## Properties of quadratic multi-soliton generation near phase-match in periodically poled potassium titanyl phosphate

Optics Express, Vol. 11, Issue 11, pp. 1328-1337 (2003)

http://dx.doi.org/10.1364/OE.11.001328

Acrobat PDF (603 KB)

### Abstract

The properties of the multi-quadratic-soliton generation process have been investigated both theoretically and experimentally near and on phase-match in non-critically-phase-matched, periodically poled, potassium titanyl phosphate (PPKTP). It was found that multi-soliton generation occurs primarily due to asymmetry in the input beam and at phase-matching. The number of solitons generated depended on the input intensity in a non-trivial way.

© 2003 Optical Society of America

## 1. Introduction

^{(2)}[1]. They have been studied extensively by inputting a fundamental field in second harmonic generation geometries for which the stationary quadratic soliton consists of in-phase fundamental and harmonic fields with a specific amplitude ratio [1–10

10. For a comprehensive review, see 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]

11. S. Polyakov, R. Malendevich, L. Jankovic, G. Stegeman, Ch. Bosshard, and P. Gunter, “Effects of Anisotropic Diffraction on Quadratic Multi Soliton Excitation in Non-critically Phase-matched Crystals,” Opt. Lett. **27**, 1049 (2002). [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]

13. S. Carrasco, L. Torner, J. P. Torres, D. Artigas, E. López-Lago, V. Couderc, and A. Barthélémy, “Quadratic Solitons: Existence versus Excitation,” IEEE J. Sel. Top. Quantum Elect. **8**, 497 (2002). [CrossRef]

7. R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. I. 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]

8. H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic Spatial Solitons in Periodically Poled KTiOPO_{4},” Opt. Lett. **28**, 640 (2003). [CrossRef] [PubMed]

11. S. Polyakov, R. Malendevich, L. Jankovic, G. Stegeman, Ch. Bosshard, and P. Gunter, “Effects of Anisotropic Diffraction on Quadratic Multi Soliton Excitation in Non-critically Phase-matched Crystals,” Opt. Lett. **27**, 1049 (2002). [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]

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]

11. S. Polyakov, R. Malendevich, L. Jankovic, G. Stegeman, Ch. Bosshard, and P. Gunter, “Effects of Anisotropic Diffraction on Quadratic Multi Soliton Excitation in Non-critically Phase-matched Crystals,” Opt. Lett. **27**, 1049 (2002). [CrossRef]

**67**, 046616 (2003). [CrossRef]

## 2. Experimental aspects

16. G. Rosenman, Kh. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO4 and isomorphic crystals,” Appl. Phys. Lett. **73**, 865 (1998). [CrossRef]

16. G. Rosenman, Kh. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO4 and isomorphic crystals,” Appl. Phys. Lett. **73**, 865 (1998). [CrossRef]

**67**, 046616 (2003). [CrossRef]

^{2}≈1.0. The input beam waist (1/e

^{2}of intensity) at focus was about 17 µm which corresponds to 6.4 diffraction length in the PPKTP sample. For both input and output pulses, their energies were monitored on a shot-to-shot basis by energy detectors, as well as their spatial distribution by cameras. The typical shot-to-shot variation in the pulse energies was ±2%, rms.

## 3. Numerical analyses

17. S. V. Polyakov and G. I. Stegeman, “Quadratic Solitons in Anisotropic Media: Variational Approach,” Phys. Rev. E **66**, 046622-1 (2002). [CrossRef]

*A*

_{1}and

*A*

_{2}are the fundamental (ω) and harmonic (2ω) amplitudes and the α

_{2}are the two photon absorption (TPA) coefficients (which are known to be large primarily at the harmonic wavelength). In these equations

*D*

_{11},

*D*

_{12},

*D*

_{21}and

*D*

_{22}stand for the diffraction of a fundamental wave (FW, first index is 1) and second harmonic (SH, first index is 2) along the z (second index is 1) and y (second index is 2) axes respectively, Γ is proportional to the coefficient of quadratic nonlinearity and Δ

*k*is the wavevector mismatch between the fundamental (

*k*

_{i}, i=1) and second harmonic (

*k*

_{i}, i=2) wavevectors, i.e. Δ

*k*=2

*k*

_{1}-

*k*

_{2}. Note that since these equations have different diffraction coefficients along the beam transverse dimensions z and y they do not preserve circular symmetry, as opposed to an isotropic medium in which

*D*

_{11}=

*D*

_{12}and

*D*

_{21}=

*D*

_{22}.

