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.

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References

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  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 and C. V. Shank and 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]
  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. 44, 13-17 (1986).
  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]
  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. 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]
  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 Schrodinger 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]
  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]
  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]

Other

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]

Supplementary Material (3)

» Media 1: MOV (583 KB)     
» Media 2: MOV (582 KB)     
» Media 3: MOV (329 KB)     

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Figures (4)

Figure 1.
Figure 1.

(a) Animated surface plot of I(r, t) of a self-focusing femtosecond pulse in fused silica for an input power of 4 MW (583kB QuickTime movie). (b) Calculated intensity profile I(r, t) in the far field (z = 1.5 m) for the field shown in (a).

Figure 2.
Figure 2.

(a) Animated surface plot of I(r, t) for input power of 4.7 MW (583kB QuickTime movie). (b) Calculated intensity profile I(r, t) in the far field for the field shown in (a). Early times are at the back of the figure.

Figure 3.
Figure 3.

(a) Axial intensity (black) and phase (red) measured with SHG-FROG. (b) Calculated axial intensity and phase (c) Measured axial spectrum of field shown in (a). The blue line is the square modulus of the Fourier transform the data of (a), while the red points are measured with a spectrometer. (d) Calculated axial spectrum. Frame (a) of this figure is linked to an animated graphic with sound (330 kB QuickTime movie with sound) that allows the reader to see and hear the frequency variations associated field.

Figure 4.
Figure 4.

(a) Intensity distribution I(r,t) after 30 mm of propagation in fused silica. The input power in this case is 5.5 MW, corresponding to an intensity of 100GW/cm2. (b) Axial intensity corresponding to (a). (c) I(r,t) in the far field for z = 1.5 m beyond the exit face of the fused silica. (d) Axial intensity corresponding to (c)

Equations (9)

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ε ( r , z , t ) = E ( r , z , t ) exp ( ikz i ω 0 t ) + c . c
i z E + ( 1 i ω t ) 2 E 2 t 2 E i 3 3 t 3 E
+ ( 1 + i ω t ) g ( E 2 ) E = 0
g ( E 2 ) = 2 π n 2 l D λ [ ( 1 α ) E ( t ) 2 + α t f ( t τ ) E ( τ ) 2 ] ,
f ( t ) = 1 + ( ω r τ r ) 2 ω r τ r 2 exp ( t / τ r ) sin ( ω r t ) .
i 1 n z E + 2 ( 1 i ω t ) E = 0
E vac ( r , L , t ) =
i 4 πLn dωdr r′ 1 ω ω E ( r′ , 0 , ω ) J 0 ( rr′ 2 ( 1 ω ω ) Ln ) exp [ i ( r 2 + r′ 2 ) 4 ( 1 ω ω ) Ln iωt ]
E vac ( 0 , , t ) 0 dr r exp ( iωt ) E ( r′ , 0 , ω ) 1 ω ω 0 dr r E ( r , 0 , t )

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