## Dynamics of self-focused femtosecond laser pulses in the near and far fields

Optics Express, Vol. 4, Issue 9, pp. 336-343 (1999)

http://dx.doi.org/10.1364/OE.4.000336

Acrobat PDF (904 KB)

### Abstract

We investigate the propagation of femtosecond pulses in a nonlinear, dispersive medium at powers several times greater than the critical power for self focusing. The combined effects of diffraction, normal dispersion and cubic nonlinearity lead to pulse splitting. We show that detailed theoretical description of the linear propagation of the pulse from the exit face of the nonlinear medium (near field) to the measuring device (far field) is crucial for quantitative interpretation of experimental data.

© Optical Society of America

1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. **20**, 73–75 (1995). [CrossRef] [PubMed]

2. E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt. Lett. **21**, 62–64 (1996). [CrossRef] [PubMed]

3. D. Strickland and P. B. Corkum, “Resistance of short pulses to self-focusing,” J. Opt. Soc. Am. B **11**, 492–497 (1994). [CrossRef]

4. 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]

5. 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]

6. R. L. Fork, C. V. Shank, C. Hirlimann, and R. Yen, “Femtosecond white-light continuum pulses,” Opt. Lett. **8**, 1–3 (1983). [CrossRef] [PubMed]

7. P. B. Corkum, C. Rolland, and T. Srinivasan-Rao, “Supercontinuum generation in gases,” Phys. Rev. Lett. **57**, 2268–2271 (1986). [CrossRef] [PubMed]

3. D. Strickland and P. B. Corkum, “Resistance of short pulses to self-focusing,” J. Opt. Soc. Am. B **11**, 492–497 (1994). [CrossRef]

9. P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. **17**, 172–174 (1992). [CrossRef] [PubMed]

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

11. G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. **19**, 862–864 (1994). [CrossRef] [PubMed]

4. 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]

5. 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]

12. 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]

13. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Inst. **68**, 3277–3295 (1997). [CrossRef]

## 1. Model

*E*(

*r*,

*z*,

*t*) of the radially-symmetric field

12. 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]

14. 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). This work and Ref. [12] contain extensive references to related theoretical work with modified nonlinear SchrÖdinger equations. [CrossRef]

*t*is measured in the frame moving at the group velocity of the pulse. The transverse Laplacian in cylindrical coordinates ∇

^{2}=

*∂*

^{2}/

*∂r*

^{2}+ (1/

*r*)

*∂*/

*∂r*accounts for diffraction, while the second and third time derivatives describe group velocity dispersion (GVD) and third order dispersion (TOD). The temporal, longitudinal, and transverse coordinates are normalized to the characteristic pulse duration

*τ*, the dispersion length

*l*= 2

_{D}*τ*

^{2}/|

*k′′*|, and the characteristic transverse length

*∊*

_{3}=

*k′′′*/(3

*k′′τ*), and

*k*= 2

*πn*/λ, with

*n*being the linear index of refraction at the central wavelength λ. The dispersion coefficients

*k′′*and

*k′′′*are the second and third derivatives of

*k*with respect to frequency, evaluated at the central frequency

*ω*

_{0}. Space-time focusing [15

15. J. Rothenberg,“Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Opt. Lett. **17**, 1340–1342 (1992). [CrossRef] [PubMed]

17. F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, “Self-steepening of light pulses,” Phys. Rev. **164**, 312–323 (1967). [CrossRef]

*g*(|

*E*|

^{2})]. As shown in Eq. (2), both of these terms are proportional to

*∊*= 1/

_{ω}*ω*

_{0}

*τ*. These two terms act together to shift energy towards the trailing edge of the pulse [14

14. 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). This work and Ref. [12] contain extensive references to related theoretical work with modified nonlinear SchrÖdinger equations. [CrossRef]

15. J. Rothenberg,“Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Opt. Lett. **17**, 1340–1342 (1992). [CrossRef] [PubMed]

16. J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. **23**, 534–536 (1998). [CrossRef]

18. R. H. Stolen and W. J. Tomlinson, “Effect of the Raman part of the nonlinear refractive index on propagation of ultrashort optical pulses in fibers,” J. Opt. Soc. Am. B **9**, 565–573 (1992) [CrossRef]

*n*

_{2}is the nonlinear index of refraction, and

*α*denotes the fractional amount of the nonlinearity due to the Raman effect.

