Page 287 - Phase Space Optics Fundamentals and Applications

268 Chapter Eight

(recalling that ˙ = 0) into this expression:
2 2
1 ∂ n 2
X C 22 + Y C 10 = uX + + Y ˙ 2

2 ∂x 2
˙

+ Y [2i˙uH + u(− − Y ˙ X + i ˙ H)]

2
1 ∂n

˙ ˙
= u + ( X − Y ) + Y(Y X − ˙ XY )
2 ∂x

+ iY (2˙uH + u ˙ H) (8.93)
By using Eq. (8.19b) as well as the fact that Y = X+iP, this expression
can be simplified to

˙

X C 22 + Y C 10 = u[(H ˙ P) − i P + iY( ˙ PX − ˙ XP )] + iY (2˙uH + u ˙ H)
(8.94)
By noticing that
2

1 ∂n
2

H = ( n (X, z) − P ) = X − 2PP = ˙ PX − ˙ XP
2

2H ∂x
(8.95)
where Eqs. (8.19a) and (8.19b) were used in the last step, Eq. (8.94) can
be simplified further as

˙

X C 22 + Y C 10 = u[(H ˙ P − i P) + iY H ] + iY (2˙uH + u ˙ H)

= iu[Y H − ( H ˙ X + iH ˙ P) ] + iY (2˙uH + u ˙ H)

˙
˙
˙

= iu[Y H − (HY) ] + iY (2˙uH + u ˙ H)

˙
=−iuHY + iY (2˙uH + u ˙ H)

∂ H
= 2i HY 3 u (8.96)
∂z Y
where Eq. (8.19a) was used in the second step. Therefore, Eq. (8.92)
is satisfied up to the two leading orders if the result of Eq. (8.96)
is asymptotically negligible. This can be enforced by writing u as a
Debye series of the form
∞
k Y (z, ) , a j

u = (8.97)
2 H(z, ) (ik) j
j=0
where the dominant term of the sum, that is, a 0 ( ), is independent of
z, and the constant factor in front was added for convenience. That is,
for SAFE, the transport equation takes the simple form ˙ 0 = 0.
a