Abstract

We investigate experimentally the light evolution inside a two-dimensional finite periodic array of weakly coupled optical waveguides with a disordered boundary. For a completely localized initial condition away from the surface, we find that the disordered boundary induces an asymptotic localization in the bulk, centered around the initial position of the input beam.

© 2012 Optical Society of America

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  1. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  2. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
    [CrossRef]
  3. R. L. Weaver, Wave Motion 12, 129 (1990).
    [CrossRef]
  4. R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
    [CrossRef]
  5. J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
    [CrossRef]
  6. S. John, Phys. Rev. Lett. 53, 2169 (1984).
    [CrossRef]
  7. V. D. Freilikher and S. A. Gredeskul, Prog. Opt. 30, 137 (1992).
    [CrossRef]
  8. D. S. Wiersma, P. Bartolini, A. Legendijk, and R. Righini, Nature 390, 671 (1997).
    [CrossRef]
  9. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef]
  10. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
    [CrossRef]
  11. A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tnnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
    [CrossRef]
  12. 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]
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    [CrossRef]
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    [CrossRef]
  15. A. García-Martín, J. J. Sáenz, and M. N. Vesperinas, Phys. Rev. Lett. 84, 3578 (2000).
    [CrossRef]
  16. V. D. Freilikher, N. M. Makarov, and I. V. Yurkevich, Phys. Rev. B 41, 8033 (1990).
    [CrossRef]
  17. J. A. Sánchez-Gil, V. D. Freilikher, I. V. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80, 948 (1998).
    [CrossRef]
  18. J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. V. Yurkevich, Phys. Rev. B 59, 5915 (1999).
    [CrossRef]
  19. M. Rendón, N. M. Makarov, and F. M. Izrailev, Phys. Rev. E 83, 051124 (2011).
    [CrossRef]
  20. M. I. Molina, Phys. Lett. A 375, 2056 (2011).
    [CrossRef]
  21. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef]
  22. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, Opt. Express 15, 1579 (2007).
    [CrossRef]
  23. A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
    [CrossRef]

2011 (3)

2010 (2)

2008 (2)

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

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

2007 (2)

2003 (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

2000 (1)

A. García-Martín, J. J. Sáenz, and M. N. Vesperinas, Phys. Rev. Lett. 84, 3578 (2000).
[CrossRef]

1999 (1)

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. V. Yurkevich, Phys. Rev. B 59, 5915 (1999).
[CrossRef]

1998 (2)

J. A. Sánchez-Gil, V. D. Freilikher, I. V. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80, 948 (1998).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Phys. Rev. Lett. 80, 4165 (1998).
[CrossRef]

1997 (2)

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Appl. Phys. Lett. 71, 1912 (1997).
[CrossRef]

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

1992 (1)

V. D. Freilikher and S. A. Gredeskul, Prog. Opt. 30, 137 (1992).
[CrossRef]

1991 (1)

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
[CrossRef]

1990 (2)

R. L. Weaver, Wave Motion 12, 129 (1990).
[CrossRef]

V. D. Freilikher, N. M. Makarov, and I. V. Yurkevich, Phys. Rev. B 41, 8033 (1990).
[CrossRef]

1984 (1)

S. John, Phys. Rev. Lett. 53, 2169 (1984).
[CrossRef]

1979 (1)

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

1958 (1)

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

Abouraddy, A. F.

Abrahams, E.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

Anderson, P. W.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

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

Armstrong, J. P.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
[CrossRef]

Aspect, A.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[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]

Bartolini, P.

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

Bernard, A.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Billy, J.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Bouyer, P.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Christodoulides, D. N.

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]

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

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Clement, C.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Dalichaouch, R.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
[CrossRef]

Di Giuseppe, G.

Dreisow, F.

Fishman, S.

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

Freilikher, V. D.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. V. Yurkevich, Phys. Rev. B 59, 5915 (1999).
[CrossRef]

J. A. Sánchez-Gil, V. D. Freilikher, I. V. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80, 948 (1998).
[CrossRef]

V. D. Freilikher and S. A. Gredeskul, Prog. Opt. 30, 137 (1992).
[CrossRef]

V. D. Freilikher, N. M. Makarov, and I. V. Yurkevich, Phys. Rev. B 41, 8033 (1990).
[CrossRef]

García-Martín, A.

A. García-Martín, J. J. Sáenz, and M. N. Vesperinas, Phys. Rev. Lett. 84, 3578 (2000).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Phys. Rev. Lett. 80, 4165 (1998).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Appl. Phys. Lett. 71, 1912 (1997).
[CrossRef]

Gredeskul, S. A.

V. D. Freilikher and S. A. Gredeskul, Prog. Opt. 30, 137 (1992).
[CrossRef]

Hambrecht, B.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Heinrich, M.

Izrailev, F. M.

M. Rendón, N. M. Makarov, and F. M. Izrailev, Phys. Rev. E 83, 051124 (2011).
[CrossRef]

John, S.

S. John, Phys. Rev. Lett. 53, 2169 (1984).
[CrossRef]

Josse, V.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Kartashov, Y. V.

Keil, R.

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.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Legendijk, A.

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

Licciardello, D. C.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

Lugan, P.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Makarov, N. M.

