Abstract

We show that Anderson localization is possible in waveguide arrays with periodically spaced defect waveguides having a lower refractive index. Such localization is mediated by Bragg reflection, and it takes place even if diagonal or off-diagonal disorder affects only defect waveguides. For off-diagonal disorder the localization degree of the intensity distributions monotonically grows with increasing disorder. In contrast, under appropriate conditions, increasing diagonal disorder may result in weaker localization.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (6)

2011 (1)

2010 (2)

2009 (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
[CrossRef]

2008 (3)

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

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (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 (1)

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

2005 (1)

2004 (1)

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

1997 (1)

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 (1)

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

1989 (1)

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

1984 (1)

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

1980 (1)

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

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]

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.

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]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (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]

Belic, M. R.

D. M. Jović, M. R. Belić, and C. Denz, Phys. Rev. A 85, 031801(R) (2012).
[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]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

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

Conti, C.

Dalichaouch, R.

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

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]

Denz, C.

D. M. Jović, M. R. Belić, and C. Denz, Phys. Rev. A 85, 031801(R) (2012).
[CrossRef]

Di Giuseppe, G.

Dreisow, F.

Efremidis, N. K.

Fishman, S.

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

Folli, V.

Freilikher, V. D.

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

Hizanidis, K.

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]

Jovic, D. M.

D. M. Jović, M. R. Belić, and C. Denz, Phys. Rev. A 85, 031801(R) (2012).
[CrossRef]

Kartashov, Y. V.

Keil, R.

Kivshar, Y. S.

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

Konotop, V. V.

Lagendijk, A.

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]

Lazarides, N.

M. I. Molina, N. Lazarides, and G. P. Tsironis, Phys. Rev. E 85, 017601 (2012).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Legendijk, A.

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

Lewenstein, M.

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]

Martin, L.

McCal, S. L.

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

Meyer, J. M.

Molina, M. I.

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]

Naether, U.

Nolte, S.

Pelinovsky, D. E.

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

Perez-Leija, A.

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]

Righini, R.

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

Saleh, B. E. A.

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.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

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

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

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]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Stützer, S.

Sukhorukov, A. A.

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

Szameit, A.

Torner, L.

Tsironis, G. P.

M. I. Molina, N. Lazarides, and G. P. Tsironis, Phys. Rev. E 85, 017601 (2012).
[CrossRef]

Tünnermann, A.

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]

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]

Nature (4)

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. (6)

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (1)

D. M. Jović, M. R. Belić, and C. Denz, Phys. Rev. A 85, 031801(R) (2012).
[CrossRef]

Phys. Rev. E (2)

M. I. Molina, N. Lazarides, and G. P. Tsironis, Phys. Rev. E 85, 017601 (2012).
[CrossRef]

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

Phys. Rev. Lett. (4)

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

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

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

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

Prog. Opt. (2)

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

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Prog. Opt. 52, 63 (2009).
[CrossRef]

Sov. J. Radiofizika (1)

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

Wave Motion (1)

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

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Figures (5)

Fig. 1.
Fig. 1.

(a) Bandgap spectrum of a uniform lattice and (b) propagation constant of the defect mode versus pd for pa=7. Red circles indicate gap borders.

Fig. 2.
Fig. 2.

Dynamics of light propagation in disordered lattices with negative defects, averaged over Q=103 disorder realizations. Disorder with (a) Sd=0.15, (b) 0.25, and (c) 0.4 is introduced into positions of the defect channels. In all cases, defect waveguides are separated by two waveguides; pd=4, pa=7, and σ=0. The plots show the field modulus.

Fig. 3.
Fig. 3.

Averaged output intensity distributions at ξ=600 for (a) Sd=0.25 and (b) Sd=0.35. (c) Averaged inverse form factor versus propagation distance at Sd=0.15 (curve 1), Sd=0.2 (curve 2), and Sd=0.3 (curve 3). In all cases pd=4, pa=7, σ=0.

Fig. 4.
Fig. 4.

(a) Inverse averaged form factor at ξ=600 versus disorder strength for pd=4, when disorder affects only the positions of the defect waveguides (circles) and when disorder affects the positions of all waveguides (pentagons) and (b) averaged form factor at ξ=600 versus defect depth for Sd=0.4. In all cases pa=7, σ=0.

Fig. 5.
Fig. 5.

(a) Inverse averaged form factor versus propagation distance at A2=0.2 in focusing (σ=1) and defocusing (σ=+1) media and (b) inverse averaged form factor at ξ=600 versus input peak intensity. In panels (a) and (b), pd=4, pa=7, Sd=0.3. (c) Inverse averaged form factor versus strength of diagonal disorder at ξ=1000 for pd=5, pa=7, σ=0.

Equations (1)

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iqξ=122qη2R(η)q+σq|q|2.

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