Page 267 - Phase Space Optics Fundamentals and Applications

248 Chapter Eight

the ray(s) passing through this point. That is, if we only integrate the
ray equations over a thin bundle of rays, we can estimate the field
along this bundle over a long distance, regardless of what the rest of
the rays do, as long as rays do not cross. Notice also that if rays from
a point source are considered, and the medium is an ABCD system in
the paraxial approximation, the formula in Eq. (8.32) can be shown to
give the point-spread function in Eq. (1.42) after solving for in terms
of the rays’ final position.
Notice, however, that the expression for the amplitude of this es-
timate given in Eq. (8.31) diverges when either of the two following
conditions is met:

H(z, ) = 0 (8.33)
[X(z, )]
= 0 (8.34)
( )
The first of these conditions happens when rays turn around in z.As
mentioned at the outset, it is assumed that this does not occur. (If it
does, this problem can be alleviated by using a different ray parame-
terization.) The second condition, on the other hand, occurs when the
bundle collapses, i.e., when the rays that make up the bundle cross.
This crossing of contiguous rays is what is known as a caustic. While
the field at a caustic is indeed large, it is not infinite as Eq. (8.31) pre-
dicts. This signals a problem in this formalism, which arises from the
fact that in the vicinity of the caustic, the right-hand side of Eq. (8.26b)
for j = 1 becomes large due to the fast variation of A 0 . This causes A 1 ,
as well as the rest of the A j , to be large too, up to a point where the
Debye series cannot be approximated by its leading term. The field
estimate in Eq. (8.32) therefore breaks down in the vicinity of caustics.
Remarkably, though, if after the caustic the rays become sufficiently
spread and uniform, this field estimate becomes valid again. Note,
however, that the passage through the caustic gives rise to changes
in sign for the Jacobian inside the square root in Eq. (8.31). Due to
the square root, these sign changes cause phase shifts that are integer
multiples of /2, where the integer is called the Maslov or Morse
index. The determination of these indices can be complicated, 11,12 as
the appropriate sign of the square root must be chosen. (Note: This
phase shift is consistent with the Gouy phase shift undergone by a
focused beam.) In many situations, after a caustic or caustics, there
are several bundles of rays occupying the same volume. The field
estimate then results from summing their contributions, each with a
suitable Maslov phase. (This is one example of how ray optics can
account for interference effects.) Also, there might be regions to one
side of a caustic where no rays arrive. The simple estimate presented
above then offers no access to the field in these regions, suggesting