Vector solution of the diffraction task using the Hertz vector

A large class of diffraction problems can be solved on the basis of the Huygens principle. However, methods of solving diffraction problems based on this principle exhibit narrow boundaries of applicability. The goal of the present work is to offer a relatively simple physically based and mathematically strict "dipole wave" vector theory of non-paraxial diffraction of electromagnetic radiation which allows analytical solutions of typical diffraction problems. The suggested theory logically retains the wave approach used in the Kirchhoff method and does not exhibit strict limitations to applicability inherent in the Kirchhoff integral. The diffraction problem is solved by using the Hertz vector in the Kirchhoff integral instead of the field vector. The method efficiency is illustrated in several examples. Analytical solutions of diffraction base problems have been obtained for linearly polarized radiation on an infinite slit and on various-shaped holes at an arbitrary angle of incidence and polarization. It was shown the possibility of vector addition particular solutions to obtain diffraction patterns from several holes. The diffraction of radiation with azimuthal and radial directions of polarization on a ring slit is also considered. The main qualitative feature of the obtained solutions is the presence of "poles" one or two points of zero field in the diffraction pattern which are superimposed on the common system of light and dark fringes. The poles are located along electrical field vector directions. The vector analytical formulas describing the propagation of some laser beams in the free space have been obtained too. The solutions of the diffractive problems satisfy the Maxwell equations and the reciprocity principle.