We propose a discretization strategy for systems with axial symmetry. This strategy replaces the continuous position coordinates by a discrete set of sensor points, on which the discrete wave fields transform covariantly with the group of 2×2 symplectic matrices. We examine polar arrays of sensors (i.e., numbered by radius and angle) and find the complete, orthonormal sets of discrete-waveguide Meixner functions; when the sensors come closer together, these tend to the Laguerre eigenmodes of the continuous waveguide. In particular, the fractional Hankel transforms are discretized in order to define the fractional Hankel–Meixner transforms and similarly for all axis-symmetric linear optical maps. Coherent states appear in the discrete cylindrical waveguide. Covariant discretization leads to the same Wigner phase-space function for both the discrete and the continuum cases. This reinforces a Lie-theoretical model for the phase space of discrete systems.
© 2000 Optical Society of America
(000.3860) General : Mathematical methods in physics
(070.4560) Fourier optics and signal processing : Data processing by optical means
(200.4560) Optics in computing : Optical data processing
(350.6980) Other areas of optics : Transforms
Natig M. Atakishiyev, Shakir M. Nagiyev, Luis Edgar Vicent, and Kurt Bernardo Wolf, "Covariant discretization of axis-symmetric linear optical systems," J. Opt. Soc. Am. A 17, 2301-2314 (2000)