Abstract

Defect modes are studied in parity-time (PT) symmetric periodic complex potentials for both positive and negative defects. Such new kinds of linear localized modes may conserve their energy or endure gain or loss upon propagating. The existence domain of the conserved modes would be prolonged by positive defects or shortened by negative defects, and the dissipative modes would appear before the phase transition point of the PT symmetric potential at stronger negative defects.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).
  2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  3. S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474(2001).
    [CrossRef] [PubMed]
  4. U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
    [CrossRef]
  5. R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, Opt. Lett. 28, 834 (2003).
    [CrossRef] [PubMed]
  6. F. Fedele, J. K. Yang, and Z. G. Chen, Stud. Appl. Math. 115, 279 (2005).
    [CrossRef]
  7. J. Wang, J. Yang, and Z. Chen, Phys. Rev. A 76, 013828 (2007).
    [CrossRef]
  8. K. Y. Zhou, Z. Y. Guo, A. A. Muhammad, and S. T. Liu, Chin. Phys. B 19, 014201 (2010).
    [CrossRef]
  9. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  10. B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
    [CrossRef]
  11. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef] [PubMed]
  12. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef] [PubMed]
  13. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
    [CrossRef] [PubMed]
  14. L. Jin and Z. Song, Phys. Rev. A 80, 052107 (2009).
    [CrossRef]
  15. 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] [PubMed]
  16. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature Phys. 6, 192 (2010).
    [CrossRef]
  17. N.Akhmediev and A.Ankiewicz, eds., Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).
    [CrossRef]

2010 (2)

K. Y. Zhou, Z. Y. Guo, A. A. Muhammad, and S. T. Liu, Chin. Phys. B 19, 014201 (2010).
[CrossRef]

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

2009 (3)

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

L. Jin and Z. Song, Phys. Rev. A 80, 052107 (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] [PubMed]

2008 (2)

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

2007 (1)

J. Wang, J. Yang, and Z. Chen, Phys. Rev. A 76, 013828 (2007).
[CrossRef]

2005 (1)

F. Fedele, J. K. Yang, and Z. G. Chen, Stud. Appl. Math. 115, 279 (2005).
[CrossRef]

2003 (2)

2001 (2)

B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
[CrossRef]

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474(2001).
[CrossRef] [PubMed]

1999 (1)

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

1998 (1)

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[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] [PubMed]

Aitchison, J. S.

Bagchi, B.

B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
[CrossRef]

Bender, C. M.

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Bendix, O.

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

Boettcher, S.

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Chen, Z.

J. Wang, J. Yang, and Z. Chen, Phys. Rev. A 76, 013828 (2007).
[CrossRef]

Chen, Z. G.

F. Fedele, J. K. Yang, and Z. G. Chen, Stud. Appl. Math. 115, 279 (2005).
[CrossRef]

Christodoulides, D. N.

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

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

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

Dandelik, D.

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] [PubMed]

Eisenberg, H. S.

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature 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] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

Fedele, F.

F. Fedele, J. K. Yang, and Z. G. Chen, Stud. Appl. Math. 115, 279 (2005).
[CrossRef]

Fleischmann, R.

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

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] [PubMed]

Guo, Z. Y.

K. Y. Zhou, Z. Y. Guo, A. A. Muhammad, and S. T. Liu, Chin. Phys. B 19, 014201 (2010).
[CrossRef]

Jin, L.

L. Jin and Z. Song, Phys. Rev. A 80, 052107 (2009).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

Kip, D.

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

Kivshar, Y. S.

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474(2001).
[CrossRef] [PubMed]

Kottos, T.

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

Lederer, F.

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

Liu, S. T.

K. Y. Zhou, Z. Y. Guo, A. A. Muhammad, and S. T. Liu, Chin. Phys. B 19, 014201 (2010).
[CrossRef]

Makris, K. G.

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

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

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

Mingaleev, S. F.

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474(2001).
[CrossRef] [PubMed]

Modotto, D.

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] [PubMed]

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, Opt. Lett. 28, 834 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

Muhammad, A. A.

K. Y. Zhou, Z. Y. Guo, A. A. Muhammad, and S. T. Liu, Chin. Phys. B 19, 014201 (2010).
[CrossRef]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

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

Peschel, U.

