Abstract

This Letter shows that bound states in the continuum (BIC), which are normalizable modes with energy embedded in the continuous spectrum of scattered states, exist in certain optical waveguide lattices with PT-symmetric defects. There are two distinct types of BIC modes: BIC states that exist in the broken PT phase, and correspond to exponentially localized modes with either exponentially damped or amplified optical power; and BIC modes with sub-exponential spatial localization that also can exist in the unbroken PT phase. These two types of BIC modes at the PT symmetry breaking point behave differently: in the former case, spatial localization is lost and the defect coherently radiates outgoing waves with an optical power that linearly increases with the propagation distance; in the latter case, localization is maintained and the optical power increase is quadratic.

© 2014 Optical Society of America

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  1. J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929).
  2. E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 78, 075105 (2008).
  3. D. C. Marinica and A. G. Borisov, Phys. Rev. Lett. 100, 183902 (2008).
    [CrossRef]
  4. N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009).
    [CrossRef]
  5. M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012).
    [CrossRef]
  6. Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
    [CrossRef]
  7. G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
    [CrossRef]
  8. S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
    [CrossRef]
  9. C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
    [CrossRef]
  10. C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
    [CrossRef]
  11. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
    [CrossRef]
  12. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef]
  13. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  14. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
    [CrossRef]
  15. S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
    [CrossRef]
  16. S. Longhi, Phys. Rev. A 82, 031801 (2010).
    [CrossRef]
  17. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
    [CrossRef]
  18. L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
    [CrossRef]
  19. T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
    [CrossRef]
  20. K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928 (2010).
    [CrossRef]
  21. A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
    [CrossRef]
  22. S. Longhi, Phys. Rev. Lett. 105, 013903 (2010).
    [CrossRef]
  23. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
    [CrossRef]
  24. Y. N. Joglekar, D. Scott, and M. Babbey, Phys. Rev. A 82, 030103(R) (2010).
    [CrossRef]
  25. A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
    [CrossRef]
  26. S. Longhi, Phys. Rev. A 82, 032111 (2010).
    [CrossRef]
  27. S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A, submitted for publication.
  28. A. A. Andrianov and A. V. Sokolov, SIGMA 7, 111 (2011).

2013

G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
[CrossRef]

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

2012

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012).
[CrossRef]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

2011

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

A. A. Andrianov and A. V. Sokolov, SIGMA 7, 111 (2011).

2010

S. Longhi, Phys. Rev. A 82, 032111 (2010).
[CrossRef]

Y. N. Joglekar, D. Scott, and M. Babbey, Phys. Rev. A 82, 030103(R) (2010).
[CrossRef]

S. Longhi, Phys. Rev. Lett. 105, 013903 (2010).
[CrossRef]

K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928 (2010).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

S. Longhi, Phys. Rev. A 82, 031801 (2010).
[CrossRef]

2009

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009).
[CrossRef]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
[CrossRef]

2008

E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 78, 075105 (2008).

D. C. Marinica and A. G. Borisov, Phys. Rev. Lett. 100, 183902 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

2007

1929

J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929).

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Andrianov, A. A.

A. A. Andrianov and A. V. Sokolov, SIGMA 7, 111 (2011).

Babbey, M.

Y. N. Joglekar, D. Scott, and M. Babbey, Phys. Rev. A 82, 030103(R) (2010).
[CrossRef]

Bender, C. M.

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Bersch, C.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Borisov, A. G.

D. C. Marinica and A. G. Borisov, Phys. Rev. Lett. 100, 183902 (2008).
[CrossRef]

Bulgakov, E. N.

E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 78, 075105 (2008).

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Christodoulides, D. N.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Chua, S.-L.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Corrielli, G.

G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
[CrossRef]

Crespi, A.

G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
[CrossRef]

Della Valle, G.

G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
[CrossRef]

S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A, submitted for publication.

Dreisow, F.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

Eichelkraut, T.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Guo, Z.

Heilmann, R.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

Heinrich, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

Hsu, C. W.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Joglekar, Y. N.

Y. N. Joglekar, D. Scott, and M. Babbey, Phys. Rev. A 82, 030103(R) (2010).
[CrossRef]

Johnson, S. G.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Keil, R.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Kivshar, Y. S.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012).
[CrossRef]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Lee, J.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Liu, S.

Longhi, S.

G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
[CrossRef]

S. Longhi, Phys. Rev. A 82, 031801 (2010).
[CrossRef]

S. Longhi, Phys. Rev. A 82, 032111 (2010).
[CrossRef]

S. Longhi, Phys. Rev. Lett. 105, 013903 (2010).
[CrossRef]

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A, submitted for publication.

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Marinica, D. C.

D. C. Marinica and A. G. Borisov, Phys. Rev. Lett. 100, 183902 (2008).
[CrossRef]

Milanovic, V.

N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009).
[CrossRef]

Miri, M.-A.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Miroshnichenko, A. E.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012).
[CrossRef]

Molina, M. I.

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012).
[CrossRef]

Mostafazadeh, A.

A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
[CrossRef]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Näger, J.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

Nolte, S.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

Oliveira, J. E. B.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Onishchukov, G.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Osellame, R.

G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
[CrossRef]

Peleg, O.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

Peschel, U.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Plotnik, Y.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

Prodanovic, N.

N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009).
[CrossRef]

Radovanovic, J.

