Abstract
In classical optics the Wolf function is the natural analogue of the quantum Wigner function and like the latter it may be negative in some regions. We discuss the implications this negativity has on the generalized ray interpretation of free-space paraxial wave evolution. Important examples include two classes of beams carrying optical orbital angular momentum—Laguerre–Gaussian (LG) and Bessel beams. We formulate their defining eigenfunction properties as phase–space symmetries of their Wolf functions, whose analytical form is shown, and discuss their interpretation in the ray picture. By moving to a more general picture of partly coherent fields, we find that new solutions displaying the same symmetries appear. In particular, we find that mixtures of Gaussian beams (thus fully describable using classical ray optics) can mimic the basic properties of LG beams without the need for negativity, and are not restricted to quantized values of angular momentum. The quantization of both the l and p parameters and negativity of the Wolf function are both inevitable and, indeed, arise naturally when a requirement on the purity of the solution is added. This work is supplemented by a set of computer animations, graphically illustrating the interpretative aspects of the described model.
Highlights
We know that most phenomena of light can only be explained when its wave nature is taken into account
LG and Bessel beams are two important classes of scalar optical beams carrying orbital angular momentum (OAM). They are characterized by rotational symmetry along with preservation of spatial structure under propagation (LG beams) or a full symmetry under free-space propagation (Bessel beams)
We presented their respective Wolf functions and how these two symmetries naturally manifest themselves in phase–space
Summary
We know that most phenomena of light can only be explained when its wave nature is taken into account. Geometric ray optics remains a good compromise between clarity and exactness in many real-world situations. Its domain had been restricted to the limit of short wavelength, applicable to a good extent to many important fields including photography, microscopy, and telescopy, but failing at scales where interference phenomena can no longer be neglected. The concept of generalized ray description of light waves, providing an exact mathematical model of light propagation and detection accounting even for interference phenomena, but keeping the intuitiveness of a geometric picture, dates back to the 1970ʼs. First attempts can be identified in [2], after which the topic has been broadly—and to a good extent independently—developed by Bastiaans [3,4,5] and Sudarshan [6,7,8]. Simon [9] has noted that the ray picture becomes plausible and well-behaved in the paraxial approximation, which was an ad hoc initial assumption in [2]
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