The study of focal shift in focused beams using unapodized apertures has been well documented. However, not much work has been done on apodized apertures. In this thesis we use a Fourier-Optic approach to analyze the field distribution of a focused beam around the region of geometrical focus. The analytical formulation developed is general in nature as it is valid for any arbitrary aperture functions. This is then applied to some specific cases. Two cases of interest that are considered are the unapodized and the Gaussian apodized annular apertures. In order to study the intensity distributions around the geometrical focus, simulation results are presented using closed form analytical expressions and approximate integral forms. Specific emphasis is placed on the focal shift in the two apertures and on the effect of changing various parameters. A prognosis for future work using a-Modulation on Gaussian apodized annular apertures is also presented. / Master of Science