^{th}order Runge-Kutta scheme for solving of the nonlinear part was used to numerically calculate the evolution of a CW and a fully 3D spatio-temporal input beam under the influence of the equations above [11

**27**, 1049 (2002). [CrossRef]

**67**, 046616 (2003). [CrossRef]

*A*

_{1}(

*z, y*)=

*A*

_{1}exp[-

*z*

^{2}/

*y*

^{2}/

*w*

_{1}≠

*w*

_{2}for an elliptical input beam and

*w*

_{1}=

*w*

_{2}for a circular symmetric beam.

## 4. Multi-soliton generation: On phase-match

### 4.1 Simulations

^{(2)}parametric processes to generate the appropriate harmonic component. This is quite different from inputting the stationary solutions which contain in-phase fundamental and harmonic components of a specific amplitude ratio. Simulations have shown that both the fundamental and harmonic oscillate with distance with the peak oscillations dying out with distance from the input facet [9

9. For an overview, see L. Torner and G.I. Stegeman, “Multicolor Solitons,” Opt. Photon. News **12**, (2), 36 (2001). [CrossRef]

10. For a comprehensive review, see 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**, 1049 (2002). [CrossRef]

**67**, 046616 (2003). [CrossRef]

^{2}for a beam waist of about 17 µm) [8

8. H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic Spatial Solitons in Periodically Poled KTiOPO_{4},” Opt. Lett. **28**, 640 (2003). [CrossRef] [PubMed]

*D*

_{11}/

*D*

_{12}) and 13% (

*D*

_{21}/

*D*

_{22}) for the SH. Therefore, to a good approximation

*D*

_{11}

*D*

_{22}≈

*D*

_{12}

*D*

_{21}and hence a rescaling of one or both of the transverse co-ordinates can return the equations to circular symmetry. This rescaling also affects a perfectly circular incident beam transforming it into an elliptical beam in the rescaled co-ordinates. Naturally, the reverse operation, i.e. rescaling of the coordinates to bring an elliptical input beam to a circular shape would introduce an effective diffraction anisotropy. Therefore within this approximation, the evolution of the multiple solitons patterns for an elliptical incident beam with its axes being different by 5% in an optically isotropic medium will be same as for anisotropic diffraction with a circular symmetric incident beam with a difference in the ratio of major to minor diffraction coefficients of about 11%. Such a beam asymmetry is very small and difficult to achieve and control experimentally. Hence with typical input beams one expects the beam asymmetry to dominate the multi-soliton generation process. In the absence of anisotropic diffraction, the solitons are aligned along the major axis of the input beam. When both mechanisms are present, the anisotropic diffraction of PPKTP and beam shape contributions can either interfere constructively or destructively, depending on the detailed circumstances [12

**67**, 046616 (2003). [CrossRef]

**67**, 046616 (2003). [CrossRef]

**27**, 1049 (2002). [CrossRef]

**67**, 046616 (2003). [CrossRef]

21. A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A **52**, 4099 (1995). [CrossRef] [PubMed]

### 4.2 Experiments

^{2}. The input beam ellipticity for this case was about 5±2%. Furthermore, the orientation of this ellipticity was appropriate for reducing the effects of the crystal’ s diffraction anisotropy. Based on the simulations, the net asymmetry is too small to produce multisoliton generation for the crystal length employed with the incident peak intensities up to 20 GW/cm

^{2}. Shown for comparison in Fig. 5(b) is a demonstration of multi-soliton generation for an incident beam with a larger ellipticity. Clearly multi-soliton generation occurs.

**67**, 046616 (2003). [CrossRef]

^{2}three solitons first appear in Fig. 6. Note that the three soliton formation disappears at high intensities, although there is still some elongation of the single remaining soliton along the direction in which the multi-solitons occurred. This is in excellent agreement with the theoretical predictions in view of the fact that pulsed lasers were used so that each experiment involves a distribution of intensities in time and no sharp thresholds are expected.

## 5. Multi-soliton generation: Off phase-match

20. G. Assanto and G. Stegeman, “The Simple Physics of Quadratic Spatial Solitons,” Opt. Express **10**, 388 (2002). [CrossRef] [PubMed]

## 6. Summary

## References and links

1. | Yu.N. Karamzin and A.P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Zh. Eksp. Teor. Phys |

2. | 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. |

3. | R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in plannar waveguides,” Phys. Rev. E |

4. | P. Di Trapani, G. Valiulis, W. Chianglia, and A. Adreoni, “Two-dimensional spatial solitary waves from traveling-wave parametric amplification of the quantum noise,” Phys. Rev. Lett. |

5. | X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. |

6. | 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. |

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

8. | H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, “Quadratic Spatial Solitons in Periodically Poled KTiOPO |

9. | For an overview, see L. Torner and G.I. Stegeman, “Multicolor Solitons,” Opt. Photon. News |

