High dispersive electrostatic mirrors of rotational symmetry with the third order time-of-flight focusing by energy

Expressions describing the time of flight of charged particles taking into account time-of-flight chromatic aberrations in electrostatic mirrors of rotational symmetry are presented. The conditions of time-of-flight focusing by particle energy up to the third order inclusively in the three-electrode mirrors, the electrodes of which are coaxial cylinders of equal diameter, were determined by numerical calculations. Various modes of operation of such mirrors are considered: 1) space-time-of-flight focusing in the Gaussian plane of the mirror; 2) space-time-of-flight focusing in the focal plane of the mirror; 3) time-of-flight focusing in telescopic mode; 4) time-of-flight focusing in collimator mode. The results of calculations are presented for two fundamental solutions of the equation of paraxial trajectories. One of these solutions describes trajectories whose direct and reverse branches coincide, and the second solution describes trajectories whose direct and reverse branches are symmetrical with respect to the axis of symmetry of the field. It is shown that the time-of-flight dispersion of the mirror by mass, determined on the basis of the second solution, is several times higher than that obtained on the basis of the first solution for all modes of its operation.