Diffraction of femtosecond pulses; nonparaxial regime

We present a systematic study of linear propagation of ultrashort laser pulses in media with dispersion, dispersionless media, and vacuum. The applied method of amplitude envelopes makes it possible to estimate the limits of the slowly varying amplitude approximation and to describe an amplitude integrodifferential equation governing propagation of optical pulses in the single-cycle regime in solids. The well-known slowly varying amplitude equation and the amplitude equation for the vacuum case are written in dimensionless form. Three parameters are obtained defining different linear regimes of optical pulse evolution. In contrast to previous studies we demonstrate that in the femtosecond region the nonparaxial terms are not small and can dominate over the transverse Laplacian. The normalized amplitude nonparaxial equations are solved using the method of Fourier transforms. Fundamental solutions with spectral kernels different from those according to Fresnel are found. Exact unidirectional analytical solution of the nonparaxial amplitude equations and the 3D wave equations with initial conditions compatible with Gaussian light bullets are obtained also. One unexpected new result is the relative stability of light bullets (pulses with spherical and spheroidal spatial form) when we compare their transverse enlargement with paraxial diffraction of light beams 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 a thousand kilometers. A new formula that calculates the diffraction length of optical pulses is suggested. Finally, propagation of single-cycle pulses in air and vacuum was investigated, and a coronal (semispherical) form of diffraction at short distances was observed.