12. 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]

*μ*m, respectively. Furthermore, the beam waist is located at the entrance face of the sample. The linear index of refraction is

*n*= 1.45 at the center wavelength of λ = 0.8

*μ*m. The nonlinear index of refraction is

*n*

_{2}= 2.5 × 10

^{-16}GW/cm

^{2}, resulting in a self-focusing critical power of

*P*

_{crit}= (0.61λ)

^{2}

*π*/(8

*nn*

_{2}) = 2.6 MW. The GVD and TOD coefficients are

*k′′*= 360 fs

^{2}/cm, and

*k′′′*= 275 fs

^{3}/cm. For the Raman response of fused silica we use

*α*= 0.15,

*τ*= 50 fs, and

_{r}*ω*= 4.2[18

_{r}τ_{r}18. R. H. Stolen and W. J. Tomlinson, “Effect of the Raman part of the nonlinear refractive index on propagation of ultrashort optical pulses in fibers,” J. Opt. Soc. Am. B **9**, 565–573 (1992) [CrossRef]

5. 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]

**82**, 1430–1433 (1999). [CrossRef]

*n*in the first term of Eq. (5) is due to the fact that the wave vector of the field in air (assumed to be vacuum) is smaller than in the medium by a factor of

*n*. The amplitude of the radially-symmetric field at distance

*L*from the nonlinear medium follows from the solution of Eq. (5) and is given by the equation

*J*

_{0}is the zeroth-order Bessel function. Equation (6) is evaluated numerically using the frequency domain representation

*E*(

*r*′,0,

*ω*) of the field at the output of the nonlinear medium. Finally we note that the on-axis in the far field (

*r*= 0,

*L*→ ∞), the amplitude of the pulse simplifies to the expression

*t*can be approximated by the integral over the cross-section of the near field at the same local time. This result will be used below when discussing the physics of pulse splitting.

## 2. Results

*I*(

*r*,

*t*) is shown as a function of the propagation distance

*z*(

*z*runs from 0 to 30 mm). The spatial dimension is radially-symmetric, with position

*r*= 0 at the center of the axis. The orientation of the figure is such that early times are at the back of the figure. The peak intensity at

*z*= 0 cm is 72 GW/cm

^{2}, corresponding to a peak power of 4 MW.

*t*= 0; however, it is not sufficient to cause pulse splitting inside the nonlinear medium. Nevertheless, splitting occurs as the pulse propagates in vacuum to the far field [Figure 1(b)]. As discussed above, the free propagation of the field from the output of the fused silica to the measuring apparatus involves the evaluation of Eq. (6). Figure 1(b) shows the result of this calculation of the intensity distribution

*I*(

*r*,

*t*) at

*z*= 1500 mm beyond the output of the 30 mm fused silica media. This far-field spatio-temporal shape of the pulse clearly demonstrates splitting. Numerical simulations show that for the above input intensity the pulse splitting becomes noticeable several centimeters after the exit face of the nonlinear medium.

*t*= 0), but is more constant in the wings of the field. As a result, the differently-phased portions of the field about

*t*= 0 in the near field may destructively interfere in the far field leading to a local minimum and the observed temporal pulse splitting of Figure 1(b).

