## Pulse splitting in the anomalous group-velocity-dispersion regime |

Optics Express, Vol. 19, Issue 10, pp. 9309-9314 (2011)

http://dx.doi.org/10.1364/OE.19.009309

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### Abstract

We investigate experimentally the role that the initial temporal profile of ultrashort laser pulses has on the self-focusing dynamics in the anomalous group-velocity dispersion (GVD) regime. We observe that pulse-splitting occurs for super-Gaussian pulses, but not for Gaussian pulses. The splitting does not occur for either pulse shape when the GVD is near-zero. These observations agree with predictions based on the nonlinear Schrödinger equation, and can be understood intuitively using the method of nonlinear geometrical optics.

© 2011 OSA

*P*of a laser pulse is greater than a certain critical power

*P*. Although much of the initial work dealt with spatial effects, the dynamics of spatio-temporal wave collapse has generated significant interest [1

_{cr}1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. **13**, 479–482 (1964). [CrossRef]

4. A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. **84**, 3582–3585 (2000). [CrossRef] [PubMed]

5. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. **441**, 47–189 (2007). [CrossRef]

6. J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. **17**, 583–585 (1992). [CrossRef] [PubMed]

7. G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of a small normal time dispersion,” Phys. Rev. A **52**, 4218–4228 (1995). [CrossRef] [PubMed]

8. J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. **77**, 3783–3786 (1996). [CrossRef] [PubMed]

13. K. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. **29**, 995–997 (2004). [CrossRef] [PubMed]

14. M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A **57**, 4791–4803 (1998). [CrossRef]

16. J. Liu, R. Li, and Z. Xu, “Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laser pulse in materials with anomalous dispersion,” Phys. Rev. A **74**, 043801 (2006). [CrossRef]

17. G. Fibich, N. Gavish, and X. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D **231**, 55–86 (2007). [CrossRef]

18. N. Gavish, G. Fibich, L. T. Vuong, and A. L. Gaeta, “Predicting the filamentation of high-power beams and pulses without numerical integration: a nonlinear geometrical optics method,” Phys. Rev. A **78**043807 (2008). [CrossRef]

19. T. D. Grow, A. A. Ishaaya, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Collapse dynamics of super-Gaussian beams,” Opt. Express **14**, 5468–5475 (2006). [CrossRef] [PubMed]

18. N. Gavish, G. Fibich, L. T. Vuong, and A. L. Gaeta, “Predicting the filamentation of high-power beams and pulses without numerical integration: a nonlinear geometrical optics method,” Phys. Rev. A **78**043807 (2008). [CrossRef]

*P*. We observe that temporal super-Gaussian input pulses undergo pulse-splitting, whereas Gaussian ones do not. To the best of our knowledge, no previous experimental work has shown that the temporal dynamics depend on whether or not the input pulse is temporally flat-top. We also find that no pulse-splitting occurs at the zero-GVD regime, regardless of the initial pulse shape.

_{cr}*A*(

*η*,

*ξ*,

*τ*) centered at frequency

*ω*, where

*w*

_{0}is the spot size,

*L*= (

_{nl}*n*

_{2}

*n*

_{0}

*ω*|

*A*

_{0}|

^{2}/2

*π*)

^{−1}is the nonlinear length, |

*A*

_{0}| is the magnitude of the input laser field,

*η*=

*x*/

*w*

_{0}and

*ξ*=

*y*/

*w*

_{0}are the normalized coordinates,

*ζ*=

*z*/

*L*is the normalized propagation length,

_{df}*τ*is the temporal pulse width,

_{p}*τ*= [

*t*– (

*z*/

*v*)]/

_{g}*τ*is the normalized retarded time for the pulse traveling at the group velocity

_{p}*v*. Figure 1 shows simulations of the NLSE [Eq. (1)]. All simulations have a Gaussian spatial profile exp(−

_{g}*r*

^{2}) as the input, while the temporal profile is varied. The value of

*L*/

_{df}*L*for all simulations is 0.04. Two cases of the collapse of a Gaussian pulse exp(−

_{ds}*t*

^{2}) are shown in the top part of the figure. Figure 1(a) is 2

*P*and Fig. 1(b) is 3

_{cr}*P*. Overall the temporal dynamics result in 3-D collapse as predicted by previous work. However, for a super-Gaussian temporal profile exp(−

_{cr}*t*

^{4}), the pulse undergoes splitting, as shown in Fig. 1(c) for 2

*P*and Fig. 1(d) for 3

_{cr}*P*. In the spatial domain all input profiles are a Guassian, and therefore they evolve into a peak-type profile.