M. Rendón, N. M. Makarov, and F. M. Izrailev, Phys. Rev. E 83, 051124 (2011).
[CrossRef]

V. D. Freilikher, N. M. Makarov, and I. V. Yurkevich, Phys. Rev. B 41, 8033 (1990).
[CrossRef]

Maradudin, A. A.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. V. Yurkevich, Phys. Rev. B 59, 5915 (1999).
[CrossRef]

J. A. Sánchez-Gil, V. D. Freilikher, I. V. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80, 948 (1998).
[CrossRef]

Martin, L.

McCal, S. L.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
[CrossRef]

Molina, M. I.

M. I. Molina, Phys. Lett. A 375, 2056 (2011).
[CrossRef]

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.

Platzman, P. M.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
[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.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

Rendón, M.

M. Rendón, N. M. Makarov, and F. M. Izrailev, Phys. Rev. E 83, 051124 (2011).
[CrossRef]

Righini, R.

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

Sáenz, J. J.

A. García-Martín, J. J. Sáenz, and M. N. Vesperinas, Phys. Rev. Lett. 84, 3578 (2000).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Phys. Rev. Lett. 80, 4165 (1998).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Appl. Phys. Lett. 71, 1912 (1997).
[CrossRef]

Saleh, B. E. A.

Sánchez-Gil, J. A.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. V. Yurkevich, Phys. Rev. B 59, 5915 (1999).
[CrossRef]

J. A. Sánchez-Gil, V. D. Freilikher, I. V. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80, 948 (1998).
[CrossRef]

Sanchez-Palencia, L.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Schulz, S.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
[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]

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]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[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]

Szameit, A.

Tnnermann, A.

Torner, L.

Torres, J. A.

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Phys. Rev. Lett. 80, 4165 (1998).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Appl. Phys. Lett. 71, 1912 (1997).
[CrossRef]

Tuennermann, A.

Vesperinas, M. N.

A. García-Martín, J. J. Sáenz, and M. N. Vesperinas, Phys. Rev. Lett. 84, 3578 (2000).
[CrossRef]

Vesperinas, M. Nieto.

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Phys. Rev. Lett. 80, 4165 (1998).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Appl. Phys. Lett. 71, 1912 (1997).
[CrossRef]

Vysloukh, V. A.

Weaver, R. L.

R. L. Weaver, Wave Motion 12, 129 (1990).
[CrossRef]

Wiersma, D. S.

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

Yurkevich, I. V.

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. V. Yurkevich, Phys. Rev. B 59, 5915 (1999).
[CrossRef]

J. A. Sánchez-Gil, V. D. Freilikher, I. V. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80, 948 (1998).
[CrossRef]

V. D. Freilikher, N. M. Makarov, and I. V. Yurkevich, Phys. Rev. B 41, 8033 (1990).
[CrossRef]

Zeil, P.

Zuo, Z. C.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Appl. Phys. Lett. 71, 1912 (1997).
[CrossRef]

J. Phys. B (1)

A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
[CrossRef]

Nature (5)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, Nature 354, 53 (1991).
[CrossRef]

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

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

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

Opt. Express (2)

Opt. Lett. (1)

Phys. Lett. A (1)

M. I. Molina, Phys. Lett. A 375, 2056 (2011).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. B (2)

V. D. Freilikher, N. M. Makarov, and I. V. Yurkevich, Phys. Rev. B 41, 8033 (1990).
[CrossRef]

J. A. Sánchez-Gil, V. D. Freilikher, A. A. Maradudin, and I. V. Yurkevich, Phys. Rev. B 59, 5915 (1999).
[CrossRef]

Phys. Rev. E (1)

M. Rendón, N. M. Makarov, and F. M. Izrailev, Phys. Rev. E 83, 051124 (2011).
[CrossRef]

Phys. Rev. Lett. (6)

J. A. Sánchez-Gil, V. D. Freilikher, I. V. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80, 948 (1998).
[CrossRef]

A. García-Martín, J. A. Torres, J. J. Sáenz, and M. Nieto. Vesperinas, Phys. Rev. Lett. 80, 4165 (1998).
[CrossRef]

A. García-Martín, J. J. Sáenz, and M. N. Vesperinas, Phys. Rev. Lett. 84, 3578 (2000).
[CrossRef]

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[CrossRef]

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

S. John, Phys. Rev. Lett. 53, 2169 (1984).
[CrossRef]

Prog. Opt. (1)

V. D. Freilikher and S. A. Gredeskul, Prog. Opt. 30, 137 (1992).
[CrossRef]

Wave Motion (1)

R. L. Weaver, Wave Motion 12, 129 (1990).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Scheme of a finite 2D coupled array of elliptical waveguides with disordered boundary.

Fig. 2.
Fig. 2.

Comparison between simulations of an ordered 13×5 array (right column) and the disordered-boundary waveguide array of Eq. (1) (left column) with Cα=2, Cβ=1.7, ωα=11/28, and ωβ=4/17, averaged over 100 realizations.

Fig. 3.
Fig. 3.

Experimental results for the 5×5 arrays: (a) mean output of 30 realizations, λ=800nm; (b), (c) logarithmic plots of |un,mc|, |unc,m| respectively.

Fig. 4.
Fig. 4.

Experimental results for the 13×5 arrays: (a) microscope image of one configuration of a disordered boundary; (b) ordered array; (c) mean output of 30 realizations of the disordered-boundary waveguide array, λ=840nm; (d) output of the ordered array; (e), (f) logarithmic plots of |un,mc|, |unc,m|.

Equations (1)

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j=p±1Cvjuj,q+j=q±1Chjup,j=idupqdz,

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