ROOTS OF THE IDENTITY OPERATOR AND OPTICS

Fresnel diffraction and lens transformations can be combined to implem ent optical transformations corresponding to operators that are roots of the identity operator I. Since their nature remains invariant under unitary transformations, new roots can be obtained by Fresnel diffrac tion. A correspondence is established with ray matrices in periodic le ns systems with unit magnification. Consecutive powers of particular r oots of I form groups from which the projectors can be obtained by mea ns of optical discrete Fourier transforms (DFT's), conveniently perfor med by networks of beam splitters and mirrors. One can then obtain add itional roots of I from the projectors by reassigning eigenvalues amon g them, which is done by introducing suitable phase shifts, and reusin g the DFT network backward.