Page 282 - Phase Space Optics Fundamentals and Applications

Rays and Waves 263

where = x − X and H = ˙ L − P ˙ X. Also, it is convenient to expand
the refractive index in a Taylor series around the ray position as

∞ j 2
, 1 ∂ n
2 j
n (x, z) = j (X, z) (8.80)
j! ∂x
j=0
Equations (8.79a), (8.79b), and (8.80) are now substituted into the
Helmholtz equation for U G . The result can be grouped in powers of
k, with each coefficient itself being separated into powers of :
2
2
[k (C 20 +C 21 +C 22 +· · ·)+k(C 10 +C 11 +···)+C 00 ] g = 0 (8.81)
where the first few coefficients of each subseries are
2
2
2
C 20 = u[n (X, z) − P − H ] (8.82a)

∂n 2
C 21 = u (X, z) − 2H ˙ P + 2i (H ˙ X − P) (8.82b)
∂x
2 2
1 ∂ n 2 2
C 22 = u 2 (X, z) + + ( ˙ X + i ˙ P) − iH ˙ (8.82c)
2 ∂x
2
C 10 = 2i˙uH + u(− − ˙ X − i ˙ X ˙ P + i ˙ H) (8.82d)
C 11 = 2˙u( ˙ X + i ˙ P) + u( ¨ X + i ¨ P + 2˙ ˙ X) (8.82e)
C 00 = ¨u (8.82f)

We now assume that k is large, and we go on to perform the asymp-
totic treatment. Notice, however, that the coefficients of each power
of k involve terms with different powers of , which cannot be mixed
to lead to an asymptotic constraint, because depends on the spatial
√
variable x. Since g is a Gaussian of width 1/ k centered at x = X,
and = x − X, the importance of each term in Eq. (8.81) decreases
with increasing powers of . The main contribution, then, is the one
2
with the largest power of k and the smallest power of ,or k C 20 .As
can be seen from Eq. (8.82a), this contribution vanishes if X, P, and H
are chosen to follow the rules of ray optics, i.e., if the two-dimensional
version of Eq. (8.16) is satisfied. The ray propagation equations for X
and P, given in Eqs. (8.19a) and (8.19b), are also found from setting to
2
zero the next contribution in importance, or k C 21 , as can be easily
seen from Eq. (8.82b). That is, as in the position- and momentum-
based derivations presented earlier, forcing the leading orders of the
asymptotic form of the wave equation to vanish leads to the laws of
ray optics.