Root and power transformations in optics

If the performance of an optical system A can be executed by a cascade of n identical optical systems B, we term the system B the nth root of A. At the same time A is the nth power of B. It is shown that, in principle, any optical system can be decomposed into its roots of any order. The procedure is facilitated by a merger of the ray matrix representation and the canonical operator representation of first-order optical systems. The results are demonstrated by several examples, including the fractional Fourier transform, which is just one special case in a complete group structure. Moreover, it is shown that the root and power transformations themselves represent special cases of a much more general family of transformations. Application in optical design, optical signal processing, and resonator theory can be envisaged.