## Dark and bright blocker soliton interaction in defocusing waveguide arrays

Optics Express, Vol. 14, Issue 23, pp. 11248-11255 (2006)

http://dx.doi.org/10.1364/OE.14.011248

Acrobat PDF (531 KB)

### Abstract

We experimentally demonstrate the interaction of an optical probe beam with both bright and dark blocker solitons formed with low optical light power in a saturable defocusing waveguide array in photorefractive lithium niobate. A phase insensitive interaction of the beams is achieved by means of counterpropagating light waves. Partial and full reflection (blocking) of the probe beam on the positive or negative light-induced defect is obtained, respectively, in good agreement with numerical simulations.

© 2006 Optical Society of America

## 1. Introduction

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

5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. **83**, 4756 (1999). [CrossRef]

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled wave-guides,” Opt. Lett. **13**, 794 (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 (1998). [CrossRef]

5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. **83**, 4756 (1999). [CrossRef]

8. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**, 1863 (2000). [CrossRef] [PubMed]

9. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. **90**, 53902 (2003). [CrossRef]

14. H. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisions of photorefractive solitons,” Opt. Lett. **22**, 448 (1997). [CrossRef] [PubMed]

17. D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. **87**, 233901 (2001). [CrossRef] [PubMed]

18. A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays”, Phys. Rev. E **53**, 1172 (1996). [CrossRef]

19. I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrodinger equation,” Phys. Rev. E **68**, 046604 (2003). [CrossRef]

20. J. Meier, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, M. Sorel, H. Yang, G. Salamo, J. S. Aitchison, and Y. Silberberg, “Nonlinear beam interactions in 1D discrete Kerr systems,” Opt. Express **13**, 1797 (2005). [CrossRef] [PubMed]

21. J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. **93**, 093903 (2004). [CrossRef] [PubMed]

22. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Beam interactions with a blocker soliton in one-dimensional arrays,” Opt. Lett. **30**, 1027 (2005). [CrossRef] [PubMed]

23. J. Meier, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, G. Salamo, H. Yang, M. Sorel, Y. Silberberg, and J. S. Aitchison, “Incoherent blocker soliton interaction in Kerr waveguide arrays,” Opt. Lett. **30**, 3174 (2005). [CrossRef] [PubMed]

24. D. Kip, M. Wesner, C. Herden, and V. Shandarov, “Interaction of spatial photorefractive solitons in a planar waveguide,” Appl. Phys. B **68**, 971 (1999). [CrossRef]

25. W. Królikowski, B. Luther-Davies, C. Denz, and T. Tschudi, “Annihilation of photorefractive solitons,” Opt. Lett. **23**, 97 (1998). [CrossRef]

26. D. Kip, M. Soljačić, M. Segev, S. M. Sears, and D. N. Christodoulides, “(1+1) Dimensional modulation instability of spatially-incoherent light,” J. Opt. Soc. Am. B **19**, 502 (2002). [CrossRef]

13. Lj. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, “Power controlled soliton stability and steering in lattices with saturable nonlinearity,” Phys. Rev. Lett. **93**, 33901 (2004). [CrossRef]

_{3}. The transverse electrically (TE), extraordinarily polarized narrow soliton beam and broader probe beam are mutually coherent but are propagating in opposite directions in the array, thus forming an interference pattern with a grating vector directed along the propagation direction (y axis). As a consequence, a modulated space charge electric field builds up, however, no appropriate electrooptic tensor element (here

*r*=

_{eff}*r*

_{32}=0) exists for crystals with point symmetry 3m, and the resulting nonlinear index change is zero. In this way we achieve an almost complete, phase-insensitive reflection (blocking) of the probe beam on either positive or negative light-induced defects formed by bright and dark (blocker) solitons, respectively. Numerical simulations based on a nonlinear beam propagation method (BPM) are in fairly good agreement with our measurements and show the suitability of this nonlinear process for the realization of beam splitters with adjustable splitting ratios.

