## Quadratic soliton collisions

Optics Express, Vol. 12, Issue 22, pp. 5562-5576 (2004)

http://dx.doi.org/10.1364/OPEX.12.005562

Acrobat PDF (3435 KB)

### Abstract

The details of two soliton collision processes were investigated in detail in a 1 cm long periodically poled KTP crystal for the case when the solitons were excited by inputting only the fundamental beam. The effects on the collision outcomes of the distance of the collision into the sample, collision angle and phase mismatch were measured for different relative phases between the input beams. At small angles (around 0.40) fusion, repulsion and energy transfer processes were observed, while at the collision angles approaching 3.20 the two output soliton beams were essentially unaffected by the interaction. The phase mismatch was varied from 3.5 to -1.5π for the 0.40 collision angle case. The output soliton separation at π input phase difference showed strongly asymmetric behavior with phase mismatch. In general, the measurements indicate a decrease in the interaction strength with increasing phase mismatch. All collision processes were performed in the vicinity of a non-critical phase matching.

© 2004 Optical Society of America

## 1. Introduction

1. G. I. Stegeman and M. Segev “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science **286**, 1518 (1999). [CrossRef] [PubMed]

1. G. I. Stegeman and M. Segev “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science **286**, 1518 (1999). [CrossRef] [PubMed]

2. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. **16**, 15 (1991). [CrossRef] [PubMed]

7. E. A. Ultanir, G. I. Stegeman, C. H. Lange, and F. Lederer, “Interactions of Dissipative Spatial Solitons,” Opt. Lett. **29**, 283 (2004). [CrossRef] [PubMed]

8. W.E. Torruellas, Z. Wang, D.J. Hagan, E.W. VanStryland, G.I. Stegeman, L. Torner, and C.R. Menyuk, “Observation of Two-Dimensional Spatial Solitary Waves in a Quadratic Medium,” Phys. Rev. Lett. **74**, 5036 (1995). [CrossRef] [PubMed]

9. L. Torner, C. R. Menyuk, and G.I. Stegeman, “Excitation of soliton-like waves with cascaded nonlinearities,” Opt. Lett. **19**, 1615 (1994). [CrossRef] [PubMed]

11. R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, Ch. Bosshard, and P. Gunter, “Two-Dimensional Type I Quadratic Spatial Solitons in KNbO_{3} Near Non-Critical Phase-Matching,” Opt. Lett. **27**, 631 (2002). [CrossRef]

14. L. Jankovic, S. Polyakov, G. Stegeman, S. Carrasco, L. Torner, C. Bosshard, and P. Gunter, “Complex soliton-like pattern generation in Potassium Niobate due to noisy, high intensity, input beams,” Opt. Express **11**, 2206 (2003). [CrossRef] [PubMed]

*χ*

^{(2)}soliton interactions see for example ref. 34) have been observed in both 1D (waveguides) and 2D (bulk media) [5

5. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. **27**, 1460 (2002). [CrossRef]

15. B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between Type-II two dimensional quadratic solitons,” Opt. Lett. **23**, 424 (1998). [CrossRef]

18. G. I. Stegeman, L. Jankovic, H. Kim, S. Polyakov, S. Carrasco, L. Torner, Ch. Bosshard, P. Gunter, M. Katz, and D. Eger, “Generation of, and Interactions Between Quadratic Spatial Solitons in Non-Critically-Phase-Matched Crystals,” Journal of Nonlinear Optical Physics & Materials **12**, 447, (2003). [CrossRef]

16. C. Simos, V. Couderc, A. Barthelemy, and A. V. Buryak, “Phase-dependent interactions between three-wave spatial solitons in bulk quadratic media,” J. Opt. Soc. Am. B **20**, 2133 (2003). [CrossRef]

17. L. Jankovic, H. Kim, S. Polyakov, G. I. Stegeman, Ch. Bosshard, and P. Gunter, “Soliton Birth In Quadratic Spatial Soliton Collisions,” Opt. Lett. **28**,1037 (2003). [CrossRef] [PubMed]

## 2. Relevant properties of quadratic spatial solitons and their interactions

19. Y. N. Karamzin and A. P. Sukhorukov“Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency convertors,” JETP Lett.20, 339 (1974);Yu.N. Karamzin and A.P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Zh. Eksp. Teor. Phys.68, 834 (1975) (Sov. Phys.-JETP41, 414 (1976)).

