Page 262 - Phase Space Optics Fundamentals and Applications

Rays and Waves 243

By using this result, the transverse part of Eq. (8.12a) gives
P
˙
X = (8.18a)
H
where the overdot now denotes a derivative with respect to z. Sim-
ilarly, the transverse and longitudinal parts of Eq. (8.12b) become,
respectively,

n ∂n 1 ∂n 2
˙
P = (X,z) = (X,z) (8.18b)
H ∂x 2H ∂x
n ∂n 1 ∂n 2
˙ H = (X,z) = (X,z) (8.18c)
H ∂z 2H ∂z
where ∂/∂x is the transverse (or x, y) part of the gradient. Finally,
Eq. (8.12c) becomes
2
n (X,z)
˙ L = (8.18d)
H

8.2.3 Ray-Optical Phase Space and the
Lagrange Manifold
At this point, it is convenient to recall that rays can be represented by
points in phase space. For simplicity, let us consider fields propagating
inonlytwodimensions.Thatis,thereisonlyonetransversecoordinate
x plus the longitudinal coordinate z. In this case, the ray equations
become

P(z, )
˙ X(z, ) = (8.19a)
H(z, )
1 ∂n 2
˙ P(z, ) = [X(z, ),z] (8.19b)
2H(z, ) ∂x
1 ∂n 2
˙ H(z, ) = [X(z, ),z] (8.19c)
2H(z, ) ∂z
Notice that there is now only one parameter that labels the rays. The
equations for the path length L become

2
n (X, z)
˙ L(z, ) = (8.20)
H

L (z, ) = PX (8.21)
where the primes denote derivatives in .