ABSTRACT
In the case of the periodic transport over long distances, the desire is not so much to give a special shape to the beam as the beam exits, but, even much more simply, to just contain the beam. Actually it is rather straightforward to formulate a necessary condition on the linear matrix: it is not allowed to have any eigenvalue of magnitude greater than unity. If the eigenvalue is real, the argument is simple: if this were the case, any particle that has its coordinates lined up with the corresponding real eigenvector will after one period end up on the same line, but all its coordinates would have increased by a factor equal to the eigenvalue. For many practical purposes it is particularly important to know in detail the parameters of the ellipse that is invariant under stable linear motion.