23. A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic Applications,” Phys. Rep. **370**, 63 (2002). [CrossRef]

24. B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of Quadratic Spatial Solitons in Periodically Poled Lithium Niobate,” Opt. Lett. **24**, 1410 (1999). [CrossRef]

25. H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic Spatial Solitons in Periodically Poled KTiOPO4,” Opt. Lett. **28**, 640 (2003). [CrossRef] [PubMed]

11. R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, Ch. Bosshard, and P. Gunter, “Two-Dimensional Type I Quadratic Spatial Solitons in KNbO_{3} Near Non-Critical Phase-Matching,” Opt. Lett. **27**, 631 (2002). [CrossRef]

26. M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, Silvia Carrasco, and L. Torner, “Second Harmonic Generation Tuning Curves In Quasi-Phase-Matched KTP With Narrow, High Intensity Beams,” J. Appl. Phys. **93**, 8852 (2003). [CrossRef]

_{1}) and SH (wavevector k

_{2}) components near NCPM are determined by the input power and the low power wavevector mismatch Δk=2k

_{1}-k

_{2}[27

27. similar to the case discussed inL. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. **121**, 149 (1995). [CrossRef]

_{s}=0 and this is achieved by additional nonlinear phase shifts Δϕ

_{1}and Δϕ

_{2}that are linear in the propagation distance z so that Δk-(2Δϕ

_{1}-Δϕ

_{2})/z=Δk

_{s}=0 [23

23. A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic Applications,” Phys. Rep. **370**, 63 (2002). [CrossRef]

29. G. Assanto and G. I. Stegeman, “The Simple Physics of Quadratic Spatial Solitons,” Opt. Exp. **10**, 388–96 (2002). [CrossRef]

_{s}=Δk only on phase-match.) As a result, the threshold power needed for stationary solitons is minimum at Δk=0 where 2Δϕ

_{1}=Δϕ

_{2}.[27

27. similar to the case discussed inL. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. **121**, 149 (1995). [CrossRef]

23. A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic Applications,” Phys. Rep. **370**, 63 (2002). [CrossRef]

27. similar to the case discussed inL. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. **121**, 149 (1995). [CrossRef]

**121**, 149 (1995). [CrossRef]

9. L. Torner, C. R. Menyuk, and G.I. Stegeman, “Excitation of soliton-like waves with cascaded nonlinearities,” Opt. Lett. **19**, 1615 (1994). [CrossRef] [PubMed]

12. S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, and M. Katz, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E **67**, 046616 (2003). [CrossRef]

_{s}=0. There are two repercussions to FW only excitation for varying Δk. A larger input FW intensity is needed to generate the required SH for soliton formation as Δk decreases, i.e. the soliton generation threshold is higher for FW only excitation. [29

29. G. Assanto and G. I. Stegeman, “The Simple Physics of Quadratic Spatial Solitons,” Opt. Exp. **10**, 388–96 (2002). [CrossRef]

30. R.A. Fuerst, M.T.G. Canva, D. Baboiu, and G.I. Stegeman, “Properties of Type II Quadratic Solitons Excited by Unbalanced Fundamental Waves,” Opt. Lett. **22**, 1748 (1997). [CrossRef]

12. S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, and M. Katz, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E **67**, 046616 (2003). [CrossRef]

_{4}(PPKTP), the crystal used in the present experiments, are given in references [12

12. S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, and M. Katz, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E **67**, 046616 (2003). [CrossRef]

13. S. Polyakov, L. Jankovic, H. Kim, G. Stegeman, S. Carrasco, L. Torner, and M. Katz, “Properties of Quadratic Multi-Soliton Generation Near Phase-Match in Periodically Poled Potassium Titanyl Phosphate,” Opt. Express **11**, 1328 (2003). [CrossRef] [PubMed]

25. H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic Spatial Solitons in Periodically Poled KTiOPO4,” Opt. Lett. **28**, 640 (2003). [CrossRef] [PubMed]

26. M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, Silvia Carrasco, and L. Torner, “Second Harmonic Generation Tuning Curves In Quasi-Phase-Matched KTP With Narrow, High Intensity Beams,” J. Appl. Phys. **93**, 8852 (2003). [CrossRef]

