Abstract

We study nonlinear binary arrays composed of parity-time-symmetric optical waveguides with gain and loss. We demonstrate that such nonlinear binary lattices support stable discrete solitons, which can be adiabatically tuned and switched through nonlinear symmetry breaking by varying gain and loss parameters.

© 2010 Optical Society of America

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  1. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
    [CrossRef]
  2. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
    [CrossRef] [PubMed]
  3. S. Klaiman, U. Guenther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008).
    [CrossRef] [PubMed]
  4. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef] [PubMed]
  5. M. V. Berry, J. Phys. A 41, 244007 (2008).
    [CrossRef]
  6. S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
    [CrossRef] [PubMed]
  7. S. Longhi, Phys. Rev. A 81, 022102 (2010).
    [CrossRef]
  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. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature Phys. 6, 192 (2010).
    [CrossRef]
  11. A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
    [CrossRef] [PubMed]
  12. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  13. R. A. Vicencio and M. Johansson, Phys. Rev. A 79, 065801 (2009).
    [CrossRef]

2010

S. Longhi, Phys. Rev. A 81, 022102 (2010).
[CrossRef]

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

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

2009

R. A. Vicencio and M. Johansson, Phys. Rev. A 79, 065801 (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]

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

2008

S. Klaiman, U. Guenther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008).
[CrossRef] [PubMed]

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

M. V. Berry, J. Phys. A 41, 244007 (2008).
[CrossRef]

2007

2005

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

2003

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

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]

Berry, M. V.

M. V. Berry, J. Phys. A 41, 244007 (2008).
[CrossRef]

Christodoulides, D. N.

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

C. E. Ruter, 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]

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

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[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] [PubMed]

El-Ganainy, R.

C. E. Ruter, 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]

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

Ganainy, R. El

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

Guenther, U.

S. Klaiman, U. Guenther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008).
[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]

Johansson, M.

R. A. Vicencio and M. Johansson, Phys. Rev. A 79, 065801 (2009).
[CrossRef]

Kip, D.

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

Kivshar, Yu. S.

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Klaiman, S.

S. Klaiman, U. Guenther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008).
[CrossRef] [PubMed]

Longhi, S.

S. Longhi, Phys. Rev. A 81, 022102 (2010).
[CrossRef]

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

Makris, K. G.

C. E. Ruter, 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]

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

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

Moiseyev, N.

S. Klaiman, U. Guenther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008).
[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]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

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]

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

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Ruter, C. E.

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

Sukhorukov, A. A.

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

Vicencio, R. A.

R. A. Vicencio and M. Johansson, Phys. Rev. A 79, 065801 (2009).
[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] [PubMed]

J. Phys. A

M. V. Berry, J. Phys. A 41, 244007 (2008).
[CrossRef]

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Nature Phys.

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

Opt. Lett.

Phys. Rev. A

R. A. Vicencio and M. Johansson, Phys. Rev. A 79, 065801 (2009).
[CrossRef]

S. Longhi, Phys. Rev. A 81, 022102 (2010).
[CrossRef]

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

Phys. Rev. Lett.

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]

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

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

S. Klaiman, U. Guenther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008).
[CrossRef] [PubMed]

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

Other

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of a binary waveguide array with gain and loss. (b), (c) 3D plots of | a j ( z ) | 2 (dots) and | b j ( z ) | 2 (open circles) showing linear diffraction of small-amplitude Gaussian beam (initial amplitude | a 0 ( z = 0 ) | 2 = 0.05 and inverse width 0.02) when PT-symmetry condition is (b) not satisfied ( C 1 = C 2 = 3 ) and (c) satisfied ( C 1 = 3 , C 2 = 2 ). For both plots, ρ a = ρ b = 0.8 .

Fig. 2
Fig. 2

(a) Approximate soliton profiles for ρ = 0.8 , shown at different propagation distances z = 0 and z = 18.3 to illustrate beating with internal mode. (b) Total intensity I ( z ) for approximate soliton solutions at different values of ρ, indicated for each curve. The values of C 1 and C 2 correspond to Fig. 1c, and ρ a = ρ b = ρ .

Fig. 3
Fig. 3

Soliton tuning by balanced variation of gain and loss along the waveguides. (a) Introduced dependencies of ρ = ρ a = ρ b on z; two different rates of increase of ρ are compared. (b) Corresponding evolution of total intensity I ( z ) , when the input profile corresponds to the soliton solution for a conservative lattice with ρ = 0 . The inset shows the intensity profiles at z = 0 and z = 400 corresponding to curve 1. For all the plots, the values of C 1 and C 2 correspond to Fig. 1c.

Fig. 4
Fig. 4

Soliton tuning by gain-loss imbalance. (a) Total intensity versus z for ρ a = ρ + ρ + and ρ b = ρ + ρ + , where the imbalance value ρ + for curves 1 to 4 is 1.6 × 10 4 , 0.8 × 10 4 , 0.8 × 10 4 , and 1.6 × 10 4 , respectively. (b) Soliton profiles for different z for ρ + = 1.6 × 10 4 shown by curve 1 in (a). For all the plots, ρ = 0.8 , and the values of C 1 and C 2 correspond to Fig. 1c.

Equations (4)

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

i d a j d z + i ρ a a j + C 1 b j + C 2 b j 1 + G a ( | a j | 2 ) a j = 0 , i d b j d z + i ρ b b j + C 1 a j + C 2 a j + 1 + G b ( | b j | 2 ) a j = 0 ,
β ± ( k ) = i ρ + ± D ( k ) , D ( k ) = C 1 2 + C 2 2 + 2 C 1 C 2 cos K ρ 2 ,
| | C 1 | | C 2 | | | ρ | = | ρ a ρ b | / 2 .
{ a j ( z ) , b j ( z ) } = { a j ( z = 0 ) , b j ( z = 0 ) } exp ( i β z ) ,

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