Page 256 - Phase Space Optics Fundamentals and Applications

CHAPTER8

Rays and Waves

Miguel A. Alonso
The Institute of Optics, University of Rochester, New York, USA

8.1 Introduction
From a conceptual point of view, the ray model for the propagation of
light is an outdated theory. Yet, it is still perhaps the most important
tool for the design and modeling of imaging and illumination optical
instruments due to its simplicity, intuitiveness, and often sufficient
accuracy. (An analogous although significantly more extreme situa-
tion occurs for mechanical systems: machines and tools are designed
and modeled using classical mechanics, which is also conceptually an
outdated theory; quantum effects are important only for very small or
very special mechanical systems.) It turns out that even when wave
effects are important, they can often be modeled based on the ray-
optical description of the system in question. There are a variety of
methods for modeling wave propagation based on rays. Phase space
is a natural framework for studying the link between the ray and
wave models. Using phase-space representations, a wave field can be
described as a function of both position and direction of propagation.
In general, the use of rays leads only to approximate wave propaga-
tion models. However, the laws of wave propagation can be expressed
exactly in terms of rays in three limits. The first is the paraxial limit
for the case of propagation through the so-called ABCD or first-order
systems. (See Chaps. 1 and 3 by Martin Bastiaans and Tatiana Alieva,
respectively.) These systems include free-space and homogeneous
media, thin quadratic lenses, and transversely linear and quadratic
gradient-index media. The propagation of waves in these systems can
be described exactly in ray terms, either by employing a point-spread
function like that in Eq. (1.42), or in terms of the Wigner function as
in Eq. (1.44). The second limit is the so-called quasi-homogeneous

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