Page 276 - Phase Space Optics Fundamentals and Applications

Rays and Waves 257

of Eq. (8.57a), i.e.,

2
∂ ∂ ∂n 2 ∂ ∂
2p + 2 =− · (8.61)
∂z ∂p ∂z ∂x ∂p ∂p
¯
The evaluation of this expression at p = P(z, ) gives, after reordering
and the use of Eq. (8.59),

2
P ¯ 1 ∂n 2 ∂ ∂ ∂
=− · − (8.62)
¯ H 2 ¯ H ∂x ∂p ∂p ∂p ∂z
From the comparison of Eqs. (8.60) and (8.62), we see that it is conve-
nient to choose
P ¯
˙ ¯
X = (8.63a)
¯ H
1 ∂n 2
˙ ¯ ¯
P = (X,z) (8.63b)
2 ¯ H ∂x
These equations are identical to Eqs. (8.18a) and (8.18b). Also, notice
that the substitution of Eqs. (8.58) and (8.59) into Eq. (8.57a) evaluated
¯
at p = P(z, ) implies that

2 ¯ ¯ 2
¯ H = n (X,z) −|P| (8.64)
so, remarkably, the parameterized trajectories that result from the
asymptotic treatment in the momentum representation are the stan-
dard rays. From now on, the bars over X, P, and H are dropped.
The phase function is again obtained parameterically. Let us define
T(z, ) = (P,z) (8.65)

The evolution of this function results from considering its derivative
with respect to z:

∂ ∂ 1 ∂n 2
˙
˙ T = · P + =− X · (X,z) + H (8.66)
∂p ∂z 2H ∂x
where Eqs. (8.58), (8.59), and (8.63b) were used in the second step.
The relation between the values of T for contiguous rays is found
similarly by taking the partial derivative with respect to j of both
sides of Eq. (8.65):
∂T ∂ ∂P ∂P
= · =−X · (8.67)
∂ j ∂p ∂ j ∂ j
This relation, combined with Eq. (8.13), implies that
T(z, ) = L(z, ) − X(z, ) · P(z, ) (8.68)