Abstract

We study defect solitons (DSs) in a parity-time (PT) symmetric superlattice with focusing Kerr nonlinearity. The properties of the DSs with a PT symmetrical potential are obviously different from those in a superlattice with a real refractive index. Unusual features stemming from PT symmetry can be found. Research results show that the solitons with a zero defect or a positive defect can exist and stably propagate in the semi-infinite gap, but they cannot exist in the first gap. For the case of a negative defect, the soliton can stably exist in both the semi-infinite gap and the first gap.

© 2011 OSA

1. Introduction

Due to the advanced experimental techniques, it is possible to create a new class of materials, parity-time symmetric media, which admire the combined PT symmetry but do not obey parity and time symmetry separately. In 1998, it had already been reported by Bender and Co-workers [1] that non-Hermitian Hamiltonians with PT symmetry can still show real spectra. Some unusual features stemming from PT symmetry will be found in such kinds of new media. In recent years, there has became more and more interest in PT symmetrical systems because of these unusual features such as double refraction and PT soliton [26]. On the topic of PT solitons, Z.H.Musslimani et al. [2] studied optical solitons in PT periodic potentials. The necessary condition of a Hamiltonian with PT symmetry has also been pointed out in their research work: when the real part of a PT complex potential is an even function of position and the imaginary part of that is an odd function of position, the Hamiltonian is PT symmetric.

The nonlinear defect modes, defect solitons, have been extensively studied in some systems in the past decades [713]. Lots of research results indicate that lattice defects can significantly modify the solitons properties. A variety of interesting phenomena resulting from DSs can be found and can further be applied for fundamental studies and practical applications such as controllable filtering [14], switching [15], and steering of optical beams in lattices [16]. Recently, the existence and stability properties of linear and nonlinear modes in a PT periodic potential were studied [17,18]. In this paper, we will study defect solitons in a superlattice with a PT symmetric potential. The existence and stability of DSs will be discussed analytically and numerically.

2. Theory

We consider a light beam propagating in a PT symmetric superlattice in the Kerr nonlinear media described by the following Schörding equation [2]:

iUz+Uxx+[V(x)+iW(x)]U+|U|2U=0,
where x (in unit of a) and z (in unit of 2k 0 nea 2) is the transverse and longitudinal scale, respectively, in which a is the input beam width, k 0 = 2π/λ 0, and ne is the unperturbed refractive index. In Eq. (1), V(x) and W(x) are related to the real part and imaginary part of the complex refractive-index distribution [2,7]:
V(x)={V0{ε1sin2[(x+π2)]+(1ε1)sin2[2(x+π2)]},(x>π2orx<π2),0.78125V0sin2[(x+π2)][1+εexp(x8128)],(π2xπ2),
and
W(x)=W0sin(2x),
where V 0 (in unit of 1/(2k02nea2)) is the peak intensity of superlattices. ε 1 and ε are the modulation parameter of superlattice intensity and defect intensity, respectively. The parameter W 0 (in unit of 1/(2k02nea2)) represents the strength of gain or loss compared with the real index distribution. In our paper, we will choose reasonable parameters: ε 1 = 0.5, λ 0 = 1μm, ne = 3, a 0 = 10μm, V 0 = 6, and W 0 = 0.4. For these parameters, x = 1 is corresponding to 10μm and z = 1 is corresponding to propagation distance 3.77mm [18]. A solution different from that in Ref. [17] will be obtained for considering the effect of nonlinearity. Although we justly study the conserved solitons and do not considered about the case of the symmetry breaking, we will analyze the stability of solitons using the perturbation growth rate. We search the stationary solitons numerical solution of Eq. (1) in the form of U = f(x)exp(−iμz), where μ is the propagation constant and f(x) is the complex function which satisfies the following equation
fxx+[V(x)+iW(x)]f+|f|2f+μf=0.
Equation (4) can be solved numerically [19].

Figure 1(a) shows Blcoh band structure of PT superlattice. From this figure, we can find that the region of semi-infinite gap is μ ≤ −3.26 and the first, second gap is −2.72 ≤ μ ≤ −1.60 and −0.17 ≤ μ ≤ 1.85, respectively. The intensity distributions of PT superlattice with defect: ε = 0, ε = 0.5, and ε = −0.5 are displayed in Fig. 1(b)–1(d), respectively.

