Realization of first-order optical systems using thin convex lenses of fixed focal length

A general axially symmetric first-order optical system is realized using free propagation and thin convex lenses of fixed focal length. It is shown that not more than five convex lenses of fixed focal length are required to realize the most general first-order optical system, with the required number of lenses depending on the situation. The free propagation distances are evaluated explicitly in each situation. The optimality of the decomposition obtained in each situation is brought out. Decompositions for some familiar subgroups of SL2(R) are also worked out. Convex or concave lenses of arbitrary focal length are realized using three or two convex lenses of fixed focal length, respectively. It is further shown that three convex lenses of arbitrary focal length are sufficient to realize the most general first-order optical system.