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 OSA

1. Introduction

In quantum mechanics, physical observables require the corresponding operators must be Hermitian, i.e. the operator shows a real spectrum. Intriguingly, because of the pseudo-Hermitian, a class of non-Hermitian Hamiltonian can also still show entirely real spectra, and they exhibit parity-time (PT) symmetry [1]. The definition of PT operator and its properties were discussed in [15]. The real part of a PT complex potentials must be an even function of position whereas the imaginary component is odd. In the PT symmetric potentials, there exists a critical threshold above which the system undergoes a sudden phase transition, i.e., the spectrum is no longer real but instead becomes a complex one [1, 68]. In optical field, PT symmetric potentials can be also constructed by introducing a complex refractive-index distribution into the wave-guided system: n(x) = nr(x) + ini(x), where nr(x) = nr(−x), ni(x) = −ni(−x), and x is the normalized transverse coordinate. The imaginary part in n(x) represent gain and loss region of the medium, i.e., negative and positive ni(x) stand for gain and loss, respectively [9]. Therefore, the light propagating in such system will experience a PT periodic potential.

When light propagates in periodic optical lattice with a local defect, both linear and nonlinear defect modes can be formed due to the bandgap guidance [10, 11]. Defect guiding phenomena of light in photonic crystals [12], fabricated waveguide arrays [1315], and optically induced photonic lattices [1623], have been demonstrated both theoretically and experimentally in the past few years. Light propagating in PT periodic potentials will exhibit some unique characteristics such as double refraction, power oscillations etc [8]. Very recently, existing properties of linear defect modes in a PT periodic potential were also studied [24]. In this article, nonlinear defect modes (defect solitons) in PT periodic lattices with a single-sited defect are studied, and their stability properties are analyzed.

2. Theoretical model

The normalized equation describing light propagating in a PT lattice with Kerr nonlinear response is [7]:

iUz+Uxx+V0[V(x)+iW(x)]+|U|2U=0.
Here
V(x)=cos2(x)[1+ɛfD(x)],W(x)=W0sin(2x),
are related to the real part nr and imaginary part ni of the complex refractive-index distribution (nrni). Function fD = exp(−x8/128) represents the single-sited defect of the lattice, and the strength of the defect upon the real part of the PT lattice is determined by parameter ɛ. The spatial transverse and longitudinal coordinates x and z are scaled to the input beam width a and diffraction length Ldiff = 2k0n0a2, respectively, where k0 = 2π/λ0 and n0 is the background refractive index. The depth of the PT lattice V0 is scaled to the parameter 1/(2k02n0a2). Therefore, from a physical perspective, for example, if λ0 = 1μm, n0 = 3, and a = 10μm, then x = 1 corresponds to 10μm, z = 1 means the propagation distance of 3.77mm, and V0=1 corresponds to a maximum variation of the refractive index nrmax4.22×105. The parameter W0 represents the strength of gain or loss compared with the real index variation nr.

When ɛ = 0, the PT periodic lattice in Eq.(1) admits Bloch band structure. However, differing from the band structure of real periodic lattice, there exists a threshold W0 = 0.5 for the PT lattice below which all of its eigenvalue spectrum are pure real; once W0 > 0.5, its eigenvalue spectrum becomes a complex one, and the first two bands start to merge together [6]. In this paper, only PT lattice with its Bloch spectrum below the phase transition point is considered, and without of generality, parameters W0 = 0.1 and V0 = 3 are adopted throughout the paper. Typical PT lattice profile and its Bloch band structure are shown in 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).

Download Full Size | PPT Slide | PDF

Defect soliton solutions to Eq.(1) are sought in the form of U(x,z) = u(x)eiμz, and μ is the propagation constant. After substituting U(x,z) into Eq.(1), stationary defect soliton solution u(x) satisfies:

uxx+V0[V(x)+iW(x)]u+|u|2u+μu=0.
Eq.(3) can be solved numerically [25].

To study the stability properties of defect solitons, the stationary solutions u(x) are perturbed in the form:

U(x,z)={u(x)+[v(x)w(x)]eλz+[v(x)+w(x)]*eλ*z}eiμz,
where v,w<<1, and * represents the complex conjugation. Substituting perturbed U(x,z) into Eq.(1), after linearization, an eigen value equation about v and w is derived:
i[iV0WiIm(u2)L^+V0VRe(u2)L^+V0V+Re(u2)iV0W+iIm(u2)][vw]=λ[vw].
Here = d2/dx2 + 2|u|2 + μ.

The unstable growth rate Re(λ) can be obtained numerically [26]. If Re(λ) = 0, defect solitions are linearly stable, while when λ has a real component, they becomes linearly unstable.

