Huygens–Fresnel principle and Abbe formula

Abstract

The paper describes the approach for identification of the spatial bandwidth of an optical system in the lateral and axial directions. This approach considers application of Huygens–Fresnel principle to obtain the integral for calculation of amplitude distribution near a focal point. The replacement of integration variables leads to identification of the limits of integration in a space of spatial frequencies. These limits are the spatial cutoff frequencies for amplitude distribution. Doubling these values produces the spatial cutoff frequencies for intensity distribution, and it shows the same result as Abbe theory predicts. The proposed approach can be used for mathematical explanation why optical systems have limited spatial resolution and why the spatial harmonics with high frequencies cannot pass through optical systems. The analog of Abbe theory for axial direction is proposed. To identify the spatial bandwidth in axial direction, it has to consider the grating located in the plane with an optical axis and with slits perpendicular to this axis. Then it is possible to follow Abbe theory: to consider the diffraction on this grating, formation of a spatial spectrum, and a grating image with axial orientation in an image space.