Negative frequencies get real: a missing puzzle piece in nonlinear optics
Motivated by recent experimental results, we demonstrate that the ubiquitous pulse propagation equation based on a single generalized nonlinear Schroedinger equation is incomplete and inadequate to explain the formation of the so called negative resonant radiation emitted by optical solitons. The origin of this deficiency is due to the absence of a peculiar nonlinear coupling between the positive and negative frequency components of the ultrashort pulse spectrum during propagation, a feature that the slowly-varying envelope approximation is unable to capture. We therefore introduce a conceptually new model, based on the envelope of the analytic signal, that takes into account the full spectral dynamics of all frequency components, is prone to analytical treatment and retains the simulation efficiency of the nonlinear Schroedinger equation. We use our new equation to derive from first principles the phase-matching condition of the negative resonant radiation observed in previous experiments.