Effects of the Transverse-Longitudinal Coupling on the Strong Focusing of Charged Beams for ICF

A key issue in the conceptual design of a ion driven ICF reactor is the final focusing system. Both target requirements and accelerator constraints demand for a highly efficient final focusing system. Such requirement has stimulated new effort in the magnetic lenses design and some solutions have been proposed in order improve the present days tecniques. Among these the plasma lens seems to be particularly appropriate for ICF for various reasons. In particular it has been recently proposed a new attractive scheme, called adiabatic plasma lens focusings, for the collection of multiple converging beams in the reactor chamber. In the strong focusing of a beam, when large convergence angles and small spot sizes have to be reached, a fundamental limitation arises from the effects of transverse-longitudinal motion coupling. These effects have, in fact, to be carefully considered in order to establish the related limitations in the focusing process. This point has been often disregarded because of minor importance in traditional focusing systems, as for example quadrupoles multiplets, where the so called para-axial approximation holds. In this work we develop a model, starting from the complete equation of motion for the beam particle in an axial symmetric azimuthal magnetic field (like that produced in a plasma lens), which takes into account in exact way the transverse-longitudinal coupling effects. We derive exact solution by of the trajectory equation by means of elliptic integrals and discuss some general properties of the coupled dynamic. A simpler model, based on the thin lens approximation is also developed and its results compared to the general one. Finally dimensionless parameters are related to cases of practical interest and the aberration effects are evaluated in those cases.