Page 279 - Phase Space Optics Fundamentals and Applications

260 Chapter Eight

Notice that the contribution for each ray now extends over all the
configuration space. That is, in this picture, rays do not contribute to
the field as infinitesimally thin conduits of power. Instead, they are
infinitely extended waves that interfere to make up the wave field. In
fact, for the case of homogeneous media, these waves are plane waves.
This estimate has no problems at position caustics (focal points), but
fails near momentum caustics.
While in this section we considered Fourier transforms over both
transverse coordinates, it is also possible to find estimates where
Fourier transforms over only one transverse coordinate are per-
formed. These could be useful in problems with specific asymmetries,
or in the implementation of the method discussed next.

8.7 Maslov’s Canonical Operator Method
The ray-based schemes discussed in previous sections present prob-
lems at either position or momentum caustics. These problematic situ-
ationshavegeometricalinterpretationsintermsofphasespace.Again,
for simplicity, let us consider the case of two-dimensional propaga-
tion, where there is only one transverse position and one transverse
momentum, so that phase space is a plane. In this case, the formulas
for the amplitudes of the estimates in Eqs. (8.31) and (8.73) become,
respectively,

H(z 0 , ) X (z 0 , )

A 0 [X(z, ),z] = A 0 [X(z 0 , ),z 0 ] (8.75a)
H(z, ) X (z, )

H(z 0 , ) P (z 0 , )

B 0 [X(z, ),z] = B 0 [P(z 0 , ),z 0 ] (8.75b)
H(z, ) P (z, )

The field estimate resulting from using the position-dependent ap-
proach fails at caustics, i.e., when X = 0. In phase space, caustics cor-

respond to segments of the PSC that are locally vertical (see Fig. 8.5).
Ontheotherhand,themomentum-representation-basedestimatefails

at momentum caustics, when P = 0, i.e., at segments of the PSC that
are locally horizontal. One could formulate field estimates based on
other representations associated, e.g., with a fractional Fourier trans-
form over the transverse variable of the field. 22 These field estimates
would be well behaved at both position and momentum caustics, but
would fail around rays associated with segments of the PSC with a
given inclination (depending on the degree of the fractional Fourier
transform).
When a PSC is sufficiently complicated, the caustic problems are un-
avoidable, regardless of representation. Based on this fact, Maslov 11