Abstract

The paraxial wave equation, as is well known, predicts the catastrophic collapse of self-focusing beams. It is pointed out that this collapse is due to the loss of validity of the paraxial wave equation in the neighborhood of a self-focus. If nonparaxiality of the beam propagation is taken into account, on the other hand, a lower limit of the order of one optical wavelength is imposed on the diameter of a self-focus. A nonparaxial algorithm for the Helmholtz equation is applied to the self-focusing of Gaussian and ring-shaped beams. The self-focusing is noncatastrophic, and the results give insight into filament formation and beam breakup resulting from the self-focusing of optical beams.

© 1988 Optical Society of America

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  1. G. A. Askar’yan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088 (1962) [Zh. Eksp. Teor. Fiz. 42, 1567 (1962)].
  2. R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 469 (1964); E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347 (1966).
    [CrossRef]
  3. For a comprehensive review of theoretical work on self-focusing, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1 (1975).
    [CrossRef]
  4. For a comprehensive review of theoretical work on self-focusing, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35 (1975).
    [CrossRef]
  5. V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].
  6. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  7. J. H. Marburger, E. L. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556 (1968); E. L. Dawes, J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862 (1969).
    [CrossRef]
  8. C. C. Wang, “Length-dependent threshold for stimulated Raman effect and self-focusing of laser beams in liquids,” Phys. Rev. Lett. 16, 334 (1966); “Nonlinear susceptibility constants and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
    [CrossRef]
  9. J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of light beams,” Appl. Phys. Lett. 15, 313 (1969).
    [CrossRef]
  10. J. A. Fleck, R. L. Carman, “Effect of relaxation on small-scale filament formation by ultrashort light pulses,” Appl. Phys. Lett. 20, 290 (1974).
    [CrossRef]
  11. F. Shimizu, “Numerical calculation. of self-focusing and trapping of a short light pulse in Kerr liquids,” IBM J. Res. Dev. 17, 286 (1973).
    [CrossRef]
  12. A. L. Dyshko, V. N. Lugovoi, A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146 (1967) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 6, 655 (1967)].
  13. A. L. Dyshko, V. N. Lugovi, A. M. Prokhorov, “Multifocus structure of a beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235 (1972) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 61,2305].
  14. E. Yablonovitch, B. Bloembergen, “Avalanche ionization and the limiting diameter of filaments induced by light pulses in transparent media,” Phys. Rev. Lett. 29, 907 (1972).
    [CrossRef]
  15. J. A. Fleck, J. R. Morris, M. D. Feit, “Propagation of high-energy laser beams in the atmosphere,” Appl. Phys. 10, 129 (1976).
    [CrossRef]
  16. M. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3390 (1979).
  17. This derivation of Eq. (2) in Ref. (16) is due to D. J. Thomson, N. R. Chapman, “A wide angle split-step operator for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848 (1983).
    [CrossRef]
  18. The self-focusing of a laser beam in a plasma, which is governed by electron-density gradients and hydrodynamics, is also characterized by multiple foci. See M. D. Feit, J. A. Fleck, “Self-trapping of a laser beam in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121 (1976).
    [CrossRef]
  19. V. N. Lugovoi, A. M. Prokhorov, “A possible explanation of the small-scale self-focusing filaments,” JETP Lett. 7, 117 (1968) [Zh. Eksp. Teor. Fiz. 7, 153 (1968)]; M. T. Loy, Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994 (1969).
    [CrossRef]
  20. J. A. Fleck, C. Layne, “Study of self-focusing damage in a high-power Nd:glass-rod amplifier,” Appl. Phys. Lett. 22, 467 (1973).
    [CrossRef]
  21. A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. 23, 628 (1973).
    [CrossRef]
  22. V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 3, 471 (1966)].
  23. A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178 (1974).
    [CrossRef]
  24. A dust-particle perturbation was used to study beam instability due to self-focusing in glass laser amplifiers. The dust particle was modeled by setting the field to zero at the center of the dust particle. See J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978).
    [CrossRef]

1983 (1)

This derivation of Eq. (2) in Ref. (16) is due to D. J. Thomson, N. R. Chapman, “A wide angle split-step operator for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848 (1983).
[CrossRef]

1979 (1)

M. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3390 (1979).

