Abstract

Using a nonstationary state solution to the nonlinear Schrödinger equation (NLSE), we report the results of our numerical investigation on the swing behavior of spatial solitons propagating along waveguides whose refractive indices in the transverse direction are perturbed by Scarf II type parity–time symmetric profiles. We show that solitons, after entering parity–time symmetry (PTS) cells with Scarf II profiles will, in general, swing along the waveguide with nonuniform amplitude and period. Nevertheless, it is demonstrated that when the average incident position in the transverse direction is set at the center of the profile symmetry and the amplitude of the incident soliton exceeds a specific value, the soliton behavior could be approximated by a stationary-state solution to the NLSE and say it is almost self-trapped. Simulation also shows that, depending on the soliton’s initial average transverse input position, the swing behavior could be greatly influenced by the nonreciprocity of PTS cells.

© 2012 Optical Society of America

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  1. T. Kato, Perturbation Theory for Linear Operators (Springer Verlag, 1966).
  2. C. M. Bender and S. Bottcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [CrossRef]
  3. C. M. Bender, S. Bottcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
    [CrossRef]
  4. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, 171–176 (2005).
    [CrossRef]
  5. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
    [CrossRef]
  6. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [CrossRef]
  7. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  8. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
    [CrossRef]
  14. Z. Lu and Z. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011).
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    [CrossRef]
  16. D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic PT-symmetric potential,” Phys. Rev. A 85, 043840(2012).
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  17. M. Ebnali-Heidary, M. K. Moravvej-Farshi, and A. Zarifkar, “Multichannel wavelength conversion using fourth-order soliton decay,” J. Lightwave Technol. 25, 2571–2578(2007).
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    [CrossRef]
  22. A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light beam propagation at nonlinear interfaces. I. Equivalent particle theory for a single interface,” Phys. Rev. A 39, 1809(1989).
    [CrossRef]
  23. A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264–274 (2006).
    [CrossRef]
  24. F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of spatial soliton,” Opt. Commun. 139, 193–198 (1997).
    [CrossRef]
  25. M. Ebnali-Heidari, M. K. Moravvej-Farshi, and A. Zarifkar, “Swing effect of spatial solitons propagating through Gaussian and triangular waveguides,” Appl. Opt. 48, 5005–5014 (2009).
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    [CrossRef]

2012 (1)

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic PT-symmetric potential,” Phys. Rev. A 85, 043840(2012).
[CrossRef]

2011 (4)

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[CrossRef]

Z. Lu and Z. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011).
[CrossRef]

F. Nazari, M. Nazari, and M. K. Moravvej-Farshi, “A 2×2 spatial optical switch based on PT-symmetry,” Opt. Lett. 36, 4368–4370 (2011).
[CrossRef]

2010 (4)

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinearsuppression of time-reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

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

2009 (2)

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

M. Ebnali-Heidari, M. K. Moravvej-Farshi, and A. Zarifkar, “Swing effect of spatial solitons propagating through Gaussian and triangular waveguides,” Appl. Opt. 48, 5005–5014 (2009).
[CrossRef]

2008 (2)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

2007 (1)

2006 (1)

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264–274 (2006).
[CrossRef]

2005 (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, 171–176 (2005).
[CrossRef]

2001 (2)

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “New phase modulation technique based on spatial soliton switching,” J. Lightwave Technol. 19, 1036–1042 (2001).
[CrossRef]

1999 (1)

C. M. Bender, S. Bottcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

1998 (1)

C. M. Bender and S. Bottcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[CrossRef]

1997 (2)

F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of spatial soliton,” Opt. Commun. 139, 193–198 (1997).
[CrossRef]

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photon. Technol. Lett. 9, 1364–1366 (1997).
[CrossRef]

1994 (1)

1989 (1)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light beam propagation at nonlinear interfaces. I. Equivalent particle theory for a single interface,” Phys. Rev. A 39, 1809(1989).
[CrossRef]

1987 (1)

Abdullaev, F. K.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[CrossRef]

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light beam propagation at nonlinear interfaces. I. Equivalent particle theory for a single interface,” Phys. Rev. A 39, 1809(1989).
[CrossRef]

Ahmed, Z.

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001).
[CrossRef]

Barthelemy, A.

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photon. Technol. Lett. 9, 1364–1366 (1997).
[CrossRef]

Bender, C. M.

C. M. Bender, S. Bottcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

C. M. Bender and S. Bottcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[CrossRef]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Bertolotti, M.

Bottcher, S.

C. M. Bender, S. Bottcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

C. M. Bender and S. Bottcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[CrossRef]

Cao, X. D.

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

Christodoulides, D. N.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

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

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, 171–176 (2005).
[CrossRef]

Doran, N. J.

Ebnali-Heidari, M.

Ebnali-Heidary, M.

El-Ganainy, R.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Garzia, F.

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

Kartashov, Y. V.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[CrossRef]

Kato, T.