 

Fig. 1 (Color online) (a) Band structure of the superlattice with V 0 = 6 and W 0 = 0.4. Lattices intensity profile of the PT superlattice with (b) ε = 0, (c) ε = 0.5, and (d) ε = −0.5. Blue line: real part, red line: imaginary part.

Download Full Size | PPT Slide | PDF

To examine the stability of defect solitons, we search the perturbed solutions of Eq. (1) in the from

U=exp(iμz){f(x)+[v(x)w(x)]exp(δz)+[v(x)+w(x)]*exp(δ*z)}
where v,w << 1, and * represents the complex conjugation. Substituting Eq. (5) into Eq. (1) and then linearing Eq. (1), we arrive at an eigenvalue equation
{δv=i[d2w/dx2+μw+VwiWv+2|f|2w12(f2f*2)v12(f2+f*2)w],δw=i[d2v/dx2+μv+VwiWw+2|f|2v+12(f2f*2)w+12(f2+f*2)v].

This equation can be solved numerically to get the perturbation growth rate Re(δ)[20]. For Re(δ) = 0, the slitons are linearly stable; otherwise, they are linearly unstable.

3. Numerical results and discussion

Random-noise perturbation whose relative amplitude is set at 10% is added to the initial input light to simulate the soliton propagation. We choose ε = −0.5 as a case of the negative defect in the PT superlattices and the corresponding power P of defect solitons versus the propagation constant μ is showed in Fig. 2(a). It can be found in this figure that defect solitons can exist in both the semi-infinite gap and the first gap. In the semi-infinite gap, the stable region of defect solitons is μ ≤ −4.03 and the unstable region is −4.03 < μ ≤ −3.26. In the first gap, the stable region of defect solitons is −2.72 ≤ μ ≤ −1.75 and the unstable region is −1.75 < μ ≤ −1.61. Two stable examples: μ = −6.0 (point A in Fig. 2(a)) and μ = −2.0 (point B in Fig. 2(a)) have been used to certify soliton stability in the semi-infinite gap and the first gap. Their soliton profiles are showed in Fig. 2(c) and 2(d) and the corresponding soliton propagations are showed in Fig. 2(f) and 2(g), respectively. In the first gap, we select an unstable example: μ = −1.74 (point C in Fig. 2(a)) and plot its soliton profile in Fig. 2(e). In Ref. [18], the real part of soliton solution in the first gap is odd and the imaginary part is even. In Fig. 2(e), we can see an opposite result that the real part of f(x) is even and the imaginary part is odd. The corresponding soliton propagation is showed in Fig. 2(h). Figure 2(b) shows the change of Re(δ) with propagation constant μ. In the regions: μ ≤ −4.03 and −2.72 ≤ μ ≤ −1.75, the slope of power curve is negative and the Re(δ) is equal to zero; while in the regions: −4.03 < μ ≤ −3.26 and −1.75 < μ ≤ −1.61, the slope of power curve is positive and the Re(δ) is not equal to zero. From these results, we can conclude that the stability or instability of defect soliton in the both semi-infinite gap and the first gap is in accordance with the Vakhitov-Kolokolov (VK) criterion.

 

Fig. 2 (Color online) (ε = −0.5) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Re(δ) versus the propagation constant. (c) Stable soliton with μ = −6.0 (point A in Fig. 2(a)). (d) Stable soliton with μ = −2.0 (point B in Fig. 2(a)). (e) Unstable soliton with μ = −1.74 (point C in Fig. 2(a)). In (c)–(e): black lines plot the real part of f and red lines plot the imaginary part of f. (f) Soliton propagation for (c). (g) Soliton propagation for (d). (h) Soliton propagation for (e).

Download Full Size | PPT Slide | PDF

Power P versus propagation constant μ for a positive defect (ε = 0.5) in the PT superlattices is presented in Fig. 3(a). In this figure, we can see that the defect solitons can only exist in the semi-infinite gap. In these regions which we have considered, defect solitons can stably transmit. We choose two stable example:μ = −6.5, −5.0 (points A and B in Fig. 3(a)) and show their soliton profiles in Fig. 3(b) and 3(c), respectively. Figure 3(d) and 3(e) present their corresponding soltion propagations. In the regions of the semi-infinite gap which we have considered, the negative slope of power curve and the Re(δ) = 0 indicate that the solitons stability is in accordance with the VK criterion.

