Abstract

We report on the observation of Anderson wave localization in one-dimensional waveguide arrays with off-diagonal disorder. The waveguide elements are inscribed in silica glass, and a uniform random distribution of coupling parameters is achieved by a precise variation of the relative waveguide positions. In the absence of disorder we observe ballistic transport as expected from discrete diffraction in periodic arrays. When off-diagonal disorder is deliberately introduced into the array we observe Anderson localization. The strength of the localization signature increases with higher levels of disorder.

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    [CrossRef]
  2. A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2008).
    [CrossRef]
  3. C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702 (1981).
    [CrossRef]
  4. P. Erdös and R. C. Herndon, “Theories of electrons in one-dimensional disordered systems,” Adv. Phys. 31, 65–163 (1982).
    [CrossRef]
  5. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
    [CrossRef] [PubMed]
  6. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 100, 170506 (2008).
    [PubMed]
  7. T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
    [CrossRef]
  8. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  18. D. Blömer, A. Szameit, F. Dreisow, T. Schreiber, S. Nolte, and A. Tünnermann, “Nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Express 14, 2151–2157 (2006).
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    [CrossRef] [PubMed]
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    [CrossRef]

2010 (2)

2009 (1)

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

2008 (4)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2008).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 100, 170506 (2008).
[PubMed]

2007 (2)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef] [PubMed]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, “Control of directional evanescent coupling in fs laser written waveguides,” Opt. Express 15, 1579–1587 (2007).
[CrossRef] [PubMed]

2006 (3)

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

D. Blömer, A. Szameit, F. Dreisow, T. Schreiber, S. Nolte, and A. Tünnermann, “Nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Express 14, 2151–2157 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

2003 (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

1999 (1)

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

1997 (1)

F. M. Izrailev, T. Kottos, A. Politi, and G. P. Tsironis, “Evolution of wave packets in quasi-one-dimensional and one-dimensional random media: diffusion versus localization,” Phys. Rev. E 55, 4951–4963 (1997).
[CrossRef]

1982 (2)

P. Erdös and R. C. Herndon, “Theories of electrons in one-dimensional disordered systems,” Adv. Phys. 31, 65–163 (1982).
[CrossRef]

J. B. Pendry, “Off-diagonal disorder and 1D localization,” J. Phys. C: Solid State Phys. 15, 5773–5778 (1982).
[CrossRef]

1981 (1)

C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702 (1981).
[CrossRef]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 100, 170506 (2008).
[PubMed]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef] [PubMed]

Blömer, D.

Braeuer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Christodoulides, D. N.

A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber–Fock photonic lattices,” Opt. Lett. 35, 2409–2411 (2010).
[CrossRef] [PubMed]

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

Cianci, E.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Desyatnikov, A. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Dreisow, F.

Economou, E. N.

C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702 (1981).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 100, 170506 (2008).
[PubMed]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Erdös, P.

P. Erdös and R. C. Herndon, “Theories of electrons in one-dimensional disordered systems,” Adv. Phys. 31, 65–163 (1982).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef] [PubMed]

Foglietti, V.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Heinrich, M.

Herndon, R. C.

P. Erdös and R. C. Herndon, “Theories of electrons in one-dimensional disordered systems,” Adv. Phys. 31, 65–163 (1982).
[CrossRef]

Izrailev, F. M.

F. M. Izrailev, T. Kottos, A. Politi, and G. P. Tsironis, “Evolution of wave packets in quasi-one-dimensional and one-dimensional random media: diffusion versus localization,” Phys. Rev. E 55, 4951–4963 (1997).
[CrossRef]

Kartashov, Y. V.

Keil, R.

Kip, D.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

Kivshar, Yu. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Kottos, T.

F. M. Izrailev, T. Kottos, A. Politi, and G. P. Tsironis, “Evolution of wave packets in quasi-one-dimensional and one-dimensional random media: diffusion versus localization,” Phys. Rev. E 55, 4951–4963 (1997).
[CrossRef]

Krolikowski, W.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Lagendijk, A.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2008).
[CrossRef]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Laporta, P.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Lederer, F.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Lobino, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Longhi, S.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Makris, K. G.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

Marangoni, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Morandotti, R.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 100, 170506 (2008).
[PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Moya-Cessa, H.

Neshev, D. N.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Nolte, S.

Peleg, O.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Off-diagonal disorder and 1D localization,” J. Phys. C: Solid State Phys. 15, 5773–5778 (1982).
[CrossRef]

Perets, H. B.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Perez-Leija, A.

Pertsch, T.

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, “Control of directional evanescent coupling in fs laser written waveguides,” Opt. Express 15, 1579–1587 (2007).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 100, 170506 (2008).
[PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Politi, A.

F. M. Izrailev, T. Kottos, A. Politi, and G. P. Tsironis, “Evolution of wave packets in quasi-one-dimensional and one-dimensional random media: diffusion versus localization,” Phys. Rev. E 55, 4951–4963 (1997).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Ramponi, R.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef] [PubMed]

Rüter, C. E.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

Schreiber, T.

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef] [PubMed]

Segev, M.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef] [PubMed]

Shandarova, K.

K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009).
[CrossRef] [PubMed]

Silberberg, Y.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 100, 170506 (2008).
[PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702 (1981).
[CrossRef]

Sukhorukov, A. A.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Szameit, A.

Torner, L.

Trompeter, H.

