Analysis of a Symmetric Triplet and its Application to Ring Insertions

It is interesting to study the matching capability of a triplet for a large range of parameters and to cover all the possible solutions. Fully analytical treatments can be used for this purpose, when working with the thin lens approximation. Closed solutions and the conditions in which they exist are described for the special case of triplets with a symmetric geometrical arrangement. They define all possible matchings, which correspond to the requests or limits on the Twiss parameters that are specified, and then define the range of values that are obtainable for the betatron functions. Numerical extension to thick lenses gives the complete solutions for the retained cases. Applications are presented on insertion and lattice problems, with emphasis on the design for a possible isochronous ring that is part of the injection chain of the CLIC drive beam.