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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Vol. 17, Iss. 10 — Oct. 1, 2000
  • pp: 1749–1758

Self-focusing of elliptic beams: an example of the failure of the aberrationless approximation

G. Fibich and B. Ilan  »View Author Affiliations


JOSA B, Vol. 17, Issue 10, pp. 1749-1758 (2000)
http://dx.doi.org/10.1364/JOSAB.17.001749


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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

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.5940) Nonlinear optics : Self-action effects
(260.5950) Physical optics : Self-focusing

Citation
G. Fibich and B. Ilan, "Self-focusing of elliptic beams: an example of the failure of the aberrationless approximation," J. Opt. Soc. Am. B 17, 1749-1758 (2000)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-17-10-1749


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References

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

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