## Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media |

JOSA B, Vol. 30, Issue 3, pp. 494-504 (2013)

http://dx.doi.org/10.1364/JOSAB.30.000494

Acrobat PDF (1709 KB)

### Abstract

We interpret the purely spectral forward Maxwell equation with up to third-order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named the nonlinear wave equation in the frequency domain, includes quadratic and cubic nonlinearities, delayed Raman effects, and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due to competition between self-steepening, Raman effects, and self-steepening-like effects from cascading originating in the group-velocity mismatch between the pump and the second harmonic. We analyze the first-order contributions, and show that this balance can be broken to create fast or slow pulses. Through further simulations we demonstrate few-cycle compressed solitons in extremely short crystals, where spectral phenomena, such as blue/red shifting, nonstationary radiation in accordance with the nonlocal phase-matching condition, and dispersive-wave generation are observed and marked, which helps improve the experimental knowledge of cascading nonlinear soliton pulse compression.

© 2013 Optical Society of America

## 1. INTRODUCTION

1. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. **17**, 28–30 (1992). [CrossRef]

2. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B **19**, 2505–2510 (2002). [CrossRef]

4. B. Zhou, A. Chong, F. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. **109**, 043902 (2012). [CrossRef]

6. T. Kartaloğlu, K. Köprülü, O. Aytür, M. Sundheimer, and W. Risk, “Femtosecond optical parametric oscillator based on periodically poled **23**, 61–63 (1998). [CrossRef]

10. C. Phillips, C. Langrock, J. Pelc, M. Fejer, I. Hartl, and M. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express **19**, 18754–18773 (2011). [CrossRef]

## 2. NONLINEAR WAVE EQUATIONS IN FREQUENCY DOMAIN

17. M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. **18**, 494–506(2012). [CrossRef]

18. R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. **20**, 364–381 (1979). [CrossRef]

20. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A **81**, 053841(4) (2010). [CrossRef]

22. M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B **28**, 1231–1237 (2011). [CrossRef]

23. M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E **70**, 036604 (2004). [CrossRef]

24. M. Bache and F. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A **81**, 053815 (2010). [CrossRef]

19. A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. **87**, 203901(4) (2001). [CrossRef]

*et al.*with only quadratic nonlinearities [22

22. M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B **28**, 1231–1237 (2011). [CrossRef]

## 3. NLS-LIKE EQUATION

11. J. Moses and F. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. **97**, 73903 (2006). [CrossRef]

*et al.*[12

12. F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B **21**, 376–383 (2004). [CrossRef]

*et al.*[13

13. C. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to **11**, 2434–2443 (1994). [CrossRef]

*et al.*[14

14. M. Bache, O. Bang, J. Moses, and F. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007). [CrossRef]

15. M. Bache, J. Moses, and F. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B **24**, 2752–2762 (2007). [CrossRef]

*et al.*[27

27. G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A **83**, 043834 (2011). [CrossRef]

16. M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express **16**, 3273–3287 (2008). [CrossRef]

15. M. Bache, J. Moses, and F. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B **24**, 2752–2762 (2007). [CrossRef]

16. M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express **16**, 3273–3287 (2008). [CrossRef]

28. G. Stegeman, D. Hagan, and L. Torner, “**28**, 1691–1740 (1996). [CrossRef]

29. M. Bache, F. Eilenberger, and S. Minardi, “Higher-order Kerr effect and harmonic cascading in gases,” Opt. Lett. **37**, 4612–4614 (2012). [CrossRef]

## 4. CONDITIONS OF CASCADING NONLINEAR SOLITON PULSE COMPRESSION

1. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. **17**, 28–30 (1992). [CrossRef]

30. M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. **27**, 1296–1309 (1991). [CrossRef]

16. M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express **16**, 3273–3287 (2008). [CrossRef]

14. M. Bache, O. Bang, J. Moses, and F. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007). [CrossRef]

15. M. Bache, J. Moses, and F. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B **24**, 2752–2762 (2007). [CrossRef]

4. B. Zhou, A. Chong, F. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. **109**, 043902 (2012). [CrossRef]

## 5. QPM-INDUCED INFLUENCE AND SELF-STEEPENING PERFORMANCES

11. J. Moses and F. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. **97**, 73903 (2006). [CrossRef]

12. F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B **21**, 376–383 (2004). [CrossRef]

11. J. Moses and F. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. **97**, 73903 (2006). [CrossRef]

## 6. SIMULATIONS AND DISCUSSIONS

4. B. Zhou, A. Chong, F. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. **109**, 043902 (2012). [CrossRef]

### A. Type I BBO at 1030 nm: A Clean Few-Cycle Soliton Pulse Compressor

14. M. Bache, O. Bang, J. Moses, and F. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007). [CrossRef]

2. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B **19**, 2505–2510 (2002). [CrossRef]

31. P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter **10**, 673–681 (1998). [CrossRef]

2. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B **19**, 2505–2510 (2002). [CrossRef]

3. J. Moses and F. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. **31**, 1881–1883 (2006). [CrossRef]

**97**, 73903 (2006). [CrossRef]

12. F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B **21**, 376–383 (2004). [CrossRef]

**32**, 2490–2492 (2007). [CrossRef]

**16**, 3273–3287 (2008). [CrossRef]

*a priori*be excluded.

