Analytic theory of self-similar mode-locking with rapidly varying, mean-zero dispersion

Brandon G. Bale, J. Nathan Kutz

    Research output: Chapter in Book/Published conference outputConference publication


    Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures observed in a broad range of applications and natural phenomena. Indeed, they have been predicted or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system, not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed in mode-locked laser cavities.

    Original languageEnglish
    Title of host publicationProceedings of SPIE - The International Society for Optical Engineering
    Subtitle of host publicationPhotonics North 2009
    Publication statusPublished - 20 Nov 2009
    EventPhotonics North 2009 - Laval, QC, United Kingdom
    Duration: 24 May 200927 May 2009


    ConferencePhotonics North 2009
    Country/TerritoryUnited Kingdom
    CityLaval, QC

    Bibliographical note

    Copyright 2009 SPIE. One print or electronic copy may be made for personal use only. Systematic reproduction, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.


    • Dispersion-management
    • Mode-locked lasers
    • Self-similarity
    • Similaritons


    Dive into the research topics of 'Analytic theory of self-similar mode-locking with rapidly varying, mean-zero dispersion'. Together they form a unique fingerprint.

    Cite this