## 2. Experimental methods

*T*=1040 °C. To enhance the photorefractive nonlinearity additional doping with Fe is realized by in-diffusion of a 5.6nm Fe layer that is annealed for 24 hours at

*T*=1060 °C. The final waveguide array has a pitch of 8.4

*µ*m with a distance of 4.4

*µ*m between adjacent channels.

_{4}laser with a wavelength

*λ*=532 nm. The light is split into three beams, where two of them are formed by a Michelson interferometer. For the excitation of bright solitons one of these two beams is blocked, while the other remaining beam is focused onto the left hand input facet of the sample. On the other hand, dark solitons may be formed by partially superimposing both beams of the interferometer (having a phase difference of

*π*) under a small angle on the input facet of the 1D waveguide array. In this way a broad beam covering about 15 channels with a small dark notch caused by destructive interference in the overlap region is formed. A third beam that enters the sample from the right hand side (rear facet) under a small angle is used to form the counterpropagating probe beam. This beam is launched at half the Bragg angle of the array (transverse wave vector

*k*=

_{z}*π*/2Λ), thus it experiences almost zero diffraction during propagation. Input and output light intensities can be monitored by two CCD cameras. For the imaging of input light distributions reflection from the polished input facet is used.

## 3. Experimental results and discussion

29. M. Matuszewski, C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, A. Mitchell, M. Trippenbach, M. W. Austin, W. Królikowski, and Yu. S. Kivshar, “Crossover from self-defocusing to discrete trapping in nonlinear waveguide arrays,” Opt. Express **14**, 254 (2006). [CrossRef] [PubMed]

*z*=25

*µ*m, covering roughly 3 channels of the lattice, i.e. it is slightly broader then the input for the bright soliton beam. Finally, in part (c) the evolution of the probe beam (propagating from top to bottom with transverse wave vector

*k*=

_{z}*π*/2Λ) is given. The probe beam covers about 5 channels and shows only negligible diffraction. As can be seen, in all three cases a rather good agreement of experiment and numerical simulation is achieved.

*n*=Δ

_{nl}*n*

_{0}

*I*/(

*I*+

*I*), with fixed amplitude Δ

_{d}*n*

_{0}=-2×10

^{-4}and an intensity ratio

*r*=

*I*/

*I*, where Id is the so-called dark irradiance and

_{d}*I*is amplitude of the input light intensity.

*P*=6

*µ*W covers mainly one channel of the array, with a build-up time of about 60 minutes. The input power of the probe beam is chosen to be very low (

*P*=10nW), therefore nonlinear index changes induced by the probe beam itself can be ruled out. As can be seen in (a), during the build-up of the defect state an increasing part of the incident probe beam is reflected to the reverse side of the array, with almost total energy reflection in steady-state. Because of the low power of the probe beam no lateral shift of the bright gap soliton is observed here, as has been recently reported for Kerr media [23

_{in}23. J. Meier, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, G. Salamo, H. Yang, M. Sorel, Y. Silberberg, and J. S. Aitchison, “Incoherent blocker soliton interaction in Kerr waveguide arrays,” Opt. Lett. **30**, 3174 (2005). [CrossRef] [PubMed]

*P*=1

*µ*W nearly 60% of the probe beam intensity is guided in the channel adjacent to the defect state, and for slightly lower input power partial trapping of the probe beam’s intensity inside the soliton channel is observed.

*P*=6

_{in}*µ*W, however this power is now distributed among roughly 12 input channels. Consequently, the resulting intensity ratio is

*r*≈0.5. In steady-state, only partial reflection of the probe beam’s intensity is achieved, which may be attributed to the different cross-section of the defect and in particular to the lower input power (per channel) of the used dark soliton when compared to the situation of the bright blocker beam. Again, a good agreement of numerical simulations with the experimental situation os obtained. Furthermore, in a recent submission [27] we have demonstrated the formation of localized nonlinear dark modes displaying a phase jump that is located in-between two channels (mode B). For such off-channel dark solitons reflection of a probe beam is observed in a similar manner as for the on-channel case discussed before.

*n*=0. For the simulation we used again the corresponding input profiles (narrow single-channel bright beam and three-channel dark input beam) of the experiments from Figs. 3 and 4. As can be seen, for both cases and higher input power, reflected light intensities saturate at a reflectivity of almost 90 percent. Note that here narrow bright solitons only exist for input power ratios

*r*>1.1 [30], whereas dark solitons can be excited also at significantly lower input power. In the low input power region (small ratios

*r*) reflection by the dark soliton is more effective when compared to the bright case, which may be attributed to the larger spatial extension of the induced (positive) defect state. Again, the experimentally obtained reflection of about 50% for the dark blocker soliton with

*r*=0.5 is in reasonable agreement with our simulations.

## 4. Summary

## References and links

1. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled wave-guides,” 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. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

4. | S. G. Johnson and J. D. Joannopoulos, in “Photonic Crystals: The Road from Theory to Practice”, (Kluwer, Boston, 2002). |

5. | R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. |

6. | 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. |

7. | F. Chen, M. Stepić, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays,” Opt. Express |

8. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. |

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

10. | J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express |

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

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. | Lj. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, “Power controlled soliton stability and steering in lattices with saturable nonlinearity,” Phys. Rev. Lett. |

14. | H. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisions of photorefractive solitons,” Opt. Lett. |

15. | M. Shih, M. Segev, and G. Salamo, “Three dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. |

16. | G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science |

17. | D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. |

18. | A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays”, Phys. Rev. E |

19. | I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrodinger equation,” Phys. Rev. E |

20. | J. Meier, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, M. Sorel, H. Yang, G. Salamo, J. S. Aitchison, and Y. Silberberg, “Nonlinear beam interactions in 1D discrete Kerr systems,” Opt. Express |

21. | J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. |

22. | J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Beam interactions with a blocker soliton in one-dimensional arrays,” Opt. Lett. |

23. | J. Meier, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, G. Salamo, H. Yang, M. Sorel, Y. Silberberg, and J. S. Aitchison, “Incoherent blocker soliton interaction in Kerr waveguide arrays,” Opt. Lett. |

24. | D. Kip, M. Wesner, C. Herden, and V. Shandarov, “Interaction of spatial photorefractive solitons in a planar waveguide,” Appl. Phys. B |

25. | W. Królikowski, B. Luther-Davies, C. Denz, and T. Tschudi, “Annihilation of photorefractive solitons,” Opt. Lett. |

26. | D. Kip, M. Soljačić, M. Segev, S. M. Sears, and D. N. Christodoulides, “(1+1) Dimensional modulation instability of spatially-incoherent light,” J. Opt. Soc. Am. B |

27. | E. Smirnov, C. E. Rüter, M. Stepić, V. Shandarov, and D. Kip, “Formation and light guiding properties of dark solitons in one-dimensional waveguide arrays,” arXiv:nlin.PS/0607030 (2006). |

28. | M. Stepić, E. Smirnov, C. E. Rüter, V. Shandarov, and D. Kip, “Beam interactions in one-dimensional saturable waveguide arrays,” will appear in PRE, arXiv:physics/0607028 (2006). |

29. | M. Matuszewski, C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, A. Mitchell, M. Trippenbach, M. W. Austin, W. Królikowski, and Yu. S. Kivshar, “Crossover from self-defocusing to discrete trapping in nonlinear waveguide arrays,” Opt. Express |

30. |
For |

**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: September 6, 2006

Revised Manuscript: October 9, 2006

Manuscript Accepted: October 9, 2006

Published: November 13, 2006

**Citation**

Eugene Smirnov, Christian E. Rüter, Milutin Stepić, Vladimir Shandarov, and Detlef Kip, "Dark and bright blocker soliton interaction in defocusing waveguide arrays," Opt. Express **14**, 11248-11255 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11248

