Calculation of a triode field-emission system with a modulator

The electrostatic potential of a triode field-emission system with a modulator that represents a circular diaphragm is calculated. A tip field-emission cathode serves as an emitter. The inner part of the system under study is filled with two insulators. In the calculation of the potential distribution, a real field-emission cathode is changed by a virtual cathode whose surface coincides with an equipotential surface, so that the effect of the cathode on the distribution of the electrostatic potential is simulated as the effect of a finite charged wire that is located at the axis of the system. The separation of variables is used to solve the problem. The potential is represented as an expansion in terms of eigenfunctions, and the expansion coefficients are found from the solution to a system of linear algebraic equations.