T. Kato, Perturbation Theory for Linear Operators (Springer Verlag, 1966).

Kip, D.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinearsuppression of time-reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic PT-symmetric potential,” Phys. Rev. A 85, 043840(2012).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[CrossRef]

Kottos, T.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Lefort, L.

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photon. Technol. Lett. 9, 1364–1366 (1997).
[CrossRef]

Lu, Z.

Makris, K. G.

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

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Meisinger, P. N.

C. M. Bender, S. Bottcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

Meyerhofer, D. D.

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light beam propagation at nonlinear interfaces. I. Equivalent particle theory for a single interface,” Phys. Rev. A 39, 1809(1989).
[CrossRef]

Moravvej-Farshi, M. K.

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, 171–176 (2005).
[CrossRef]

Musslimani, Z. H.

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Nazari, F.

Nazari, M.

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light beam propagation at nonlinear interfaces. I. Equivalent particle theory for a single interface,” Phys. Rev. A 39, 1809(1989).
[CrossRef]

Ramezani, H.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, 171–176 (2005).
[CrossRef]

Ruter, C. E.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

Segev, M.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Sibilia, C.

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinearsuppression of time-reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Suryanto, A.

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264–274 (2006).
[CrossRef]

van Groesen, E.

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264–274 (2006).
[CrossRef]

Wood, D.

Xu, Z.

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinearsuppression of time-reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Zarifkar, A.

Zezyulin, D. A.

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic PT-symmetric potential,” Phys. Rev. A 85, 043840(2012).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[CrossRef]

Zhang, Z.

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (1)

L. Lefort and A. Barthelemy, “All-optical demultiplexing of a signal using collision and waveguiding of spatial solitons,” IEEE Photon. Technol. Lett. 9, 1364–1366 (1997).
[CrossRef]

J. Lightwave Technol. (2)

J. Math. Phys. (1)

C. M. Bender, S. Bottcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, 171–176 (2005).
[CrossRef]

Nat. Phys. (1)

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[CrossRef]

Opt. Commun. (2)

A. Suryanto and E. van Groesen, “Self-splitting of multisoliton bound states in planar Kerr waveguides,” Opt. Commun. 258, 264–274 (2006).
[CrossRef]

F. Garzia, C. Sibilia, and M. Bertolotti, “Swing effect of spatial soliton,” Opt. Commun. 139, 193–198 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Lett. A (1)

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001).
[CrossRef]

Phys. Rev. A (6)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light beam propagation at nonlinear interfaces. I. Equivalent particle theory for a single interface,” Phys. Rev. A 39, 1809(1989).
[CrossRef]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinearsuppression of time-reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic PT-symmetric potential,” Phys. Rev. A 85, 043840(2012).
[CrossRef]

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

Phys. Rev. Lett. (5)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[CrossRef]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

C. M. Bender and S. Bottcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[CrossRef]

Other (1)

T. Kato, Perturbation Theory for Linear Operators (Springer Verlag, 1966).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Real and imaginary parts of waveguides of the same V1=0.05 but various V2=0, 0.1, 0.2, and 0.3. (b) Z dependence of the phase parameters for normally incident solitons of the same η=1, entering the waveguides of Fig. 1(a), all at X0=0. (c) The corresponding average transverse positions Xav(Z). (d)–(f) Top views of the soliton trajectories inside V2=0.1, 0.2, and 0.3.

Fig. 2.
Fig. 2.

(a) The transverse phase factor, (b) the average transverse position, and (c) the field intensity of the soliton of η=1.75, V1=0.05, and V2=0.1 after entering at X0=0.

Fig. 3.
Fig. 3.

(a) Variation of the phase parameters and (b) the corresponding average transverse positions for η=1, entering the four waveguides V2=0, 0.1, 0.2, and 0.3, all at X0=0.85. (c)–(e) Top view of the trajectories of the solitons inside the three given PTS cells [Fig. 1(a)].

Fig. 4.
Fig. 4.

(a) Variation of phase parameters and (b) the corresponding average transverse positions for η=1, entering the four waveguides V2=0, 0.1, 0.2, and 0.3, all at X0=+0.85. (c)–(e) Top view of the trajectories of the solitons inside the three given PTS cells [Fig. 1(a)].

Equations (8)

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n(x)=n0(1+ΔnR(x)+iΔnI(x))n0(1+Δn(x)).
iΨZ+122ΨX2+[VR(X)+iVI(X)]Ψ+|Ψ|2Ψ=0,
Ψ(X,Z)=ηsech[η(XXav(Z))]×exp{i[β(Z)X+γ(X,Z)]},
Xav(Z)X|Ψ|2dX|Ψ|2dX
β(Z)=dXav/dZ,
dγ(X,Z)/dZ(η2β2(Z))/2.
VR(X)V1sech2(X),
VI(X)=V2sech(X)tanh(X),

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