## Incoherent multi-gap optical solitons in nonlinear photonic lattices

Optics Express, Vol. 13, Issue 8, pp. 2916-2923 (2005)

http://dx.doi.org/10.1364/OPEX.13.002916

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### Abstract

We demonstrate numerically that partially incoherent light can be trapped in the spectral band gaps of a photonic lattice, creating partially incoherent multi-component spatial optical solitons in a self-defocusing nonlinear periodic medium. We find numerically such incoherent multi-gap optical solitons and discuss how to generate them in experiment by interfering incoherent light beams at the input of a nonlinear periodic medium.

© 2005 Optical Society of America

## 1. Introduction

2. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. **58**, 160–163 (1987). [CrossRef] [PubMed]

4. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. **76**, 1627–1630 (1996). [CrossRef] [PubMed]

5. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. **90**, 053902–4 (2003). [CrossRef] [PubMed]

6. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. **92**, 093904-4 (2004). [CrossRef]

7. J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147–150 (2003). [CrossRef] [PubMed]

8. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. **90**, 023902–4 (2003). [CrossRef] [PubMed]

9. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. **93**, 083905-4 (2004). [CrossRef]

10. O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. **91**, 113901–4 (2003). [CrossRef] [PubMed]

11. A. A. Sukhorukov and Yu. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. **91**, 113902–4 (2003). [CrossRef] [PubMed]

12. D. E. Pelinovsky, A. A. Sukhorukov, and Yu. S. Kivshar, “Bifurcations and stability of gap solitons in periodic potentials,” Phys. Rev. E **70**, 036618-17 (2004). [CrossRef]

14. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. **90**, 253902–4 (2003). [CrossRef] [PubMed]

15. H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. **92**, 223901–4 (2004). [CrossRef] [PubMed]

*a self-focusing nonlinear periodic medium*by a single partially coherent beam incident at normal angle on a periodic photonic structure, as was recently demonstrated in experiment for optically-induced photonic lattices [16

16. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons”, Nature **433**, 500–503 (2005). [CrossRef] [PubMed]

10. O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. **91**, 113901–4 (2003). [CrossRef] [PubMed]

11. A. A. Sukhorukov and Yu. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. **91**, 113902–4 (2003). [CrossRef] [PubMed]

12. D. E. Pelinovsky, A. A. Sukhorukov, and Yu. S. Kivshar, “Bifurcations and stability of gap solitons in periodic potentials,” Phys. Rev. E **70**, 036618-17 (2004). [CrossRef]

*self-defocusing nonlinear periodic media*, but this is only possible when all the modes are localized in the Bragg-reflection gaps of the lattice spectrum.

## 2. Model and multi-gap optical solitons

*A*of the electric field,

*z*and

*x*are the propagation and transverse coordinates, respectively,

*D*=1/(2

*k*

_{0}

*n*) is the diffraction coefficient,

_{e}*k*

_{0}is the vacuum wavenumber,

*n*is the average refractive index of the unperturbed crystal,

_{e}*ν*(

*x*) is the periodic modulation of the refractive index, and

*δn*(

*I*) is the nonlinearity-induced change of the refractive index that depend on the time-averaged light intensity

*I*. We consider the case of slow

*self-defocusing nonlinearity*,

*δ*

^{2}

*n*(

*I*)≃-

*γI*, where

*γ*>0 is the nonlinear coefficient, and we neglect the effect of the nonlinearity saturation.

*A*(

*x*,

*t*) changes very rapidly (i.e. on a time scale much faster than the response time of the crystal). There exist several equivalent approaches to analyze such an incoherent light theoretically. The coherent density approach [17

17. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. **78**, 646–649 (1997). [CrossRef]

18. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. **79**, 4990–4993 (1997). [CrossRef]

*A*(

*x*,

*t*)=∑

*(*

_{m}ϕ_{m}*x*)exp[

*iγ*(

_{m}*t*)], which are mutually incoherent due to the random phase factors

*γ*(

_{i}*t*).

15. H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. **92**, 223901–4 (2004). [CrossRef] [PubMed]

*ϕ*(

_{m}*x*) should correspond to the modes of the soliton-induced waveguide. These may include both fundamental and higher-order modes localized in multiple band-gaps. To find incoherent solitons numerically, we use the following procedure. First, we start with a refractive index distribution and calculate, by means of linear algebra, all localized and radiation modes of this refractive index distribution, using Eq. (1). Second, we construct the solutions based on a set number of lowest order localized modes in the first one or two Bragg-reflection band-gaps. We scale the mode amplitudes to match the chosen intensity and power levels, and calculate the time-average refractive index distribution for the multi-mode light field. Then, we repeat the calculations again until a self-consistent multi-component solution is reached with a certain accuracy.

