We investigate a general form of the Wigner function for wave fields that satisfy the Helmholtz equation in two-dimensional free space. The momentum moment of this Wigner function is shown to correspond to the flux of the wave field. For a forward-propagating wave field, the negative regions of the Wigner function are seen to be associated with small regions of backward flux in the field. We also study different projections of the Wigner function, each corresponding to a distribution in a reduced phase space that fully characterizes the wave field. One of these projections is the standard Wigner function of the field at a screen. Another projection introduced by us has the added property of being conserved along rays and is better suited to the description of nonparaxial wave fields.
© 1999 Optical Society of America
(000.3860) General : Mathematical methods in physics
(030.5630) Coherence and statistical optics : Radiometry
(070.2590) Fourier optics and signal processing : ABCD transforms
(350.6980) Other areas of optics : Transforms
(350.7420) Other areas of optics : Waves
Kurt Bernardo Wolf, Miguel Angel Alonso, and Gregory W. Forbes, "Wigner functions for Helmholtz wave fields," J. Opt. Soc. Am. A 16, 2476-2487 (1999)