Abstract

We study numerically the interaction of spatially localized modes in strongly scattering two-dimensional (2D) media. We move eigenvalues in the complex plane by changing gradually the index of a single scatterer. When spatial and spectral overlap is sufficient, localized states couple, and avoided level crossing is observed. We show that local manipulation of the disordered structure can couple several localized states to form an extended chain of hybridized modes crossing the entire sample, thus changing the nature of certain modes from localized to extended in a nominally localized disordered system. We suggest such a chain in 2D random systems is the analog of one-dimensional necklace states, the occasional open channels predicted by Pendry [Physics1, 20 (2008).10.1103/Physics.1.20PYCSAP0554-128X] through which the light can sneak through an opaque medium.

© 2012 Optical Society of America

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  1. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  2. M. Y. Azbel, Solid State Commun. 45, 527 (1983).
    [CrossRef]
  3. M. Y. Azbel, Phys. Rev. B 28, 4106 (1983).
    [CrossRef]
  4. J. B. Pendry, J. Phys. C 20, 733 (1987).
    [CrossRef]
  5. J. B. Pendry, Adv. Phys. 43, 461 (1994).
    [CrossRef]
  6. J. Bertolotti, S. Gottardo, and D. S. Wiersma, Phys. Rev. Lett. 94, 113903 (2005).
    [CrossRef]
  7. J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
    [CrossRef]
  8. P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
    [CrossRef]
  9. P. Sebbah, B. Hu, V. I. Kopp, and A. Z. Genack, J. Opt. Soc. Am. B 24, A77 (2007).
    [CrossRef]
  10. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
    [CrossRef]
  11. D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
    [CrossRef]
  12. U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000).
    [CrossRef]
  13. U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Phys. Rev. Lett. 100, 126402 (2008).
    [CrossRef]
  14. H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
    [CrossRef]
  15. C. Vanneste and P. Sebbah, Phys. Rev. A 79, 041802(R) (2009).
    [CrossRef]
  16. J. Jin, The Finite Element Method in Electromagnetics(Wiley, 1993).
  17. See for instance C. Cohen-Tannoudji, B. Diu, and F. Laloe, Mécanique Quantique (Herman, 1973), Chap. 4.
  18. P. von Brentano and M. Philipp, Phys. Lett. B 454, 171 (1999), and references therein.
    [CrossRef]
  19. M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, Phys. Rev. E 62, 1922 (2000).
    [CrossRef]
  20. J. Pendry, Physics 1, 20 (2008).
    [CrossRef]
  21. Y. Imry, Europhys. Lett. 1, 249 (1986).
    [CrossRef]

2011 (1)

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

2009 (1)

C. Vanneste and P. Sebbah, Phys. Rev. A 79, 041802(R) (2009).
[CrossRef]

2008 (3)

J. Pendry, Physics 1, 20 (2008).
[CrossRef]

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
[CrossRef]

U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Phys. Rev. Lett. 100, 126402 (2008).
[CrossRef]

2007 (1)

2006 (2)

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

2005 (1)

J. Bertolotti, S. Gottardo, and D. S. Wiersma, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

2000 (2)

U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000).
[CrossRef]

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, Phys. Rev. E 62, 1922 (2000).
[CrossRef]

1999 (1)

P. von Brentano and M. Philipp, Phys. Lett. B 454, 171 (1999), and references therein.
[CrossRef]

1994 (1)

J. B. Pendry, Adv. Phys. 43, 461 (1994).
[CrossRef]

1990 (1)

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

1987 (1)

J. B. Pendry, J. Phys. C 20, 733 (1987).
[CrossRef]

1986 (1)

Y. Imry, Europhys. Lett. 1, 249 (1986).
[CrossRef]

1983 (2)

M. Y. Azbel, Solid State Commun. 45, 527 (1983).
[CrossRef]

M. Y. Azbel, Phys. Rev. B 28, 4106 (1983).
[CrossRef]

1958 (1)

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

Anderson, P. W.

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

Andreani, L. C.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

Azbel, M. Y.

M. Y. Azbel, Solid State Commun. 45, 527 (1983).
[CrossRef]

M. Y. Azbel, Phys. Rev. B 28, 4106 (1983).
[CrossRef]

Bertolotti, J.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

J. Bertolotti, S. Gottardo, and D. S. Wiersma, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

Bliokh, K. Y.

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
[CrossRef]

Bliokh, Y. P.

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
[CrossRef]

Cao, H.

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

Cohen-Tannoudji, C.

See for instance C. Cohen-Tannoudji, B. Diu, and F. Laloe, Mécanique Quantique (Herman, 1973), Chap. 4.

Diu, B.

See for instance C. Cohen-Tannoudji, B. Diu, and F. Laloe, Mécanique Quantique (Herman, 1973), Chap. 4.

Dunlap, D. H.

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Freilikher, V.

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
[CrossRef]

Galli, M.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

Genack, A. Z.

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
[CrossRef]

P. Sebbah, B. Hu, V. I. Kopp, and A. Z. Genack, J. Opt. Soc. Am. B 24, A77 (2007).
[CrossRef]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

Ghulinyan, M.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

Gottardo, S.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

J. Bertolotti, S. Gottardo, and D. S. Wiersma, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

Hu, B.

