Pierce electrodes for a multigap accelerating system

A well-known Pierce's solution that allows to focus a beam of charged particles using properly shaped electrodes outside the beam is generalized to the case of multigap accelerating system. Simple parametric formulae for Pierce electrodes are derived for an accelerating system with current density, limited either by space charge or by emitting property of the cathode. As an example of general approach, Pierce electrodes shape is analyzed for a system with two accelerating gaps. It is shown that precise Pierce's solution exists if acceleration rate within second gap is lower than within first gap. In the opposite case quasi-Pierce solution can be implemented using non-equipotential electrode between the gaps, and guidelines, based on numerical simulations, for the design of equipotential focusing electrodes are given.