### B. Type 0 MgOLN: Few-Cycle, Fast Pulses Dominated by the Material Raman Effects

**109**, 043902 (2012). [CrossRef]

*et al.*’s experiment [4

**109**, 043902 (2012). [CrossRef]

### C. Type 0 PPMgOLN: Few-Cycle, Zero-Delay Pulse Generation with Enhanced D-Wave

## 7. CONCLUSIONS

## ACKNOWLEDGMENTS

## REFERENCES

1. | R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. |

2. | S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B |

3. | J. Moses and F. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. |

4. | B. Zhou, A. Chong, F. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. |

5. | D. Nikogosian, |

6. | T. Kartaloğlu, K. Köprülü, O. Aytür, M. Sundheimer, and W. Risk, “Femtosecond optical parametric oscillator based on periodically poled |

7. | M. Sundheimer, C. Bosshard, E. Van Stryland, G. Stegeman, and J. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. |

8. | S. Ashihara, T. Shimura, K. Kuroda, N. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. |

9. | M. Arbore, O. Marco, and M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. |

10. | C. Phillips, C. Langrock, J. Pelc, M. Fejer, I. Hartl, and M. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express |

11. | J. Moses and F. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. |

12. | F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B |

13. | C. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to |

14. | M. Bache, O. Bang, J. Moses, and F. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. |

15. | M. Bache, J. Moses, and F. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B |

16. | M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express |

17. | M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. |

18. | R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. |

19. | A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. |

20. | M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A |

21. | R. Boyd, |

22. | M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B |

23. | M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E |

24. | M. Bache and F. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A |

25. | P. Banks, M. Feit, and M. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B |

26. | M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of |

27. | G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A |

28. | G. Stegeman, D. Hagan, and L. Torner, “ |

29. | M. Bache, F. Eilenberger, and S. Minardi, “Higher-order Kerr effect and harmonic cascading in gases,” Opt. Lett. |

30. | M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. |

31. | P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.5650) Nonlinear optics : Raman effect

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 1, 2012

Revised Manuscript: December 6, 2012

Manuscript Accepted: December 22, 2012

Published: February 8, 2013

**Virtual Issues**

February 12, 2013 *Spotlight on Optics*

**Citation**

Hairun Guo, Xianglong Zeng, Binbin Zhou, and Morten Bache, "Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media," J. Opt. Soc. Am. B **30**, 494-504 (2013)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-3-494

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### References

- R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992). [CrossRef]
- S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B 19, 2505–2510 (2002). [CrossRef]
- J. Moses and F. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. 31, 1881–1883 (2006). [CrossRef]
- B. Zhou, A. Chong, F. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012). [CrossRef]
- D. Nikogosian, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).
- T. Kartalo?lu, K. Köprülü, O. Aytür, M. Sundheimer, and W. Risk, “Femtosecond optical parametric oscillator based on periodically poled KTiOPO4,” Opt. Lett. 23, 61–63 (1998). [CrossRef]
- M. Sundheimer, C. Bosshard, E. Van Stryland, G. Stegeman, and J. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18, 1397–1399 (1993). [CrossRef]
- S. Ashihara, T. Shimura, K. Kuroda, N. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett. 84, 1055–1058 (2004). [CrossRef]
- M. Arbore, O. Marco, and M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. 22, 865–867 (1997). [CrossRef]
- C. Phillips, C. Langrock, J. Pelc, M. Fejer, I. Hartl, and M. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express 19, 18754–18773 (2011). [CrossRef]
- J. Moses and F. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 73903 (2006). [CrossRef]
- F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B 21, 376–383 (2004). [CrossRef]
- C. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to ?(2):?(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994). [CrossRef]
- M. Bache, O. Bang, J. Moses, and F. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007). [CrossRef]
- M. Bache, J. Moses, and F. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007). [CrossRef]
- M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008). [CrossRef]
- M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012). [CrossRef]
- R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979). [CrossRef]
- A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901(4) (2001). [CrossRef]
- M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841(4) (2010). [CrossRef]
- R. Boyd, Nonlinear Optics (Academic, 2003).
- M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011). [CrossRef]
- M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004). [CrossRef]
- M. Bache and F. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010). [CrossRef]
- P. Banks, M. Feit, and M. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002). [CrossRef]
- M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of ?-BaB2O4,” arXiv:1209.3158 [physics.optics], (2012).
- G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011). [CrossRef]
- G. Stegeman, D. Hagan, and L. Torner, “?(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]
- M. Bache, F. Eilenberger, and S. Minardi, “Higher-order Kerr effect and harmonic cascading in gases,” Opt. Lett. 37, 4612–4614 (2012). [CrossRef]
- M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991). [CrossRef]
- P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998). [CrossRef]

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