Skip to main content

Some Modern Aspects of Self-focusing Theory

  • Chapter
Self-focusing: Past and Present

Part of the book series: Topics in Applied Physics ((TAP,volume 114))

Abstract

In this chapter, we give a brief summary of the present status of self-focusing theory, while trying to highlight the fascinating evolution of this theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Y.R. Shen. Self-focusing: Experimental. Prog. Quant. Electr. 4, 1–34 (1975).

    Article  ADS  Google Scholar 

  2. J.H. Marburger. Self-focusing: theory. Prog. Quant. Electr. 4, 35–110 (1975).

    Article  ADS  Google Scholar 

  3. G. Fibich, G.C. Papanicolaou. Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension. SIAM J. Applied Math. 60, 183–240 (1999).

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  5. S.N. Vlasov, V.A. Petrishchev, V.I. Talanov. Averaged description of wave beams in linear and nonlinear media. Izv. Vuz Radiofiz (in Russian), 14, 1353–1363 (1971) Radiophys. and Quantum Electronics 14, 1062–1070 (1971) (in English)

    Google Scholar 

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

    ADS  Google Scholar 

  7. R.Y. Chiao, E. Garmire, C.H. Townes. self-trapping of optical beams. Phys. Rev. Lett. 13, 479–482 (1964).

    Article  ADS  Google Scholar 

  8. M.I. Weinstein. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87, 567–576 (1983).

    Article  ADS  MATH  Google Scholar 

  9. P.L. Kelley. Self-focusing of optical beams. Phys. Rev. Lett. 15, 1005–1008 (1965).

    Article  ADS  Google Scholar 

  10. F. Merle. On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Comm. Pure Appl. Math. 45, 203–254 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Merle. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power. Duke Math. J. 69, 427–454, (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. G. Fibich, B. Ilan. Self focusing of elliptic beams: An example of the failure of the aberrationless approximation. JOSA B 17, 1749–1758 (2000).

    Article  ADS  Google Scholar 

  14. G. Fibich. Self-focusing in the damped nonlinear Schrödinger equation. SIAM. J. Appl. Math 61, 1680–1705 (2001).

    Article  MathSciNet  MATH  Google Scholar 

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

    ADS  Google Scholar 

  16. M.J. Landman, G.C. Papanicolaou, C. Sulem et al. Stability of isotropic singularities for the nonlinear Schrödinger equation. Physica D 47, 393–415, (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. F. Merle, P. Raphael. Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13, 591–642 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  18. K.D. Moll, A.L. Gaeta, G. Fibich. Self-similar optical wave collapse: Observation of the Townes profile. Phys. Rev. Lett. 90, 203902 (2003).

    Article  ADS  Google Scholar 

  19. T.D. Grow, A.A. Ishaaya, L.T. Vuong et al. Collapse dynamics of super-gaussian beams. Opt. Express 14, 5468–5475 (2006).

    Article  ADS  Google Scholar 

  20. G. Fibich, N. Gavish, X.P. Wang. New singular solutions of the nonlinear schrodinger Schrödinger equation. Physica D 211, 193–220 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. V.R. Costich, B.C. Johnson. Apertures to shape high-power laser beams. Laser Focus 10, 43–46 (1974).

    Google Scholar 

  22. I. Kryzhanovskiĩ, B.M. Sedov, V.A. Serebryakov et al. Formation of the spatial structure of radiation in solid-state laser systems by apodizing and hard apertures. Sov. J. Quant. Electron. 13, 194–198 (1983).

    Article  ADS  Google Scholar 

  23. L. Bergé, C. Goutéduard, J. Schjodt-Erikson, H. Ward. Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation, and polarization states. Physica D 176, 181–211, (2003).

    Article  ADS  MATH  Google Scholar 

  24. V.M. Malkin. Dynamics of wave collapse in the critical case. Phys. Lett. A 151, 285–288 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  25. G.M. Fraiman. Asymptotic stability of manifold of self-similar solutions in self-focusing. Sov. Phys. JETP 61, 228–233 (1985).