 

Fig. 3 (Color online) (ε = 0.5) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Stable soliton with μ = −6.5 (point A in Fig. 3(a)). (c) Stable soliton with μ = −5.0 (point B in Fig. 3(a)). In (b) and (c): black lines plot the real part of f and red lines plot the imaginary part of f. (d) Soliton propagation for (b). (e) Soliton propagation for (c).

Download Full Size | PPT Slide | PDF

Figure 4(a) plots power P versus propagation constant μ for the case of a zero defect (ε = 0). This figure indicates that solitons only exist in the semi-infinite gap. In the regions of propagation constant: −4.07 ≤ μ ≤ −3.50, and −3.36 ≤ μ, the solitons cannot stably propagate. One example: μ = −3.3 (point B in Fig. 4(a)) has been introduced to certify the instability of soliton in these regions. Its solitons profile is showed in Fig. 4(d) and the corresponding soliton propagations is showed in Fig. 4(f). In the regions of propagation constant: μ < −4.07 and −3.50 < μ < −3.36, the solitons can stably transmit. The soliton profile of a stable example: μ = −5.0 (point A in Fig. 4(a)) is shown in Fig. 4(c) and its corresponding propagation is showed in Fig. 4(e). To further verify the instability, we plot the change of the Re(δ) with propagation constant μ in Fig. 4(b). For the negative slope of power curve (dP/dμ < 0) and Re(δ) = 0 in the regions: μ < −4.07 and the positive slope of power (dP/dμ > 0) and Re(δ) > 0 in the regions: −3.36 ≤ μ, we can conclude that the solitons stability or instability in these regions is in accordance with the VK criterion.

 

Fig. 4 (Color online) (ε = 0) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Re(δ) versus the propagation constant. (c) Stable soliton with μ = −5.0 (point A in Fig. 4(a)). (d) Unstable soliton with μ = −3.3 (point B in Fig. 4(a)). In (c) and (d): black lines plot the real part of f and red lines plot the imaginary part of f. (e) Soliton propagation for (c). (f) Soliton propagation for (d).

Download Full Size | PPT Slide | PDF

Figure 5 shows stable and unstable domains (μ, ε) of the PT solitons according to the relation between defect depth ε and the propagation constant μ. When increasing the defect depth of positive defect from ε = 0.1 to 0.8 or negative defect from ε = −0.1 to −0.7, the stable region of defect solitons in the semi-infinite gap will become narrow (see Fig. 5(a) and 5(b)). For the case of ε = −0.8, the stable region will disappear. In the first gap, the stable region of defect soliton will firstly become large and then become narrow when increasing the defect depth from ε = −0.1 to −0.6. The stable region is vanished for the case of ε = −0.7 (see Fig. 5(c)). The stability of solitons propagation is mainly affected by light diffraction and self-focusing resulting from nonlinearity.Beacuse of the counteraction between light diffraction and nonlinearity, the soliton pulses can stably propagate. When diffraction cannot be suppressed by nonlinearity, soliton pulses will finally decay into linear diffractive waves. In the case of a negative defect, the defect site, which has lower light intensity, can leads to that the repusion from the negative defect will increase the light diffraction and then change the existence and stability of PT solitons. Thus, with the effect of a negative defect mode, the PT gap soliton cannot only exist, but also stably propagate in the first gap. In the case of a positive defect, the defect site, which has a higher intensity, is attractive to the light field. The attraction from the positive defect will decrease the light diffraction. As a result, the existence and stability of PT solitons will also be altered, justly like the case of the negative defect.

 

Fig. 5 (Color online) Stable (green) and unstable domains (μ, ε) of PT solitons in the semi-infinite gap (a) and (b), and the first gap (c).

Download Full Size | PPT Slide | PDF

4. Conclusion

To summarize, we have demonstrated that stable defect solitons can be formed in a parity-time symmetrical superlattice with focusing Kerr nonlinearity. These solitons properties are remarkably different from those in a conventional superlattice which has not considered a parity-time potential. The parity-time potential can drastically affect the existence and stability of defect solitons. Numerical results in this paper show that the soliton can stably exist in the semi-infinite gap but cannot exist in the first gap for the cases of a zero defect or a positive defect. For the case of a negative defect, the stable soliton can exist in both the semi-infinite gap and the first gap.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 60978009) and the National Basic Research Program of China (Grant Nos. 2009CB929604 and 2007CB925204).

