Abstract

The self-focusing of femtosecond optical pulses in a normally dispersive medium is studied numerically. This situation represents a general problem that may be modeled by a 3 + 1-dimensional nonlinear Schrödinger equation, where two dimensions are self-focusing and the third is self-defocusing. The numerical simulations show that the dispersion causes the splitting of a pulse before it self-focuses into two temporally separated pulses, which then continue to self-focus and compress rapidly. The calculated behavior results in periodic modulation of the generated continuum spectrum, as was recently observed in continuum generation by focused femtosecond pulses in gases.

© 1992 Optical Society of America

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References

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  1. Y. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
    [Crossref]
  2. J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
    [Crossref]
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    [Crossref]
  5. F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967); A. C. Cheung, D. M. Rank, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 20, 786 (1968); R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 592 (1970); A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973); N. Bloembergen, Opt. Commun. 8, 285 (1973).
    [Crossref]
  6. J. H. Glownia, G. Arjavalingam, P. P. Sorokin, J. E. Rothenberg, Opt. Lett. 11, 79 (1986); J. H. Glownia, J. Misewich, P. P. Sorokin, J. Opt. Soc. Am. B 3, 1573 (1986).
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    [Crossref] [PubMed]
  8. P. B. Corkum, C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
    [Crossref]
  9. T. R. Gosnell, A. J. Taylor, D. P. Greene, Opt. Lett. 15, 130 (1990).
    [Crossref] [PubMed]
  10. D. Strickland, P. B. Corkum, Proc. Soc. Photo-Opt. Instrum. Eng. 1413, 54 (1991).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  14. J. H. Marburger, W. G. Wagner, IEEE J. Quantum Electron. QE-3, 415 (1967); G. L. McAllister, J. H. Marburger, L. G. DeShazer, Phys. Rev. Lett. 21, 1648 (1968).
    [Crossref]
  15. F. DeMartini, C. H. Townes, T. K. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
    [Crossref]
  16. M. D. Feit, J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
    [Crossref]
  17. A. Dalgarno, A. E. Kingston, Proc. R. Soc. London Ser. A 259, 424 (1961).

1991 (1)

D. Strickland, P. B. Corkum, Proc. Soc. Photo-Opt. Instrum. Eng. 1413, 54 (1991).

1990 (2)

1989 (1)

P. B. Corkum, C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[Crossref]

1988 (1)

1986 (2)

1984 (1)

1975 (2)

Y. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
[Crossref]

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[Crossref]

1969 (2)

M. M. T. Loy, Y. R. Shen, Phys. Rev. Lett. 22, 994 (1969).
[Crossref]

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969); F. Shimizu, IBM J. Res. Dev. 17, 286 (1973); J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
[Crossref] [PubMed]

1968 (1)

V. N. Lugovi, A. M. Prokhorov, Zh. Eksp. Teor. Fiz. 7, 153 (1968) [JETP Lett. 7, 117 (1968)].

1967 (3)

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967); A. C. Cheung, D. M. Rank, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 20, 786 (1968); R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 592 (1970); A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973); N. Bloembergen, Opt. Commun. 8, 285 (1973).
[Crossref]

J. H. Marburger, W. G. Wagner, IEEE J. Quantum Electron. QE-3, 415 (1967); G. L. McAllister, J. H. Marburger, L. G. DeShazer, Phys. Rev. Lett. 21, 1648 (1968).
[Crossref]

F. DeMartini, C. H. Townes, T. K. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

1961 (1)

A. Dalgarno, A. E. Kingston, Proc. R. Soc. London Ser. A 259, 424 (1961).

Arjavalingam, G.

Corkum, P. B.

D. Strickland, P. B. Corkum, Proc. Soc. Photo-Opt. Instrum. Eng. 1413, 54 (1991).

P. B. Corkum, C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[Crossref]

P. B. Corkum, C. Rolland, T. Srinivasan-Rao, Phys. Rev. Lett. 57, 2268 (1986).
[Crossref] [PubMed]

Dalgarno, A.

A. Dalgarno, A. E. Kingston, Proc. R. Soc. London Ser. A 259, 424 (1961).

DeMartini, F.

F. DeMartini, C. H. Townes, T. K. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Feit, M. D.

Fleck, J. A.

Glownia, J. H.

Gosnell, T. R.

Greene, D. P.

Gustafson, T. K.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969); F. Shimizu, IBM J. Res. Dev. 17, 286 (1973); J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
[Crossref] [PubMed]

F. DeMartini, C. H. Townes, T. K. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Haus, H. A.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969); F. Shimizu, IBM J. Res. Dev. 17, 286 (1973); J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
[Crossref] [PubMed]

Kelley, P. L.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969); F. Shimizu, IBM J. Res. Dev. 17, 286 (1973); J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
[Crossref] [PubMed]

F. DeMartini, C. H. Townes, T. K. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Kingston, A. E.

A. Dalgarno, A. E. Kingston, Proc. R. Soc. London Ser. A 259, 424 (1961).

Lifsitz, J. R.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969); F. Shimizu, IBM J. Res. Dev. 17, 286 (1973); J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
[Crossref] [PubMed]

Loy, M. M. T.

M. M. T. Loy, Y. R. Shen, Phys. Rev. Lett. 22, 994 (1969).
[Crossref]

Lugovi, V. N.

