## Different discrete soliton states in periodic optical induced waveguide lattice

Optics Express, Vol. 15, Issue 10, pp. 6232-6240 (2007)

http://dx.doi.org/10.1364/OE.15.006232

Acrobat PDF (365 KB)

### Abstract

Different discrete soliton states in optical waveguide lattices are studied theoretically for the different external conditions such as the biased field, lattice spacing and the position of the input pulse. It is demonstrated that the discrete solitons can be formed in an appropriate biased field and exhibit different discrete properties in different biased field. The results also indicate that lattice spacing and the position of input pulse can affect apparently the discreteness of discrete solitons. If changing the ratio of the waist width of input beam to lattice period, the discrete solitons will be excited in more or less channels. In addition, the increase of lattice intensity almost does not affect the propagation behavior of soliton.

© 2007 Optical Society of America

## 1. Introduction

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794–796 (1988). [CrossRef] [PubMed]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**, 3383–3386 (1998). [CrossRef]

3. D. N. Christodoulides, F. Lederer, and Y. Siberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

4. D. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today , **57**, 43–49 (2004). [CrossRef]

5. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. **28**, 1942–1944 (2003). [CrossRef] [PubMed]

7. Y.-D. Wu, “New all-optical switch based on the spatial soliton repulsion,” Opt. Express , **14**, 4005–4012 (2006). [CrossRef] [PubMed]

9. K. Staliunas and R. Herrero, “Nondiffractive propagation of light in photonic crystals,” Phys. Rev. E **73**, 016601 1–6 (2006). [CrossRef]

10. F. Lederer and Y. Silberberg, “Discrete solitons,” Opt. Photon. News **13**, 48–53 (2002). [CrossRef]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**, 3383–3386 (1998). [CrossRef]

12. 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]

14. T. J. Alexander and Y. S. Kivshar, “Soliton complexes and flat-top nonlinear modes in optical lattices,” Appl. Phys. B **82**, 203–206 (2006). [CrossRef]

14. T. J. Alexander and Y. S. Kivshar, “Soliton complexes and flat-top nonlinear modes in optical lattices,” Appl. Phys. B **82**, 203–206 (2006). [CrossRef]

15. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E. **66**, 046602, 1–5 (2002). [CrossRef]

16. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. **93**, 083905, 1–4 (2004). [CrossRef]

## 2. Theory expression

17. I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Soliton control in modulated optically-induced photonic lattices,” Opt. Express **13**, 5704–5710 (2005). [CrossRef] [PubMed]

*x*axis. When the plane waves linearly polarized along the ordinary

*y*axis are input in the incident face, the transverse periodic refractive index modulation is achieved for creating the optical lattices. They are highly nonlinear for extraordinary light that is polarized along

*x*axis [15

15. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E. **66**, 046602, 1–5 (2002). [CrossRef]

*I*

_{0}is the maximum lattice intensity, and

*D*is the lattice spacing.

15. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E. **66**, 046602, 1–5 (2002). [CrossRef]

*k*=

*k*

_{0}

*n*and

_{e}*n*is the refractive index for extraordinary polarized beam,

_{e}*r*

_{33}represents the electro-optical coefficient and

*E*

_{0}stands for the applied field intensity.

*E*is the steady state space charge-field given by

_{sc}*K*is the Boltzmann constant,

_{B}*T*is the temperature, and

*e*is the electron charge. In Eq. (3), the first term describes dominant screening nonlinearity of the photorefractive crystal, and the second term arises in weak diffusion effects. However, it does not affect the formation of discrete solitons so that it is ignored here. In photorefractive optical lattices,

*I*(

*x*) =

*I*(

_{g}*x*)+|

*u*|

^{2}is total normalized light intensity.

*ξ*=

*z*/(

*k*

_{0}

*n*

_{e}x_{0}

^{2}) and

*η*=

*x*/

*x*

_{0}are used, where

*x*

_{0}is the characteristic beam width, and the following normalized equation is obtained from Eq. (2) and Eq. (3):

*V*

_{0}is proportional to the biased field, exactly,

*V*

_{0}=

*k*

_{0}

^{2}

*n*

_{e}^{4}

*r*

_{33}

*E*

_{0}

*x*

_{0}

^{2}/2. And for convenience, we set

*d*=

*D*/

*x*

_{0}, then Eq. (1) is transformed to the following:

*η*

_{0}and Ω characterize the center position and the waist width of the input pulse, respectively.

## 3. Results and discussions

*η*

_{0}=0, the maximum lattice intensity

*I*

_{0}=1, the lattice spacing

*d*=2 and waist width Ω=1. In all of simulation, the propagation distance is considered from

*ξ*=0 to

*ξ*=20. Firstly, we consider the cases of low biased field at

*V*

_{0}=1. The simulation results are shown in Fig. 2. In the presence of lattice, the beam reaches a steady localized state after a short distance with weak diffraction, the discreteness cannot be seen even the existence of the lattice potential as shown in Fig. 2(a). When we ignore the lattice, which means the beam propagation only in a photorefractive crystal without the periodic refractive index modulation of transverse scalar, the profile of amplitude always keep a steady state that can be seen in Fig. 2(b), namely soliton state.

