Calculation of the axially symmetric eigen functions of the finite propagation operator in the near-field diffraction

The propagation of axially symmetric laser beams in the near diffraction (at a distance in the order of the wavelength) can be described by means of an expansion in plane waves, which after considering axial symmetry reduces to an axisymmetric propagation operator involving Fourier-Hankel transforms. The eigenfunctions of the operator, when eigenvalues are close to one, determine the characteristics of the signals (information) transmitted lossless (without distortion). The beam propagation distance and the region of spatial frequency limitation are parameters of the operator and essentially change the set of eigenvalues and functions. We calculate the axisymmetric eigenfunctions of the finite propagation operator in the near diffraction zone and investigate their qualitative and quantitative characteristics depending on the propagation distance and the constraints imposed in the object and spectral domains.