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Rays and Waves 271

p
p
√ (γ –1 + μ )/2k
–1

P x
√ (γ –1 + μ )/2k
–1
x
X
(a) (b)

FIGURE 8.6 (a) The rms widths in phase space of the Gaussian-windowed
Fourier transform of a field contribution in Eq. (8.102) due to a ray. (b) Picture
of the Gaussian-windowed Fourier transform of SAFE’s field estimate as the
result of spray-painting over the PSC, where the footprint of the spray can
has the shape shown in part a.

For example, near a position caustic, the size of the footprint in the p
direction should not be excessively large, placing an upper bound on
. On the other hand, near a momentum caustic, the x direction width
of the footprint should be sufficiently small to resolve the correspond-
ing curvature, and this results in a lower bound on . This implies that
the geometry of the PSC (or the Lagrange manifold) dictates the valid
ranges for the width of the contributions.
It is worth noticing that, in the limit → 0, SAFE’s estimate re-
duces to the momentum representation estimate in Eq. (8.74), as the
contributions become infinitely wide. On the other hand, in the limit
→∞, the Gaussian contributions become delta functions, and
the estimate reduces to that resulting from the position representa-
tion treatment, given in Eq. (8.32). The fact that, in the presence of
position or momentum caustics, these limiting values of violate
the limitations outlined in the previous paragraph is consistent with
the failure of the estimates discussed in Secs. 8.3 and 8.6 in these
situations.

8.10 A Simple Example
Some of the methods described earlier are applied here to a simple ex-
ample, corresponding to a mirage like reflection in two-dimensional
space. Consider a hot surface at x = 0 heating up a transpar-
ent medium in the half-space x > 0. Away from this surface, the
medium’s refractive index is n 0 , while near it, the refractive index de-
creases smoothly due to thermal expansion. Let us use the following