_{cr}18. N. Gavish, G. Fibich, L. T. Vuong, and A. L. Gaeta, “Predicting the filamentation of high-power beams and pulses without numerical integration: a nonlinear geometrical optics method,” Phys. Rev. A **78**043807 (2008). [CrossRef]

*T*is the temporal coordinate of the ray at the propagation distance

*z*,

*ψ*

_{0}is the input field and

*C*is the

*z*-dependent amplitude. Equations (2) and (3) show that for a high-power temporal profile of the form exp(−

*t*

^{2}

*), the pulse will undergo splitting if*

^{m}*m*> 1, but will focus to a single peak if

*m*= 1 [18

**78**043807 (2008). [CrossRef]

*m*= 2 (temporal super-Gaussian), shows a remarkable agreement between the solutions of the NLSE [Eq. (1)] and of the NGO equations [Eqs. (2) and (3)], see Fig. 2. Note that the peaks of the split pulses are at the same temporal position

*τ*= ±0.2.

*μ*J pulses at 1510 nm or 125-

*μ*J pulses at 1275 nm. These two wavelengths are chosen since they correspond to the anomalous-GVD and the zero-GVD regimes, respectively, for the fused-silica sample. We spatially filter the output and temporally shape the pulse using a standard 4-

*f*pulse shaper [20

20. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B **5**, 1563–1572 (1998). [CrossRef]

*μ*m and 185

*μ*m for 1510 nm and for 1275 nm, respectively. We use a 30-mm fused-silica sample, for which the physical parameters are as follows: at 1510 nm

*β*

_{2}= −279 fs

^{2}/cm and

*n*

_{2}= 2.2×10

^{−16}cm

^{2}/W, and at 1275 nm

*β*

_{2}= 1 fs

^{2}/cm and

*n*

_{2}= 2.5×10

^{−16}cm

^{2}/W [15

15. S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma generation versus chromatic dispersion,” Physica D **220**, 14–30 (2006). [CrossRef]

21. J. K. Ranka, A. L. Gaeta, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Autocorrelation measurement of 6-fs pulses based on the two-photon-induced photocurrent in a GaAsP photodiode,” Opt. Lett. **22**, 1344–1346 (1997). [CrossRef]

*β*

_{2}= 1 fs

^{2}/cm). As expected, we do not observe pulse-splitting at this wavelength for either a Gaussian pulse [Fig. 4(a)] or for a super-Gaussian pulse [Fig. 4(b)] for the range of powers studied, which indicates that the pulse-splitting of super-Gaussian pulses should only occur in the anomalous-GVD regime, as predicted.

17. G. Fibich, N. Gavish, and X. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D **231**, 55–86 (2007). [CrossRef]

19. T. D. Grow, A. A. Ishaaya, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Collapse dynamics of super-Gaussian beams,” Opt. Express **14**, 5468–5475 (2006). [CrossRef] [PubMed]

7. G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of a small normal time dispersion,” Phys. Rev. A **52**, 4218–4228 (1995). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. |

2. | N. A. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovskii, “Multiple fractionation of wave structure in a nonlinear medium,” Pis’ma Zh. Eksp. Teor. Fiz. |

3. | S. N. Vlasov, L. V. Piskunova, and V. I. Talanov, “Three-dimensional wave collapse in the nonlinear Schrödinger equation model,” Zh. Eksp. Teor. Fiz. |

4. | A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. |

5. | A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. |

6. | J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. |

7. | G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of a small normal time dispersion,” Phys. Rev. A |