*µ*m and varying in the value between

*ν*

_{+}=7.20×10

^{-3}and

*ν*

_{-}=-

*ν*

_{+}. As for other parameters, we assume light of a vacuum wavelength of 532nm and an unperturbed refractive index

*n*=2.3. The bandgap spectrum of such a periodic structure is shown in Fig. 1. Note that we choose a rather strong modulation of the refractive index (‘a deep lattice’). This results in a considerable width of the first two band gaps. The third band gap, however, is still rather narrow and can only support weakly localized modes. In the following, we consider solitons based on strongly localized modes in the first and second gaps.

_{e}*coherent gap solitons*studied earlier. The first feature of the incoherence is that the gap soliton is very broad with a rather flat top. The soliton has to be broad, otherwise it could not support so many localized modes, especially in the second bandgap where localized modes tend to have a large spatial extent. The second feature introduced by the incoherence is that there is some light intensity even in the regions of lower refractive index. Near the center of the soliton the lowest intensity in the lower index region is still about 3–4% of the maximum intensity of the soliton. Although this might be a small effect, it is still a qualitative difference to the coherent case, where the intensity inevitably drops to zero somewhere in the lower index region. We note that coherent gap solitons can form stable bound states for a particular phase difference [19

19. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Exp. **12**, 2831, (2004). [CrossRef]

20. T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Yu. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. **94**, 063904-4 (2005). [CrossRef]

## 3. Generation of incoherent gap solitons

*d*=(

_{i}*k*

_{0}

*θ*)

^{-1}but propagating at slightly tilted angles

*α*. These patterns will excite Bloch waves at a particular band edge which symmetry is selected by a choice of the parameters

_{j}*d*

*and*

_{i}*x*. The suggested generation scheme provides a generalization of the approach which was recently used for generating coherent spatial gap solitons [6

_{i}6. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. **92**, 093904-4 (2004). [CrossRef]

9. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. **93**, 083905-4 (2004). [CrossRef]

20. T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Yu. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. **94**, 063904-4 (2005). [CrossRef]

*d*=

_{i}*d*/2 and

*x*=0. The soliton dynamics is summarized in Fig. 5, where we use a shallow lattice (

_{i}*ν*

_{+}=-

*ν*

_{-}=1.2×10

^{-3}) to make all effects more visible. The input and output (after 16mm) intensity profiles are presented in in Fig. 5(a). We observe highly efficient generation of immobile multi-gap soliton with minimal amount of radiation, in a sharp contrast to the case of a homogeneous self-defocusing medium.

16. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons”, Nature **433**, 500–503 (2005). [CrossRef] [PubMed]

## 4. Conclusions

## References and links

1. | Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz.51, 902–907 (1981) [English translation: Tech. Phys. 26, 541–544 (1981)]. |

2. | W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. |

3. | C. M. de Sterke and J. E. Sipe
, “Gap solitons,” in |

4. | B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. |

5. | D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. |

6. | D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide arrays,” Phys. Rev. Lett. |

7. | J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

8. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

9. | D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. |

10. | O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. |

11. | A. A. Sukhorukov and Yu. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. |

12. | D. E. Pelinovsky, A. A. Sukhorukov, and Yu. S. Kivshar, “Bifurcations and stability of gap solitons in periodic potentials,” Phys. Rev. E |

13. | B. Hanna, D. Neshev, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Experimental observation of multi-gap vector solitons,” In |

14. | D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. |

15. | H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, “Random-phase solitons in nonlinear periodic lattices,” Phys. Rev. Lett. |

16. | O. Cohen, G. Bartal, H. Buljan, T. Carmon, J.W. Fleischer, M. Segev, and D.N. Christodoulides, “Observation of random-phase lattice solitons”, Nature |

17. | D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, “Theory of incoherent self-focusing in biased photorefractive media,” Phys. Rev. Lett. |

18. | M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of self-trapped spatially incoherent light beams,” Phys. Rev. Lett. |

19. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Exp. |

20. | T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Yu. S. Kivshar, “Coherence controlled soliton interactions,” Phys. Rev. Lett. |