V. N. Lugovi, A. M. Prokhorov, Zh. Eksp. Teor. Fiz. 7, 153 (1968) [JETP Lett. 7, 117 (1968)].

Marburger, J. H.

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[Crossref]

J. H. Marburger, W. G. Wagner, IEEE J. Quantum Electron. QE-3, 415 (1967); G. L. McAllister, J. H. Marburger, L. G. DeShazer, Phys. Rev. Lett. 21, 1648 (1968).
[Crossref]

Prokhorov, A. M.

V. N. Lugovi, A. M. Prokhorov, Zh. Eksp. Teor. Fiz. 7, 153 (1968) [JETP Lett. 7, 117 (1968)].

Rolland, C.

P. B. Corkum, C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[Crossref]

P. B. Corkum, C. Rolland, T. Srinivasan-Rao, Phys. Rev. Lett. 57, 2268 (1986).
[Crossref] [PubMed]

Rothenberg, J. E.

Shen, Y. R.

G. Yang, Y. R. Shen, Opt. Lett. 9, 510 (1984).
[Crossref] [PubMed]

Y. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
[Crossref]

M. M. T. Loy, Y. R. Shen, Phys. Rev. Lett. 22, 994 (1969).
[Crossref]

Shimizu, F.

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967); A. C. Cheung, D. M. Rank, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 20, 786 (1968); R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 592 (1970); A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973); N. Bloembergen, Opt. Commun. 8, 285 (1973).
[Crossref]

Silberberg, Y.

Sorokin, P. P.

Srinivasan-Rao, T.

P. B. Corkum, C. Rolland, T. Srinivasan-Rao, Phys. Rev. Lett. 57, 2268 (1986).
[Crossref] [PubMed]

Strickland, D.

D. Strickland, P. B. Corkum, Proc. Soc. Photo-Opt. Instrum. Eng. 1413, 54 (1991).

Taran, J. P.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969); F. Shimizu, IBM J. Res. Dev. 17, 286 (1973); J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
[Crossref] [PubMed]

Taylor, A. J.

Townes, C. H.

F. DeMartini, C. H. Townes, T. K. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

Wagner, W. G.

J. H. Marburger, W. G. Wagner, IEEE J. Quantum Electron. QE-3, 415 (1967); G. L. McAllister, J. H. Marburger, L. G. DeShazer, Phys. Rev. Lett. 21, 1648 (1968).
[Crossref]

Yang, G.

IEEE J. Quantum Electron. (2)

P. B. Corkum, C. Rolland, IEEE J. Quantum Electron. 25, 2634 (1989).
[Crossref]

J. H. Marburger, W. G. Wagner, IEEE J. Quantum Electron. QE-3, 415 (1967); G. L. McAllister, J. H. Marburger, L. G. DeShazer, Phys. Rev. Lett. 21, 1648 (1968).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Lett. (4)

Phys. Rev. (2)

F. DeMartini, C. H. Townes, T. K. Gustafson, P. L. Kelley, Phys. Rev. 164, 312 (1967).
[Crossref]

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969); F. Shimizu, IBM J. Res. Dev. 17, 286 (1973); J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

P. B. Corkum, C. Rolland, T. Srinivasan-Rao, Phys. Rev. Lett. 57, 2268 (1986).
[Crossref] [PubMed]

M. M. T. Loy, Y. R. Shen, Phys. Rev. Lett. 22, 994 (1969).
[Crossref]

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967); A. C. Cheung, D. M. Rank, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 20, 786 (1968); R. R. Alfano, S. L. Shapiro, Phys. Rev. Lett. 24, 592 (1970); A. Penzkofer, A. Laubereau, W. Kaiser, Phys. Rev. Lett. 31, 863 (1973); N. Bloembergen, Opt. Commun. 8, 285 (1973).
[Crossref]

Proc. R. Soc. London Ser. A (1)

A. Dalgarno, A. E. Kingston, Proc. R. Soc. London Ser. A 259, 424 (1961).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. Strickland, P. B. Corkum, Proc. Soc. Photo-Opt. Instrum. Eng. 1413, 54 (1991).

Prog. Quantum Electron. (2)

Y. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
[Crossref]

J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

V. N. Lugovi, A. M. Prokhorov, Zh. Eksp. Teor. Fiz. 7, 153 (1968) [JETP Lett. 7, 117 (1968)].

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Figures (4)

Fig. 1
Fig. 1

Calculated intensity on axis versus time of a self-focusing pulse at the propagation distances indicated. The peak intensity (×1012 W/cm2) at each distance is (a) 0.85, (b) 6.3, (c) 11, (d) 14, (e) 20, (f) 34, and (g) 150.

Fig. 2
Fig. 2

Peak axial intensity of the split pulses (solid curve) and the axial intensity at the original pulse center (t = 0, dashed curve) versus the propagation distance.

Fig. 3
Fig. 3

Spatially integrated power (solid curves) at the same distances as in Fig. 1. The dashed curves are the axial intensities displayed in Fig. 1. The dotted lines indicate the critical power level (15 MW).

Fig. 4
Fig. 4

Spectral intensity versus the relative frequency at the same distances as in Fig. 1.

Equations (1)

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E z + i k ( 2 ) 2 2 E t 2 - i 2 k ( 2 E x 2 + 2 E y 2 ) = i n 2 k 0 E 2 E ,

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