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.

© 1999 Optical Society of America

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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, 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]
  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. 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]
  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]
  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]

1999 (1)

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]

1998 (3)

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]

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]

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]

1997 (1)

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]

1996 (2)

1995 (1)

1994 (2)

1992 (4)

1986 (2)

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).

1983 (1)

1967 (1)

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

Braun, A.

Chernev, P.

Clement, T. S.

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

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]

Corkum, P. B.

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

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

Curley, P. F.

DeLong, K. W.

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]

DeMartini, F.

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

Diddams, S. A.

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

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]

Du, D.

Eaton, H. K.

Fittinghoff, D. N.

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]

Fork, R. L.

Franco, M. A.

Gaeta, A. L.

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]

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]

Grillon, G.

Gustafson, T. K.

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

Hirlimann, C.

Kelley, P. L.

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

Korn, G.

Krumbügel, M. A.

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]

Litvak, A. G.

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).

Liu, X.

Luther, G. G.

Moloney, J. V.

Mourou, G.

Mysy-rowicz, A.

Newell, A. C.

Nibbering, E. T. J.

Petrov, V.

Petrova, T. A.

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).

Prade, B. S.

Ranka, J. K.

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]

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]

Richman, B. A.

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]

Rolland, C.

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

Rothenberg, J.

Salin, F.

Schirmer, R. W.

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]

Sergeev, A. M.

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).

Shank, C. V.

Squier, J.

Srinivasan-Rao, T.

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

Stolen, R. H.

Strickland, D.

Sweetser, J. N.

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]

Tomlinson, W. J.

Townes, C. H.

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

Trebino, R.

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]

Van Engen, A. G.

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]

Wright, E. M.

Yanukoviskii, A. D.

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).

Yen, R.

Zharova, N. A.

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).

Zozulya, A. A

Zozulya, A. A.

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

J. Opt. Soc. Am. B (2)

JETP Lett. (1)

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).

Opt. Lett. (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]

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. 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]

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

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]

Phys. Rev. (1)

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

Phys. Rev. A (1)

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]

Phys. Rev. Lett. (3)

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

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]

Rev. Sci. Inst. (1)

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]

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)

Equations on this page are rendered with MathJax. Learn more.

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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