P. Sebbah, B. Hu, V. I. Kopp, and A. Z. Genack, J. Opt. Soc. Am. B 24, A77 (2007).
[CrossRef]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

Imry, Y.

Y. Imry, Europhys. Lett. 1, 249 (1986).
[CrossRef]

Izrailev, F. M.

U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Phys. Rev. Lett. 100, 126402 (2008).
[CrossRef]

U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics(Wiley, 1993).

Klosner, J. M.

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

Kopp, V. I.

Krokhin, A. A.

U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Phys. Rev. Lett. 100, 126402 (2008).
[CrossRef]

U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000).
[CrossRef]

Kuhl, U.

U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Phys. Rev. Lett. 100, 126402 (2008).
[CrossRef]

U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000).
[CrossRef]

Laloe, F.

See for instance C. Cohen-Tannoudji, B. Diu, and F. Laloe, Mécanique Quantique (Herman, 1973), Chap. 4.

Liew, S. F.

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

Noh, H.

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

Pascovici, G.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, Phys. Rev. E 62, 1922 (2000).
[CrossRef]

Pavesi, L.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

Pendry, J.

J. Pendry, Physics 1, 20 (2008).
[CrossRef]

Pendry, J. B.

J. B. Pendry, Adv. Phys. 43, 461 (1994).
[CrossRef]

J. B. Pendry, J. Phys. C 20, 733 (1987).
[CrossRef]

Philipp, M.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, Phys. Rev. E 62, 1922 (2000).
[CrossRef]

P. von Brentano and M. Philipp, Phys. Lett. B 454, 171 (1999), and references therein.
[CrossRef]

Phillips, P. W.

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Richter, A.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, Phys. Rev. E 62, 1922 (2000).
[CrossRef]

Rooks, M. J.

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

Sapienza, R.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

Sebbah, P.

C. Vanneste and P. Sebbah, Phys. Rev. A 79, 041802(R) (2009).
[CrossRef]

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
[CrossRef]

P. Sebbah, B. Hu, V. I. Kopp, and A. Z. Genack, J. Opt. Soc. Am. B 24, A77 (2007).
[CrossRef]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

Solomon, G. S.

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

Stöckmann, H.-J.

U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000).
[CrossRef]

Vanneste, C.

C. Vanneste and P. Sebbah, Phys. Rev. A 79, 041802(R) (2009).
[CrossRef]

von Brentano, P.

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, Phys. Rev. E 62, 1922 (2000).
[CrossRef]

P. von Brentano and M. Philipp, Phys. Lett. B 454, 171 (1999), and references therein.
[CrossRef]

Wiersma, D. S.

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

J. Bertolotti, S. Gottardo, and D. S. Wiersma, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

Wu, H.-L.

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Yang, J.-K.

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

Adv. Phys. (1)

J. B. Pendry, Adv. Phys. 43, 461 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 633 (2000).
[CrossRef]

Europhys. Lett. (1)

Y. Imry, Europhys. Lett. 1, 249 (1986).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. C (1)

J. B. Pendry, J. Phys. C 20, 733 (1987).
[CrossRef]

Phys. Lett. B (1)

P. von Brentano and M. Philipp, Phys. Lett. B 454, 171 (1999), and references therein.
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (1)

C. Vanneste and P. Sebbah, Phys. Rev. A 79, 041802(R) (2009).
[CrossRef]

Phys. Rev. B (1)

M. Y. Azbel, Phys. Rev. B 28, 4106 (1983).
[CrossRef]

Phys. Rev. E (2)

J. Bertolotti, M. Galli, R. Sapienza, M. Ghulinyan, S. Gottardo, L. C. Andreani, L. Pavesi, and D. S. Wiersma, Phys. Rev. E 74, 035602 (2006).
[CrossRef]

M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, Phys. Rev. E 62, 1922 (2000).
[CrossRef]

Phys. Rev. Lett. (6)

U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Phys. Rev. Lett. 100, 126402 (2008).
[CrossRef]

H. Noh, J.-K. Yang, S. F. Liew, M. J. Rooks, G. S. Solomon, and H. Cao, Phys. Rev. Lett. 106, 183901 (2011).
[CrossRef]

J. Bertolotti, S. Gottardo, and D. S. Wiersma, Phys. Rev. Lett. 94, 113903 (2005).
[CrossRef]

P. Sebbah, B. Hu, J. M. Klosner, and A. Z. Genack, Phys. Rev. Lett. 96, 183902 (2006).
[CrossRef]

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, and P. Sebbah, Phys. Rev. Lett. 101, 133901 (2008).
[CrossRef]

D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev. Lett. 65, 88 (1990).
[CrossRef]

Physics (1)

J. Pendry, Physics 1, 20 (2008).
[CrossRef]

Solid State Commun. (1)

M. Y. Azbel, Solid State Commun. 45, 527 (1983).
[CrossRef]

Other (2)

J. Jin, The Finite Element Method in Electromagnetics(Wiley, 1993).

See for instance C. Cohen-Tannoudji, B. Diu, and F. Laloe, Mécanique Quantique (Herman, 1973), Chap. 4.

Supplementary Material (1)

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