1978 (1)

A dust-particle perturbation was used to study beam instability due to self-focusing in glass laser amplifiers. The dust particle was modeled by setting the field to zero at the center of the dust particle. See J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978).
[CrossRef]

1976 (2)

The self-focusing of a laser beam in a plasma, which is governed by electron-density gradients and hydrodynamics, is also characterized by multiple foci. See M. D. Feit, J. A. Fleck, “Self-trapping of a laser beam in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121 (1976).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, “Propagation of high-energy laser beams in the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

1975 (2)

For a comprehensive review of theoretical work on self-focusing, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

For a comprehensive review of theoretical work on self-focusing, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

1974 (2)

J. A. Fleck, R. L. Carman, “Effect of relaxation on small-scale filament formation by ultrashort light pulses,” Appl. Phys. Lett. 20, 290 (1974).
[CrossRef]

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178 (1974).
[CrossRef]

1973 (3)

J. A. Fleck, C. Layne, “Study of self-focusing damage in a high-power Nd:glass-rod amplifier,” Appl. Phys. Lett. 22, 467 (1973).
[CrossRef]

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. 23, 628 (1973).
[CrossRef]

F. Shimizu, “Numerical calculation. of self-focusing and trapping of a short light pulse in Kerr liquids,” IBM J. Res. Dev. 17, 286 (1973).
[CrossRef]

1972 (2)

A. L. Dyshko, V. N. Lugovi, A. M. Prokhorov, “Multifocus structure of a beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235 (1972) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 61,2305].

E. Yablonovitch, B. Bloembergen, “Avalanche ionization and the limiting diameter of filaments induced by light pulses in transparent media,” Phys. Rev. Lett. 29, 907 (1972).
[CrossRef]

1969 (1)

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of light beams,” Appl. Phys. Lett. 15, 313 (1969).
[CrossRef]

1968 (2)

V. N. Lugovoi, A. M. Prokhorov, “A possible explanation of the small-scale self-focusing filaments,” JETP Lett. 7, 117 (1968) [Zh. Eksp. Teor. Fiz. 7, 153 (1968)]; M. T. Loy, Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994 (1969).
[CrossRef]

J. H. Marburger, E. L. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556 (1968); E. L. Dawes, J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862 (1969).
[CrossRef]

1967 (1)

A. L. Dyshko, V. N. Lugovoi, A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146 (1967) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 6, 655 (1967)].

1966 (2)

C. C. Wang, “Length-dependent threshold for stimulated Raman effect and self-focusing of laser beams in liquids,” Phys. Rev. Lett. 16, 334 (1966); “Nonlinear susceptibility constants and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
[CrossRef]

V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 3, 471 (1966)].

1965 (2)

V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 469 (1964); E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

1962 (1)

G. A. Askar’yan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088 (1962) [Zh. Eksp. Teor. Fiz. 42, 1567 (1962)].

Askar’yan, G. A.

G. A. Askar’yan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088 (1962) [Zh. Eksp. Teor. Fiz. 42, 1567 (1962)].

Bespalov, V. I.

V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 3, 471 (1966)].

Bliss, E. S.

A dust-particle perturbation was used to study beam instability due to self-focusing in glass laser amplifiers. The dust particle was modeled by setting the field to zero at the center of the dust particle. See J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978).
[CrossRef]

Bloembergen, B.

E. Yablonovitch, B. Bloembergen, “Avalanche ionization and the limiting diameter of filaments induced by light pulses in transparent media,” Phys. Rev. Lett. 29, 907 (1972).
[CrossRef]

Campillo, A. J.

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178 (1974).
[CrossRef]

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. 23, 628 (1973).
[CrossRef]

Carman, R. L.

J. A. Fleck, R. L. Carman, “Effect of relaxation on small-scale filament formation by ultrashort light pulses,” Appl. Phys. Lett. 20, 290 (1974).
[CrossRef]

Chapman, N. R.

This derivation of Eq. (2) in Ref. (16) is due to D. J. Thomson, N. R. Chapman, “A wide angle split-step operator for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848 (1983).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 469 (1964); E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

Dawes, E. L.

J. H. Marburger, E. L. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556 (1968); E. L. Dawes, J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862 (1969).
[CrossRef]

Dyshko, A. L.

A. L. Dyshko, V. N. Lugovi, A. M. Prokhorov, “Multifocus structure of a beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235 (1972) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 61,2305].

A. L. Dyshko, V. N. Lugovoi, A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146 (1967) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 6, 655 (1967)].

Feit, M. D.

M. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3390 (1979).

