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 o f A. At the same time A is the nth power of B. It is shown that, in pr inciple, any optical system can be decomposed into its roots of any or der. The procedure is facilitated by a merger of the ray matrix repres entation and the canonical operator representation of first-order opti cal systems. The results are demonstrated by several examples, includi ng the fractional Fourier transform, which is just one special case in a complete group structure. Moreover, it is shown that the root and p ower transformations themselves represent special cases of a much more general family of transformations. Application in optical design, opt ical signal processing, and resonator theory can be envisaged. (C) 199 5 Optical Society of America