N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009).
[CrossRef]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Regensburger, A.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Sadreev, A. F.

E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 78, 075105 (2008).

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Scott, D.

Y. N. Joglekar, D. Scott, and M. Babbey, Phys. Rev. A 82, 030103(R) (2010).
[CrossRef]

Segev, M.

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Sokolov, A. V.

A. A. Andrianov and A. V. Sokolov, SIGMA 7, 111 (2011).

Soljacic, M.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Stützer, S.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

Sukhorukov, A. A.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

Szameit, A.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

von Neumann, J.

J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929).

Wang, J.

Weimann, S.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

Wigner, E.

J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929).

Xu, Y.

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

Xu, Y.-L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Zhen, B.

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Zhou, K.

J. Phys. A

N. Prodanovic, V. Milanovic, and J. Radovanovic, J. Phys. A 42, 415304 (2009).
[CrossRef]

Nat. Commun.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Nat. Commun. 4, 2533 (2013).
[CrossRef]

Nat. Mater.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. 12, 108 (2012).
[CrossRef]

Nat. Phys.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Nature

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, Nature 499, 188 (2013).
[CrossRef]

Opt. Lett.

Phys. Rev. A

Y. N. Joglekar, D. Scott, and M. Babbey, Phys. Rev. A 82, 030103(R) (2010).
[CrossRef]

S. Longhi, Phys. Rev. A 82, 032111 (2010).
[CrossRef]

S. Longhi, Phys. Rev. A 82, 031801 (2010).
[CrossRef]

Phys. Rev. B

E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 78, 075105 (2008).

Phys. Rev. Lett.

D. C. Marinica and A. G. Borisov, Phys. Rev. Lett. 100, 183902 (2008).
[CrossRef]

M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, Phys. Rev. Lett. 108, 070401 (2012).
[CrossRef]

Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, and M. Segev, Phys. Rev. Lett. 107, 183901 (2011).
[CrossRef]

G. Corrielli, G. Della Valle, A. Crespi, R. Osellame, and S. Longhi, Phys. Rev. Lett. 111, 220403 (2013).
[CrossRef]

S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, S. Nolte, A. A. Sukhorukov, A. Szameit, and Y. S. Kivshar, Phys. Rev. Lett. 111, 240403 (2013).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
[CrossRef]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

S. Longhi, Phys. Rev. Lett. 105, 013903 (2010).
[CrossRef]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Rep. Prog. Phys.

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

SIGMA

A. A. Andrianov and A. V. Sokolov, SIGMA 7, 111 (2011).

Z. Phys.

J. von Neumann and E. Wigner, Z. Phys. 30, 465 (1929).

Other

S. Longhi and G. Della Valle, “Optical lattices with exceptional points in the continuum,” Phys. Rev. A, submitted for publication.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Schematic of a homogeneous lattice with two site impurities at waveguides n=±1, [Eq. (2) in the text]. (b) Behavior of the energy spectrum (eigenvalues of the matrix normalized to the coupling constant κ) for Δ/κ=0.3 and for increasing values of g/κ. Shaded region depicts the energy band of scattered states (real eigenvalues). Solid and dashed curves are the real and imaginary parts, respectively, of bound state energies. For small values of g/κ (below <0.457), there is one BOC state with a real-valued energy that falls just outside the upper band edge. PT symmetry breaking occurs at gth/κ0.752, above which two type-I BIC modes with complex conjugate energies emerge. (c) Amplitude distribution (|c¯n|) of the amplified type-I BIC mode for g/gth=1.08 (curve 1), g/gth=1.2 (curve 2), and g/gth=1.5 (curve 3). (d) Lattice spectral transmittance for g/gth=0.5 (curve 1), g/gth=0.8 (curve 2) and g/gth=0.9 (curve 3). Resonance peak occurs at q=q00.161π. (e) Evolution of beam intensity and (f) of normalized total beam power P(z)/P(0) for initial excitation of waveguide n=0 at the symmetry breaking transition.

Fig. 2.
Fig. 2.

(a) Behavior of the hopping rates κn (for n0,1) of the inhomogeneous lattice defined by Eq. (4). (b) Behavior of the energy spectrum of for increasing values of g/κ. The shaded region depicts the energy band of scattered states (real eigenvalues). The solid and dashed curves are the real and imaginary parts, respectively, of bound state energies. PT symmetry breaking occurs at gth/κ=1, above which two type-I BIC modes with complex conjugate energies emerge. The type-II BIC mode with energy E=0 exists for any value of g. (c) Amplitude distribution (|c¯n|) of type-II BIC mode at g=gth. (d) Lattice spectral transmittance T(q) for g/gth=0.5 (curve 1), g/gth=0.9 (curve 2) and g/gth=0.99 (curve 3). (e) Evolution of beam intensity and (f) Evolution of square root of normalized total beam power (i.e., P(z)/P(0)) for initial excitation of waveguide n=0 at the symmetry breaking transition.

Equations (6)

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idcndz=κncn1+κn+1cn+1+Vncncn,
κn=κ,Vn=σδn,1+σ*δn,1,
t(q)=isinqexp(2iq)Δ+isinq|σ|2cosqexp(2iq).
κnκ={(n+1)/(n1)neven,n0κ0=ig(n2)/nnodd,n1κ1=ig,
c¯n={0noddκ/gn=0n|n|in+1n21neven,n0.
cn(z)=(1iεz)c¯n+εfn

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