## 3. Experimental conditions

_{eff}=9.5pm/V [31

31. A. Englander, R. Lavi, M. Katz, M. Oron, D. Eger, E. Lebiush, G. Rosenman, and A. Skliar, “Highly efficient doubling of a high-repetition-rate diode-pumped laser with bulk periodically poled KTP,” Opt. Lett. **22**, 1598 (1997). [CrossRef]

11. R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, Ch. Bosshard, and P. Gunter, “Two-Dimensional Type I Quadratic Spatial Solitons in KNbO_{3} Near Non-Critical Phase-Matching,” Opt. Lett. **27**, 631 (2002). [CrossRef]

^{2}(soliton threshold) for the given beam parameters.[25

25. H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic Spatial Solitons in Periodically Poled KTiOPO4,” Opt. Lett. **28**, 640 (2003). [CrossRef] [PubMed]

## 4. Collision processes and soliton formation

^{2}, were kept slightly above the soliton threshold (~3GW/cm

^{2}) [25

**28**, 640 (2003). [CrossRef] [PubMed]

26. M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, Silvia Carrasco, and L. Torner, “Second Harmonic Generation Tuning Curves In Quasi-Phase-Matched KTP With Narrow, High Intensity Beams,” J. Appl. Phys. **93**, 8852 (2003). [CrossRef]

32. D.-M. Baboiu, G.I. Stegeman, and L. Torner, “Collision of solitary waves in quadratic media,” Opt. Lett. **20**, 2282 (1995). [CrossRef] [PubMed]

^{0}relative phase case shows at the output a collapse into a single, high intensity beam, around π two well-separated beams result, and at other phase angles energy has been transferred preferentially to one soliton at the expense of the other with a reversal of the energy flow direction occurring in passing through π (see Ref. [29

29. G. Assanto and G. I. Stegeman, “The Simple Physics of Quadratic Spatial Solitons,” Opt. Exp. **10**, 388–96 (2002). [CrossRef]

^{0}phase difference, a beam leaves the collision with sufficient intensity to eventually evolve into a soliton. However, at π relative phase, the output beam is barely visible for the 4.1mm case, representing strongly diffracted beams with a peak intensity value an order of magnitude lower than for the corresponding 6.6mm output. In fact the input beams interfere with each other soon after entering the crystal for the 4.1mm case, resulting in quasi-linear interference effects. Evolving beams, not yet having formed solitons, are strongly influenced by these interference effects. At intermediate phase differences, the effects of the limited soliton formation are smaller. Intermediate collision distances showed results intermediate between the two cases discussed. Clearly, there is a minimum propagation distance before the beams collide required in order to perform “soliton” collisions. For the current case this distance is around 6mm.

## 5. Collisions at different phase mismatches

6. R. Schiek, Y. Baek, G. Stegeman, and W. Sohler, “Interactions between one dimensional quadratic soliton-like beams,” Opt. Quant. Elect. **30**, 861 (1998). [CrossRef]

**121**, 149 (1995). [CrossRef]

^{0}and repulsion at π for both cases. Note that the solitons are better confined at the higher intensities associated with Δk≠0. However the detailed behavior is different. At a phase mismatch of 3.5π the generated soliton is surrounded by an enhanced radiation pattern (bath) relative to the PM case, indicating that stronger coupling to radiation fields occurs for collisions with Δk≠0. Furthermore, the separation between output solitons at π phase difference is larger on phase mismatch. For phase differences away from 0 or π, the energy exchange between the two colliding solitons is less efficient away from phase match as evidenced by the existence of the second soliton in many cases. This indicates weaker interactions off phase match.

_{PM}). The output pictures show significant differences in the soliton separation with temperature. Clearly the separation is the smallest for the phase matched configuration and ranges from ~23µm at the phase match to ~30µm at 27°C. In fact, both the input intensity and the separation increase together with increasing phase mismatch. At negative phase mismatch there is a high intensity background consisting of vertical fringes. The interference comes from the radiation associated with the solitons’ generation. Because of this background the transverse soliton mobility is increased and influences the final distribution of the solitons. Note that in the 50°C result, the fringe separations are larger and that the solitons appear to be “pulled apart” by the fringes on which they “sit”.