    MathSciNet  Google Scholar 

  26. M.J. Landman, G.C. Papanicolaou, C. Sulem et al. Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A 38, 3837–3843, (1988).

    Article  ADS  MathSciNet  Google Scholar 

  27. B.J. LeMesurier, G.C. Papanicolaou, C. Sulem, P.L. Sulem. Local structure of the self-focusing singularity of the nonlinear Schrödinger equation. Physica D 32, 210–226 (1988).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. G. Fibich. An adiabatic law for self-focusing of optical beams. Opt. Lett. 21, 1735–1737 (1996).

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  30. E.L. Dawes, J.H. Marburger. Computer studies in self-focusing. Phys. Rev. 179, 862–868 (1969).

    Article  ADS  Google Scholar 

  31. A.J. Campillo, S.L. Shapiro, B.R. Suydam. Relationship of self-focusing to spatial instability modes. Apple. Phys. Lett. 24, 178–180 (1974)

    Article  ADS  Google Scholar 

  32. G. Fibich, S. Eisenmann, B. Ilan et al. Self-focusing distance of very high power laser pulses. Opt. Express 13, 5897–5903 (2005).

    Article  ADS  Google Scholar 

  33. G. Fibich, Y. Sivan, Y. Ehrlich et al. Control of the collapse distance in atmospheric propagation. Opt. Express 14, 4946–4957 (2006).

    Article  ADS  Google Scholar 

  34. V.I. Bespalov, V.I. Talanov. Filamentary structure of light beams in nonlinear media. Hz. Eksper. Tenor. Fizz. - Pis’ma Redact. (U.Sis.R. JET), 3, 471–476, 1966. Transf. in JET Lett. 3, 307–310 (1966).

    Google Scholar 

  35. G. Fibich, B. Ilan. Vectorial and random effects in self-focusing and in multiple filamentation. Physica D 157, 112–146 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. K. Konno, H. Suzuki. Self-focusing of a laser bean in nonlinear media. Physica Scripta 20, 382–386 (1979).

    Article  ADS  Google Scholar 

  37. J.M. Soto-Crespo, E.M. Wright, N.N. Akhmediev. Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium. Phys. Rev. A 45, 3168–3174 (1992).

    Article  ADS  Google Scholar 

  38. G. Fibich, B. Ilan. Self-focusing of circularly polarized beams. Phys. Rev. E 67, 036622 (2003).

    Article  ADS  Google Scholar 

  39. L.T. Vuong, T.D. Grow, A. Ishaaya et al. Collapse of optical vortices. Phys. Rev. Lett. 96, 133901 (2006).

    Article  ADS  Google Scholar 

  40. G. Fibich, B. Ilan. Multiple filamentation of circularly polarized beams. Phys. Rev. Lett. 89, 013901 (2002)

    Article  ADS  Google Scholar 

  41. J. W. Grantham, H.M. Gibbs, G. Khitrova et al. Kaleidoscopic spatial instability: Bifurcations of optical transverse solitary waves. Phys. Rev. Lett. 66, 1422–1425 (1991).

    Article  ADS  Google Scholar 

  42. G. Fibich, B. Ilan. Vectorial effects in self-focusing and multiple filamentation. Opt. Lett. 26, 840–842 (2001).

    Article  ADS  Google Scholar 

  43. A. Dubietis, G. Tamošauskas, G. Fibich, et al. Multiple filamentation induced by input-beam ellipticity. Opt. Lett. 29, 1126–1128 (2004).

