Page 258 - Phase Space Optics Fundamentals and Applications

Rays and Waves 239

The goal now is to separate this equation into two or more equations
that are amenable to a simple solution or that at least lead to an in-
tuitive interpretation. Two alternative approaches are considered in
what follows:
1. Assume that the wave number k is large, and use an asymptotic
treatment.
2. Assume that both A and are real, and separate Eq. (8.3) into
real and imaginary parts.

In this section and the next we explore the first approach, which
leads to geometrical optics. The second approach is discussed in
Sec. 8.4.

8.2.1 The Eikonal and Geometrical Optics
Since k is assumed to be large, the leading part of Eq. (8.3) is the term
2
proportional to k . Therefore, as a first step toward enforcing Eq. (8.3),
we choose to make the coefficient of this leading part vanish. This
results in the expression

2
2
r
|∇ ( )| = n ( r) (8.4)
This equation is the well-known eikonal equation, which is formally
equivalent to other formulations of geometrical optics. The function
is called the eikonal function or simply the eikonal. This formulation
6
was proposed by Bruns in 1895, although an equivalent formalism
7
was proposed by Hamilton almost seventy years earlier.
The eikonal equation can be solved (or at least written in a form
that is better suited for numerical solution) by parameterizing the
position vector in terms of three independent parameters , 1 , 2 as
r = R( , 1 , 2 ). These parameters must be chosen so that R moves in
three different directions with variations in each of them (i.e., the three
vectors corresponding to the partial derivatives of R with respect to
each of the parameters must be linearly independent). To solve the
eikonal equation, the partial derivative with respect to (denoted by
an overdot) of R is chosen as parallel to the gradient of the eikonal,
i.e.,

∂ R ˙
= R = ∇ ( R) (8.5)
∂

where the proportionality function ( , 1 , 2 ) is assumed to be posi-
tive. Substituting the gradient of the eikonal as given in Eq. (8.5) into