An extension of the analytical potential expression of a three-plane parallel-electrode system to arbitrary apertures

Potential distributions in the entire space of a two-dimensional lens consisting of three parallel plates of symmetrical construction and arbitrary apertures are obtained analytically by finding special transformation functions between the real lens space and the corresponding imaginary one. The functions contain two arbitrary constants given as a function of the inner aperture, the outer aperture, and the half-lens space. The potential distribution on the optical axis exactly coincides with the one given analytically by G.D. Archard (Brit. J. Appl. Phys., vol.5, p.395, 1954)