The self-focusing of a laser beam in a plasma, which is governed by electron-density gradients and hydrodynamics, is also characterized by multiple foci. See M. D. Feit, J. A. Fleck, “Self-trapping of a laser beam in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121 (1976).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, “Propagation of high-energy laser beams in the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

Fleck, J. A.

M. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3390 (1979).

A dust-particle perturbation was used to study beam instability due to self-focusing in glass laser amplifiers. The dust particle was modeled by setting the field to zero at the center of the dust particle. See J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978).
[CrossRef]

The self-focusing of a laser beam in a plasma, which is governed by electron-density gradients and hydrodynamics, is also characterized by multiple foci. See M. D. Feit, J. A. Fleck, “Self-trapping of a laser beam in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121 (1976).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, “Propagation of high-energy laser beams in the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

J. A. Fleck, R. L. Carman, “Effect of relaxation on small-scale filament formation by ultrashort light pulses,” Appl. Phys. Lett. 20, 290 (1974).
[CrossRef]

J. A. Fleck, C. Layne, “Study of self-focusing damage in a high-power Nd:glass-rod amplifier,” Appl. Phys. Lett. 22, 467 (1973).
[CrossRef]

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of light beams,” Appl. Phys. Lett. 15, 313 (1969).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 469 (1964); E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

Kelley, P. L.

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of light beams,” Appl. Phys. Lett. 15, 313 (1969).
[CrossRef]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Layne, C.

J. A. Fleck, C. Layne, “Study of self-focusing damage in a high-power Nd:glass-rod amplifier,” Appl. Phys. Lett. 22, 467 (1973).
[CrossRef]

Lugovi, V. N.

A. L. Dyshko, V. N. Lugovi, A. M. Prokhorov, “Multifocus structure of a beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235 (1972) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 61,2305].

Lugovoi, V. N.

V. N. Lugovoi, A. M. Prokhorov, “A possible explanation of the small-scale self-focusing filaments,” JETP Lett. 7, 117 (1968) [Zh. Eksp. Teor. Fiz. 7, 153 (1968)]; M. T. Loy, Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994 (1969).
[CrossRef]

A. L. Dyshko, V. N. Lugovoi, A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146 (1967) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 6, 655 (1967)].

Marburger, J. H.

For a comprehensive review of theoretical work on self-focusing, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

J. H. Marburger, E. L. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556 (1968); E. L. Dawes, J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862 (1969).
[CrossRef]

Morris, J. R.

A dust-particle perturbation was used to study beam instability due to self-focusing in glass laser amplifiers. The dust particle was modeled by setting the field to zero at the center of the dust particle. See J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, “Propagation of high-energy laser beams in the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

Prokhorov, A. M.

A. L. Dyshko, V. N. Lugovi, A. M. Prokhorov, “Multifocus structure of a beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235 (1972) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 61,2305].

V. N. Lugovoi, A. M. Prokhorov, “A possible explanation of the small-scale self-focusing filaments,” JETP Lett. 7, 117 (1968) [Zh. Eksp. Teor. Fiz. 7, 153 (1968)]; M. T. Loy, Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994 (1969).
[CrossRef]

A. L. Dyshko, V. N. Lugovoi, A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146 (1967) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 6, 655 (1967)].

Shapiro, S. L.

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178 (1974).
[CrossRef]

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. 23, 628 (1973).
[CrossRef]

Shen, Y. R.

For a comprehensive review of theoretical work on self-focusing, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

Shimizu, F.

F. Shimizu, “Numerical calculation. of self-focusing and trapping of a short light pulse in Kerr liquids,” IBM J. Res. Dev. 17, 286 (1973).
[CrossRef]

Suydam, B. R.

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178 (1974).
[CrossRef]

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. 23, 628 (1973).
[CrossRef]

Talanov, V. I.

V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 3, 471 (1966)].

V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].

Thomson, D. J.

This derivation of Eq. (2) in Ref. (16) is due to D. J. Thomson, N. R. Chapman, “A wide angle split-step operator for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848 (1983).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 469 (1964); E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

Wang, C. C.

