Application of the effective potentials technique for investigation of two-dimensional electron mirrors with curvilinear axial trajectory

Using a method of effective potentials the authors reduced the equations of motion for two-dimensional electron mirrors with curvilinear axial trajectory to the form analogous to that for the mirrors with a rectilinear axial trajectory. On the basis of this analogy general expressions describing time-of-flight and trajectory in the middle plane in two-dimensional electron mirrors with curvilinear axial trajectory to an accuracy of the second order of infinitesimal were obtained. l and time-of-flight focusing conditions as well as conditions of elimination of some types of aberrations were determined. By numerical calculations the conditions of stigmatic focusing and elimination of the second-order spherical aberrations in two- and three-electrode mirror electrostatic energy analyzers, whose electrodes were pairs of parallel plates separated by straight slits, were obtained. The conditions of time-of-flight focusing for a three-electrode flat mirror up to the second-order inclusive were also found.