Approximate analytical description of the underdense plasma lens

The perturbative approach for describing the underdense plasma-ultrarelativistic electron bunch system is developed, using the ratio n0/nb as a small parameter (n_b-bunch, n₀-plasma electron densities). It is shown that in the zero and first approximations for the ultrarelativistic bunch case the role of positive ion column is negligible (~O(γ^-2), where γ₀ is the Lorentz factor of the bunch) for the arbitrary bunch length. Focusing of the electron bunch emerged in the first approximation of the perturbative procedure as a result of the plasma electrons redistribution. Focusing gradient and strength for ultrarelativistic, flat, uniform and short bunch are obtained and compared with the previous results, which differs drastically from the obtained ones. The plasma chamber conducting walls effect on focusing force is considered by the method of images, adopting the diffuse scattering of plasma electrons on the walls. The effect is small, when the plasma chamber width is much larger than that of bunch