Parametric amplification of spatiotemporal localized envelope waves

The process of parametric amplification of dispersionless and diffractionless spatiotemporal envelope waves in nonlinear chi((2)) media is theoretically investigated in the undepleted pump limit and assuming a continuous-wave (cw) plane wave pump. By deriving an extended paraxial envelope equation for the signal field, we show that distortionless and efficient parametric amplification is possible for different types of localized waves, including pulsed Bessel beams, envelope X waves and sinc-shaped envelope waves. An analytic expression of the spectral parametric gain for polychromatic Bessel beams is also derived beyond the paraxial approximation and takes into account higher-order dispersion effects.