OSA's Digital Library

Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 17, Iss. 5 — May. 1, 2000
  • pp: 751–757

Self-guided waves and exact solutions of the nonlinear Helmholtz equation

Timo A. Laine and Ari T. Friberg  »View Author Affiliations

JOSA B, Vol. 17, Issue 5, pp. 751-757 (2000)

View Full Text Article

Enhanced HTML    Acrobat PDF (280 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The paraxial wave theory is known to lead to inaccurate predictions in self-focusing of optical beams. The nonlinear Helmholtz equation describes more accurately wave propagation in dispersive, spatially local, Kerr-type media. We derive rigorous bright and dark solutions to the nonlinear Helmholtz equation in a full three-dimensional form. These expressions are new and unique. The solutions are obtained with a multidimensional extension of the (paraxial) nonlinear Schrödinger equation. We also establish energy conservation laws for both nonlinear wave equations in terms of spatial currents. Our results give insight, for example, into the diffraction and breakup of tightly confined nonlinear fields.

© 2000 Optical Society of America

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(240.4350) Optics at surfaces : Nonlinear optics at surfaces

Timo A. Laine and Ari T. Friberg, "Self-guided waves and exact solutions of the nonlinear Helmholtz equation," J. Opt. Soc. Am. B 17, 751-757 (2000)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. R. W. Boyd, Nonlinear Optics (London, Academic, 1992).
  2. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990). [CrossRef] [PubMed]
  3. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. 16, 15–17 (1991). [CrossRef] [PubMed]
  4. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964); E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966). [CrossRef]
  5. V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 2, 138–141 (1965); P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965); Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. PQUEAH 4, 1–34 (1975). [CrossRef]
  6. D. Anderson, M. Bonneal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979); D. Anderson and M. Bonneal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979). [CrossRef]
  7. M. Karlsson, D. Anderson, and M. Desaix, “Dynamics of self-focusing and self-phase modulation in a parabolic index optical fiber,” Opt. Lett. 17, 22–24 (1992). [CrossRef] [PubMed]
  8. M. Karlsson, D. Anderson, M. Desaix, and M. Lisak, “Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses,” Opt. Lett. 16, 1373–1375 (1991). [CrossRef] [PubMed]
  9. A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).
  10. M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the selffocusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988). [CrossRef]
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Eq. (5.2-14).
  12. L. Gagnon, “Exact solutions for optical wave propagation including transverse effects,” J. Opt. Soc. Am. B 7, 1098–1102 (1990). [CrossRef]
  13. See Ref. 11, Eqs. (19.1–6) and (19.1–7) at frequency ω. Since the linear refractive index (n0) is constant and the nonlinearity is weak (γ≪1), Eq. (9) remains valid as long as the vector property of the electromagnetic field is not dominant.
  14. S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited