Fresnel region approximations for wide angles and large fresnel numbers

A general approximation is developed which permits the accurate computation of radiation patterns from circular apertures and reflectors by single integration both at wide angles and for large Fresnel numbers. Both the far-field and small angle Fresnel region approximations are obtained as special cases. The approximation is applicable to regions where the ratio of the aperture diameter to the distance to the point of observation is small and where this distance is many wavelengths. A comparison is made with the values determined by precise numerical integration for a certain range of parameters and shows good agreement for both the amplitudes and phases of the integrals within the expected range of validity of the approximation.