Abstract

In this article, properties of solitons in a parity-time periodical lattices with a single-sited defect are investigated. Both of the negative and positive defects are considered. Linear stability analyses show that, when the defect is positive, in the semi-infinite gap, the solitons are always stable, while in the first gap, the solitons are unstable in most of their existence region except for those near the edge of the second band; when the defect is negative, in the semi-infinite gap, other than those near the edge of the first band, most solitons are stable, but in the first gap, all solitons are unstable. Such stability analyses are corroborated by numerical simulations.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2010 (6)

2009 (1)

2008 (4)

W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect linear modes in one-dimensional photonic lattices,” Phys. Lett. A 372, 3525–3530 (2008).
[CrossRef]

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (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] [PubMed]

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] [PubMed]

2007 (3)

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76, 013828 (2007).
[CrossRef]

J. Yang and T. I. Lakoba, “Universally convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[CrossRef]

2006 (5)

2005 (1)

F. Fedele, J. Yang, and Z. Chen, “Properties of defect modes in one-dimensional optically induced photonic lattices,” Stud. Appl. Math. 115, 279–301 (2005).
[CrossRef]

2003 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[CrossRef]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[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]

A. A. Sukhorukov and Yu. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

1998 (1)

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

Belicev, P. P.

Bender, C. M.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[CrossRef]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

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

Boettcher, S.

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

Brody, D. C.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[CrossRef]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Chen, F.

Chen, W. H.

Chen, Z.

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76, 013828 (2007).
[CrossRef]

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 026609 (2006).
[CrossRef]

X. Wang, J. Young, Z. Chen, D. Weinstein, and J. Yang, “Observation of lower to higher bandgap transition of one-dimensional defect modes,” Opt. Express 14, 7362–7367 (2006).
[CrossRef] [PubMed]

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

F. Fedele, J. Yang, and Z. Chen, “Properties of defect modes in one-dimensional optically induced photonic lattices,” Stud. Appl. Math. 115, 279–301 (2005).
[CrossRef]

Christodoulides, D. N.

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]

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] [PubMed]

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] [PubMed]

Dreisow, F.

El-Ganainy, R.

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] [PubMed]

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] [PubMed]

Fedele, F.

F. Fedele, J. Yang, and Z. Chen, “Properties of defect modes in one-dimensional optically induced photonic lattices,” Stud. Appl. Math. 115, 279–301 (2005).
[CrossRef]

Guo, Z.

He, Y. J.

W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect linear modes in one-dimensional photonic lattices,” Phys. Lett. A 372, 3525–3530 (2008).
[CrossRef]

W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt. Express 14, 11271–11276 (2006).
[CrossRef] [PubMed]

Heinrich, M.

Ilic, I.

Jones, H. F.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[CrossRef]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Kartashov, Y. V.

Keil, R.

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.

Kivshar, Yu. S.

A. A. Sukhorukov and Yu. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Lakoba, T. I.

J. Yang and T. I. Lakoba, “Universally convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[CrossRef]

Lederer, F.

Li, R. H.

Liu, S.

Makasyuk, I.

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

Makris, K. G.

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]

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] [PubMed]

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] [PubMed]

Maluckov, A.

Meister, B. K.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

Molina, L. M.

Molina, M. I.

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] [PubMed]

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] [PubMed]

Nolte, S.

Pertsch, T.

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]

Stepic, M.

Sukhorukov, A. A.

A. A. Sukhorukov and Yu. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Szameit, A.

Tan, Y.

Torner, L.

Tunnermann, A.

Vicencio, R. A.

Vysloukh, V. A.

Wang, H. Z.

W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect linear modes in one-dimensional photonic lattices,” Phys. Lett. A 372, 3525–3530 (2008).
[CrossRef]

W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt. Express 14, 11271–11276 (2006).
[CrossRef] [PubMed]

Wang, J.

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010).
[CrossRef] [PubMed]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76, 013828 (2007).
[CrossRef]

Wang, X.

Weinstein, D.

Wu, T. W.

Yang, J.