8. | J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. |

9. | S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. |

10. | A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. |

11. | A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A |

12. | S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. |

13. | K. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. |

14. | M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A |

15. | S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma generation versus chromatic dispersion,” Physica D |

16. | J. Liu, R. Li, and Z. Xu, “Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laser pulse in materials with anomalous dispersion,” Phys. Rev. A |

17. | G. Fibich, N. Gavish, and X. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D |

18. | N. Gavish, G. Fibich, L. T. Vuong, and A. L. Gaeta, “Predicting the filamentation of high-power beams and pulses without numerical integration: a nonlinear geometrical optics method,” Phys. Rev. A |

19. | T. D. Grow, A. A. Ishaaya, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Collapse dynamics of super-Gaussian beams,” Opt. Express |

20. | A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B |

21. | J. K. Ranka, A. L. Gaeta, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Autocorrelation measurement of 6-fs pulses based on the two-photon-induced photocurrent in a GaAsP photodiode,” Opt. Lett. |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.5940) Nonlinear optics : Self-action effects

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 18, 2011

Manuscript Accepted: April 17, 2011

Published: April 26, 2011

**Citation**

Samuel E. Schrauth, Bonggu Shim, Aaron D. Slepkov, Luat T. Vuong, Alexander L. Gaeta, Nir Gavish, and Gadi Fibich, "Pulse splitting in the anomalous group-velocity-dispersion regime," Opt. Express **19**, 9309-9314 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9309

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### References

- R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964). [CrossRef]
- N. A. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yunakovskii, “Multiple fractionation of wave structure in a nonlinear medium,” Pis’ma Zh. Eksp. Teor. Fiz. 44, 13–17 (1986).
- S. N. Vlasov, L. V. Piskunova, and V. I. Talanov, “Three-dimensional wave collapse in the nonlinear Schrödinger equation model,” Zh. Eksp. Teor. Fiz. 95, 1945–1950 (1989).
- A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000). [CrossRef] [PubMed]
- A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007). [CrossRef]
- J. E. Rothenberg, “Pulse splitting during self-focusing in normally dispersive media,” Opt. Lett. 17, 583–585 (1992). [CrossRef] [PubMed]
- G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of a small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995). [CrossRef] [PubMed]
- J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996). [CrossRef] [PubMed]
- S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, “Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media,” Opt. Lett. 23, 379–381 (1998). [CrossRef]
- A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999). [CrossRef]
- A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998). [CrossRef]
- S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001). [CrossRef] [PubMed]
- K. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29, 995–997 (2004). [CrossRef] [PubMed]
- M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998). [CrossRef]
- S. Skupin and L. Bergé, “Self-guiding of femtosecond light pulses in condensed media: Plasma generation versus chromatic dispersion,” Physica D 220, 14–30 (2006). [CrossRef]
- J. Liu, R. Li, and Z. Xu, “Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laser pulse in materials with anomalous dispersion,” Phys. Rev. A 74, 043801 (2006). [CrossRef]
- G. Fibich, N. Gavish, and X. Wang, “Singular ring solutions of critical and supercritical nonlinear Schrödinger equations,” Physica D 231, 55–86 (2007). [CrossRef]
- N. Gavish, G. Fibich, L. T. Vuong, and A. L. Gaeta, “Predicting the filamentation of high-power beams and pulses without numerical integration: a nonlinear geometrical optics method,” Phys. Rev. A 78043807 (2008). [CrossRef]
- T. D. Grow, A. A. Ishaaya, L. T. Vuong, A. L. Gaeta, N. Gavish, and G. Fibich, “Collapse dynamics of super-Gaussian beams,” Opt. Express 14, 5468–5475 (2006). [CrossRef] [PubMed]
- A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1998). [CrossRef]
- J. K. Ranka, A. L. Gaeta, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Autocorrelation measurement of 6-fs pulses based on the two-photon-induced photocurrent in a GaAsP photodiode,” Opt. Lett. 22, 1344–1346 (1997). [CrossRef]

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