Natureʹs infinite books of secrecy

In this paper we present an approach for the calculation of aberration coefficients using accurate ray tracing. For a given optical system, intersections of a large number of trajectories with a given plane are computed. In the Gaussian image plane the imaging with the selected optical system can be described by paraxial and aberration coefficients (geometric and chromatic) that can be calculated by least-squares fitting of the analytical model on the computed trajectory positions. An advantage of such a way of computing the aberration coefficients is that, in comparison with the aberration integrals and the differential algebra method, it is relatively easy to use and its complexity stays almost constant with the growing complexity of the optical system. This paper shows a tested procedure for choosing proper initial conditions and computing the coefficients of the fifth-order geometrical and third-order, first-degree chromatic aberrations by ray tracing on an example of a weak electrostatic lens. The results are compared with the values for the same lens from a paper Liu [Ultramicroscopy 106 (2006) 220–232].