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Rays and Waves 261

p

x

FIGURE 8.5 Vertical segments (indicated by black dots) of the PSC
correspond to position caustics, while horizontal segments (indicated by
open dots) correspond to momentum caustics.

proposed a scheme where the PSC is subdivided into several seg-
ments, each free of at least one type of caustic (position or momentum).
A small transition region is left between the segments to avoid errors
introduced by the abrupt cuts. A field estimate is then performed for
each individual segment, using the appropriate prescription. The total
estimate is found by adding the contributions due to all the segments.
Notice that one must be careful in choosing the correct Maslov in-
dex phase for each contribution. This approach is known as Maslov’s
canonical operator method, or simply Maslov’s method. 5,11,12 Its im-
plementation can be quite complicated, however, especially for three-
dimensional fields where instead of a PSC one has a two-dimensional
Lagrange manifold embedded in a four-dimensional phase space. In
this case, the global estimate can involve contributions of not only the
position- and momentum-representation-based estimates, but also of
those in mixed representations mentioned at the end of Sec. 8.6.

8.8 Gaussian Beams and Their Sums
In this section, a different scheme for connecting rays and waves is
discussed. Here, field contributions with finite effective extent in both
position and momentum are considered. For simplicity, the analysis
is performed in two-dimensional space.

8.8.1 Parabasal Gaussian Beams
According to the uncertainty relation, a field contribution cannot be
simultaneously arbitrarily localized in position and direction (i.e.,
momentum), since the product of the rms widths in these two