Abstract

We report the existence and stability of gap solitons in parity-time (PT) complex periodic optical lattices with the real part of superlattices. These solitons can stably exist in the semi-infinite gap. We have studied the effects of different relative strengths of the superlattices and different amplitudes of the imaginary part on soliton propagation. It was found that the relative strength of the superlattices and the amplitude of the imaginary part significantly affect the PT symmetry and the stability of solitons in the PT complex periodic optical lattices.

© 2011 Optical Society of America

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  1. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243(1998).
    [CrossRef]
  2. Z. Ahmed, Phys. Lett. A 282, 343 (2001).
    [CrossRef]
  3. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
    [CrossRef]
  4. C. M. Bender, D. C. Brody, and H. Jones, Am. J. Phys. 71, 1095 (2003).
    [CrossRef]
  5. C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007).
    [CrossRef] [PubMed]
  6. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef] [PubMed]
  7. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef] [PubMed]
  8. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  9. 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]
  10. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature Phys. 6, 192 (2010).
    [CrossRef]
  11. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
    [CrossRef] [PubMed]
  12. H. Wang and J. Wang, Opt. Express 19, 4030 (2011).
    [CrossRef] [PubMed]
  13. K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928(2010).
    [CrossRef] [PubMed]
  14. P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. A 71, 023612 (2005).
    [CrossRef]
  15. J. Yang, J. Comput. Phys. 227, 6862 (2008).
    [CrossRef]
  16. Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, Opt. Lett. 32, 1390 (2007).
    [CrossRef] [PubMed]
  17. J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
    [CrossRef]
  18. M. J. Ablowitz, and Z. H. Musslimani, Opt. Lett. 30, 2140(2005).
    [CrossRef] [PubMed]

2011

2010

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

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. A 81, 063807 (2010).
[CrossRef]

2009

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]

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

2008

J. Yang, J. Comput. Phys. 227, 6862 (2008).
[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]

2007

Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, Opt. Lett. 32, 1390 (2007).
[CrossRef] [PubMed]

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

2005

M. J. Ablowitz, and Z. H. Musslimani, Opt. Lett. 30, 2140(2005).
[CrossRef] [PubMed]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. A 71, 023612 (2005).
[CrossRef]

2003

C. M. Bender, D. C. Brody, and H. Jones, Am. J. Phys. 71, 1095 (2003).
[CrossRef]

2002

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

2001

Z. Ahmed, Phys. Lett. A 282, 343 (2001).
[CrossRef]

1998

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

Ablowitz, M. J.

Ahmed, Z.

Z. Ahmed, Phys. Lett. A 282, 343 (2001).
[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]

Bender, C. M.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

C. M. Bender, D. C. Brody, and H. Jones, Am. J. Phys. 71, 1095 (2003).
[CrossRef]

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

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]

Brody, D. C.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

C. M. Bender, D. C. Brody, and H. Jones, Am. J. Phys. 71, 1095 (2003).
[CrossRef]

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Chen, W. H.

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]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (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]

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]

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (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]

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]

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.

He, Y. J.

Jones, H.

C. M. Bender, D. C. Brody, and H. Jones, Am. J. Phys. 71, 1095 (2003).
[CrossRef]

Jones, H. F.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

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.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. A 71, 023612 (2005).
[CrossRef]

Kottos, T.

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

Lakoba, T. I.

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

Liu, S.

Louis, P. J. Y.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. A 71, 023612 (2005).
[CrossRef]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (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]

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]

Malomed, B. A.

Meister, B. K.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

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]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (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]

M. J. Ablowitz, and Z. H. Musslimani, Opt. Lett. 30, 2140(2005).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. A 71, 023612 (2005).
[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]

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]

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, H.

Wang, H. Z.

Wang, J.

Yang, J.

J. Yang, J. Comput. Phys. 227, 6862 (2008).
[CrossRef]

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

Zhou, K.

Am. J. Phys.

C. M. Bender, D. C. Brody, and H. Jones, Am. J. Phys. 71, 1095 (2003).
[CrossRef]

J. Comput. Phys.

J. Yang, J. Comput. Phys. 227, 6862 (2008).
[CrossRef]

Nature Phys.

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

Opt. Express

Opt. Lett.

Phys. Lett. A

Z. Ahmed, Phys. Lett. A 282, 343 (2001).
[CrossRef]

Phys. Rev. A

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. A 71, 023612 (2005).
[CrossRef]

Phys. Rev. Lett.

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

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

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

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]

Stud. Appl. Math.

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

PT complex periodic OLs (the blue curves represent the real part of the OLs and the red curves represent the imaginary part of the OLs) with ω 0 = 0.1 and (a)  ε = 0.2 , (b)  ε = 0.3 , and (c)  ε = 0.7 . Accordingly, (d), (e), and (f) are the band structures corresponding to (a), (b), and (c).

Fig. 2
Fig. 2

For ε = 0.3 and ω 0 = 0.1 , (a) power versus μ (the solid curve represents the stable region and the dashed curve represents the unstable region) and (b)  Re ( δ ) versus μ.

Fig. 3
Fig. 3

Soliton profile (the blue curve is the real part and the red curve is the the imaginary part) and soliton propagation for ω 0 = 0.1 . (a), (d) μ = 0.55 and ε = 0.3 , (b), (e) μ = 1.4 and ε = 0.3 , (c), (f) μ = 1.8 and ε = 0.3 , (g), (j) μ = 0.55 and ε = 0.7 , (h), (k) μ = 3 and ε = 0.7 , and (i), (l) μ = 1.3 and ε = 0.2 .

Fig. 4
Fig. 4

(a) PT complex periodic OLs and (b) the corresponding band structure. (c) The soliton profile of μ = 0.55 and (e) the soliton propagation. (d) The soliton profile of μ = 1.3 and (f) the soliton propagation. The other parameters are ε = 0.3 and ω 0 = 0.16

Equations (8)

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

i q z + 2 q x 2 + [ v ( x ) + i w ( x ) ] q + | q | 2 q = 0 ,
2 f x 2 + [ v ( x ) + i w ( x ) ] f + | f | 2 f = μ f .
h x x + v h w e + e 2 h + h 3 = μ h ,
e x x + v e + w h + h 2 e + e 3 = μ e ,
q ( x , z ) = exp ( i μ z ) { f ( x ) + [ g ( x ) t ( x ) ] exp ( δ z ) + [ g ( x ) + t ( x ) ] * exp ( δ * z ) }
{ δ g = i [ 2 t x 2 μ t + v t i w g + 2 | f | 2 t 1 2 ( f 2 - f * 2 ) g 1 2 ( f 2 + f * 2 ) t ] δ t = i [ 2 g x 2 μ g + v g i w t + 2 | f | 2 g + 1 2 ( f 2 + f * 2 ) g + 1 2 ( f 2 f * 2 ) t ] .
2 f x 2 + [ v ( x ) + i w ( x ) ] f = μ f .
( 2 x 2 + 2 i k x k 2 ) F k + [ v ( x ) + i w ( x ) ] F k = μ F k .

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