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Extended criterion for the modulation instability
Modulation instability, following the classical Lighthill criterion, appears if nonlinearity and dispersion make opposite contributions to the wave frequency, e.g. in the framework of the one-dimensional nonlinear Schrödinger equation (NLSE). Several studies of the wave instabilities in optical fibers revealed four wave mixing instabilities that are not covered by the Lighthill criterion and require use of the generalized NLSE. We derive an extended criterion, which applies to all four wave interactions, covers arbitrary dispersion, and depends neither on the propagation equation nor on the slowly varying envelope approximation. © 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.
Generalized Lighthill criterion for the modulation instability
An @universal modulation instability is subject to Lighthill criterion: nonlinearity and dispersion should make opposite contributions to the wave frequency. Recent studies of wave instabilities in optical fibers with the minimum chromatic dispersion revealed situations in which the criterion is violated and fast unstable modulations appear due to the four wave mixing process. We derive a generalized criterion, it applies to an arbitrary dispersion and to both slow and fast unstable modulations. Since the fast modulations depend on nonlinear dispersion, we also demonstrate how to describe them in the framework of a single generalized nonlinear Schrödinger equation.
Unusual ways of four-wave mixing instability
A pump carrier wave in a dispersive system may decay by giving birth to blue- and red-shifted satellite waves due to modulation or four-wave mixing instability. We analyse situations where the satellites are so different from the carrier wave, that the red-shifted satellite either changes its propagation direction (k < 0, ω > 0) or even gets a negative frequency (k, ω < 0). Both situations are beyond the envelope approach and require application of Maxwell equations.