Page 273 - Phase Space Optics Fundamentals and Applications

254 Chapter Eight

8.6 Small-Wavelength Limit in the
Momentum Representation
In the asymptotic approach presented in Secs. 8.2 and 8.3, the field es-
timate at a point is only due to the rays that go through that point. That
is, rays are completely local entities. This approach leads to problems
at caustics, where there is an infinite density of rays. An alternative
asymptotic approach is presented in this section, where instead of
working with the field as a function of r, we use its Fourier transform
over the transverse coordinates, defined as

k
˜ U( p x ,p y ,z) = U(x, y, z) exp(−ikx · p) dx dy (8.50)
2
where x = (x, y) and p = ( p x ,p y ). While the derivation presented in
the next few pages is in many ways analogous to that in Secs. 8.2 and
8.3, it is appreciably more cumbersome. Those readers who prefer to
do so can jump directly to the final result, given in Eq. (8.74).

8.6.1 The Helmholtz Equation in the
Momentum Representation
In free space, the transverse Fourier transform in Eq. (8.50) is known
21
as the angular spectrum representation. In a smoothly inhomogeneous
medium, ˜ U satisfies an equation corresponding to the Fourier trans-
formation of both sides of Eq. (8.1):

∂ 2 i ∂
2 2 2 2
− k |p| + + k n ,z ˜ U = 0 (8.51)
∂z 2 k ∂p
Here a function evaluated at a derivative is to be interpreted in terms
of its Taylor expansion:

j
∞
i ∂ , 1 i
˜
2
n ,z U(p,z) =
k ∂p j! k
j=0
j
←− −→
∂
∂
2
× n (x,z) · ˜ U(p,z) (8.52)
∂x ∂p
x=(0,0)
where the arrows indicate the direction in which the derivatives act.
2
Notice that it is assumed here that n is an analytic function and, for
convenience, the Taylor expansion is carried out around x = (0, 0).
It turns out, however, that the asymptotic results of this section are
independent of the point of expansion, and they hold as long as n is
continuous.