^{0}phase difference, again as expected. The small variations of the nearly flat response around π phase (typically 3–5µm variations) occur quite consistently in the data shown and are not understood at this time. The solitons with a relative phase close to 0

^{0}undergo strong energy transfer along their propagation. If the energy transfer is strong enough the solitons eventually collapse into one soliton and the remaining energy is either captured by the existing soliton or it appears as radiation. If the solitons do not fuse they propagate along approximately the same paths as those for the π phase case. In some cases solitons were observed to perform small spiraling (the 43.6°C case in Fig. 10) indicating non-coplanar interactions. This would be expected to cause only a small deviation.

^{0}relative phase at T=27°C.

## 6. Soliton collisions at “small” and “large” angles

1. G. I. Stegeman and M. Segev “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science **286**, 1518 (1999). [CrossRef] [PubMed]

**28**, 640 (2003). [CrossRef] [PubMed]

^{2}, slightly lower than twice the single soliton threshold at the given phase mismatch. As a result of this high intensity the solitons were generated within a short propagation distance into the crystal. The measurements corresponding to the collision angles 0.2, 0.35 1.1 and 3.2 degrees (the collision points are 5.8, 6, 5 and 4mm respectively) are shown in Fig. 11 for a few selected phase differences. The numbers on the left side indicate the relative phase difference between the initially launched FW beams. The large magnification scans for the 0.20 and 0.350 collision angles show features similar to Fig. 6. The fusion and the inter-soliton energy transfer processes are clearly visible at small angles. The output pattern changes dramatically from small to large collision angles even when the difference in magnification is factored in. As the angle increases to 1.10, the phase dependence decreases significantly. At 0

^{0}and 2π relative phase the two beams tend to attract, and as seen from Fig. 11 they collapse towards each other. The resulting beam is elongated and due to the smaller magnification of the imaging system it is not clear if the beams only attract or if they are already partially fused. At the other phase differences the solitons go through the energy exchange processes but their efficiency is significantly smaller than for the small collision angle case. For example, while the weaker output soliton carries around 25% of the total energy for the 0.350 case at the π/2 relative phase, it contains almost 45% of the total energy for the 1.10 case indicating a weak interaction.

^{0}relative phase and repulsion over a wide range of relative phase around π phase difference. The soliton collisions at large angles 1.10 and 3.20 show very different behavior from the 0.350 case. For the 1.10 case there is still a significant drop in the soliton separation at 0

^{0}relative phase, indicating that the interaction process still influences the output solitons. The soliton separation achieves approximately a constant value (~100µm) over a very large region of the relative phase. The asymmetric shape is believed to be associated with the data processing procedure that is limited by the imaging system magnification and resolution. For the 3.20 configuration the interaction processes have negligible influence on the colliding solitons due to the short interaction distance. The curve is featureless with only small stochastic oscillations around an approximately constant 320µm soliton separation. Finally we note that to first order the plateau separations, ~35µm, 100µm and 310µm reflect the increase in collision angles of 0.350, 1.10 and 3.20.

## 7. Radiative losses on collision

17. L. Jankovic, H. Kim, S. Polyakov, G. I. Stegeman, Ch. Bosshard, and P. Gunter, “Soliton Birth In Quadratic Spatial Soliton Collisions,” Opt. Lett. **28**,1037 (2003). [CrossRef] [PubMed]

## 8. Conclusions

^{0}and a strengthening for π relative phase. In addition, at larger collision angles, the interaction efficiency decreased due to reduced interaction length and finally vanished at around 3

^{0}collision angle, as expected.

## References and links

1. | G. I. Stegeman and M. Segev “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science |

2. | J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. |

3. | J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “Nonlinear Optical Beam Interactions In Waveguide Arrays,” Phys. Rev. Lett. in press |

4. | for exampleM.F. Shih and M. Segev, “Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. |

5. | M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. |

6. | R. Schiek, Y. Baek, G. Stegeman, and W. Sohler, “Interactions between one dimensional quadratic soliton-like beams,” Opt. Quant. Elect. |

7. | E. A. Ultanir, G. I. Stegeman, C. H. Lange, and F. Lederer, “Interactions of Dissipative Spatial Solitons,” Opt. Lett. |

8. | W.E. Torruellas, Z. Wang, D.J. Hagan, E.W. VanStryland, G.I. Stegeman, L. Torner, and C.R. Menyuk, “Observation of Two-Dimensional Spatial Solitary Waves in a Quadratic Medium,” Phys. Rev. Lett. |

