Powers of transfer matrices determined by means of eigenfunctions

The parameters of the transfer matrix describing a first-order optical system that is a cascade of k identical subsystems defined by the transfer matrix M, are determined from considering the subsystem's eigenfunctions. A condition for the cascade to be cyclic is derived. Particular examples of cyclic first-order optical systems are presented. Structure and properties of eigenfunctions of cyclic transforms are considered. A method of optical signal encryption by using cyclic first-order systems is proposed.