ζ)q,

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