Abstract

We show that the increase in critical power for elliptic input beams is only 40% of what had been previously estimated based on the aberrationless approximation. We also find a theoretical upper bound for the critical power, above which elliptic beams always collapse. If the power of an elliptic beam is above critical, the beam self-focuses and undergoes partial beam blowup, during which the collapsing part of the beam approaches a circular Townesian profile. As a result, during further propagation additional small mechanisms, which are neglected in the derivation of the nonlinear Schrödinger equation (NLS) from Maxwell’s equations, can have large effects, which are the same as in the case of circular beams. Our simulations show that most predictions for elliptic beams based on the aberrationless approximation are either quantitatively inaccurate or simply wrong. This failure of the aberrationless approximation is related to its inability to capture neither the partial beam collapse nor the subsequent delicate balance between the Kerr nonlinearity and diffraction. We present an alternative two-stage approach and use it to analyze the effect of nonlinear saturation, nonparaxiality, and time dispersion on the propagation of elliptic beams. The results of the two-stage approach are found to be in good agreement with NLS simulations.

© 2000 Optical Society of America

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  1. C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).
    [CrossRef]
  2. F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
    [CrossRef]
  3. S. Konar and A. Sengupta, “Propagation of an elliptic Gaussian laser beam in a medium with saturable nonlinearity,” J. Opt. Soc. Am. B 11, 1644–1646 (1994).
    [CrossRef]
  4. G. Cerullo, A. Dienes, and V. Magni, “Space–time coupling and collapse threshold for femtosecond pulses in dispersive nonlinear media,” Opt. Lett. 21, 65–67 (1996).
    [CrossRef] [PubMed]
  5. T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).
  6. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).
  7. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 85–110 (1975).
    [CrossRef]
  8. D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
    [CrossRef]
  9. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
    [CrossRef]
  10. M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726–2734 (1992).
    [CrossRef] [PubMed]
  11. B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
    [CrossRef]
  12. G. Fibich and A. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
    [CrossRef]
  13. G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
    [CrossRef]
  14. C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation (Springer, New York, 1999).
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    [CrossRef]
  16. V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).
  17. The values of e can be recovered by use of the equation e=1±h2−1.
  18. The relative error of relation (9) is less than 1.5% in the range 1/2.5≤e≤2.5.
  19. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].
    [CrossRef]
  20. E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
    [CrossRef]
  21. The value of zc here is twice that given in Eq. (13) below, because there ψ0=c exp(−r2).
  22. I.e., if ∫x2+y2≤ri(z)|ψ|2dxdy=Ni, then Vi(z)=∫x2+y2≤ri(z)r2|ψ|2dxdy.
  23. M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
    [CrossRef]
  24. G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. 76, 4356–4359 (1996).
  25. G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).
    [CrossRef] [PubMed]
  26. V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
    [CrossRef]
  27. I.e., the location of blowup in the unperturbed NLS [Eq. (4)].
  28. G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).
    [CrossRef]
  29. J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
    [CrossRef]
  30. M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  31. N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).
    [CrossRef] [PubMed]
  32. J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
    [CrossRef]
  33. S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1560 (1995).
    [CrossRef] [PubMed]
  34. G. G. Luther, A. C. Newell, J. V. Moloney, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862–864 (1994).
    [CrossRef] [PubMed]
  35. G. Fibich and G. C. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).
    [CrossRef]
  36. 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]
  37. 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]
  38. Clearly, the critical power for collapse of ultrashort elliptic pulses should be higher than for circular pulses.
  39. V. Magni, G. Cerullo, S. De Silvesti, and A. Monguzzi, “Astigmatism in Gaussian-beam self-focusing and in resonators for Kerr-lens mode locking,” J. Opt. Soc. Am. B 12, 476–485 (1995).
    [CrossRef]

2000 (1)

1999 (2)

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

1998 (3)

1997 (1)

1996 (2)

1995 (3)

1994 (2)

1993 (3)

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
[CrossRef]

1992 (2)

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726–2734 (1992).
[CrossRef] [PubMed]

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

1991 (2)

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

1990 (1)

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

1988 (1)

1983 (1)

M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates,” Commun. Math. Phys. 87, 567–576 (1983).
[CrossRef]

1979 (1)

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

1975 (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 85–110 (1975).
[CrossRef]

1974 (1)

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
[CrossRef]

1972 (1)

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).
[CrossRef]

1971 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].
[CrossRef]

1970 (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

1969 (1)

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

1966 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

Akhmediev, N. N.