*V*

_{0}is increased to 30, as shown in Fig. 3, both the case with the lattice and without the lattice are very different from that of

*V*

_{0}=1. Discrete behavior of the beam was observed distinctly in optical lattice. After light has propagated a short distance, due to the interaction between the lattice potential and probe beam

*u*, the beam trapped quickly in the four center lattice sites and the energy of the soliton in the two inner sites are much greater than that in the two outer sites [as shown in Fig. 3(a) and Fig. 3(c)]. However, if without the lattice, the compression of beam and the slight increase of the beam peak are observed due to the increase of nonlinearity [as shown in Fig. 3(b) and Fig. 3(d)].

*V*

_{0}= 80 is checked for comparing with that of above, and the results are similar to those of

*V*

_{0}=30. The beams are all localized in the four center lattice sites, however, comparing with that of

*V*

_{0}=30, there is a more power concentrated in the two outer lattices after propagation a short distance as shown in Fig. 4(a) and Fig. 4(c). Therefore, solitons are excited in more channels with the increase of applied voltage as long as the incident sech-type beam covers all of these lattices. That is the discreteness of soliton is more obviously than that of the weak nonlinearity under low applied voltage. For that of without the lattice, the evolution of the beam is shown in Fig. 4(b) and Fig. 4(d). It can be see from Fig. 4(b) and Fig. 4(d) that the nonlinear focusing effect is so strengthening that the beam is compressed more due to the high nonlinearity at the high biased field.

*η*= 0) at three sites (

*ξ*= 1,8,16) with biased field

*V*

_{0}. Figure 5(a) shows that of light beam propagation in the optical waveguide lattice with

*d*=2. Figure 5(b) shows that of without the lattice. It can be seen clearly that the amplitude of the beam at the site

*η*=0 varies with the external biased field. When

*V*

_{0}is increased, the energy of the beam trapped into the lattices and formed discrete solitons in optical lattice, but it is compressed for the case of without lattice. The results are consistent with the facts that were shown in Fig. 3 and Fig. 4.

*d*= 5 at

*V*

_{0}=30 and

*V*

_{0}=80, respectively. The hyperbolic secant beam is input at

*η*= 0 and the results are shown in Fig. 6. It is found that the beam trapped in the center lattice and the two nearest lattices dependence on the

*V*

_{0}, which does not like the cases of

*d*= 2. At the small lattice spacing (

*d*= 2), the beam easily trapped in the neighboring lattices (left and right). As shown in Fig. 6(a), the beam trapped strongly in the centric lattice and relatively weaker in the adjacent two lattices. However, with the increase of

*V*

_{0}, then the beam mostly trapped into the two adjacent lattices [as shown in Fig. 6(b)].

*η*

_{0}. By fixed

*d*= 2, and the incident beam is input at the first left lattice where

*η*

_{0}= -1, the evolution of the beam at

*V*

_{0}=30 and

*V*

_{0}=80 is shown in Fig. 7. Compared Fig. 7 with Fig. 3(a) and Fig. 4(a), it can be found that their discrete behavior is different, but the symmetry of patterns is same in all of these cases. When the beam is input in one lattice, the propagation of energy is mainly as discrete solitons by trapped in the lattice and the adjacent lattices (see Fig. 7). When the beam is input in the position between two lattices, then the beam trapped in the left and right adjacent lattices and thus formed the discrete solitons [Fig. 3(a) and Fig. 4(a)].

*I*

_{0}. Since one of the meaningful factors that determine physically the propagation regime of laser beam in periodic structure is not the width of beam itself but rather the ratio of the beam width to lattice period [19

19. Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, “Spatial soliton switching in quasicontinuous optical arrays,” Opt. Lett. **29**, 766–768 (2004). [CrossRef] [PubMed]

*d*=2 and the waist width of input soliton is varied in our numerical simulations. We found that the solitons are excited in more channels, which is illustrated in Fig.8(a) and Fig. 8(b), with the increase of the waist width of the soliton by decreasing the value of Ω. However, when only the lattice intensity is increased by increasing the value of

*I*

_{0}, there is hardly changing for the propagation behavior of discrete soliton (as shown in Fig. 9) except for the distance they trapped in lattices [compared Fig. 4(a) with Fig. 9(a) and Fig. 9(b)]).

