Skip to main content
SpringerLink
Log in
Book cover

Instabilities and Chaos in Quantum Optics pp 139–173Cite as

Global Bifurcations and Turbulence in a Passive Optical Resonator

  • Chapter

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 34))

Abstract

Dynamical instabilities in an externally-pumped passive nonlinear optical ring resonator are reviewed. Starting with the simple plane-wave model, phase portraits are used to explain a new type of bifurcation associated with the formation of homoclinic orbits in the phase plane. The plane-wave map is shown to be more unstable to perturbations with a short-scale transverse structure than to plane-wave perturbations. This latter result has important ramifications, one of which is that period doubling cascades should be unlikely in high-finesse optical resonators. Instead, a modulational type of chaos is predicted to occur. Transverse effects are, in fact, inevitable and the study is extended to include pump beams with Gaussian spatial profiles in one and two transverse dimensions. The role of transverse solitons and solitary waves as steady states or as coherent spatial structures undergoing temporally chaotic oscillations is discussed. A self-focusing induced route to optical turbulence is identified and contrasted with plane-wave and self-defocusing models. The importance of transverse effects in determining the final asymptotic state of the optical field envelope will be emphasized.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A useful review of this topic may be found in “Nonlinear Dynamics”, 357, Annals of the New York Academy of Sciences, ed. by R.G. Helleman (1980)

    Google Scholar 

  2. A. Libchaber, S. Fauve, C. Laroche: Physica 7D, 73 (1984) and references therein

    ADS  Google Scholar 

  3. H.L. Swinney: Physica 7D, 3 (1984) and references therein

    ADS  Google Scholar 

  4. H.L. Swinney, J.P. Gollub (eds.): Hydrodynamic Instabilities and the Transition to Turbulence, 2nd. ed., Topics Appl. Phys., Vol. 45 (Springer, Berlin, Heidelberg 1985)

    MATH  Google Scholar 

  5. S.E. Newhouse, D. Ruelle, F. Takens: Commun. Math. Phys. 64, 35 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. M.J. Feigenbaum: J. Stat. Phys. 19, 25 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Y. Pomeau, P. Manneville: Commun. Math. Phys. 74, 189 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  8. E.N. Lorenz: J. Atmos. Sci. 20, 130 (1963)

    Article  ADS  Google Scholar 

  9. A.J. Lichtenberg, M.A. Lieberman: Regular and Stochastic Motion, (Springer, Berlin, Heidelberg 1983) Chap. 7

    MATH  Google Scholar 

  10. H. Haken: Phys. Lett. 53A, 77 (1975)

    ADS  Google Scholar 

  11. Other contributions to this volume. See in particular the articles by N.B. Abraham, et al. and F.T. Arrechi (Chaps. 2 and 3)

    Google Scholar 

  12. K. Ikeda: Opt. Commun. 30, 256 (1979)

    Article  ADS  Google Scholar 

  13. Y. Silberberg, I. Bar Joseph: Phys. Rev. Lett. 48, 1541 (1982)

    Article  ADS  Google Scholar 

  14. H.G. Winful, G.D. Cooperman: Appl. Phys. Lett. 40, 298 (1982)

    Article  ADS  Google Scholar 

  15. H.M. Gibbs, F.A. Hopf, D.L. Kaplan, R.L. Shoemaker: Phys. Rev. Lett. 46, 474 (1981)

    Article  ADS  Google Scholar 

  16. R.G. Harrison, W.J. Firth, C.A. Emshary, I.A. Al-Saidi: Phys. Rev. Lett., 51, 562 (1983)

    Article  ADS  Google Scholar 

  17. R.G. Harrison, W.J. Firth, I.A. Al-Saidi: Phys. Rev. Lett. 53, 258 (1984)

    Article  ADS  Google Scholar 

  18. M. LeBerre, E. Ressayre, A. Tallet, K. Tai, F.A. Hopf, H.M. Gibbs, J.V. Moloney: “Optical bistability and instabilities via diffration-free-encoding and a single feedback mirror”, postdeadline presentation IQEC ’84, Los Angeles. The first observation of period-2 and evidence for period-6 oscillations in a CW experiment was reported.

    Google Scholar 

  19. J.V. Moloney, F.A. Hopf, H.M. Gibbs: Phys. Rev. A25, 3442 (1982); and Appl. Phys. B28, 98 (1982)

    Article  ADS  Google Scholar 

  20. J.V. Moloney, F.A. Hopf: J. Opt. Soc. Am. 71, 1634 (1981)

    ADS  Google Scholar 

  21. W.J. Firth, E.M. Wright: Phys. Lett. 92A, 211 (1982)

    ADS  Google Scholar 

  22. W.J. Firth, E. Abraham, E.M. Wright: Applied Phys. B28, 170 (1982); a number of analytic models that assume mode matched Fabry-Perot resonators are discussed and referenced in Optical Bistability II, ed. by C.M. Bowden, H.M. Gibbs, S.L. McCall (Plenum, New York 1984)

    Google Scholar 

  23. S.M. Hammel, C.K.R.T. Jones, J.V. Moloney: “Global Dynamical Behavior of the Optical Field in a Ring Cavity”, J. Opt. Soc. Am. B2, 552 (1985)