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### References

- D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled wave-guides," Opt. Lett. 13, 794 (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 (1998). [CrossRef]
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817 (2003). [CrossRef] [PubMed]
- S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice, (Kluwer, Boston, 2002).
- R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756 (1999). [CrossRef]
- 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, 23902 (2003). [CrossRef]
- F. Chen, M. Stepić, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express 13, 4314 (2005). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863 (2000). [CrossRef] [PubMed]
- D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti and J. S. Aitchison, "Band structure of waveguide arrays and excitation of Floquet-Bloch solitons," Phys. Rev. Lett. 90, 53902 (2003). [CrossRef]
- J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780 (2005). [CrossRef] [PubMed]
- D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap solitons in waveguide arrays," Phys. Rev. Lett. 92, 93904 (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 (2003). [CrossRef] [PubMed]
- Lj. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93, 33901 (2004). [CrossRef]
- H. Meng, G. Salamo, M. Shih, and M. Segev, "Coherent collisions of photorefractive solitons," Opt. Lett. 22, 448 (1997). [CrossRef] [PubMed]
- M. Shih, M. Segev, and G. Salamo, "Three dimensional spiraling of interacting spatial solitons," Phys. Rev. Lett. 78, 2551 (1997). [CrossRef]
- G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518 (1999). [CrossRef] [PubMed]
- D. N. Christodoulides and E. D. Eugenieva, "Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays," Phys. Rev. Lett. 87, 233901 (2001). [CrossRef] [PubMed]
- A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, "Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172 (1996). [CrossRef]
- I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, "Soliton collisions in the discrete nonlinear Schrodinger equation," Phys. Rev. E 68, 046604 (2003). [CrossRef]
- J. Meier, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, M. Sorel, H. Yang, G. Salamo, J. S. Aitchison, and Y. Silberberg, "Nonlinear beam interactions in 1D discrete Kerr systems," Opt. Express 13, 1797 (2005). [CrossRef] [PubMed]
- J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Nonlinear optical beam interactions in waveguide arrays," Phys. Rev. Lett. 93, 093903 (2004). [CrossRef] [PubMed]
- J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, "Beam interactions with a blocker soliton in one-dimensional arrays," Opt. Lett. 30, 1027 (2005). [CrossRef] [PubMed]
- J. Meier, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, G. Salamo, H. Yang, M. Sorel, Y. Silberberg, and J. S. Aitchison, "Incoherent blocker soliton interaction in Kerr waveguide arrays," Opt. Lett. 30, 3174 (2005). [CrossRef] [PubMed]
- D. Kip, M. Wesner, C. Herden, and V. Shandarov, "Interaction of spatial photorefractive solitons in a planar waveguide," Appl. Phys. B 68, 971 (1999). [CrossRef]
- W. Królikowski, B. Luther-Davies, C. Denz, and T. Tschudi, "Annihilation of photorefractive solitons," Opt. Lett. 23, 97 (1998). [CrossRef]
- D. Kip, M. Soljačić, M. Segev, S. M. Sears, and D. N. Christodoulides, "(1+1) Dimensional modulation instability of spatially-incoherent light," J. Opt. Soc. Am. B 19, 502 (2002). [CrossRef]
- E. Smirnov, C. E. Rüter, M. Stepić, V. Shandarov, and D. Kip, "Formation and light guiding properties of dark solitons in one-dimensional waveguide arrays," arXiv:nlin.PS/0607030 (2006).
- M. Stepić, E. Smirnov, C. E. Rüter, V. Shandarov, and D. Kip, "Beam interactions in one-dimensional saturable waveguide arrays," will appear in PRE, arXiv:physics/0607028 (2006).
- M. Matuszewski, C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, A. Mitchell, M. Trippenbach, M. W. Austin, W. Królikowski, and Yu. S. Kivshar, "Crossover from self-defocusing to discrete trapping in nonlinear waveguide arrays," Opt. Express 14, 254 (2006). [CrossRef] [PubMed]
- For r < 1.1 only partial focusing of the narrow input beam is achieved. Nevertheless for such small intensity ratios (i.e. in the Kerr regime) bright solitons may still be obtained using larger soliton (input) widths.

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