Page 261 - Phase Space Optics Fundamentals and Applications

242 Chapter Eight

point source in free space, the rays are straight lines radiating away
from the source, and the normal surfaces are spheres centered at the
source. The reference surface is usually chosen to be the sphere of zero
radius, i.e., the source itself.

8.2.2 Choosing z as the Parameter
The equations for the rays given above are general in the sense that
the parameterization along the rays is arbitrary. There are, however,
particular parameterizations that are convenient in certain situations,
leading to several different forms for the ray equations. 8–10 Some com-
mon choices are the ones that make equal to the arclength of the ray
(so v = 1), the optical path length (so v = 1/n), or the length divided
by the local refractive index (so v = n). In what follows, we con-
centrate on a fourth particular parameterization which, while more
limited in application, is convenient for the type of problems studied
in this book. This parameterization is only valid when one can choose
a “main direction of propagation” such that the component of the
momentum in this direction for all rays in the family is always pos-
itive. Let us align the z axis with this direction of propagation. Then
the condition for the application of this parameterization is that the
rays do not turn around in z, so that their positions are single-valued
functions of z. Under these circumstances, z itself can be used as the
parameter of propagation.
For this parameterization, it is convenient to separate the z and
transverse components of the position and momentum vectors as

R(z, ) = [X(z, ),Y(z, ),z] = [X(z, ),z] (8.14)
P(z, ) = [P x (z, ),P y (z, ),H(z, )] = [P(z, ),H(z, )] (8.15)

where X = (X, Y) and P = (P x ,P y ). As stated earlier, it is assumed that
the longitudinal component of the momentum, namely H, is always
positive. As we know from the eikonal equation [i.e., Eq. (8.4)], this
function is related to P and the refractive index by

2 2
H(z, ) = n (X,z) −|P| (8.16)
Let us now find the form that the ray equations take for this parameter-
ization. First, the longitudinal part (i.e., the z component) of Eq. (8.12a)
gives 1 = vH/n, that is,

n
v = (8.17)
H