Abstract

One of the principal goals in the design of photonic crystals is the engineering of band gaps and defect states. Here I describe the formation of topologically protected localized midgap states in systems with spatially distributed gain and loss. These states can be selectively amplified, which finds applications in the beam dynamics along a photonic lattice and in the lasing of quasi-one-dimensional photonic crystals.

© 2013 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. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
    [CrossRef]
  4. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
    [CrossRef]
  5. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
    [CrossRef]
  6. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
    [CrossRef]
  7. H. Schomerus and N. Yunger Halpern, Phys. Rev. Lett. 110, 013903 (2013).
    [CrossRef]
  8. M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
    [CrossRef]
  9. K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
    [CrossRef]
  10. T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
    [CrossRef]
  11. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
    [CrossRef]
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    [CrossRef]
  13. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
    [CrossRef]
  14. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  15. M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
    [CrossRef]
  16. H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
    [CrossRef]
  17. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef]
  18. H. Schomerus, Phys. Rev. Lett. 104, 233601 (2010).
    [CrossRef]
  19. S. Longhi, Phys. Rev. A 82, 031801 (2010).
    [CrossRef]
  20. Y. D. Chong, L. Ge, and A. D. Stone, Phys. Rev. Lett. 106, 093902 (2011).
    [CrossRef]
  21. S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
    [CrossRef]

2013 (2)

H. Schomerus and N. Yunger Halpern, Phys. Rev. Lett. 110, 013903 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

2012 (4)

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
[CrossRef]

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

2011 (2)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Y. D. Chong, L. Ge, and A. D. Stone, Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

2010 (5)

H. Schomerus, Phys. Rev. Lett. 104, 233601 (2010).
[CrossRef]

S. Longhi, Phys. Rev. A 82, 031801 (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]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[CrossRef]

2009 (2)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

2008 (1)

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

2003 (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

2002 (1)

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

1987 (2)

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

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

1979 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Aspuru-Guzik, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Berg, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Broome, M. A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
[CrossRef]

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

Christodoulides, D. N.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

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

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

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

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Demler, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[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]

Fan, S.

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

Fang, K.

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

Fedrizzi, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Fleischmann, R.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Hafezi, M.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[CrossRef]

Hatsugai, Y.

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

Heeger, A. J.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Joannopoulos, J. D.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
[CrossRef]

John, S.

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

Kane, C. L.

M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[CrossRef]

Kassal, I.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[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]

Kitagawa, T.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Kottos, T.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

Kovanis, V.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

Kraus, Y. E.

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
[CrossRef]

Lahini, Y.

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
[CrossRef]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Longhi, S.

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

Lukin, M. D.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[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]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (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]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

Ramezani, H.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

Ringel, Z.

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
[CrossRef]

Rudner, M. S.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (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]

Ryu, S.

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Schomerus, H.

H. Schomerus and N. Yunger Halpern, Phys. Rev. Lett. 110, 013903 (2013).
[CrossRef]

H. Schomerus, Phys. Rev. Lett. 104, 233601 (2010).
[CrossRef]

Schrieffer, J. R.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

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

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Soljacic, M.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
[CrossRef]

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

Su, W. P.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

Taylor, J. M.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Tünnermann, A.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

Verbin, M.

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
[CrossRef]

Vitebskiy, I.

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
[CrossRef]

White, A. G.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Yablonovitch, E.

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

Yu, Z.

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

Yunger Halpern, N.

H. Schomerus and N. Yunger Halpern, Phys. Rev. Lett. 110, 013903 (2013).
[CrossRef]

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

Zheng, M. C.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

Zilberberg, O.

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
[CrossRef]

Nat. Comms. (1)

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nat. Comms. 3, 882 (2012).
[CrossRef]

Nat. Photonics (2)

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

Nat. Phys. (2)

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

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Nature (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
[CrossRef]

Phys. Rev. A (2)

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

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

Phys. Rev. Lett. (11)

H. Ramezani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 109, 033902 (2012).
[CrossRef]

Y. D. Chong, L. Ge, and A. D. Stone, Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).
[CrossRef]

H. Schomerus, Phys. Rev. Lett. 104, 233601 (2010).
[CrossRef]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
[CrossRef]

H. Schomerus and N. Yunger Halpern, Phys. Rev. Lett. 110, 013903 (2013).
[CrossRef]

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

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

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[CrossRef]

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

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Rev. Mod. Phys. (1)

M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Complex Su–Schrieffer–Heeger (cSSH) chain with alternating couplings ta and tb and alternating imaginary onsite potential iγA=i(γ¯+γ) and iγB=i(γ¯γ) (describing loss or gain). For n<0 the system is in the α configuration; for n>0 it is in the β configuration. (b) Dispersion Reε(k) of the extended states, for tb=0.6ta and γ=0.3ta. These states have Imε(k)=γ¯. (c) Dispersion in the complex eigenvalue plane, including the midgap state at ε0=iγA, which forms due to the coupling defect. (d) The midgap state is exponentially localized and confined to the A sublattice.

Fig. 2.
Fig. 2.

(a) Realization of the cSSH model in a photonic lattice of single-mode waveguides with intrinsic propagation constants γA and γB as well as alternating spacings a and b, and a defect in that spacing sequence (around x=0). (b) Beam propagation of an initially broad wave packet in a lattice of 101 waveguides with tb=0.2ta, γA=0, and γB=0.1ta. (c), (d) Beam propagation with light fed into an A or B fiber close to x=0, for a lattice with tb=0.6ta, γA=0, and γB=0.1ta. (e) Adiabatic pumping of light: waveguide geometry close to the center of the system. (f) Beam propagation in a lattice of 101 waveguides, with tb=0.2ta, γA=0, and γB=0.2ta.

Fig. 3.
Fig. 3.

(a) Realization of the cSSH model in a quasi-one-dimensional photonic laser with a staggered arrangement of active (A) and lossy components (B) in a unit cell of size a0. (b) Midgap state in a dielectric medium with regions of refractive index nA=20.01i (gain) and nB=2+0.01i (loss). The regions have lengths a0/3 and are separated by gaps (refractive index n=1) of alternating size a0/12 and a0/4 (system length 40 a0). The state is predominantly localized in the gain medium. (c) Bands 8 and 9 of the system, in increasing order of Reεi. The depicted midgap state has index i=308.

Equations (7)

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εψn(A)=iγAψn(A)+tnψn1(B)+tnψn(B),
εψn(B)=iγBψn(B)+tnψn(A)+tn+1ψn+1(A),
(k)=(iγAf(k)f(k)iγB),f(k)={ta+tbeik(α)tb+taeik(β),
ε±(k)=iγ¯±ta2+tb2+2tatbcoskγ2
φ(k)=N(f(k)ε(k)iγA)(φ(A)(k)φ(B)(k)),
S=(σx,σy,σz)=(Sx,Sy,Sz).
Sx+iSy(ε(k)iγA)f(k)g(k).

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