Abstract

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.

© 2007 Optical Society of America

Geometry and dynamics in the Fresnel transforms of discrete systems

Kurt Bernardo Wolf and Guillermo Krötzsch
J. Opt. Soc. Am. A 24(9) 2568-2577 (2007)

Geometry and dynamics in the fractional discrete Fourier transform

Kurt Bernardo Wolf and Guillermo Krötzsch
J. Opt. Soc. Am. A 24(3) 651-658 (2007)

Rotation and gyration of finite two-dimensional modes

Kurt Bernardo Wolf and Tatiana Alieva
J. Opt. Soc. Am. A 25(2) 365-370 (2008)

Fractional Fourier transforms in two dimensions

R. Simon and Kurt Bernardo Wolf
J. Opt. Soc. Am. A 17(12) 2368-2381 (2000)

Finite signals in planar waveguides

Juvenal Rueda-Paz and Kurt Bernardo Wolf
J. Opt. Soc. Am. A 28(4) 641-650 (2011)