Spatiotemporal modulation instability as off-axis parametric amplification: insights from the phase

The canonical linear-stability analysis of spatiotemporal modulation instability (MI), which treats MIs as instabilities of the steady-state solution of the field evolution equation with respect to weak complex harmonic perturbations, is shown to be physically equivalent to a coupled-wave analysis of off-axis parametric amplification of the Stokes and anti-Stokes fields by an intense pump wave. With an appropriate space-evolution transfer matrix, a complex harmonic trial function used in the standard MI model translates into a pair of coupled off-axis waves, of which one is exponentially growing, while the other is exponentially decreasing, thus recovering a two-wave field structure that is inherent in the fields undergoing parametric amplification through four-wave mixing. Analysis of the phase of these fields offers useful physical insights into high-order dispersion effects in spatiotemporal MIs, suggests a physically transparent and accurate method to include high-order dispersion in MI gain calculations, and reveals new spatiotemporal MI effects induced by high-order dispersion.