3. Numerical results

The defect solitons’ P vs μ curves are shown in the first row of Fig.2, where the soliton power P is defined as P=+|U|2dx. Here two typical defect strength are considered, i.e, ɛ=0.5 for the positive defect and ɛ=−0.5 for the negative defect. For comparison, power curves of the solitons in uniform (ɛ=0) PT lattice are also shown in Fig.2 (left column).

 

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.

Download Full Size | PPT Slide | PDF

As can be seen, both positive and negative defect solitons exist in the opened Bloch gaps of the PT lattice. For positive defect solitons, unlike solitons in uniform lattice, their power curves can not approach the corresponding band edges; in the semi-infinite and the first gap, their power curves terminate at μ = −2.6 and μ = 0.02, respectively. For the same propagation constant μ, to form a soliton, in the lattice with negative defect more light power is needed, while in the lattice with positive defect less light power is needed, comparing to the case in the uniform lattice. As known that a positive defect likes a higher index waveguide which can even guide linear modes without nonlinearity, and that’s why the power curves of positive defect soliton become zero before they approach the band edges. While negative defect likes a lower index waveguide, which needs light power to compensate the index difference (by nonlinearity) to guide light. In the semi-infinite gap, for negative defect solitons, there exists a power threshold P = 3.857 (corresponding μ = −2.118), below which the solitons do not exist. Therefore, the power slope upon the propagation constant, i.e. dP/dμ changes sign at this point, and according to the V-K criterion, the stability property will also change at this point. In the first gap, the numerical results show that power curve of negative defect solitons terminates at μ = 0 (P = 4.45).

Soliton solutions in PT lattices are complex. Fig.2 (d)–(f) and (g)–(i) show some typical soliton profiles in the semi-infinite gap and the first gap, respectively. In the semi-infinite gap, the real part of the solition solution is even and the imaginary part is odd; while in the first gap, the real part is odd and the imaginary part becomes even.

The most unstable growth rate max{Re(λ)} of solitons are shown in Fig.3. Without defect, fundamental solitons in the semi-infinite gap are stable [10]. Similarly, positive defect solitons in the semi-infinite gap shown in the second column of Fig.2 are stable throughout their existence region. For negative defect solitons, their stability properties obey the V-K criterion, i.e. when dP/dμ<0 (μ<−2.116), they are stable, and when when dP/dμ>0 (−2.118<μ<−2.04), they are unstable. Their maximum unstable growth rate max{Re(λ)} vs μ curve is shown in Fig.3 (c). In the first gap, negative defect solitons shown in the right column of Fig.2 are unstable throughout their existence region (see Fig.3(d)). For solitons in the uniform lattice and positive defect solitons, in the first gap, their stable region are very narrow which are 0.13 < μ < 0.142 and −0.07 < μ < 0.02, respectively, i.e. they are stable only when they are very near the right side of their existence region. Their maximum unstable growth rate max{Re(λ)} vs μ curves are shown Fig.3(a) and (b).

 

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.

Download Full Size | PPT Slide | PDF

 

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.

Download Full Size | PPT Slide | PDF

To test the stability robustness and validate the above stability analyses, the dynamical Eq.(1) is simulated numerically with the stationary soliton solutions u(x) plus small random noise as the initial input beam at z = 0. The simulation results are shown in Fig.4. In Fig.4, the first row display three propagation examples of solitons in the uniform PT lattice, the second row display three propagation examples of positive defect solitons, and the third row shows propagation examples of negative defect solitons. Fig.4 (a) shows the stable propagation of uniform soliton (profile was displayed in Fig.2 (d)) in the semi-infinite gap. Fig.4(b) shows unstable propagation of uniform soliton in the first gap with μ=−0.7. The spatial profile of this soliton was shown in Fig.2 (g), and it’s maximum unstable growth rate was marked by a red circle in Fig.3 (a). Panel (c) in Fig.4 shows stable propagation of uniform soliton in the first gap with propagation const μ=0.13. Stable propagation of positive defect soliton in the semi-infinite gap with μ=−4 is shown in Fig.4 (d). Fig.4 (e) and (f) display unstable and stable propagations of positive defect solitons in the first gap with μ=−0.7 and μ=0, respectively. The profile of this unstable soliton was plotted in Fig.2 (h), and the maximum unstable growth rate of this soliton was marked by a circle in Fig.3 (b). In Fig.4 (g) and (h), stable and unstable propagations of negative defect solitons in the semi-infinite gap with μ=−4 and μ=−2.06 are shown. Soliton profile of Fig.4 (g) was plotted in Fig.2 (f). The maximum unstable growth rate of soliton in Fig.4 (h) was marked in Fig.3 (c). Unstable propagation of negative defect soliton in the first gap with μ=−0.7 is displayed in Fig.4 (i), with its profile plotted in Fig.2 (i) and its maximum unstable growth rate marked in Fig.3 (d). As can be seen from Fig.4, the simulation results are in good agreement with the linear stability analyses.