*z*= 0 cm is 85 GW/cm

^{2}, corresponding to a peak power of 4.7 MW. In Figure 2(a), we observe that an initially uniform Gaussian input self-focuses in both space and time before splitting into two separate pulses. The pulse splitting results when self-focusing moves off-axis energy towards the peak of the pulse, while positive dispersion acts to pull this energy away from

*t*= 0. As this process continues, the peak intensity drops, stopping the collapse at

*t*= 0. However, off-axis energy continues to focus at

*t*≠ 0 such that two pulses are resolved.

16. J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. **23**, 534–536 (1998). [CrossRef]

14. 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). This work and Ref. [12] contain extensive references to related theoretical work with modified nonlinear SchrÖdinger equations. [CrossRef]

*z*= 19 mm in Figure 2(a). The resulting shock front causes extensive broadening on the blue side of the spectrum via self phase modulation. Although this shock formation initially results in a trailing pulse with higher intensity and shorter duration, this pulse spreads much faster than the longer and lower intensity leading pulse as seen for

*z*> 20 mm in Figure 2(a). We also note that the Raman nonlinearity, which is included Eq. (2), counteracts the nonlinear shock and space-time focusing terms by transferring energy to the leading (red-shifted) pulse. However, this is a smaller effect.

*z*= 30 mm frame of Figure 2(a)], there are now three pulses near

*r*= 0 in the far field. This is shown in more detail by the data of Figure 3. In this figure, we present a comparison of the measured and calculated axial field for the same conditions shown in Figure 2. Figure 3(a) is the measured axial (

*r*= 0) intensity and phase of the complex envelope

*E*(

*r*= 0,

*z*= 1500 mm,

*t*). This data was acquired using the SHG-FROG technique under experimental conditions corresponding to those of the calculation of Figure 2. For comparison, Figure 3(b) is the calculated axial intensity and phase, demonstrating good agreement with the measurement. The measured and calculated frequency domain representation of the field is shown in Figure 3(c) and (d), respectively. In Figure 3(c) the blue line is the square modulus of the Fourier transform of the measured field of Figure 3(a), while the red points are an independently measured spectrum (using a 0.27 m spectrometer). The discrepancy seen on the short wavelength side of the spectrum is due to bandwidth limitations in the SHG-FROG measurement. Nonetheless, this data verifies the fidelity of the FROG measurement over several orders of magnitude.

^{-3}level. As means of further illustrating the self-phase modulation that results in the spectral broadening, we present animated video with sound that is linked to Figure 3(a). In this multimedia clip the intensity (blue dots) and phase (black line) of Figure 3(a) are again shown; however, the pulse phase is additionally presented as a “sonogram.” The pitch of the sound one hears when playing the movie is proportional to the local frequency (indicated by the moving red point). This local, or instantaneous, frequency is defined as

*ω*=

_{inst}*ω*

_{0}-

*∂ϕ*/

*∂t*with

*ϕ*(

*t*) being the phase of the complex envelope. One will note that sharp negative slopes of the phase correspond to higher pitch and therefore correspond to blue shifts. On the other hand, positive slopes in the pulse phase correspond to red shifts as indicated by the lower pitch.

*I*(

*r*,

*t*) at the output of 30 mm of fused silica for an input power if 5.5 MW. The axial intensity in this case is shown in Figure 4(b). Of significance here is the absence of repeated multiple splittings of the form seen in Figure 2. This implies that the original two pulses of Figure 2(a) do not simply create two additional daughter pulses of their own. In the far field, shown in Figure 4(c), we see the field beginning to coalesce towards a single dominant peak. Our measurements at this and higher powers show good agreement with these calculations[12

**82**, 1430–1433 (1999). [CrossRef]

## 3. Acknowledgement

## References

1. | A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. |

2. | E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysy-rowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt. Lett. |

3. | D. Strickland and P. B. Corkum, “Resistance of short pulses to self-focusing,” J. Opt. Soc. Am. B |

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

5. | 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. |

6. | R. L. Fork, C. V. Shank, C. Hirlimann, and R. Yen, “Femtosecond white-light continuum pulses,” Opt. Lett. |

