The airy disc formula for systems of high relative aperture

The light distribution near the focus of high-aperture systems is investigated, account being taken of two factors which are neglected in the derivation of the Airy disc formula for low-aperture systems. These are: (a) an inhomogeneity of amplitude which is shown to be present over the emergent spherical wave-surface, and (b) the fact that light vectors at the focus of the wave are perpendicular to the ray-paths associated with them, it being shown that it is only necessary to consider the components of such light vectors in the direction of the light vector of the incident polarized wave. A further consideration is that it is not necessary to assume that the emergent spherical wave-surface is coincident with the plane of the exit pupil. The inhomogeneity of amplitude is shown to be different for the case of a Gaussian system and for one obeying the sine condition: the problem is solved for each of these cases, and conclusions are drawn concerning the limits within which the low-aperture formula is valid, together with some remarks on the resolving power of high-aperture systems.