Far-field spectral characteristics of a space-time-dependent Gaussian pulsed beam from an annular aperture

The far-field spectral characteristics of a space-time-dependent Gaussian pulsed beam passing through an annular aperture are studied. Based on the expansion of a hard aperture function into a finite sum of complex Gaussian functions and starting from the Fresnel diffraction integral, some analytical expressions for the spectral distribution of the diffraction field are derived. The expressions for a circular black screen and a circular aperture are given as special cases of the general results. The red and blue shifts of the spectral intensity maximum and the spectral intensity shape distortion and distribution are studied and illustrated with numerical calculations.