21. | M. Segev and D. N. Christodoulides
, “Incoherent solitons: self-trapping of weakly-correlated wave-packets,” in |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(190.3270) Nonlinear optics : Kerr effect

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 8, 2005

Revised Manuscript: March 29, 2005

Published: April 18, 2005

**Citation**

Kristian Motzek, Andrey Sukhorukov, Friedemann Kaiser, and Yuri Kivshar, "Incoherent multi-gap optical solitons in nonlinear photonic lattices," Opt. Express **13**, 2916-2923 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-2916

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### References

- Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, "Stationary waves in nonlinear, periodically modulated media with higher group retardation," Zh. Tekh. Fiz. 51, 902�??907 (1981) [English translation: Tech. Phys. 26, 541�??544 (1981)].
- W. Chen and D. L. Mills, �??Gap solitons and the nonlinear optical-response of superlattices,�?? Phys. Rev. Lett. 58, 160�??163 (1987). [CrossRef] [PubMed]
- C. M. de Sterke and J. E. Sipe, �??Gap solitons,�?? in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1994), Vol. XXXIII, pp. 203�??260.
- B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, �??Bragg grating solitons,�?? Phys. Rev. Lett. 76, 1627�??1630 (1996). [CrossRef] [PubMed]
- D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,�?? Phys. Rev. Lett. 90, 053902�??4 (2003). [CrossRef] [PubMed]
- D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, �??Gap solitons in waveguide arrays,�?? Phys. Rev. Lett. 92, 093904�??4 (2004). [CrossRef]
- J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, �??Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,�?? Nature 422, 147�??150 (2003). [CrossRef] [PubMed]
- J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, �??Observation of discrete solitons in optically induced real time waveguide arrays,�?? Phys. Rev. Lett. 90, 023902�??4 (2003). [CrossRef] [PubMed]
- D. Neshev, A. A. Sukhorukov, B. Hanna,W. Krolikowski, and Yu. S. Kivshar, �??Controlled generation and steering of spatial gap solitons,�?? Phys. Rev. Lett. 93, 083905�??4 (2004). [CrossRef]
- O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D. N. Christodoulides, �??Multiband vector lattice solitons,�?? Phys. Rev. Lett. 91, 113901�??4 (2003). [CrossRef] [PubMed]
- A. A. Sukhorukov and Yu. S. Kivshar, �??Multigap discrete vector solitons,�?? Phys. Rev. Lett. 91, 113902�??4 (2003). [CrossRef] [PubMed]
- D. E. Pelinovsky, A. A. Sukhorukov, and Yu. S. Kivshar, �??Bifurcations and stability of gap solitons in periodic potentials,�?? Phys. Rev. E 70, 036618�??17 (2004). [CrossRef]
- B. Hanna, D. Neshev, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, �??Experimental observation of multi-gap vector solitons,�?? In Nonlinear Guided Waves and Their Applications, Postconference ed. OSA pp. PD�??5 (Optical Society of America, Washington DC, 2004).
- D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Observation of mutually trapped multiband optical breathers in waveguide arrays,�?? Phys. Rev. Lett. 90, 253902�??4 (2003). [CrossRef] [PubMed]
- H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, �??Random-phase solitons in nonlinear periodic lattices,�?? Phys. Rev. Lett. 92, 223901�??4 (2004). [CrossRef] [PubMed]
- O. Cohen, G. Bartal, H. Buljan, T. Carmon, J.W. Fleischer, M. Segev, and D.N. Christodoulides, �??Observation of random-phase lattice solitons�??, Nature 433, 500�??503 (2005). [CrossRef] [PubMed]
- D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, �??Theory of incoherent self-focusing in biased photorefractive media,�?? Phys. Rev. Lett. 78, 646�??649 (1997). [CrossRef]
- M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, �??Theory of self-trapped spatially incoherent light beams,�?? Phys. Rev. Lett. 79, 4990�??4993 (1997). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, �??Soliton trains in photonic lattices,�?? Opt. Exp. 12, 2831, (2004). [CrossRef]
- T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Yu. S. Kivshar, �??Coherence controlled soliton interactions,�?? Phys. Rev. Lett. 94, 063904�??4 (2005). [CrossRef]
- M. Segev and D. N. Christodoulides, �??Incoherent solitons: self-trapping of weakly-correlated wave-packets,�?? in Spatial Optical Solitons, Vol. 82 of Springer Series in Optical Sciences, S. Trillo and W. E. Torruellas, eds., (Springer-Verlag, New York, 2001), pp. 87�??125.

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