References and links

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

2. 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]  

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

4. 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]   [PubMed]  

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]   [PubMed]  

6. C. E. Rüter, 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 (2010). [CrossRef]  

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

8. Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. 32, 1390–1392 (2007). [CrossRef]   [PubMed]  

9. X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express 18(20), 20786–20792 (2010). [CrossRef]   [PubMed]  

10. X. Zhu, H. Wang, T. W. Wu, and L. X. Zheng, “Defect solitons in triangular optical lattices,” J. Opt. Soc. Am. B 28(3), 521 (2011). [CrossRef]  

11. Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009). [CrossRef]  

12. Z. Lu and Z. Zhang, “Surface line defect solitons in square optical lattice,” Opt. Express 19(3), 2410 (2011). [CrossRef]   [PubMed]  

13. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009). [CrossRef]   [PubMed]  

14. A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch wave filtering in periodic photonic lattices,” Opt. Lett. 30(14), 1849–1851 (2005). [CrossRef]   [PubMed]  

15. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008). [CrossRef]  

16. A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008). [CrossRef]  

17. 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]  

18. H. Wang and J. Wang, “Defect solitons in parity-time periodic potenticals,” Opt. Express 19(5), 4030 (2011). [CrossRef]   [PubMed]  

19. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007). [CrossRef]  

20. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008). [CrossRef]  

References

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  1. C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  2. 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]
  3. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  4. 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] [PubMed]
  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] [PubMed]
  6. C. E. Rüter, 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 (2010).
    [CrossRef]
  7. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
    [CrossRef]
  8. Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. 32, 1390–1392 (2007).
    [CrossRef] [PubMed]
  9. X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express 18(20), 20786–20792 (2010).
    [CrossRef] [PubMed]
  10. X. Zhu, H. Wang, T. W. Wu, and L. X. Zheng, “Defect solitons in triangular optical lattices,” J. Opt. Soc. Am. B 28(3), 521 (2011).
    [CrossRef]
  11. Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
    [CrossRef]
  12. Z. Lu and Z. Zhang, “Surface line defect solitons in square optical lattice,” Opt. Express 19(3), 2410 (2011).
    [CrossRef] [PubMed]
  13. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
    [CrossRef] [PubMed]
  14. A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch wave filtering in periodic photonic lattices,” Opt. Lett. 30(14), 1849–1851 (2005).
    [CrossRef] [PubMed]
  15. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
    [CrossRef]
  16. A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
    [CrossRef]
  17. 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]
  18. H. Wang and J. Wang, “Defect solitons in parity-time periodic potenticals,” Opt. Express 19(5), 4030 (2011).
    [CrossRef] [PubMed]
  19. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
    [CrossRef]
  20. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008).
    [CrossRef]

2011 (3)

2010 (4)

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. Rüter, 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 (2010).
[CrossRef]

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]

X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express 18(20), 20786–20792 (2010).
[CrossRef] [PubMed]

2009 (3)

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009).
[CrossRef] [PubMed]

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] [PubMed]

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

2008 (5)

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
[CrossRef]

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (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]

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008).
[CrossRef]

2007 (2)

Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. 32, 1390–1392 (2007).
[CrossRef] [PubMed]

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

2006 (1)

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

2005 (1)

1998 (1)

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

Assanto, G.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Bender, C. M.

C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (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] [PubMed]

Böttcher, S.

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

Chen, W. H.

Chen, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Chen, Z.

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

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. Rüter, 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 (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]

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] [PubMed]

Guo, Z.

He, Y. J.

Heinrich, M.

Kartashov, Y. V.

Kip, D.

C. E. Rüter, 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 (2010).
[CrossRef]

Kivshar, Y. S.

Kottos, T.

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] [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(2), 153–197 (2007).
[CrossRef]

Lederer, F.

Li, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Liu, S.

Lu, Z.

Lucchetti, L.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Makris, K. G.

C. E. Rüter, 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 (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]

Malomed, B. A.

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]

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]

Nolte, S.

Pang, W.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Pertsch, T.

Piccardi, A.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Rüter, C. E.

C. E. Rüter, 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 (2010).
[CrossRef]

Segev, M.

C. E. Rüter, 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 (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] [PubMed]

Simoni, F.

