Page 268 - Phase Space Optics Fundamentals and Applications

Rays and Waves 249

that the field must vanish there. This is not true, however, since the
field is small but not zero in these regions. There are generalizations of
the scheme presented here that give accurate estimates in these dark
regions by considering rays that leave the real space (the so-called
13
complex rays). This more complicated approach is beyond the scope
of this chapter. As will be shown later, the methods described in Secs.
8.4, 8.6, 8.7, 8.8, and 8.9 do lead to nonvanishing field estimates in the
dark regions.
The type of ray-based field estimation presented in this section is
useful, e.g., for modeling standard imaging systems, as long as it is
complemented appropriately by a simple wave-based computation.
Consider the case where the object is a point source. Then the two-
parameter family of rays corresponds to the rays emanating from
this point. The field at the image plane cannot be estimated with this
scheme, as the image is a caustic. The field must instead be recon-
structed at the exit pupil, where the rays are uniformly spread. Then a
wave-based computation must be used for the final stage of free-space
propagation from the exit pupil to the image plane. This computation
can be performed in a numerically efficient fashion through the use of
fast Fourier transforms. An additional advantage of this approach is
that the effects of diffraction from the aperture stop are automatically
accounted for in this last step by setting the field outside the exit pupil
to zero.

8.4 Flux Lines versus Rays
The second approach mentioned at the beginning of Sec. 8.2, which
consists of the assumption that both A and are purely real, is dis-
cussed in this section. In this case, Eq. (8.3) can be separated into two
equations, corresponding to its real and imaginary parts, which can
be written as

2
∇ A
2 2
|∇ | = n − 2 (8.35a)
k A
2
∇· (A ∇ ) = 0 (8.35b)
Notice that Eq. (8.35b) is identical to Eq. (8.27), except for the fact that it
includes the full amplitude Ainstead of A 0 , which is only the leading
term in a series. That is, in this case it is not necessary to express A
as a Debye series, and the full A can, in principle, be found by inte-
grating Eq. (8.35b) over an infinitesimal bundle of trajectories. These
trajectories must be found by solving Eq. (8.35a) parameterically. The
nature of these trajectories becomes apparent from the substitution of