    Article  ADS  Google Scholar 

  44. G. Fibich, S. Eisenmann, B. Ilan et al.: Control of multiple filamentation in air. Opt. Lett. 29, 1772–1774 (2004).

    Article  ADS  Google Scholar 

  45. G. Mechain, A. Couairon, M. Franco et al. Organizing multiple femtosecond filaments in air. Phys. Rev. Lett. 93, 035003 (2004).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  47. G. Fibich, V.M. Malkin, G.C. Papanicolaou. Beam self-focusing in the presence of small normal time dispersion. Phys. Rev. A 52, 4218–4228 (1995).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  49. N.A. Zharova, A.G. Litvak, T.A. Petrova et al. Multiple fractionation of wave structures in a nonlinear medium. JETP Lett., 44 13–17 (1986).

    ADS  Google Scholar 

  50. P. Chernev, V. Petrov. Self-focusing of light pulses in the presence of normal group-velocity dispersion. Opt. Lett. 17, 172–174 (1992).

    Article  ADS  Google Scholar 

  51. S.A. Diddams, H.K. Eaton, A.A. Zozulya, et al. Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media. Opt. Lett. 23, 379–381 (1998).

    Article  ADS  Google Scholar 

  52. G.G. Luther, A.C. Newell, J.V. Moloney. The effects of normal dispersion on collapse events. Physica D 74, 59–73 (1994).

    Article  ADS  MATH  Google Scholar 

  53. J.K. Ranka, R.W. Schirmer et al. A.L. Gaeta. Observation of pulse splitting in nonlinear dispersive media. Phys. Rev. Lett. 77, 3783–3786 (1996).

    Article  ADS  Google Scholar 

  54. J.E. Rothenberg. Pulse splitting during self-focusing in normally dispersive media. Opt. Lett., 17 583–585, (1992).

    Article  ADS  Google Scholar 

  55. K. Germaschewski, R. Grauer, L. Berge et al. Splittings, coalescence, bunch and snake patterns in the 3D nonlinear Schrödinger equation with anisotropic dispersion. Physica D 151, 175–198 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. G. Fibich, W. Ren, X.P. Wang. Numerical simulations of self-focusing of ultrafast laser pulses. Phys. Rev. E 67, 056603 (2003).

    Article  ADS  Google Scholar 

  57. J. Coleman, C. Sulem. Numerical simulations of blow-up solutions of the vector nonlinear Schrödinger equation. Phys. Rev. E 66, 036701 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  58. S.N. Vlasov. Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium. Sov. J. Quantum Electron. 17, 1191–1193 (1987).

    Article  ADS  Google Scholar 

  59. M.D. Feit, 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).

    Article  ADS  Google Scholar 

  60. N. Akhmediev, 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).

    Article  ADS  Google Scholar 

  61. N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo. Does the nonlinear Scharödinger equation correctly describe beam propagation? Opt. Lett. 18, 411–413 (1993).

    Article  ADS  Google Scholar 

  62. G. Fibich, S.V. Tsynkov. High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering. J. Comput. Phys. 171, 1–46 (2001).

    Article  MathSciNet  Google Scholar 

  63. G. Fibich, S.V. Tsynkov. Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions. J. Comput. Phys. 210, 183–224 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. G. Fibich, B. Ilan, S.V. Tsynkov. Computation of nonlinear backscattering using a high-order numerical method. J. Sci. Comput. 17, 351–364 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  65. G. Fibich, B. Ilan, S.V. Tsynkov. Backscattering and nonparaxiality arrest collapse of nonlinear waves. SIAM. J. Appl. Math. 63, 1718–1736 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  66. M. Sever. An existence theorem for some semilinear elliptic systems. J. Differnetial Equations 226, 572–593 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  67. S. Chi, Q. Guo. Vector theory of self-focusing of an optical beam in Kerr media. Opt. Lett. 20, 1598–1560 (1995).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Fibich, G. (2009). Some Modern Aspects of Self-focusing Theory. In: Boyd, R.W., Lukishova, S.G., Shen, Y. (eds) Self-focusing: Past and Present. Topics in Applied Physics, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-0-387-34727-1_17

Download citation

Publish with us

Policies and ethics