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 [Physics 1, 20 (2008). [CrossRef]  ] through which the light can sneak through an opaque medium.

© 2012 Optical Society of America

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References

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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)

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

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

J. Pendry, Physics 1, 20 (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)

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]

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]

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

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

» Media 1: MOV (544 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

Monitoring of a. the real part ν and b. the imaginary part ν of three localized states as the refractive index, ns, of a particular scatterer (shown in Fig. 2) is varied from 1.05 to 2.70. The Y axis has been split in b to span a broad interval of ν.

Fig. 2.
Fig. 2.

Spatial distribution of the field amplitude for Mode1 and Mode2 of Fig. 1 for three different values of the refractive index of the scatterer located at the white spot. Media 1 displays the evolution of the mode profiles as ns increases from 1.05 to 2.40.

Fig. 3.
Fig. 3.

a. Hybridization of four localized states by setting the index of refraction to 2.9 and 2.8 for left and right scatterers, respectively, shown as white spots; all other scatters set at nc=2.0. b. Transmittance spectra for plane wave impinging from left on perturbed (solid line) and unperturbed (dotted line) systems. c. Field amplitude spatial distribution for wave impinging on perturbed system at λ=439.60nm. d. Field amplitude spatial distribution for wave impinging on unperturbed system at λ=440.63nm.

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