The spatially averaged electric field in the near field and far field of a circular aperture

This paper presents a theoretical and numerical investigation of the spatially averaged electric field in the beam of a circular aperture. The investigation leads to closed-form analytical expressions, based on scalar diffraction theory, which describe the spatially averaged electric field in the Fresnel region of a circular aperture excited by a spatially uniform, harmonic plane wave. The expressions ultimately permit rapid, practical, and efficient prediction of certain routine electromagnetic measurements. Because the expressions are valid in the Fresnel region, they are also valid in the near field, the far field, and the Fraunhofer region of a circular aperture. In fact, it is shown that the closed-form expressions contain, as special cases, classic on-axis and far-field results associated with a circular aperture. The analytical expressions are based on a generalization of Fresnel diffraction originally developed by Lommel in the late 1800s. Hence, a thorough review of the literature on the Lommel diffraction formulation is presented. Finally, it is shown that results obtained from the closed-form expressions compare quite favorably to results obtained from the exact solution computed via the dyadic Green's function approach.