## 4. Conclusions

*V*

_{0}, lattice period and the position of the input pulse. With appropriate

*V*

_{0}, the generation of the discrete solitons is observed in optical waveguide lattice and the discrete localized states are also varies with

*V*

_{0}. If without the lattice in the photorefractive crystal, the propagation of the beam exhibits different phenomena with that of the lattice for variant

*V*

_{0}. The lattice spacing and the input position of the light pulse also affect the discreteness of the beam, but they cannot determine whether the discrete solitons can be formed. Though it is found that the discrete behavior are different at different input position of original pulse, the symmetry of patterns is remained. In addition, the soliton propagation in the waveguide lattice is also affected by the ratio of the waist width of beam to lattice period, for example, the wider waist of input beam, the more lattices localized in. However, the lattice intensity has hardly affection on the propagation of discrete soliton except that the generation of localization happens quickly or slowly. Therefore, these results are useful reference for observing discrete solitons experimentally in optical induced waveguide lattice. It is also significant to control propagation of light in the optical induced waveguide lattices for practical application.

## Acknowledgments

## References and links

1. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

2. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

3. | D. N. Christodoulides, F. Lederer, and Y. Siberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

4. | D. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today , |

5. | R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. |

6. | A. A. Sukhorukov and Y. S. Kivshar, “Soliton switching and Bloch-wave filtering in periodic photonic lattices,” arXiv Physics/ 041215123, Dec 2004 |

7. | Y.-D. Wu, “New all-optical switch based on the spatial soliton repulsion,” Opt. Express , |

8. | Yu. S. Kivshar and G. P. Agrawal, |

9. | K. Staliunas and R. Herrero, “Nondiffractive propagation of light in photonic crystals,” Phys. Rev. E |

10. | F. Lederer and Y. Silberberg, “Discrete solitons,” Opt. Photon. News |

11. | T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schiek, R. Iwanow, G. I. Stegeman, Y. Min, and W. Sohler, “Discrete Solitons in |

12. | 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 |

13. | F. Lederer, S. Damanyan, and A. Kobyakov, Discrete solitons, in: Spatial Solitons, S. Trillo, and W. Torruellas, eds., (Springer, Berlin2001) pp. 269–276. |

14. | T. J. Alexander and Y. S. Kivshar, “Soliton complexes and flat-top nonlinear modes in optical lattices,” Appl. Phys. B |

15. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E. |

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

17. | I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Soliton control in modulated optically-induced photonic lattices,” Opt. Express |

18. | G. P. Agrawal, |

19. | Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, “Spatial soliton switching in quasicontinuous optical arrays,” Opt. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5330) Nonlinear optics : Photorefractive optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 20, 2006

Revised Manuscript: April 23, 2007

Manuscript Accepted: April 24, 2007

Published: May 7, 2007

**Citation**

Jun Zhou, Yihong Qi, Chunhua Xue, Senyue Lou, and Yuntuan Fang, "Different discrete soliton states in periodic optical induced waveguide lattice," Opt. Express **15**, 6232-6240 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6232

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### References

- D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998). [CrossRef]
- D. N. Christodoulides, F. Lederer and Y. Siberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- D. Campbell, S. Flach and Y. S. Kivshar, "Localizing energy through nonlinearity and discreteness," Phys. Today, 57, 43-49 (2004). [CrossRef]
- R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, "Controlled switching of discrete solitons in waveguide arrays," Opt. Lett. 28, 1942-1944 (2003). [CrossRef] [PubMed]
- A. A. Sukhorukov and Y. S. Kivshar, "Soliton switching and Bloch-wave filtering in periodic photonic lattices," arXiv Physics/ 0412151 23, Dec 2004
- Y.-D. Wu, "New all-optical switch based on the spatial soliton repulsion," Opt. Express, 14, 4005-4012 (2006). [CrossRef] [PubMed]
- Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
- K. Staliunas and R. Herrero, "Nondiffractive propagation of light in photonic crystals," Phys. Rev. E 73, 016601 1-6 (2006). [CrossRef]
- F. Lederer and Y. Silberberg, "Discrete solitons," Opt. Photon. News 13,48-53 (2002). [CrossRef]
- T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schiek, R. Iwanow, G. I. Stegeman, Y. Min, and W. Sohler, "Discrete Solitons in χ(2)-waveguide arrays," in Proc. Nonlinear Guided Waves and their Applications (Stresa, Italy, Sept. 2002), paper NLTuA1.
- 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]
- F. Lederer, S. Damanyan, A. Kobyakov, Discrete solitons, in: Spatial Solitons, S. Trillo and W. Torruellas, eds., (Springer, Berlin 2001) pp. 269-276.
- T. J. Alexander and Y. S. Kivshar, "Soliton complexes and flat-top nonlinear modes in optical lattices," Appl. Phys. B 82, 203-206 (2006). [CrossRef]
- N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E. 66, 046602, 1-5 (2002). [CrossRef]
- D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, "Controlled generation and steering of spatial gap solitons," Phys. Rev. Lett. 93, 083905, 1-4 (2004). [CrossRef]
- I. L. Garanovich, A. A. Sukhorukov and Y. S. Kivshar, "Soliton control in modulated optically-induced photonic lattices," Opt. Express 13, 5704-5710 (2005). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics (London, U.K, Academic, 1989), Ch. 2 and Ch. 4.
- Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, "Spatial soliton switching in quasicontinuous optical arrays," Opt. Lett. 29, 766-768 (2004). [CrossRef] [PubMed]

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