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

Quesne, C.

B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
[CrossRef]

Rüter, C. E.

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

Segev, M.

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

Shapiro, B.

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

Silberberg, Y.

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

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, Opt. Lett. 28, 834 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[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] [PubMed]

Song, Z.

L. Jin and Z. Song, Phys. Rev. A 80, 052107 (2009).
[CrossRef]

Sorel, M.

Stanley, C. R.

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] [PubMed]

Wang, J.

J. Wang, J. Yang, and Z. Chen, Phys. Rev. A 76, 013828 (2007).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

Yang, J.

J. Wang, J. Yang, and Z. Chen, Phys. Rev. A 76, 013828 (2007).
[CrossRef]

Yang, J. K.

F. Fedele, J. K. Yang, and Z. G. Chen, Stud. Appl. Math. 115, 279 (2005).
[CrossRef]

Zhou, K. Y.

K. Y. Zhou, Z. Y. Guo, A. A. Muhammad, and S. T. Liu, Chin. Phys. B 19, 014201 (2010).
[CrossRef]

Znojil, M.

B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

Chin. Phys. B (1)

K. Y. Zhou, Z. Y. Guo, A. A. Muhammad, and S. T. Liu, Chin. Phys. B 19, 014201 (2010).
[CrossRef]

Mod. Phys. Lett. A (1)

B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
[CrossRef]

Nature (1)

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

Nature Phys. (1)

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

Opt. Lett. (1)

Phys. Rev. A (2)

L. Jin and Z. Song, Phys. Rev. A 80, 052107 (2009).
[CrossRef]

J. Wang, J. Yang, and Z. Chen, Phys. Rev. A 76, 013828 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef] [PubMed]

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

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] [PubMed]

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474(2001).
[CrossRef] [PubMed]

Stud. Appl. Math. (1)

F. Fedele, J. K. Yang, and Z. G. Chen, Stud. Appl. Math. 115, 279 (2005).
[CrossRef]

Other (2)

N.Akhmediev and A.Ankiewicz, eds., Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) Real parts of the propagation eigenvalues versus w 0 , in which the shaded regions are bands while the blank regions are gaps. (b)–(d) Bandgap structures for w 0 = 0.3 , 0.55, and 0.6, respectively. The insets in (c) and (d) denote the imaginary parts of the first two bands.

Fig. 2
Fig. 2

(a) Bifurcation of the DMs at A 0 = 10 , ϵ = 0.2 for different w 0 ; the solid and dashed curves denote the conserved DMs and dissipative DMs, respectively. (b)–(d) DMs for three typical values at (b) w 0 = 0.3 , λ = 8.25 ; (c) w 0 = 0.3 , λ = 3.18 ; and (d) w 0 = 0.75 , λ = 4.16 2.4 i . The real and imaginary parts of the DMs are plotted in red solid curves and blue dashed curves, respectively.

Fig. 3
Fig. 3

(a) Bifurcation of the DMs at A 0 = 10 , ϵ = 0.2 for different w 0 ; the solid and dashed curves denote the conserved DMs and dissipative DMs, respectively. (b)–(d) DMs for three typical values at (b) w 0 = 0.3 , λ = 4.64 ; (c) w 0 = 0.3 , λ = 0.969 ; and (d) w 0 = 0.75 , λ = 1.73 + 3.82 i . The real and imaginary parts of the DMs are plotted in red solid curves and blue dashed curves, respectively.

Fig. 4
Fig. 4

(a) Bifurcation of DMs at A 0 = 10 , ϵ = 0.8 for different w 0 ; the solid and dashed curves denote the conserved DMs and dissipative DMs, respectively. (b)–(d) DMs for three typical values at (b) w 0 = 0.24 , λ = 0.215 + 0.769 i ; (c) w 0 = 0.08 , λ = 0.685 ; and (d) w 0 = 0.08 , λ = 0.579 . The real and imaginary parts of DMs are plotted in red solid curves and blue dashed curves, respectively.

Equations (2)

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

i U ξ + 2 U η 2 + A 0 [ V ( η ) + i W ( η ) ] U = 0 ,
λ R R + λ I I + R η η + V R W I = 0 , λ R I λ I R + I η η + V I + W R = 0 .

Metrics