Page 259 - Phase Space Optics Fundamentals and Applications

240 Chapter Eight

Eq. (8.4) evaluated at R, we find
v( , 1 , 2 )
( , 1 , 2 ) = (8.6)
n[ R( , 1 , 2 )]
where

˙
v( , 1 , 2 ) =| R( , 1 , 2 )| (8.7)
is the speed of the parameterization in . The trajectories traced by the
vector R for increasing are precisely the rays of geometrical optics.
To find the equations that determine the rays, it is convenient to
define the optical momentum vector P as
n( R) ˙
P( , 1 , 2 ) =∇ [ R( , 1 , 2 )] = R (8.8)
v
where Eqs. (8.5) and (8.6) were used in the last step. The equation
for the propagation of the rays is found by considering the derivative
with respect to of the first two parts of Eq. (8.8):

2
v v ∇(∇ ·∇ ) v ∇(n )
˙ ˙
P = ( R·∇)∇ = (∇ ·∇)∇ = = = v∇n( R)
n n 2 n 2
(8.9)
where the chain rule was used in the first step, the second part of
Eq. (8.8) was used in the second step, and Eq. (8.4) was used in the
fourth step. We must also find how the eikonal evolves along a ray.
For this purpose, let L denote the eikonal evaluated along a parame-
terized ray, i.e.,

L( , 1 , 2 ) = [ R( , 1 , 2 )] (8.10)
This function increases with according to the expression

˙
˙ L = R ·∇ = n( R)v (8.11)
where the chain rule, as well as Eqs. (8.7) and (8.8), was used. The
value of the eikonal is then found by integrating this expression in .
The expressions in Eqs. (8.8), (8.9), and (8.11) are the basic equa-
tions that rule the propagation of the rays. Let us summarize these
equations:
v
˙ P
R = (8.12a)
n( R)
˙
P = v ∇n( R) (8.12b)
˙ L = vn( R) (8.12c)