9. | L. Torner, C. R. Menyuk, and G.I. Stegeman, “Excitation of soliton-like waves with cascaded nonlinearities,” Opt. Lett. |

10. | W. Torruellas, Y. Kivshar, and G.I. Stegeman, “Quadratic Solitons”, book chapter in “ |

11. | R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, Ch. Bosshard, and P. Gunter, “Two-Dimensional Type I Quadratic Spatial Solitons in KNbO |

12. | S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, and M. Katz, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E |

13. | S. Polyakov, L. Jankovic, H. Kim, G. Stegeman, S. Carrasco, L. Torner, and M. Katz, “Properties of Quadratic Multi-Soliton Generation Near Phase-Match in Periodically Poled Potassium Titanyl Phosphate,” Opt. Express |

14. | L. Jankovic, S. Polyakov, G. Stegeman, S. Carrasco, L. Torner, C. Bosshard, and P. Gunter, “Complex soliton-like pattern generation in Potassium Niobate due to noisy, high intensity, input beams,” Opt. Express |

15. | B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between Type-II two dimensional quadratic solitons,” Opt. Lett. |

16. | C. Simos, V. Couderc, A. Barthelemy, and A. V. Buryak, “Phase-dependent interactions between three-wave spatial solitons in bulk quadratic media,” J. Opt. Soc. Am. B |

17. | L. Jankovic, H. Kim, S. Polyakov, G. I. Stegeman, Ch. Bosshard, and P. Gunter, “Soliton Birth In Quadratic Spatial Soliton Collisions,” Opt. Lett. |

18. | G. I. Stegeman, L. Jankovic, H. Kim, S. Polyakov, S. Carrasco, L. Torner, Ch. Bosshard, P. Gunter, M. Katz, and D. Eger, “Generation of, and Interactions Between Quadratic Spatial Solitons in Non-Critically-Phase-Matched Crystals,” Journal of Nonlinear Optical Physics & Materials |

19. | Y. N. Karamzin and A. P. Sukhorukov“Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency convertors,” JETP Lett.20, 339 (1974);Yu.N. Karamzin and A.P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Zh. Eksp. Teor. Phys.68, 834 (1975) (Sov. Phys.-JETP41, 414 (1976)). |

20. | G.I. Stegeman, D.J. Hagan, and L. Torner, “χ |

21. | G.I. Stegeman, “Experiments on Quadratic Solitons”, book chapter for Proceedings of NATO Advanced Research Workshop on “Soliton Driven Photonics”,A.D. Boardman and A.P. Sukhorukov editors, (Kluwer Academic Publishers, Holland, 2001), pp 21–39. |

22. | W. Torruellas, Y. Kivshar, and G.I. Stegeman, “Quadratic Solitons”, book chapter in “Spatial Solitons”,S. Trillo and W. Torruellas editors (Springer-Verlag, Berlin, 2001) pp 127–168. |

23. | A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic Applications,” Phys. Rep. |

24. | B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of Quadratic Spatial Solitons in Periodically Poled Lithium Niobate,” Opt. Lett. |

25. | H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic Spatial Solitons in Periodically Poled KTiOPO4,” Opt. Lett. |

26. | M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, Silvia Carrasco, and L. Torner, “Second Harmonic Generation Tuning Curves In Quasi-Phase-Matched KTP With Narrow, High Intensity Beams,” J. Appl. Phys. |

27. | similar to the case discussed inL. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. |

28. | L. Torner and G.I. Stegeman, “Soliton Evolution in Quasi-Phase-Matched Second-Harmonic Generation,” JOSA B (special issue) |

29. | G. Assanto and G. I. Stegeman, “The Simple Physics of Quadratic Spatial Solitons,” Opt. Exp. |

30. | R.A. Fuerst, M.T.G. Canva, D. Baboiu, and G.I. Stegeman, “Properties of Type II Quadratic Solitons Excited by Unbalanced Fundamental Waves,” Opt. Lett. |

31. | A. Englander, R. Lavi, M. Katz, M. Oron, D. Eger, E. Lebiush, G. Rosenman, and A. Skliar, “Highly efficient doubling of a high-repetition-rate diode-pumped laser with bulk periodically poled KTP,” Opt. Lett. |

32. | D.-M. Baboiu, G.I. Stegeman, and L. Torner, “Collision of solitary waves in quadratic media,” Opt. Lett. |

