## Multi-stable solitons in ����-symmetric optical lattices |

Optics Express, Vol. 20, Issue 15, pp. 16823-16831 (2012)

http://dx.doi.org/10.1364/OE.20.016823

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### Abstract

We address the existence and stability properties of optical solitons in a competing cubic-quintic medium with an imprinted complex lattice featuring a parity-time (𝒫𝒯) symmetry. Various families of solitons with even and odd geometrical symmetries are found in both the semi-infinite and the first finite gaps. Linear stability analysis corroborated by direct propagation simulations reveals that solitons with different symmetries and different number of humps can propagate stably at the same propagation constants, i.e., multi-stable solitons can exist in this scheme. Interestingly enough, in sharp contrast to the stability of solitons in a conventional (real) lattice, both even and odd solitons with the same propagation constant belonging to different branches can be stable in the first gap of 𝒫𝒯 lattice, which indicates that the imaginary part of lattice plays an important role for the stabilization of solitons.

© 2012 OSA

## 1. Introduction

4. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. **32**, 2632–2634 (2007). [CrossRef] [PubMed]

10. 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**, 192–195 (2010). [CrossRef]

*n*(

*x*) =

*n*(

_{r}*x*) +

*in*(

_{i}*x*), where

*n*(

_{r}*x*) =

*n*(−

_{r}*x*) is the real index profile, and

*n*(

_{i}*x*) = −

*n*(−

_{i}*x*) stands for the imaginary one [10

10. 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**, 192–195 (2010). [CrossRef]

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

13. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. **285**, 3320–3324 (2012). [CrossRef]

14. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in *𝒫𝒯* -symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A **85**, 013831 (2012). [CrossRef]

15. 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. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A **85**, 043826 (2012). [CrossRef]

19. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with *𝒫𝒯* -symmetric potentials,” Phys. Rev. A **84**, 053855 (2011). [CrossRef]

20. S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. **285**, 1934–1939 (2012). [CrossRef]

20. S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. **285**, 1934–1939 (2012). [CrossRef]

*et al.*systematically studied the stability properties of solitons in 𝒫𝒯 lattices and found that both 1D and 2D solitons can propagate stably under appropriate conditions [22

22. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in *𝒫𝒯* -symmetric optical lattices,” Phys. Rev. A **85**, 023822 (2012). [CrossRef]

23. A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. **55**, 1291–1294 (1985). [CrossRef] [PubMed]

24. J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A **339**, 74–82 (2005). [CrossRef]

24. J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A **339**, 74–82 (2005). [CrossRef]

## 2. Model

*z*axis in a competing cubic-quintic medium with a transverse complex refractive-index modulation. Dynamics of the beam can be described by the nonlinear Schrödinger equation for the dimensionless complex field amplitude

*ψ*: Here, the longitudinal

*z*and transverse

*x*coordinates are scaled in the terms of diffraction length and beam width, respectively;

*V*(

*x*) =

*V*

_{0}sin

^{2}(

*x*) and

*W*(

*x*) =

*W*

_{0}sin(2

*x*) are the real and imaginary periodic refractive-index modulations, respectively, satisfying the 𝒫𝒯-symmetry, i.e.,

*V*(−

*x*) =

*V*(

*x*) and

*W*(−

*x*) = −

*W*(

*x*).

*V*

_{0}and

*W*

_{0}denote the amplitudes of the real lattice and imaginary lattice. Without loss of generality, through the rest of this paper, we are going to use a specific configuration by setting

*V*

_{0}= 3,

*W*

_{0}= 0.3 unless stated otherwise.

*V*

_{0}= 3,

*W*

_{0}= 0.3 and 1.55 are shown in Fig. 1(b). There exists a critical value

*W*

_{0}= 1.5 (phase transition point), above which the eigenvalue spectra become complex, and all Bloch bands merge together simultaneously. Typical profiles of the real and imaginary refractive-index modulation are displayed in Fig. 1(a). Stationary solutions of Eq. (1) can be solved by assuming

*ψ*(

*x*,

*z*) =

*ϕ*(

*x*)exp(−

*ibz*), where

*ϕ*(

*x*) is a complex function and

*b*is a real propagation constant (below the phase transition point). Substitution of the expression into Eq. (1) yields an ordinary differential equation describing the stationary solution: which can be solved numerically by applying the boundary conditions:

*ϕ*(

_{r,i}*x →*±∞) = 0.

*ψ*(

*x*,

*z*) = {

*ϕ*(

*x*) + [

*v*(

*x*) −

*w*(

*x*)]exp(

*λz*) + [

*v*(

*x*) +

*w*(

*x*)]

^{*}exp(

*λ*

^{*}

*z*)} exp(−

*ibz*), where

*v*,

*w*≪ 1 are the infinitesimal perturbations which may grow with a common complex rate

*λ*upon propagation, and * is the complex conjugate operation. Substituting the perturbed solution into Eq. (1) and linearizing it around the stationary solution

*ϕ*(

*x*), one obtains a coupled set of linear eigenvalue equations: where

*L̂*=

*d*

^{2}/

*dx*

^{2}/2 +

*b*−

*V*+ 2|

*ϕ*|

^{2}− 3|

*ϕ*|

^{4}. Obviously, if there exist nonzero real parts of

*λ*, the stationary solution is unstable, otherwise it is stable.

## 3. Multi-stable solitons in the semi-infinite gap

*ϕ*(−

*x*)] =Re[

*ϕ*(

*x*)], the symmetry of the imaginary part is Im[

*ϕ*(−

*x*)] =−Im[

*ϕ*(

*x*)] [Figs. 2(b)–2(f)]. It means that the symmetry of even solitons is consistent with that of 𝒫𝒯 lattice. The peak of the real part grows with the decrease of propagation constant. Meanwhile, the two main peaks of the imaginary part increase with a slower speed [Fig. 2(b)].

