Can Abbe theory of image formation be improved ?

Abstract

Abbe theory of image formation can be expanded from two-dimensional to three-dimensional space. It has to consider an optical system that forms an image of a grating in an object space. It is possible to introduce various inclinations of this grating relatively the optical axis round the axis in the object plane that is parallel to its slits. In this case, the minimum resolvable period and the corresponded spatial cut off frequency of the optical system may be calculated as the functions of the angle of grating inclination, the numerical apertures and the wavelength using the idea of Abbe theory. In case of axial symmetry of the optical system, the function for calculation of the spatial cut off frequencies describes a surface in the three-dimensional space of spatial frequencies that covers the three-dimensional spatial bandwidth. The spatial harmonics with spatial frequencies inside this spatial bandwidth can pass through an optical system and they participate in formation of three-dimensional images. The other spatial harmonics cannot pass through it. This approach makes understanding and explanation of formation of three-dimensional images clearer, faster and logically completed.