C. C. Wang, “Length-dependent threshold for stimulated Raman effect and self-focusing of laser beams in liquids,” Phys. Rev. Lett. 16, 334 (1966); “Nonlinear susceptibility constants and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, B. Bloembergen, “Avalanche ionization and the limiting diameter of filaments induced by light pulses in transparent media,” Phys. Rev. Lett. 29, 907 (1972).
[CrossRef]

Appl. Opt. (1)

M. D. Feit, J. A. Fleck, “Light propagation in graded index fibers,” Appl. Opt. 17, 3390 (1979).

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Propagation of high-energy laser beams in the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

Appl. Phys. Lett. (6)

J. A. Fleck, C. Layne, “Study of self-focusing damage in a high-power Nd:glass-rod amplifier,” Appl. Phys. Lett. 22, 467 (1973).
[CrossRef]

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. 23, 628 (1973).
[CrossRef]

A. J. Campillo, S. L. Shapiro, B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178 (1974).
[CrossRef]

The self-focusing of a laser beam in a plasma, which is governed by electron-density gradients and hydrodynamics, is also characterized by multiple foci. See M. D. Feit, J. A. Fleck, “Self-trapping of a laser beam in a cylindrical plasma column,” Appl. Phys. Lett. 28, 121 (1976).
[CrossRef]

J. A. Fleck, P. L. Kelley, “Temporal aspects of the self-focusing of light beams,” Appl. Phys. Lett. 15, 313 (1969).
[CrossRef]

J. A. Fleck, R. L. Carman, “Effect of relaxation on small-scale filament formation by ultrashort light pulses,” Appl. Phys. Lett. 20, 290 (1974).
[CrossRef]

IBM J. Res. Dev. (1)

F. Shimizu, “Numerical calculation. of self-focusing and trapping of a short light pulse in Kerr liquids,” IBM J. Res. Dev. 17, 286 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

A dust-particle perturbation was used to study beam instability due to self-focusing in glass laser amplifiers. The dust particle was modeled by setting the field to zero at the center of the dust particle. See J. A. Fleck, J. R. Morris, E. S. Bliss, “Small-scale self-focusing effects in a high-power glass laser amplifier,” IEEE J. Quantum Electron. QE-14, 353 (1978).
[CrossRef]

J. Acoust. Soc. Am. (1)

This derivation of Eq. (2) in Ref. (16) is due to D. J. Thomson, N. R. Chapman, “A wide angle split-step operator for the parabolic equation,” J. Acoust. Soc. Am. 74, 1848 (1983).
[CrossRef]

JETP Lett. (4)

V. N. Lugovoi, A. M. Prokhorov, “A possible explanation of the small-scale self-focusing filaments,” JETP Lett. 7, 117 (1968) [Zh. Eksp. Teor. Fiz. 7, 153 (1968)]; M. T. Loy, Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994 (1969).
[CrossRef]

A. L. Dyshko, V. N. Lugovoi, A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146 (1967) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 6, 655 (1967)].

V. I. Talanov, “Self-focusing of waves in nonlinear media,” JETP Lett. 2, 138 (1965) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 2, 218 (1965)].

V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 3, 471 (1966)].

Phys. Rev. Lett. (5)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

J. H. Marburger, E. L. Dawes, “Dynamical formation of a small-scale filament,” Phys. Rev. Lett. 21, 556 (1968); E. L. Dawes, J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862 (1969).
[CrossRef]

C. C. Wang, “Length-dependent threshold for stimulated Raman effect and self-focusing of laser beams in liquids,” Phys. Rev. Lett. 16, 334 (1966); “Nonlinear susceptibility constants and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
[CrossRef]

R. Y. Chiao, E. Garmire, C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 469 (1964); E. Garmire, R. Y. Chiao, C. H. Townes, “Dynamics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

E. Yablonovitch, B. Bloembergen, “Avalanche ionization and the limiting diameter of filaments induced by light pulses in transparent media,” Phys. Rev. Lett. 29, 907 (1972).
[CrossRef]

Prog. Quantum Electron. (2)

For a comprehensive review of theoretical work on self-focusing, see Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

For a comprehensive review of theoretical work on self-focusing, see J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

Sov. Phys. JETP (2)

G. A. Askar’yan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088 (1962) [Zh. Eksp. Teor. Fiz. 42, 1567 (1962)].

A. L. Dyshko, V. N. Lugovi, A. M. Prokhorov, “Multifocus structure of a beam in a nonlinear medium,” Sov. Phys. JETP 34, 1235 (1972) [Zh. Eksp. Teor. Fiz. Pis’ma Red. 61,2305].

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Figures (9)

Fig. 1
Fig. 1

Comparison of paraxial and nonparaxial descriptions of a self-focusing beam.

