In this work an axisymmetric diode electron-optical system based on a field emitter is simulated. The field emitter in the form of a thin filament of finite length is located on the flat substrate with the dielectric layer. The anode is a plane. The electrostatic potential distribution was found in an analytical form - in the form of Fourier-Bessel series in the whole area of the system under investigation. The coefficients of Fourier-Bessel series are the solution of the system of linear equations with constant coefficients.