7. | P. B. Corkum, C. Rolland, and T. Srinivasan-Rao, “Supercontinuum generation in gases,” Phys. Rev. Lett. |

8. | N. A. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev, and A. D. Yanukoviskii, “Multiple fractionation of wave structures in a nonlinear medium,” JETP Lett. |

9. | P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. |

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

11. | G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. |

12. | 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. |

13. | R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Inst. |

14. | 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 |

15. | J. Rothenberg,“Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Opt. Lett. |

16. | J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. |

17. | F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, “Self-steepening of light pulses,” Phys. Rev. |

18. | R. H. Stolen and W. J. Tomlinson, “Effect of the Raman part of the nonlinear refractive index on propagation of ultrashort optical pulses in fibers,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 23, 1999

Published: April 26, 1999

**Citation**

Alex Zozulya and Scott Diddams, "Dynamics of self-focused femtosecond laser pulses
in the near and far fields," Opt. Express **4**, 336-343 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-9-336

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### References

- A. Braun, G. Korn, X. Liu, D. Du, J. Squier and G. Mourou, "Self-channeling of high-peak-power femtosecond laser pulses in air," Opt. Lett. 20, 73-75 (1995). [CrossRef] [PubMed]
- E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin and A. Mysy- rowicz, "Conical emission from self-guided femtosecond pulses in air," Opt. Lett. 21, 62-64 (1996). [CrossRef] [PubMed]
- D. Strickland and P. B. Corkum, "Resistance of short pulses to self-focusing," J. Opt. Soc. Am. B 11, 492-497 (1994). [CrossRef]
- 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]
- R. L. Fork and C. V. Shank and C. Hirlimann and R. Yen, "Femtosecond white-light continuum pulses," Opt. Lett. 8, 1-3 (1983). [CrossRef] [PubMed]
- P. B. Corkum, C. Rolland and T. Srinivasan-Rao, "Supercontinuum generation in gases," Phys. Rev. Lett. 57, 2268-2271 (1986). [CrossRef] [PubMed]
- N. A. Zharova, A. G. Litvak, T. A. Petrova, A. M. Sergeev and A. D. Yanukoviskii, "Multiple fractionation of wave structures in a nonlinear medium," JETP Lett. 44, 13-17 (1986).
- P. Chernev and V. Petrov, "Self-focusing of light pulses in the presence of normal group-velocity dispersion," Opt. Lett. 17, 172-174 (1992). [CrossRef] [PubMed]
- J. Rothenberg,"Pulse splitting during self-focusing in normally dispersive media," Opt. Lett. 17, 583-585 (1992). [CrossRef] [PubMed]
- G. G. Luther, J. V. Moloney, A. C. Newell and E. M. Wright, "Self-focusing threshold in normally dispersive media" Opt. Lett. 19, 862-864 (1994). [CrossRef] [PubMed]
- 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]
- R. Trebino, K. W. DeLong, D. N. Fittingho_, J. N. Sweetser, M. A. Krumb" ugel and B. A. Richman , "Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating," Rev. Sci. Inst. 68, 3277-3295 (1997). [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). This work and Ref. [12] contain extensive references to related theoretical work with modified nonlinear Schrodinger equations. [CrossRef]
- J. Rothenberg,"Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses," Opt. Lett. 17, 1340-1342 (1992). [CrossRef] [PubMed]
- J. K. Ranka and A. L. Gaeta, "Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses," Opt. Lett. 23, 534-536 (1998). [CrossRef]
- F. DeMartini, C. H. Townes, T. K. Gustafson and P. L. Kelley, "Self-steepening of light pulses," Phys. Rev. 164, 312-323 (1967). [CrossRef]
- R. H. Stolen and W. J. Tomlinson, "Effect of the Raman part of the nonlinear refractive index on propagation of ultrashort optical pulses in fibers, " J. Opt. Soc. Am. B 9, 565-573 (1992). [CrossRef]

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