Envelope model for passive magnetic focusing of an intense proton or ion beam propagating through thin foils

Abstract

Ion beams (including protons) with low emittance and high space-charge intensity can be propagated with normal incidence through a sequence of thin metallic foils separated by vacuum gaps of order the characteristic transverse beam extent to transport/collimate the beam or to focus it to a small transverse spot. Energetic ions have sufficient range to pass through a significant number of thin foils with little energy loss or scattering. The foils reduce the (defocusing) radial electric self-field of the beam while not altering the (focusing) azimuthal magnetic self-field of the beam, thereby allowing passive self-beam focusing if the magnetic field is sufficiently strong relative to the residual electric field. Here we present an envelope model developed to predict the strength of this passive (beam generated) focusing effect under a number of simplifying assumptions including relatively long pulse duration. The envelope model provides a simple criterion for the necessary foil spacing for net focusing and clearly illustrates system focusing properties for either beam collimation (such as injecting a laser-produced proton beam into an accelerator) or for magnetic pinch focusing to a small transverse spot (for beam driven heating of materials). An illustrative example is worked for an idealization of a recently performed laser-produced proton-beam experiment to provide guidance on possible beam focusing and collimation systems. It is found that foils spaced on the order of the characteristic transverse beam size desired can be employed and that envelope divergence of the initial beam entering the foil lens must be suppressed to limit the total number of foils required to practical values for pinch focusing. Relatively modest proton-beam current at 10 MeV kinetic energy can clearly demonstrate strong magnetic pinch focusing achieving a transverse rms extent similar to the foil spacing ($20--50\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$ gaps) in beam propagation distances of tens of mm. This is a surprisingly optimistic result since placing many foils per characteristic beam radius, which one might expect to be necessary to strongly attenuate the self-electric field, would likely result in excessive scattering and loss of focusing from the current neutralization due to the beam propagating too far through solid metal. Results from the envelope model are compared with particle-in-cell simulations to help clarify limits related to envelope-model idealizations. Possible degradations of focusing in situations where strong halo can be generated and where pulse duration is short are clarified.