Page 257 - Phase Space Optics Fundamentals and Applications

238 Chapter Eight

limit, corresponding to fields of low spatial coherence. In this case,
the Wigner function and other bilinear phase-space representations
acquire all the defining properties of the radiance, which is essentially
a ray-weighting distribution whose propagation is ruled by the laws
of geometrical optics. The radiometric description of wave fields and
the quasi-homogeneous limit are discussed in Chap. 7. The third limit
is that of small wavelength. This limit is discussed in many optics
textbooks 1–3 and is the main topic of the two books by Kravtsov and
Orlov. 4,5 As it turns out, there are a variety of ways in which this third
limit can be enforced, all leading to the same laws for the rays, but
different ray-based descriptions of the wave field. These various ap-
proaches are the topic of this chapter. Also discussed briefly here is the
mathematically analogous semiclassical limit of quantum mechanics,
where instead of rays one uses classical particle trajectories to estimate
the wave aspects of particle motion.
For simplicity, the propagation of scalar fields is considered in what
follows. We also limit our attention to the case of monochromatic light
(of frequency ), where the field’s time dependence can be factored as
r
r
E( r, t) = U( ) exp(−i t), with = (x, y, z) being the position vector.
The basic equation that describes the propagation of a monochromatic
r
scalar field U( ) is the Helmholtz equation
2
2 2
[∇ + k n ( r)]U( r) = 0 (8.1)
where k = /c is the wave number (with c representing the speed of
light in vacuum) and n( ) is the position-dependent refractive index.
r
It is assumed throughout that this refractive index is real, i.e., that the
medium presents no absorption or gain.

8.2 Small-Wavelength Limit in the Position
Representation. I: Geometrical Optics
The standard procedure for studying the connection between wave
and ray optics relies on the assumption that the field U consists of a
slowlyvaryingamplitudeandarapidlyoscillatingphaseproportional
to the wave number, i.e.,

r
r
U( ) = A( ) exp[ik ( )] (8.2)
r
where it is assumed that at least is real. The substitution of Eq. (8.2)
into Eq. (8.1) gives, after some reordering and multiplication by
exp(−ik ),
2
2
2
2
2
k A(n −|∇ | ) + ik(2∇ A·∇ + A∇ ) +∇ A = 0 (8.3)