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).
[CrossRef] [PubMed]

Anderson, D.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Ashkin, A.

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
[CrossRef]

Bonnedal, M.

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Cerullo, G.

Chi, S.

Clement, T. S.

Cornolti, F.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Dawes, E. L.

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

De Silvesti, S.

Desaix, M.

Diddams, S. A.

Dienes, A.

Eaton, H. K.

Feit, M. D.

Fibich, G.

G. Fibich and A. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).
[CrossRef]

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. 76, 4356–4359 (1996).

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).
[CrossRef] [PubMed]

Fleck, J. A.

Gaeta, A.

Gaeta, A. L.

Giuliano, C. R.

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).
[CrossRef]

Gross, B.

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

Guo, Q.

Karlsson, M.

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726–2734 (1992).
[CrossRef] [PubMed]

Kaul, S. S.

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

Konar, S.

Landman, M. J.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
[CrossRef]

Lisak, M.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Lucchesi, M.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Luther, G. G.

Magni, V.

Malkin, V. M.

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).
[CrossRef] [PubMed]

V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
[CrossRef]

Manassah, J. T.

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 85–110 (1975).
[CrossRef]

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).
[CrossRef]

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

Moloney, J. V.

Monguzzi, A.

Newell, A. C.

Papanicolaou, G. C.

G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “Self-focusing in the presence of small time dispersion and nonparaxiality,” Opt. Lett. 22, 1379–1381 (1997).
[CrossRef]

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).
[CrossRef] [PubMed]

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
[CrossRef]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].
[CrossRef]

Ranka, J. K.

Sengupta, A.

Singh, T.

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

Soto-Crespo, J. M.

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

Sulem, C.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
[CrossRef]

Sulem, P. L.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
[CrossRef]

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].
[CrossRef]

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].
[CrossRef]

Wang, X. P.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
[CrossRef]

Weinstein, M. I.

M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates,” Commun. Math. Phys. 87, 567–576 (1983).
[CrossRef]

Wright, E. M.

Yariv, A.

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).
[CrossRef]

Zambon, B.

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Zozulya, A. A.

Appl. Phys. Lett. (1)

C. R. Giuliano, J. H. Marburger, and A. Yariv, “Enhancement of self-focusing threshold in sapphire with elliptical beams,” Appl. Phys. Lett. 32, 58–60 (1972).
[CrossRef]

Commun. Math. Phys. (1)

M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates,” Commun. Math. Phys. 87, 567–576 (1983).
[CrossRef]

Indian J. Pure Appl. Phys. (1)

T. Singh and S. S. Kaul, “Self-focusing and self-phase modulation of elliptic Gaussian laser beam in a graded Kerr-medium,” Indian J. Pure Appl. Phys. 11, 794–797 (1999).

J. Opt. Soc. Am. B (4)

JETP (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” JETP 23, 1025–1033 (1966).

JETP Lett. (1)

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 11, 199–201 (1970).

Opt. Commun. (2)

F. Cornolti, M. Lucchesi, and B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

J. M. Soto-Crespo and N. N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrödinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

Opt. Lett. (7)

Phys. Fluids (1)

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Phys. Lett. A (2)

B. Gross and J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

G. Fibich and G. C. Papanicolaou, “A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation,” Phys. Lett. A 239, 167–173 (1998).
[CrossRef]

Phys. Rev. (2)

G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. 76, 4356–4359 (1996).

