Wigner Optics

Classical radiometry was a framework for modeling the propagation of light which was developed prior to the understanding that light was in fact a wave. It modeled the propagation of light along rays by simply assigning to each ray a quantity thought of as the “amount of light” carried by the ray. The classical radiance function satisfies four properties that made it particularly intuitive and useful as a model:

(1) the “amount of light” carried by each ray is non-negative, so it can be loosely interpreted as intensity carried by the ray,
(2) rays originating from points outside the physical extent of the source carry zero weight,
(3) the “amount of light” carried by each ray is constant along the ray in propagation,
(4) the intensity at a point is simply given by the sum of the weights of all rays passing through that point.

As the full wave theory of light was formulated, it became clear that classical radiometry could not accurately model wave effects such as interference or diffraction since no single radiance function satisfying the above 4 properties is compatible with wave propagation. For example, the light emitted from two slits will form alternating bright and dark fringes of intensity to the right of the two slits. The classical radiance, however, models the light from the pinholes using only non-negative rays. By tracing the rays from the slits, it becomes clear that the same rays need to contribute light to the bright fringe along the central axis and to the dark fringes at points D1 and D2. But this is impossible if the above four properties are satisfied.

Rather than a single classical radiance function, a variety of generalized radiance functions can be defined, each of which only satisfies a subset of the above four properties. For numerical convenience, properties 3 and 4 may be retained in order to define a generalized radiance which remains constant along rays and which allows simple calculation of the intensity as the sum of the weights of all rays passing through a point. The trade-off is that negatively-weighted rays can exist, as well as rays which originate from points outside of the source, both of which are required in certain cases in order to accurately model wave effects. This is not a problem for the theory, since single rays are not physically measurable: computing the intensity requires summation over multiple rays, which always results in non-negative values.

It turns out that the function satisfying properties 3 and 4 for wave optics had already been defined in quantum mechanics by Eugene Wigner (for a similar purpose of describing the wave-like nature of a quantum system in terms of the behavior of classical particles). Therefore, this particular generalized radiance is known as the Wigner function.