Equilibrium and stability properties for foil-focused intense relativistic electron beam

The equilibrium and stability properties of foil-focused electron beam in terms of its envelope equation is studied.Results are as follows:(1) The parameters of the electron beam, the radius of the waveguide tube and the distance between foils must obey a relation in terms of foil-focusing; (2) Equilibrium of the electrion beam with the initial emittance is generally stable for foil-focusing; (3) The hole on foil does not affect focusing if its radius is equal or slight larger than the radius of the electron beam．