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

Sukhorukov, A. A.

Szameit, A.

Torner, L.

Tünnermann, A.

Vysloukh, V. A.

Wang, H.

Wang, H. Z.

Wang, J.

Wu, T. W.

Yang, J.

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 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(2), 153–197 (2007).
[CrossRef]

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

Ye, F.

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
[CrossRef]

Yu, Z.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zhang, H.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zhang, Z.

Zheng, L. X.

Zhou, J.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zhou, K.

Zhu, X.

Appl. Phys. Lett. (1)

A. Piccardi, G. Assanto, L. Lucchetti, and F. Simoni, “All-optical steering of soliton waveguides in dye-doped liquid crystals,” Appl. Phys. Lett. 93, 171104 (2008).
[CrossRef]

J. Comput. Phys. (1)

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008).
[CrossRef]

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

Nat. Phys. (1)

C. E. Rüter, 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 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. A (3)

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nonlinear switching of low-index modes in photonic lattices,” Phys. Rev. A 78, 013847 (2008).
[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. (4)

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] [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]

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

Stud. Appl. Math. (1)

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(Color online) (a) Band structure of the superlattice with V 0 = 6 and W 0 = 0.4. Lattices intensity profile of the PT superlattice with (b) ε = 0, (c) ε = 0.5, and (d) ε = −0.5. Blue line: real part, red line: imaginary part.

Fig. 2
Fig. 2

(Color online) (ε = −0.5) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Re(δ) versus the propagation constant. (c) Stable soliton with μ = −6.0 (point A in Fig. 2(a)). (d) Stable soliton with μ = −2.0 (point B in Fig. 2(a)). (e) Unstable soliton with μ = −1.74 (point C in Fig. 2(a)). In (c)–(e): black lines plot the real part of f and red lines plot the imaginary part of f. (f) Soliton propagation for (c). (g) Soliton propagation for (d). (h) Soliton propagation for (e).

Fig. 3
Fig. 3

(Color online) (ε = 0.5) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Stable soliton with μ = −6.5 (point A in Fig. 3(a)). (c) Stable soliton with μ = −5.0 (point B in Fig. 3(a)). In (b) and (c): black lines plot the real part of f and red lines plot the imaginary part of f. (d) Soliton propagation for (b). (e) Soliton propagation for (c).

Fig. 4
Fig. 4

(Color online) (ε = 0) (a) The power versus the propagation constant (blue regions corresponding to Bloch bands). (b) Re(δ) versus the propagation constant. (c) Stable soliton with μ = −5.0 (point A in Fig. 4(a)). (d) Unstable soliton with μ = −3.3 (point B in Fig. 4(a)). In (c) and (d): black lines plot the real part of f and red lines plot the imaginary part of f. (e) Soliton propagation for (c). (f) Soliton propagation for (d).

Fig. 5
Fig. 5

(Color online) Stable (green) and unstable domains (μ, ε) of PT solitons in the semi-infinite gap (a) and (b), and the first gap (c).

Equations (6)

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i U z + U xx + [ V ( x ) + i W ( x ) ] U + | U | 2 U = 0 ,
V ( x ) = { V 0 { ε 1 sin 2 [ ( x + π 2 ) ] + ( 1 ε 1 ) sin 2 [ 2 ( x + π 2 ) ] } , ( x > π 2 or x < π 2 ) , 0.78125 V 0 sin 2 [ ( x + π 2 ) ] [ 1 + ε exp ( x 8 128 ) ] , ( π 2 x π 2 ) ,
W ( x ) = W 0 sin ( 2 x ) ,
f xx + [ V ( x ) + iW ( x ) ] f + | f | 2 f + μ f = 0.
U = exp ( i μ z ) { f ( x ) + [ v ( x ) w ( x ) ] exp ( δ z ) + [ v ( x ) + w ( x ) ] * exp ( δ * z ) }
{ δ v = i [ d 2 w / d x 2 + μ w + Vw iWv + 2 | f | 2 w 1 2 ( f 2 f * 2 ) v 1 2 ( f 2 + f * 2 ) w ] , δ w = i [ d 2 v / d x 2 + μ v + Vw iWw + 2 | f | 2 v + 1 2 ( f 2 f * 2 ) w + 1 2 ( f 2 + f * 2 ) v ] .

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