The motion of charged particles emerging from a point source located in the middle plane of the transaxial mirror is considered. A three-electrode transaxial mirror is two parallel plates cut by straight circular cylinders of radius R₁ and R₂, whose axis coincides with the axis z. Using the methods of the theory of functions of a complex variable, we obtained expressions for calculating the harmonic component F(η,ζ). Analytical expressions for the field potential of a transaxial mirror give a good approximation for the potential φ(η,ζ ) and at the same time exactly satisfies the given Dirichlet boundary conditions and satisfies the two-dimensional Laplace equation. It is shown that as a result of reflection in a three-electrode transaxial mirror, a parallel volume beam can be formed. This property of transaxial mirrors can be used to create highly efficient time-of-flight mass spectrometers. A fairly simple analytical expression for the electrostatic potential of a three-electrode transaxial lens is also obtained, which can also used to calculate transaxial mirrors. To calculate the trajectories of particles, the dimensionless Newton equations and analytical expressions for the potential are used, which describe the field of a three-electrode transaxial mirror with good accuracy. Two modes of vertical beam focusing are calculated.

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