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Supplementary Material (4)

» Media 1: AVI (199 KB)     
» Media 2: AVI (358 KB)     
» Media 3: AVI (358 KB)     
» Media 4: AVI (356 KB)     

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Figures (7)

Fig. 1
Fig. 1

Numerical simulation of optical field propagation when light is injected into the 51 st waveguide in a 101–waveguide array. The arrays used in (a) to (d) have an increasing degree of disorder. Each plot results from averaging the intensities of 41 realizations of random disordered arrays described by a uniform probability distribution function having a mean value �� 0 = 1.8 cm−1 and width 2Δ, for disorder parameters Δ/�� 0 = 0, 0.4, 0.55, and 0.70, respectively. The average output intensity distributions for propagation lengths 35 mm (blue) and 49 mm (green), respectively, corresponding to the lengths of the two samples used, are shown on the right.

Fig. 2
Fig. 2

(a) Experimental setup. LD: laser diode (780 nm); PBS: polarizing beam splitter; OA: optical attenuator; 5× (NA = 0.1) and 10× (NA = 0.25) microscope objectives; S: waveguide array sample. (b) CCD image of the periodic waveguide array output (period = 17 μm, and distance from the top of fused silica slab ≈ 250 μm) when a single waveguide is excited at the input. (c) Measured dependence of the coupling coefficient �� on the waveguide separation for directional couplers fabricated with the same parameters as the arrays.

Fig. 3
Fig. 3

Data acquisition and analysis. Panel (a) presents data for light injected into the 50th waveguide (no = 50) of the long periodic array. The intensity at the output of the waveguide array is captured with a CCD camera (shown in the middle). The image is then post-processed to extract the discrete intensity distribution (�� n, 50) by integrating over rectangular areas 10 × 30 pixel each centered on the center of each waveguide (n) shown as the black rectangle. The central red rectangle on the CCD image in panel (a) indicates the location of the excitation site. The discrete intensity distribution, �� n, 50 is shown as the red bar-plot. The brightness image in panel (b) displays the distribution, n,no of the intensity of the light measured at the output of the waveguides (n) when only waveguide no is illuminated. The red rectangle on panel (b) indicates the output distribution for light injected into the 50th waveguide. In panel (c) the displaced distribution, n + no,no is shown. Each distribution of the measured intensity is displaced such that it is centered about the illuminated waveguide. Only the middle 41 waveguides are illuminated (one at a time) with the ordinate marking the illuminated waveguide.

Fig. 4
Fig. 4

Average displaced distribution ¯ n for long (a) and short (b) periodic waveguide arrays. The black squares represent a theoretical best-fit with ��o ≈ 1.79 cm−1 and 1.80 cm−1 for the short and long arrays, respectively. The root-mean-square (RMS) width of the experimental distributions (≈ 18.0 and ≈ 25.4) are shown in (c) as function of the array length. The line represents the best-fit to the linear ballistic expansion as a function of the array length with ��o ≈ 1.81 cm−1.

Fig. 5
Fig. 5

Effect of disorder on the propagation of light through 101-waveguide arrays in the short (first row, a–c) and long (second row, d–f) samples. The two rows correspond to disorder parameters Δ/��o ≈ 0.44, 0.69 and 0.91 for the short sample, and 0.51, 0.70 and 0.87 for the long sample. The color plot in each panel shows the displaced distributions n + no , no at the output. Each row in the plot corresponds to the output intensity distribution for a single point excitation at no after shifting it by no . Only the middle 41 waveguides are illuminated (one at a time) with the ordinate marking the illuminated waveguide. The average of the displaced distributions, ¯ n , for all 41 waveguides is plotted at the bottom of each panel with the red line showing the result of a numerical simulation with ��o and Δ as fitting parameters.

Fig. 6
Fig. 6

Single-frame excerpts from video recordings. On the left we display the recorded intensity distribution at the output of the short waveguide array when the middle 41 input waveguides are illuminated one at a time, while on the right the cumulative averaged discrete intensity distribution is updated. (a) Periodic array (Media 1); (b) array with disorder parameter Δ/��o ≈ 0.44 (Media 2); (c) array with Δ/��o ≈ 0.69 (Media 3); (c) array with Δ/��o ≈ 0.91 (Media 4).

Fig. 7
Fig. 7

RMS width as function of the disorder parameter Δ/��o for the short (red-circle symbols) and long array (blue-square symbols). The colored bands represent the range of values of the RMS-width a standard deviation around the mean value. For each value of the disorder parameter, RMS-width and its standard deviation have been evaluated by averaging over 21×40 disorder realizations for the sampling average approach (dashed lines), while 21 disorder realizations and 40 shifted input waveguides have been considered for the shifting average approach (solid lines). Inset: average displaced distribution, ¯ n for short and long arrays with disorder parameter Δ/��o ∼ 0.9. The log-scale plot highlights the exponential decay of the Anderson-localized states. The dotted-lines are a guide for the eye.

Equations (6)

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i d E n d z + β n E n + 𝒞 n , n + 1 E n + 1 + 𝒞 n , n 1 E n 1 = 0 ,
i d E n d z + 𝒞 o ( E n + 1 + E n 1 ) = 0 .
E n , n o ( z ) = A 0 i n n o J n n o ( 2 𝒞 o z ) ,
I n , n o ( z ) | J n n o ( 2 𝒞 o z ) | 2 .
i d E n d z + 𝒞 n , n + 1 E n + 1 + 𝒞 n , n 1 E n 1 = 0 .
¯ n = n 0 n + n 0 , n 0 .

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