33. | C. Etrich, F. Lederer, B.A. Malomed, T. Peschel, and U. Peschel, “Progress in Optics”, E. Wolf editor (Elsevier, New York, 2000) Vol. 41, pp. 483–568. |

34. | C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collisions of solitary waves in media with a second order nonlinearity,” Phys. Rev. B |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 15, 2004

Revised Manuscript: October 26, 2004

Published: November 1, 2004

**Citation**

Ladislav Jankovic, Pierre Aboussouan, Marco Affolter, George Stegeman, and Mordechai Katz, "Quadratic soliton collisions," Opt. Express **12**, 5562-5576 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-22-5562

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### References

- G. I. Stegeman and M. Segev �??Optical Spatial Solitons and Their Interactions: Universality and Diversity,�?? Science 286, 1518 (1999). [CrossRef] [PubMed]
- J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, �??Experimental observation of spatial soliton interactions,�?? Opt. Lett. 16, 15 (1991). [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. in press.
- for example, M.F. Shih and M. Segev, �??Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,�?? Opt. Lett. 21, 1538 (1996). [CrossRef] [PubMed]
- M. Peccianti, K. A. Brzdakiewicz and G. Assanto, �??Nonlocal spatial soliton interactions in nematic liquid crystals,�?? Opt. Lett. 27, 1460 (2002). [CrossRef]
- R. Schiek, Y. Baek, G. Stegeman and W. Sohler, �??Interactions between one dimensional quadratic solitons-like beams,�?? Opt. Quant. Elect. 30, 861 (1998). [CrossRef]
- E. A. Ultanir, G. I. Stegeman, C. H. Lange, and F. Lederer, �??Interactions of Dissipative Spatial Solitons,�?? Opt. Lett. 29, 283 (2004). [CrossRef] [PubMed]
- W.E. Torruellas, Z. Wang, D.J. Hagan, E.W. VanStryland, G.I. Stegeman, L. Torner and C.R. Menyuk, "Observation of Two-Dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036 (1995). [CrossRef] [PubMed]
- L. Torner, C. R. Menyuk and G.I. Stegeman, �??Excitation of soliton-like waves with cascaded nonlinearities,�?? Opt. Lett. 19, 1615 (1994). [CrossRef] [PubMed]
- W. Torruellas, Y. Kivshar and G.I. Stegeman, �??Quadratic Solitons�??, book chapter in Spatial Solitons, S. Trillo and W. Torruellas editors (Springer-Verlag, Berlin, 2001) pp 127-168.
- R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, Ch. Bosshard and P. Gunter, �??Two-Dimensional Type I Quadratic Spatial Solitons in KNbO3 Near Non-Critical Phase-Matching,�?? Opt. Lett. 27, 631 (2002). [CrossRef]
- S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, and M. Katz, �??Observation of multiple soliton generation mediated by amplification of asymmetries,�?? Phys. Rev. E 67, 046616 (2003). [CrossRef]
- S. Polyakov, L. Jankovic, H. Kim, G. Stegeman, S. Carrasco, L. Torner and M. Katz, �??Properties of Quadratic Multi-Soliton Generation Near Phase-Match in Periodically Poled Potassium Titanyl Phosphate,�?? Opt. Express 11, 1328 (2003). [CrossRef] [PubMed]
- L. Jankovic, S. Polyakov, G. Stegeman, S. Carrasco, L. Torner, C. Bosshard, and P. Gunter, �??Complex soliton-like pattern generation in Potassium Niobate due to noisy, high intensity, input beams,�?? Opt. Express 11, 2206 (2003). [CrossRef] [PubMed]
- B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet and V. Kermene, �??Collisions between Type-II two dimensional quadratic solitons,�?? Opt. Lett. 23, 424 (1998). [CrossRef]
- C. Simos, V. Couderc, A. Barthelemy, A. V. Buryak, �??Phase-dependent interactions between three-wave spatial solitons in bulk quadratic media,�?? J. Opt. Soc. Am. B 20, 2133 (2003). [CrossRef]
