Wave Theory for a Simple Lens

Abstract

A new wave theory describing image formation by a simple lens is formulated in the angular spectrum representation. It is closely related to Luneburg's theory of instrumental optics but is free of certain geometrical approximations made in his theory. The validity of Luneburg's essentially geometrical treatment of the lens aperture is discussed and the approximations involved are found to be justified for isoplanatic optical systems with small numerical aperture. Some concepts usually found in lens theories based on geometrical optics are seen to have analogues in the present wave theory. In particular, a connection is made between homogeneous plane waves in an angular spectrum expansion of the field and the light rays belonging to a family of rays that pass through the lens. The fundamental relations assumed in Fourier optics are shown to follow from this theory when they are applied to the special case of isoplanatic optical systems with small numerical aperture. The image field of a scalar dipole formed by a diffraction-limited lens is calculated using these results.