E. L. Dawes and J. H. Marburger, “Computer studies in self-focusing,” Phys. Rev. 179, 862–868 (1969).
[CrossRef]

Phys. Rev. A (3)

G. Fibich, V. M. Malkin, and G. C. Papanicolaou, “Beam self-focusing in the presence of small normal time dispersion,” Phys. Rev. A 52, 4218–4228 (1995).
[CrossRef] [PubMed]

N. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993).
[CrossRef] [PubMed]

M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A 46, 2726–2734 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. E. Bjorkholm and A. Ashkin, “cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
[CrossRef]

Physica D (2)

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödinger equation,” Physica D 47, 393–415 (1991).
[CrossRef]

V. M. Malkin, “On the analytical theory for stationary self-focusing of radiation,” Physica D 64, 251–266 (1993).
[CrossRef]

Prog. Quantum Electron. (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 85–110 (1975).
[CrossRef]

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 14, 1353–1363 (1971) [Radiophys. Quantum Electron. 14, 1062–1070 (1971)].
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

G. Fibich and G. C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 183–240 (1999).
[CrossRef]

Other (7)

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation (Springer, New York, 1999).

The value of zc here is twice that given in Eq. (13) below, because there ψ0=c exp(−r2).

I.e., if ∫x2+y2≤ri(z)|ψ|2dxdy=Ni, then Vi(z)=∫x2+y2≤ri(z)r2|ψ|2dxdy.

The values of e can be recovered by use of the equation e=1±h2−1.

The relative error of relation (9) is less than 1.5% in the range 1/2.5≤e≤2.5.

I.e., the location of blowup in the unperturbed NLS [Eq. (4)].

Clearly, the critical power for collapse of ultrashort elliptic pulses should be higher than for circular pulses.

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Figures (10)

Fig. 1
Fig. 1

Normalized on-axis intensity as a function of axial distance for Gaussian beams ψ0=c exp[-(x/e)2-y2] with initial ellipticity e=1.3. Blowup occurs when c=20.7872 (i.e., P0=1.015Pcircular; solid curve). When c=20.795Nc (i.e., P0=1.005Pcircular; dotted curve), no blowup occurs.

Fig. 2
Fig. 2

The increase in critical power with initial ellipticity (dotted curves with circles) is well approximated by relation (9) (solid curves). Input beam profiles are A, Gaussian; B, super-Gaussian; C, Townesian.

Fig. 3
Fig. 3

Location of blowup as predicted by the variance identity [Eq. (13), dashed curve] and by the aberrationless approximation [Eq. (14), dashed-dotted curve] is significantly larger than the actual value (circles). Also shown are the curve-fitted Eqs. (11) (dotted curve) and (12) (solid curve).

Fig. 4
Fig. 4

Dynamics of Vi for the circular input beam ψ(r, 0)=21.527Ncr exp(-r2) (i.e., P0=1.5Pcircular, H0=-2.4). Here Ni=0.75Ncr, Ncr, 1.25Ncr, N01.52Ncr, and zc0.23.

Fig. 5
Fig. 5

Same as Fig. 4 but for the elliptic input beam ψ(x, y)=21.19Ncr exp[-(x/1.3)2-y2] [i.e., P0=1.5Pcr(1.3), H0=-1.8] and zc0.30.

Fig. 6
Fig. 6

Beam astigmatism as a function of normalized on-axis intensity, according to Eqs. (16) (dotted curves) and (17) (solid curves). Here e=1.3, ψ0=2cPcr(0) exp[-(x/e)2-y2], and Pcr(e)=1.92Pcircular. A, c=0.839, P0=1.1Pcr(e); B, c=1.144, P0=1.5Pcr(e); C, c=1.525, P0=2Pcr(e); D, c=3.05, P0=4Pcr(e).

Fig. 7
Fig. 7

Beam astigmatism as a function of axial distance for the correspondingly labeled parts of Fig. 6.

Fig. 8
Fig. 8

Convergence of the collapsing part of an elliptic input beam to the circular Townesian profile L-1(z)(Rr/L(z)). Here ψ0=21.1646Ncr exp[-(3x/4)2-y2] with P0=1.5Pcr(4/3) and H0=-1.75. Spatial coordinates: x (dashed curves), y (dotted curves), and r (solid curves). A, z=0.22, L=0.52; B, z=0.28, L=0.32; C, z=0.31, L=0.165.

