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Rays and Waves 275

8.11 Concluding Remarks
The field estimates presented in this chapter were derived through the
same generic procedure: an ansatz is substituted into the Helmholtz
equation, leading to conditions that the functions in the ansatz must
satisfy. Note, however, that an alternative procedure for deriving
some of these results exists where, instead of using the (differential)
Helmholtz equation, one considers an infinite succession of prop-
agation integrals over infinitesimally short distances. For each in-
finitesimally thin slice of space, the propagation integral amounts to a
Rayleigh-Sommerfeld-like superposition of secondary waves, where
for each of these waves the refractive index is approximated as that
at the secondary source position. This description of wave propaga-
tion is analogous to Feynman’s path integral formulation of quantum
mechanics 48 as a “sum of all possible histories.” The ray-based esti-
mates can then be obtained by approximating the integrals asymptoti-
cally, where k is used as the asymptotic parameter. The position-based
estimate in particular results from applying the method of stationary
phase 49 to all the integrals. The resulting leading contribution to the
field at a point is associated with a trajectory (or a set of trajectories) ar-
riving at this point from the initial plane. The optical path length of this
trajectory is stationary with respect to infinitesimal variations, so it sat-
isfies Fermat’s theorem. These stationary paths are therefore the rays
of geometrical optics. Since the method of stationary phase consists of
approximating the rapidly varying phase as a quadratic polynomial
around a stationary point, the short-wavelength limit studied here
has something in common with the paraxial and quasi-homogeneous
limits mentioned in Sec. 8.1: they all lead to field estimates consistent
with ray optics through the quadratic approximation of phases inside
integrals, so that these integrals can be approximated analytically.
The author acknowledges support from the National Science Foun-
dation under grant 0449708, and from an anonymous donor under an
internal grant from The Institute of Optics.

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2. R. K. Luneburg, Mathematical Theory of Optics, University of California Press,
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New York, 1964.
4. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media,
Springer-Verlag, Heidelberg, 1990.
5. Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2d ed.,
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