Linear canonical transforms

Abstract

Paraxial optical setups act through linear canonical transformations between the input and output optical phase spaces. In D-dimensional scalar wave models, this action is through unitary integral transforms that form a group with kernels that are oscillating Gaussian functions, and which can represented through 2D×2D symplectic matrices to simplify their composition and other computations. This group contains the Fourier subgroup of phase space rotations that include gyrations and fractional Fourier transforms. The general linear canonical transformations can be complexified in a range of its parameters to describe coherent states, or if symmetry permits, reduce to radial canonical transforms.