- L. Jankovic, H. Kim, S. Polyakov, G. I. Stegeman, Ch. Bosshard and P. Gunter, �??Soliton Birth In Quadratic Spatial Soliton Collisions,�?? Opt. Lett. 28, 1037 (2003). [CrossRef] [PubMed]
- G. I. Stegeman, L. Jankovic, H. Kim, S. Polyakov, S. Carrasco, L. Torner, Ch. Bosshard, P. Gunter, M. Katz and D. Eger, �??Generation of, and Interactions Between Quadratic Spatial Solitons in Non-Critically-Phase-Matched Crystals,�?? Journal of Nonlinear Optical Physics & Materials 12, 447 (2003). [CrossRef]
- Y. N. Karamzin and A. P. Sukhorukov, "Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency convertors," JETP Lett. 20, 339 (1974); Yu.N.Karamzin, A.P.Sukhorukov, "Mutual focusing of high-power light beams in media with quadratic nonlinearity," Zh. Eksp. Teor. Phys. 68, 834 (1975) (Sov. Phys.-JETP 41, 414 (1976)).
- G.I. Stegeman, D.J. Hagan and L. Torner, �??x(2) Cascading Phenomena and Their Applications to All-Optical Signal Processing, Mode-Locking, Pulse Compression and Solitons,�?? J. Optical and Quant. Electron. 28, 1691 (1996). [CrossRef]
- G. I. Stegeman, �??Experiments on Quadratic Solitons�??, book chapter for Proceedings of NATO Advanced Research Workshop on Soliton Driven Photonics, A.D. Boardman and A.P. Sukhorukov editors, (Kluwer Academic Publishers, Holland, 2001), pp 21-39.
- W. Torruellas, Y. Kivshar and G.I. Stegeman, �??Quadratic Solitons�??, book chapter in Spatial Solitons, S. Trillo and W. Torruellas editors (Springer-Verlag, Berlin, 2001) pp 127-168.
- A. V. Buryak, P. Di Trapani, D. Skryabin and S. Trillo, �??Optical solitons due to quadratic nonlinearities: from basic physics to futuristic Applications,�?? Phys. Rep. 370, 63 (2002). [CrossRef]
- B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna and C. De Angelis, �??Observation of Quadratic Spatial Solitons in Periodically Poled Lithium Niobate,�?? Opt. Lett. 24, 1410 (1999). [CrossRef]
- H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger and M. Katz, �??Quadratic Spatial Solitons in Periodically Poled KTiOPO4,�?? Opt. Lett. 28, 640 (2003). [CrossRef] [PubMed]
- M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, Silvia Carrasco and L. Torner, �??Second Harmonic Generation Tuning Curves In Quasi-Phase-Matched KTP With Narrow, High Intensity Beams,�?? J. Appl. Phys. 93, 8852 (2003). [CrossRef]
- similar to the case discussed in L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas and G. I. Stegeman, "Stationary trapping of light beams in bulk second-order nonlinear media," Opt. Commun. 121, 149 (1995). [CrossRef]
- L. Torner and G.I. Stegeman, �??Soliton Evolution in Quasi-Phase-Matched Second-Harmonic Generation,�?? JOSA B (special issue) 14, 3127 (1997).
- G. Assanto and G. I. Stegeman, �??The Simple Physics of Quadratic Spatial Solitons,�?? Opt. Exp. 10, 388-96 (2002). [CrossRef]
- R.A. Fuerst, M.T.G. Canva, D. Baboiu and G.I. Stegeman, �??Properties of Type II Quadratic Solitons Excited by Unbalanced Fundamental Waves,�?? Opt. Lett. 22, 1748 (1997). [CrossRef]
- A. Englander, R. Lavi, M. Katz, M. Oron, D. Eger, E. Lebiush, G. Rosenman and A. Skliar, �??Highly efficient doubling of a high-repetition-rate diode-pumped laser with bulk periodically poled KTP,�?? Opt. Lett. 22, 1598 (1997). [CrossRef]
- D.-M. Baboiu, G.I. Stegeman and L. Torner, "Collision of solitary waves in quadratic media," Opt. Lett. 20, 2282 (1995). [CrossRef] [PubMed]
- C. Etrich, F. Lederer, B.A. Malomed, T. Peschel and U. Peschel, Progress in Optics, E. Wolf editor (Elsevier, New York, 2000) Vol. 41, pp. 483-568.
- C. Etrich, U. Peschel , F. Lederer and B. Malomed, �??Collisions of solitary waves in media with a second order nonlinearity,�?? Phys. Rev. B 52, R3444 (1995).

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