Propagation of localized optical waves in media with dispersion, in   dispersionless media and in vacuum. Low diffractive regime
  We present a systematic study on linear propagation of ultrashort laser pulses in media with dispersion, dispersionless media and vacuum. The applied method of amplitude envelopes gives the opportunity to estimate the limits of slowly warring amplitude approximation and to describe an amplitude integro-differential equation, governing the propagation of optical pulses in single cycle regime. The well known slowly varying amplitude equation and the amplitude equation for vacuum are written in dimensionless form. Three parameters are obtained defining different linear regimes of the optical pulses evolution. In contrast to previous studies we demonstrate that in femtosecond region the nonparaxial terms are not small and can dominate over transverse Laplacian. The normalized amplitude nonparaxial equations are solved using the method of Fourier transforms. Fundamental solutions with spectral kernels different from Fresnel one are found. One unexpected new result is the relative stability of light pulses with spherical and spheroidal spatial form, when we compare their transverse enlargement with the paraxial diffraction of lights beam in air. It is important to emphasize here the case of light disks, i.e. pulses whose longitudinal size is small with respect to the transverse one, which in some partial cases are practically diffractionless over distances of thousand kilometers. A new formula which calculates the diffraction length of optical pulses is suggested. 
Lubomir M. Kovachev