Abstract

We theoretically and experimentally explore the coupling behavior of defect pairs immersed in amorphous host lattices. Our observations in photonic lattice arrangements clearly demonstrate how such disordered environments can systematically accelerate the hopping dynamics between localized defect states.

© 2014 Optical Society of America

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef]
  2. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef]
  3. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
    [CrossRef]
  4. J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
    [CrossRef]
  5. J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, Opt. Lett. 23, 1573 (1998).
    [CrossRef]
  6. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, Opt. Express 3, 4 (1998).
    [CrossRef]
  7. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Opt. Lett. 30, 2575 (2005).
    [CrossRef]
  8. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef]
  9. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
    [CrossRef]
  10. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
    [CrossRef]
  11. Y. S. Kivshar, Opt. Lett. 18, 1147 (1993).
    [CrossRef]
  12. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
    [CrossRef]
  13. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
    [CrossRef]
  14. M. F. Thorpe and D. Weaire, Phys. Rev. Lett. 27, 1581 (1971).
    [CrossRef]
  15. K. Morigaki, Physics of Amorphous Semiconductors (World Scientific, 1999).
  16. M. Florescu, S. Torquato, and P. Steinhardt, Proc. Natl. Acad. Sci. USA 106, 20658 (2009).
    [CrossRef]
  17. W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.
  18. M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
    [CrossRef]
  19. A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
    [CrossRef]
  20. A. Yariv, Optical Electronics, 4th ed. (Saunders, 1991).
  21. E. P. Wigner, Ann. Math. 67, 325 (1958).
    [CrossRef]
  22. A. Szameit and S. Nolte, J. Phys. B 43, 163001 (2010).
    [CrossRef]
  23. M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
    [CrossRef]
  24. A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
    [CrossRef]

2011

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

2010

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

2009

M. Florescu, S. Torquato, and P. Steinhardt, Proc. Natl. Acad. Sci. USA 106, 20658 (2009).
[CrossRef]

2008

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

2007

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

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

2006

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

2005

2003

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

1998

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, Opt. Lett. 23, 1573 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, Opt. Express 3, 4 (1998).
[CrossRef]

1993

1988

1987

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

1973

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

1971

M. F. Thorpe and D. Weaire, Phys. Rev. Lett. 27, 1581 (1971).
[CrossRef]

1958

E. P. Wigner, Ann. Math. 67, 325 (1958).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[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]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Chaikin, P.

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Christodoulides, D. N.

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

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
[CrossRef]

Coen, S.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Dreisow, F.

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

Dudley, J.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Fan, S.

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, Opt. Lett. 23, 1573 (1998).
[CrossRef]

Fishman, S.

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

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Florescu, M.

M. Florescu, S. Torquato, and P. Steinhardt, Proc. Natl. Acad. Sci. USA 106, 20658 (2009).
[CrossRef]

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

Genty, G.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Hartung, H.

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

Haus, H. A.

Heinrich, M.

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Joannopoulos, J. D.

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Joseph, R. I.

Keil, R.

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Kivshar, Y. S.

Kuramochi, E.

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.

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

Man, W.

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

Matsuyama, K.

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Mitsugi, S.

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]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Morigaki, K.

K. Morigaki, Physics of Amorphous Semiconductors (World Scientific, 1999).

Nolte, S.

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

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

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

Notomi, M.

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]

Rechtsman, M. C.

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

Schwartz, T.

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

Segev, M.

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

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

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Shinya, A.

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]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[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]

Steinhardt, P.

M. Florescu, S. Torquato, and P. Steinhardt, Proc. Natl. Acad. Sci. USA 106, 20658 (2009).
[CrossRef]

Steinhardt, P. J.

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

Szameit, A.

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (2011).
[CrossRef]

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

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

Tanabe, T.

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Thorpe, M. F.

M. F. Thorpe and D. Weaire, Phys. Rev. Lett. 27, 1581 (1971).
[CrossRef]

Torquato, S.

