Algebraic approach to characterizing paraxial optical systems

The paraxial propagation formalism for ABCD systems is reviewed and written in terms of quantum mechanics. This formalism shows that the propagation based on the Collins integral can be generalized so that, in addition, the problem of beam quality degradation that is due to aberrations can be treated in a natural way. Moreover, because this formalism is well elaborated and reduces the problem of propagation to simple algebraic calculations, it seems to be less complicated than other approaches. This can be demonstrated with an easy and unitary derivation of several results, which were obtained with different approaches, in each case matched to the specific problem. It is first shown how the canonical decomposition of arbitrary (also complex) ABCD matrices introduced by Siegman [Lasers, 2nd ed. (Oxford U. Press, London, 1986)] can be used to establish the group structure of geometric optics on the space of optical wave functions. This result is then used to derive the propagation law for arbitrary moments in eneral ABCD systems. Finally a proper generalization to nonparaxial propagation operators that allows us to treat arbitrary aberration effects with respect to their influence on beam quality degradation is presented.