4. Summary

In this paper, the defect solitons at the single-sited defect in one-dimensional parity-time symmetric photonic lattices with focusing Kerr nonlinearity are investigated. For a positive defect, the solitons can be stable in both of the semi-infinite gap and the first gap, but in the first gap, their stability region is very narrow where the solitons are broad expanded and have lower power. For negative defect, in the semi-infinite gap, the solitons’ stability property obeys the V-K criterion, and are stable in most of their existence region; while in the first gap, the solitons are unstable in their entire existence region. The linear stability analyses are corroborated by direct numerical simulations.

Acknowledgments

We appreciate Dr. X. Zhu for his helpful discussion. This work is supported by the National Natural Science Foundation of China ( 10904009).

References and links

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

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

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

4. 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]  

5. 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]  

6. 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]  

7. 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]  

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

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

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

11. 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]  

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

13. L. M. Molina and R. A. Vicencio, “Trapping of discrete solitons by defects in nonlinear waveguide arrays,” Opt. Lett. 31, 966–968 (2006). [CrossRef]  

14. P. P. Belicev, I. Ilic, M. Stepic, A. Maluckov, Y. Tan, and F. Chen, “Observation of linear and nonlinear strongly localized modes at phase-slip defects in one-dimensional photonic lattices,” Opt. Lett. 35, 3099–3101 (2010). [CrossRef]   [PubMed]  

15. A. Szameit, M. I. Molina, M. Heinrich, F. Dreisow, R. Keil, S. Nolte, and Y. S. Kivshar, “Observation of localized modes at phase slips in two-dimensional photonic lattices,” Opt. Lett. 35, 2738–2740 (2010). [CrossRef]   [PubMed]  

16. 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]  

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

18. 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]  

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

20. W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express 18, 10956–10961 (2010). [CrossRef]   [PubMed]  

21. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34, 797–799 (2009). [CrossRef]   [PubMed]  

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

23. 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]  

24. 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]  

25. 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]  

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

References

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  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [CrossRef]
  2. C. M. Bender, D.C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
    [CrossRef]
  3. C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys. 71, 1095–1102 (2003).
    [CrossRef]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  9. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity - time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
    [CrossRef]
  10. A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
    [CrossRef] [PubMed]
  11. 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]
  12. M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, 2009).
  13. L. M. Molina and R. A. Vicencio, “Trapping of discrete solitons by defects in nonlinear waveguide arrays,” Opt. Lett. 31, 966–968 (2006).
    [CrossRef]
  14. P. P. Belicev, I. Ilic, M. Stepic, A. Maluckov, Y. Tan, and F. Chen, “Observation of linear and nonlinear strongly localized modes at phase-slip defects in one-dimensional photonic lattices,” Opt. Lett. 35, 3099–3101 (2010).
    [CrossRef] [PubMed]
  15. A. Szameit, M. I. Molina, M. Heinrich, F. Dreisow, R. Keil, S. Nolte, and Y. S. Kivshar, “Observation of localized modes at phase slips in two-dimensional photonic lattices,” Opt. Lett. 35, 2738–2740 (2010).
    [CrossRef] [PubMed]
  16. 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]
  17. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
    [CrossRef]
  18. 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]
  19. J. Wang, J. Yang, and Z. Chen, “Two-dimensional defect modes in optically induced photonic lattices,” Phys. Rev. A 76, 013828 (2007).
    [CrossRef]
  20. W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express 18, 10956–10961 (2010).
    [CrossRef] [PubMed]
  21. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tunnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34, 797–799 (2009).
    [CrossRef] [PubMed]
  22. W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt. Express 14, 11271–11276 (2006).
    [CrossRef] [PubMed]
  23. 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]
  24. 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]
  25. 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]
  26. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).
    [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 Y. 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]

Brody, D.C.

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 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,” Nature 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,” Nature 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,” Nature Phys. 6, 192–195 (2010).
[CrossRef]

Kivshar, Y. S.

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,” Nature 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,” Nature 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,” Nature Phys. 6, 192–195 (2010).
[CrossRef]

Skorobogatiy, M.

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

Stepic, M.

Sukhorukov, A. A.

A. A. Sukhorukov and Y. 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. 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. 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 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]

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

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]

Nature 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,” Nature 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 (1)

J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 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 Y. 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)

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

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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