Fig. 2
Fig. 2

On-axis intensity as a function of propagation distance for self-focusing according to the nonparaxial model: for all cases I(x = 0, y = 0, z = 0) = 1.0; (a) just above critical power, n2 = 0.002, (b) n2 = 0.01, (c) n2 = 0.015, (d) n2 = 0.015.

Fig. 3
Fig. 3

Transmitted power versus propagation distance corresponding to Fig. 2. Sudden drops in power indicate the formation of local foci.

Fig. 4
Fig. 4

Self-focusing behavior for a beam containing ~30 times critical power: (a) on-axis intensity versus propagation distance, (b) angular contours containing the indicated percentage of total power, (c) power in the beam as a function of distance.

Fig. 5
Fig. 5

Evolution of the self-focusing of a ring-shaped beam as a function of propagation distance. Self-focusing originates from a small obscuration on the right and propagates in pairs of foci successively around the ring.

Fig. 6
Fig. 6

Angular contours containing the indicated percentage of the total beam power as a function of distance. Each peak indicates the formation of a pair of foci except for the last peak, which indicates the formation of a single focus diametrically opposite to the initial perturbation.

Fig. 7
Fig. 7

Power as a function of distance corresponding to Fig. 5.

Fig. 8
Fig. 8

Self-focusing and breakup of a ring-shaped beam. Perturbation is numerical. Round-off errors introduced by a square sampling grid have the symmetry of the grid. This breaks up the circular symmetry of the beam and initiates breakup.

Fig. 9
Fig. 9

Breakup of a ring-shaped beam that has an angular momentum component. The beam shape was multiplied by the phase factor eiνθ, where ν = 6. This makes the beam somewhat more stable to breakup.

Equations (22)

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θ = sin - 1 ( κ x 2 + κ y 2 ) 1 / 2 / k .
2 E x 2 + 2 E y 2 + 2 E z 2 + ω 2 c 2 n 2 ( ω , x , y , z ) E = 0.
P = z ,
Q = ( 2 + ω 2 c 2 n 2 ) 1 / 2 ,
( P 2 + Q 2 ) E = 0 ,
{ ( P + i Q ) ( P - i Q ) + i [ P , Q ] } E = 0 ,
E z = ± i Q E .
E ( x , y , Δ z ) = exp [ i 0 Δ z ( 2 + ω 2 c 2 n 2 ) 1 / 2 d z ] E ( x , y , 0 ) .
E ( x , y , Δ z ) = exp [ i Δ z ( 2 + ω 2 c 2 n 2 ) 1 / 2 ] E ( x , y , 0 ) .
( 2 + ω 2 c 2 n 2 ) 1 / 2 = 2 ( 2 + ω 2 c 2 n 2 ) 1 / 2 + ω c n + ( ω c ) n + ω c [ n , Q ] Q + ω c n .
( 2 + ω 2 c 2 n 2 ) 1 / 2 ~ 2 ( 2 + k 2 ) 1 / 2 + k + k + k [ ( n n 0 ) - 1 ] ,
k = n 0 ω / c .
( 2 + ω 2 c 2 n 2 ) 1 / 2 ~ 1 2 k 2 + k + k [ ( n / n 0 ) - 1 ] .
E ( x , y , z ) = E ( x , y , z ) exp ( i k z ) ,
E ( x , y , Δ z ) = exp { i Δ z [ 2 ( 2 + k 2 ) 1 / 2 + k + χ ] } E ( x , y , 0 ) ,
χ = k [ ( n / n 0 ) - 1 ] .
E ( x , y , Δ z ) = exp { i Δ z 2 [ 2 ( 2 + k 2 ) 1 / 2 + k ] } exp ( i Δ z χ ) × exp { i Δ z 2 [ 2 ( 2 + k 2 ) 1 / 2 + k ] } E ( x , y , 0 ) + 0 ( Δ z ) 3 .
E ( x , y , z ) = m = - N / 2 + 1 N / 2 n = - N / 2 + 1 N / 2 E m n ( z ) exp [ 2 π i L ( m x + n y ) ] ,
E m n ( Δ z ) = E m n ( 0 ) exp { i Δ z [ - ( κ x 2 + κ y 2 ) ( - κ x 2 - κ y 2 + k 2 ) 1 / 2 + k ] } ,
n = n 0 + n 2 E 2 .
E = E 0 r 6 exp ( - r 2 / 2 σ 2 ) ,
E = [ 1 - β exp ( r - r 0 ) 2 / a 2 ) ] .

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