    Article  ADS  Google Scholar 

  24. J.V. Moloney, S.M. Hammel, C.K.R.T. Jones: “Optical Bistability II”, p. 87 (1984)

    Google Scholar 

  25. J.V. Moloney: “Solitons in Optical Bistability”, Phil. Trans. Roy. Soc. A313, 429 (1984); J. Opt. Soc. Am. B3, 467 (1984)

    Article  ADS  Google Scholar 

  26. D.W. McLaughlin, J.V. Moloney, A.C. Newell: “Two-Dimensional Transverse Solitary Waves as Asymptotic States of the Field in an Optical Resonator”, to be published

    Google Scholar 

  27. A.C. Newell: Solitons in Mathematics and Physics, CBMS Lecture Series, Vol. 48, SIAM, 1984

    Google Scholar 

  28. D.W. McLaughlin, J.V. Moloney, A.C. Newell: Phys. Rev. Lett. 51, 75 (1983)

    Article  ADS  Google Scholar 

  29. J.V. Moloney: Phys. Rev. Lett. 53, 556 (1984)

    Article  ADS  Google Scholar 

  30. D.W. McLaughlin, J.V. Moloney, A.C. Newell: “A New Class of Instabilities in Passive Optical Cavities”, Phys. Rev. Lett. 55, 168 (1985)

    Google Scholar 

  31. D.W. McLaughlin, J.V. Moloney, A.C. Newell: Physica D (to appear)

    Google Scholar 

  32. A comprehensive review of Optical Bistability and Instabilities may be found in Optical Bistability, ed. by C.M. Bowden, M. Ciftan, and H.R. Robl (Plenum, New York, 1981); “Optical Bistability II”, ed. by C.M. Bowden, H.M. Gibbs, and S.L. McCall (Plenum, New York, 1984)

    Google Scholar 

  33. R. Bonifaccio, L.A. Lugiato: Lett. Nuovo Cimento 21, 510 (1978)

    Article  Google Scholar 

  34. L. Allen, J.H. Eberly: Optical Resonance and Two Level Atoms (Wiley-Interscience, 1975)

    Google Scholar 

  35. R.Y. Chiao, E. Garmire, C.H. Townes: Phys. Rev. Lett. 13, 479 (1964)

    Article  ADS  Google Scholar 

  36. E. Garmire, R.Y. Chiao, C.H. Townes: Phys. Rev. Lett. 16, 347 (1966)

    Article  ADS  Google Scholar 

  37. A.J. Campillo, S.L. Shapiro, B.R. Suydam: Appl. Phys. Lett. 23, 628 (1973)

    Article  ADS  Google Scholar 

  38. J. Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York 1983)

    MATH  Google Scholar 

  39. J.V. Moloney: Opt. Commun. 48, 435 (1984)

    Article  ADS  Google Scholar 

  40. J.V. Moloney, H.M. Gibbs, F.A. Hopf: Optical Bistability II, ed. by C.M. Bowden, H.M. Gibbs, and S.L. McCall (Plenum, New York 1984)

    Google Scholar 

  41. J.V. Moloney, S.M. Hammel, C.K.R.T. Jones: J. Opt. Soc. Am. B3, 499 (1984)

    ADS  Google Scholar 

  42. S.M. Hammel, C.K.R.T. Jones, J.V. Moloney: to be published

    Google Scholar 

  43. S. Smale: Bull. Amer. Math. Soc. 73, 747 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  44. S.E. Newhouse: “Progress in Mathematics”, No. 8 (Birkhauser-Boston 1980) pp. 1–114

    Google Scholar 

  45. N.K. Gavrilov, L.P. Silnikov: Math. USSR S6, 88, 467 (1972)

    Article  MathSciNet  Google Scholar 

  46. J.A. Yorke: private communication

    Google Scholar 

  47. J.V. Moloney, H.M. Gibbs: Phys. Rev. Lett. 58, 1607 (1982)

    Article  ADS  Google Scholar 

  48. J.C. Roux: Physica 7D, 57 (1983)

    MathSciNet  ADS  Google Scholar 

  49. R.S. Gioggia, N.B. Abraham: Phys. Rev. Lett. 51, 650 (1983)

    Article  ADS  Google Scholar 

  50. R.S. Gioggia, A.M. Albano, C.M. Searle, T.H. Bhyba, N.B. Abraham: J. Opt. Soc. Am. B1, 499 (1984)

    ADS  Google Scholar 

  51. P. Grassberger, I. Procaccia: Phys. Rev. Lett. 50, 346 (1983); Physica 9D, 189 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  52. G. Iooss, W.F. Langford: p. 489 in Ref. 7.1

    Google Scholar 

  53. P. Davis, K. Ikeda: Phys. Lett. 100A, 455 (1984)

    ADS  Google Scholar 

  54. C. Grebogi, E. Ott, J.A. Yorke: “Attractors on an N-Torus: Quasiperiodicity versus chaos”, preprint

    Google Scholar 

Download references

Authors
  1. J. V. Moloney
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Fisica dell’ Università, Istituto Nazionale di Ottica, Largo E. Fermi 6, I-50125, Firenze, Italy

    F. Tito Arecchi

  2. Department of Physics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, UK

    Robert G. Harrison

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Moloney, J.V. (1987). Global Bifurcations and Turbulence in a Passive Optical Resonator. In: Arecchi, F.T., Harrison, R.G. (eds) Instabilities and Chaos in Quantum Optics. Springer Series in Synergetics, vol 34. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-71708-6_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-71708-6_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-71710-9

  • Online ISBN: 978-3-642-71708-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation