Properties of focused apertures in the fresnel region

The diffraction field of continuous, rectangular apertures is analyzed when the system is focused in the Fresnel region. The Fresnel region is defined by phase and amplitude considerations, and the boundary separating the near and Fresnel regions is given as well as the conventional boundary distinguishing between the Fresnel region and far field. Curves have been plotted for several square apertures of varying size for both uniform amplitude illumination and tapered illumination while the focused condition is maintained. A theorem is given proving that the far-field pattern occurs in the focal plane of a continuous, focused aperture. Also included is the expression for gain of a focused rectangular antenna, and a discussion of the concept of gain where strict 1/R^{2} dependence does not exist. A comparison of focused circular and rectangular apertures is made in regard to beamwidth, energy between 3-db points, and energy in the main beam. Consideration is also given to the problem of maximizing the field at the point of focus, and as a corollary the "depth of focus" is examined.