*b*

_{1}

*= 0.878), the peak of soliton reaches a maximum. The single-humped soliton stops to exist if the propagation constant is below the threshold value. Reversely, the two lobes beside the main lobe grow slowly with the increase of propagation constant. In this process, the peak of the main lobe still increases until it reaches another maximum value larger than 1 [Fig. 2(c)]. The 2nd branch of solitons stop to exist when the propagation constant reaches an upper threshold value (*

_{low}*b*

_{2}

*= 0.949). Now, the defocusing nonlinearity becomes dominant for the main lobe of soliton which in turn prevents the further increase of soliton peak located in the central lattice site. Thus, the two lobes beside the central one grow rapidly with the decrease of propagation constant for solitons belonging to the 3rd branch. Meanwhile, the central peak decreases due to the defocusing quintic nonlinearity. The soliton features a complete three-humped structure [Fig. 2(d)]. With the further increase of power, solitons will exhibit a structure featuring three main lobes with the same peak values (slightly larger than 1) accompanying by two lower lobes beside them [Fig. 2(e)]. The outer two lobes begin to grow if the solitons switch to the 5th branch [Fig. 2(f)]. Similar to the 3rd branch, the central peaks of the 5th branch decrease with the decrease of propagation constant. This process may be continued until the soliton solution becomes completely periodic.*

_{upp}*f*1 marked in Fig. 2(a)]. To avoid the superposition with the corresponding real part, it is enhanced by 0.1. Although the soliton includes imaginary part with odd symmetry, its amplitude profile keeps even symmetric.

8. 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]

*c*1 and

*d*1 in Fig. 2(a) are shown in Figs. 3(c) and 3(d), respectively. The direction of the flow from gain to loss regions varies across the lattice. It is negative in the lattice sites and positive in the space between lattice sites. Comparing the two curves, the slight higher amplitude of power-flow density shown in Fig. 3(d) may account for the occurrence of instability.

24. J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A **339**, 74–82 (2005). [CrossRef]

*b*1 –

*b*2 –

*d*2 –

*d*1 –

*e*1 –

*e*2 and from four humps to four humps along the path

*c*1 –

*c*2 –

*f*2 –

*f*1. Solitons belonging to the 2nd branch and the 4th branch converge together at the lower cutoff of propagation constant (left turning point) [Fig. 4(a)]. The 5th branch of solitons stop to exist when the propagation constant is smaller than 0.884.

*f*1 in Fig. 4(a). Though the left part and right part of soliton are out-of-phase, the power-flow density is an even function of

*x*. Two typical stable and unstable propagations of solitons belonging to 1st and 5th branches are illustrated in Figs. 5(c) and Fig. 5(d), respectively. The unstable soliton can survive for a long propagation distance since the instability growth rate is very small (max[

*Re*(

*λ*)] < 0.05).

## 4. Multi-stable solitons in the first gap

*b*∈ [1.16, 2.38]. The power threshold values of different families of solitons indicate that they are not a continuum of linear modes in bands.

*b*= 1.40 in Fig. 7. Here, the peak values of the humps are obviously larger than 1. It agrees with the prediction according to the slopes of power curves, i.e., solitons in the first gap experience strong defocusing nonlinearity. By comparing the real parts of even and odd solitons in the semi-infinite gap, the minima of the real parts between the neighboring lattice channels are enhanced evidently, due to the dominant defocusing nonlinearity. One can infer from Figs. 2, 4, and 7 that the imaginary parts of solitons in the first gap are very different from those in the semi-infinite gap. Note that the multi-humped solitons resemble the truncated-Bloch-wave solitons in purely real lattices [25

25. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A **79**, 043610 (2009). [CrossRef]

**339**, 74–82 (2005). [CrossRef]

*b*= 2.13) without any distortions.

## 5. Conclusions

## Acknowledgments

## References and links

1. | C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. |

2. | C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. |

3. | C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A |

4. | R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. |

5. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in |

6. | A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of |

7. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “ |

8. | Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in |

9. | K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. |

10. | 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. |

11. | F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in |

12. | D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. |

13. | Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. |

14. | Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in |

15. | K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. |

16. | H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express |

17. | Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express |

18. | S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A |

19. | Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with |

20. | S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. |

21. | L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A |

22. | S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in |

23. | A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. |

24. | J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A |

25. | J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 11, 2012

Revised Manuscript: June 28, 2012

Manuscript Accepted: June 29, 2012

Published: July 10, 2012

**Citation**

Chunyan Li, Haidong Liu, and Liangwei Dong, "Multi-stable solitons in 𝒫𝒯-symmetric optical lattices," Opt. Express **20**, 16823-16831 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16823

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### References

- 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, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999). [CrossRef]
- R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (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]
- A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef] [PubMed]
- 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. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011). [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, 192–195 (2010). [CrossRef]
- F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯 -symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011). [CrossRef]
- D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011). [CrossRef]
- Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012). [CrossRef]
- Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in 𝒫𝒯 -symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012). [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]
- H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011). [CrossRef] [PubMed]
- Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011). [CrossRef] [PubMed]
- S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012). [CrossRef]
- Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with 𝒫𝒯 -symmetric potentials,” Phys. Rev. A 84, 053855 (2011). [CrossRef]
- S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. 285, 1934–1939 (2012). [CrossRef]
- L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).
- S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A8 5, 023822 (2012). [CrossRef]
- A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985). [CrossRef] [PubMed]
- J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005). [CrossRef]
- J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009). [CrossRef]

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