M. Florescu, S. Torquato, and P. Steinhardt, Proc. Natl. Acad. Sci. USA 106, 20658 (2009).
[CrossRef]

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

Tünnermann, A.

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

Villeneuve, P. R.

Weaire, D.

M. F. Thorpe and D. Weaire, Phys. Rev. Lett. 27, 1581 (1971).
[CrossRef]

Wigner, E. P.

E. P. Wigner, Ann. Math. 67, 325 (1958).
[CrossRef]

Winn, J. N.

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, Opt. Lett. 23, 1573 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

Yadak, P.

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

A. Yariv, Optical Electronics, 4th ed. (Saunders, 1991).

Ann. Math.

E. P. Wigner, Ann. Math. 67, 325 (1958).
[CrossRef]

Appl. Phys. B

M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Appl. Phys. Lett.

A. Szameit, F. Dreisow, H. Hartung, S. Nolte, A. Tünnermann, and F. Lederer, Appl. Phys. Lett. 90, 241113 (2007).
[CrossRef]

IEEE J. Quantum Electron.

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

J. Phys. B

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

Nature

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

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

M. C. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, Phys. Rev. Lett. 106, 193904 (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]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

M. F. Thorpe and D. Weaire, Phys. Rev. Lett. 27, 1581 (1971).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Proc. Natl. Acad. Sci. USA

M. Florescu, S. Torquato, and P. Steinhardt, Proc. Natl. Acad. Sci. USA 106, 20658 (2009).
[CrossRef]

Rev. Mod. Phys.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Science

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Other

W. Man, M. Florescu, K. Matsuyama, P. Yadak, S. Torquato, P. J. Steinhardt, and P. Chaikin, in Conference on Lasers Electro-Optics 2010 CThS2 (OSA, 2010), paper CThS2.

K. Morigaki, Physics of Amorphous Semiconductors (World Scientific, 1999).

A. Yariv, Optical Electronics, 4th ed. (Saunders, 1991).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic of two defect waveguides (red) embedded within various realizations of an amorphous environment (black). The propagation constants associated with the defects (red) reside within the photonic band gap of the structure. (b) Periodic intensity transfer between two coupled waveguides. Bottom: tuned coupler; top: deterministically detuned coupler (β=c0); and center: coupler in amorphous environment, displaying a stochastic detuning, as schematically indicated by the traces of individual realizations. (c) Dependence of the hopping rate κ associated with these examples on the coupling coefficient c. For all values of c, the stochastic parameter of the amorphous environment was fixed (s=c0). All quantities are normalized with respect to the scale of an arbitrary coupling c0.

Fig. 2.
Fig. 2.

(a) Micrograph of a typical realization of an amorphous photonic lattice. The individual waveguides have elliptical cross-sections of 3μm×11μm and their positions distributed according to a liquid-like model. (b) Output pattern obtained from a coupled defect pair when one of the channels is selectively excited. To resolve ambiguities due to the periodic coupling dynamics, we directly observed the propagation by means of waveguide fluorescence microscopy. Shown are examples of evolution patterns for (c) a tuned coupler and (d) a defect pair embedded within an amorphous background lattice, each with a spacing of d=14μm. (e) Experimentally retrieved hopping rates κ for the amorphous and the deterministic (tuned) system. The random detuning of the defect guides by the amorphous environment clearly results in an accelerated hopping for large spacings.

Fig. 3.
Fig. 3.

(a) Experimentally extracted standard deviation of the hopping constant, compared to the theoretical prediction. (b) Experimental results for the mean-square and square-mean hopping coefficient, showing stochastic behavior: sm<ms at large defect spacing. The ratio between the two is a function of the probability distribution of the random detuning.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

iddz(a1a2)=(a1a2),
κdet=β24+c2.
κsto=12(s4)2+c2+[2c2sarcsinh(s4c)].
lims0κsto=c.
K=1minz|a1(z)|2maxz|a1(z)|2=c2β2+c2

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