Fig. 9
Fig. 9

Beam widths in the x direction [a(z), dotted curves] and the y direction [b(z), dashed curves] for the elliptic beam ψ0(x, y)=21.91Ncr exp[-(x/1.3)2-y2], with P0=1.5Pcr(1.3), propagating in media with saturating-nonlinearity parameter =0.01. The equations that govern propagation are A, Eq. (18); B, Eq. (19); C, Eq. (20).

Fig. 10
Fig. 10

Same as Fig. 9 but for propagation in the presence of nonparaxial effects [Eq. (22) with =0.0025].

Equations (49)

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|E|2=E02a*(z)b*(z) exp-xa*(z)2-yb*(z)2,
Pcr(e)=h(e)Pcircular,h(e)=e+1/e2,
2ik0 Az+ΔA+40ck02n2|A|2A=0,
Δ=xx+yy,
A(x, y, 0)=A0((x/a*)2+(y/b*)2)
x˜=x/r0,y˜=y/r0,z˜=z/2Ldf,
ψ(x˜, y˜, z˜)=2k0r0c0n2A(x, y, z),
P0=20n0c |A0|2dxdy=λ28π2n0n2N0,
N0=|ψ0|2dx˜dy˜.
iψz(x, y, z)+Δψ+|ψ|2ψ=0,
ψ(x, y, 0)=ψ0(x, y),
ψ0(x, y)=cf((x/a0)2+(y/b0)2),
ψ=exp(iα2z)λR(αr),r=x2+y2,
2r2+1r rR-R+R3=0,
R(0)=0,R()=0.
Ncr=2π  R2rdr2π×1.8623.
H0=|ψ0|2dx dy-½|ψ0|4dx dy<0.
c2>1a2+1b2 |f|2dxdy|f|4dxdy
|ψ0(x, y)|2dxdy>h(e)G[f],
G[f]=2|f|2dxdy |f|2dxdy|f|4dxdy.
NcrNcr[cf ((x/a0)2+(y/b0)2)]h(e)G[f].
Pcrlb=λ28π2n0n2Ncr,
Pcrub=λ28π2n0n2h(e)G[f].
Pcr(e)[0.4h(e)+0.6]Pcircular.
Vzz(z)=8H0,V= r2|ψ|2dxdy.
V(z*)=0,z*=[V0/(-4H0)]1/2.
zc=0.184[(p1/2-0.852)2-0.0219]-1/2,
p=N0/Ncr,
zc=0.317(p-1)-0.6346,
z*=1pNc/2-11/2
z*=1p-11/2.
Vi(zc)=0,N i Ncr,
Vi(zc)>0,Ni>Ncr.
|ψ|2=1a(z)b(z)Fxa(z), yb(z),
a(z)=c*|(|ψ|2)x|dxdy1/2
b(z)=c*|(|ψ|2)y|dxdy1/2,
c*=|(|ψ0|2)x|dxdy1/2|(|ψ0|2)y|dxdy1/2.
a(z)=c*Ω(z)|(|ψ|2)x|dxdy1/2,
b(z)=c*Ω(z)|(|ψ|2)y|dxdy1/2,
c*=Ω0|(|ψ0|2)x|dxdy1/2Ω0|(|ψ0|2)y|dxdy1/2.
ViL2(z)Vi(0)Vi(0)R2(0)|ψ(z, 0)|-2.
iψz(x, y, z)+Δψ+|ψ |21+|ψ |2ψ=0,0<1.
iψz(x, y, z)+Δψ+1-exp(-2|ψ|2)2ψ=0
iψz(x, y, z)+Δψ+|ψ |2ψ-|ψ |4ψ=0.
Δ+2z2E(x, y, z)+k2E=0,
k2=k021+2n2n0|E|2.
ψzz+iψz+Δψ+|ψ|2ψ=0,=λ4πr02.
iψz+Δψ+|ψ|2ψ=[Δ2ψ+4|ψ|2Δψ+4|ψ|2ψ+2(ψ)2ψ*+|ψ|4ψ].
iψz(x, y, z, t)+Δψ-Ldf kωωT2ψtt+|ψ|2ψ=0,

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