10. | For a comprehensive review, see 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. |

11. | S. Polyakov, R. Malendevich, L. Jankovic, G. Stegeman, Ch. Bosshard, and P. Gunter, “Effects of Anisotropic Diffraction on Quadratic Multi Soliton Excitation in Non-critically Phase-matched Crystals,” Opt. Lett. |

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. Carrasco, L. Torner, J. P. Torres, D. Artigas, E. López-Lago, V. Couderc, and A. Barthélémy, “Quadratic Solitons: Existence versus Excitation,” IEEE J. Sel. Top. Quantum Elect. |

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

15. | 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. |

16. | G. Rosenman, Kh. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Low temperature periodic electrical poling of flux-grown KTiOPO4 and isomorphic crystals,” Appl. Phys. Lett. |

17. | S. V. Polyakov and G. I. Stegeman, “Quadratic Solitons in Anisotropic Media: Variational Approach,” Phys. Rev. E |

18. | A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A |

19. | N. N. Akhmediev and A. Ankiewicz, |

20. | G. Assanto and G. Stegeman, “The Simple Physics of Quadratic Spatial Solitons,” Opt. Express |

21. | A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A |

**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: April 18, 2003

Revised Manuscript: May 21, 2003

Published: June 2, 2003

**Citation**

Sergey Polyakov, Ladislav Jankovic, Hongki Kim, George Stegeman, Silvia Carrasco, Lluis Torner, and Mordechai Katz, "Properties of quadratic multi-soliton generation near phase-match in periodically poled potassium titanyl phosphate," Opt. Express **11**, 1328-1337 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-11-1328

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### References

- 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.-JETP 41, 414 (1976)).
- 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]
- R. Schiek, Y. Baek and G. I. Stegeman, �??One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in plannar waveguides,�?? Phys. Rev. E 53, 1138 (1996). [CrossRef]
- P. Di Trapani, G. Valiulis, W. Chianglia and A. Adreoni, �??Two-dimensional spatial solitary waves from traveling-wave parametric amplification of the quantum noise,�?? Phys. Rev. Lett. 80, 265 (1998). [CrossRef]
- X. Liu, L. J. Qian and F. W. Wise, �??Generation of optical spatiotemporal solitons,�?? Phys. Rev. Lett. 82, 4631 (1999). [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]
- R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. I. 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]
- 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]
- For an overview, see L.Torner and G.I. Stegeman, �??Multicolor Solitons,�?? Opt. Photon. News 12, (2), 36 (2001). [CrossRef]
- For a comprehensive review , see 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]
- S. Polyakov, R. Malendevich, L. Jankovic, G. Stegeman, Ch. Bosshard and P. Gunter, �??Effects of Anisotropic Diffraction on Quadratic Multi Soliton Excitation in Non-critically Phase-matched Crystals,�?? Opt. Lett. 27, 1049 (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. Carrasco, L. Torner, J. P. Torres, D. Artigas, E. López-Lago, V. Couderc, and A. Barthélémy, �??Quadratic Solitons: Existence versus Excitation,�?? IEEE J. Sel. Top. Quantum Elect. 8, 497 (2002). [CrossRef]
- M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco and L. Torner, �??Second Harmonic Generation Tuning Curves In Quasi-Phase-Matched KTP With Narrow, High Intensity Beams,�?? J. Appl. Phys., in press
- 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]
- G. Rosenman, Kh. Garb, A. Skliar, M. Oron, D. Eger and M. Katz, �??Low temperature periodic electrical poling of flux-grown KTiOPO4 and isomorphic crystals,�?? Appl. Phys. Lett. 73, 865 (1998). [CrossRef]
- S. V. Polyakov and G. I. Stegeman, �??Quadratic Solitons in Anisotropic Media: Variational Approach,�?? Phys. Rev. E 66, 046622-1 (2002). [CrossRef]
- A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, �??Self-trapping of light beams and parametric solitons in diffractive quadratic media,�?? Phys. Rev. A 52, 1670 (1995); L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruelas, and G. I. Stegeman, �??Stationary trapping of light beams in bulk second-order nonlinear media,�?? Opt. Commun. 121, 149 (1995). [CrossRef] [PubMed]
- N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, (Chapman and Hall, London, 1997).
- G. Assanto and G. Stegeman, �??The Simple Physics of Quadratic Spatial Solitons,�?? Opt. Express, 10, 388 (2002). [CrossRef] [PubMed]
- A. D. Boardman, K. Xie and A. Sangarpaul, �??Stability of scalar spatial solitons in cascadable nonlinear media,�?? Phys. Rev. A 52, 4099 (1995). [CrossRef] [PubMed]

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