Page 264 - Phase Space Optics Fundamentals and Applications

Rays and Waves 245

8.3 Small-Wavelength Limit in the Position
Representation. II: The Transport
Equation and the Field Estimate
After we chose to satisfy the eikonal equation, the remainder of
Eq. (8.3) can be written as
1
2 2
2∇ A·∇ + A∇ + ∇ A = 0 (8.23)
ik
Again, the key to the asymptotic treatment that follows is to regard k
as a parameter that takes very large values.

8.3.1 The Debye Series Expansion
The amplitude Ais now written as a so-called Debye series of the form
∞
r
, A j ( )
r
A( ) = (8.24)
(ik) j
j=0
Then, upon substitution of Eq. (8.24), Eq. (8.23) can be written as
∞
,
2 1 2 2
2∇ A 0 ·∇ + A 0 ∇ + j 2∇ A j ·∇ + A j ∇ +∇ A j−1 = 0
(ik)
j=1
(8.25)
Since k is used as an asymptotic parameter, the coefficient of each
power of k is made to vanish independently. This gives a hierarchy of
linked equations for each of the A j of the form
2
2∇ A 0 ·∇ + A 0 ∇ = 0 (8.26a)
2
2
2∇ A j ·∇ + A j ∇ =−∇ A j−1 , j = 1, 2, ... (8.26b)

8.3.2 The Transport Equation and Its Solution
Equation (8.26a) can be solved to find A 0 , and in principle each A j
can be found successively in terms of the previous one by solving
Eq. (8.26b). For sufficientlylargek,however, A ≈ A 0 ,soonly Eq.(8.26a)
will be considered. Notice that by multiplying both sides by A 0 , this
equation can be rewritten as

2
∇· A ∇ = 0 (8.27)
0
This expression is known as the transport equation. To solve it, consider
integrating both sides over the volume occupied by a segment of an