Abstract

Fresnel diffraction and lens transformations can be combined to implement 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 diffraction. A correspondence is established with ray matrices in periodic lens systems with unit magnification. Consecutive powers of particular roots of ℐ form groups from which the projectors can be obtained by means of optical discrete Fourier transforms (DFT’s), conveniently performed by networks of beam splitters and mirrors. One can then obtain additional roots of ℐ from the projectors by reassigning eigenvalues among them, which is done by introducing suitable phase shifts, and reusing the DFT network backward.

© 1995 Optical Society of America

Root and power transformations in optics

Joseph Shamir and Nir Cohen
J. Opt. Soc. Am. A 12(11) 2415-2423 (1995)

Fractional Fourier optics

Haldun M. Ozaktas and David Mendlovic
J. Opt. Soc. Am. A 12(4) 743-751 (1995)

Optical fractional correlation: experimental results

David Mendlovic, Yigal Bitran, Rainer G. Dorsch, and Adolf W. Lohmann
J. Opt. Soc. Am. A 12(8) 1665-1670 (1995)

Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams

K. Sundar, N. Mukunda, and R. Simon
J. Opt. Soc. Am. A 12(3) 560-569 (1995)

Differential commuting operator and closed-form eigenfunctions for linear canonical transforms

Soo-Chang Pei and Chun-Lin Liu
J. Opt. Soc. Am. A 30(10) 2096-2110 (2013)