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).
[CrossRef]

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76, 013828 (2007).
[CrossRef]

J. Yang and T. I. Lakoba, “Universally convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[CrossRef]

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 026609 (2006).
[CrossRef]

X. Wang, J. Young, Z. Chen, D. Weinstein, and J. Yang, “Observation of lower to higher bandgap transition of one-dimensional defect modes,” Opt. Express 14, 7362–7367 (2006).
[CrossRef] [PubMed]

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

F. Fedele, J. Yang, and Z. Chen, “Properties of defect modes in one-dimensional optically induced photonic lattices,” Stud. Appl. Math. 115, 279–301 (2005).
[CrossRef]

Young, J.

Zhou, K.

Zhu, X.

Am. J. Phys. (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
[CrossRef]

J. Comput. Phys. (1)

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).
[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. Express (3)

Opt. Lett. (5)

Phys. Lett. A (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]

W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect linear modes in one-dimensional photonic lattices,” Phys. Lett. A 372, 3525–3530 (2008).
[CrossRef]

Phys. Rev. A (2)

J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76, 013828 (2007).
[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. E Stat. Nonlin. Soft Matter Phys. (1)

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 026609 (2006).
[CrossRef]

Phys. Rev. Lett. (7)

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

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98, 040403 (2007).
[CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

A. A. Sukhorukov and Yu. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

Stud. Appl. Math. (2)

F. Fedele, J. Yang, and Z. Chen, “Properties of defect modes in one-dimensional optically induced photonic lattices,” Stud. Appl. Math. 115, 279–301 (2005).
[CrossRef]

J. Yang and T. I. Lakoba, “Universally convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[CrossRef]

Other (1)

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, 2009).

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Figures (4)

Fig. 1
Fig. 1

(color online) (a) Profile of the PT lattice. Solid blue: real part, dashed red: imaginary part; (b) Band structure of the lattice in (a) (V0 = 3, W0 = 0.1).

Fig. 2
Fig. 2

(color online) (a)–(c) Power curves of solitons in PT lattice with ɛ=0, ɛ=0.5 and ɛ=−0.5, respectively (Solid curves: stable; dashed curves: unstable); (d)–(f) soliton solutions in the semi-infinite gap with μ=−4, which correspond to the red circle markers in (a)–(c); (g)–(i) soliton solutions in the first gap with μ=−0.7, which correspond to the blue circle markers in (a)–(c). Shaded in (a)–(c): Bloch bands; shaded in (d)–(i): the real part of the lattices (different color stripes mean the defect). In (d)–(i): solid blue lines plot the real part of u and dashed red lines plot the imaginary part of u.

Fig. 3
Fig. 3

The most unstable growth rate max{Re(λ)} versus μ of solitons in: (a) and (b) the first gap with ɛ=0 and 0.5; (c) and (d) the semi-infinite gap and the first gap with ɛ=−0.5. Shaded: Bloch bands. Unstable propagation of solitons marked with red circles are shown in Fig.4.

Fig. 4
Fig. 4

(color online) (a)–(c) Propagation of solitons in the uniform (ɛ=0) PT lattice with μ=−4, μ=−0.7, and μ=0.13, respectively; (d)–(f) Propagation of positive defect (ɛ=0.5) solitons with μ=−4, μ=−0.7, and μ=0, respectively; (g)–(i) Propagation of negative defect (ɛ=−0.5) solitons with μ=−4, μ=−2.06, and μ=−0.7, respectively. The maximum unstable growth rate of unstable solitons are marked by circles in Fig.3.

Equations (5)

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i U z + U x x + V 0 [ V ( x ) + i W ( x ) ] + | U | 2 U = 0 .
V ( x ) = cos 2 ( x ) [ 1 + ɛ f D ( x ) ] , W ( x ) = W 0 sin ( 2 x ) ,
u x x + V 0 [ V ( x ) + i W ( x ) ] u + | u | 2 u + μ u = 0 .
U ( x , z ) = { u ( x ) + [ v ( x ) w ( x ) ] e λ z + [ v ( x ) + w ( x ) ] * e λ * z } e i μ z ,
i [ i V 0 W i I m ( u 2 ) L ^ + V 0 V R e ( u 2 ) L ^ + V 0 V + R e ( u 2 ) i V 0 W + i I m ( u 2 ) ] [ v w ] = λ [ v w ] .

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