Page 292 - Phase Space Optics Fundamentals and Applications

Rays and Waves 273

where Z 2 ( ) = Z 1 ( ) + n 0 cos / is the value of z at which the
ray exits the inhomogeneous region. From the equations above,
as well as their substitution in any of the wave field estimation
formulas, it is easy to see that the solution to the problem de-
pends on the dimensionless parameters kn 0 / , x 0 , and x s ,as
well as on the dimensionless variables ( x, z). For SAFE, the
estimate also depends on / .

Field estimates can be constructed from the equations for the rays
given above, once they are supplemented with initial weights for
the rays. In this example, only the position- and momentum-based
field estimates as well as that corresponding to SAFE (with / = 1)
are calculated. Since we are considering a point source emitting light
equally in all directions, the initial ray weight for SAFE is chosen as a
constant, a 0 = U 0 . For propagation in two dimensions, the position-
representation-based estimate in Eq. (8.32) can be written as

H(z 0 , )X (z 0 , )

U[X(z, ),z] ≈ A 0 [X(z 0 , ),z 0 ] exp[ikL(z, )]
H(z, )X (z, )

(8.108)
Notice that, to plot the estimate as a function of x, the equation
x = X(z, ) has to be solved for as a function of x and z. Except for
very simple cases such as free-space propagation, this cannot be done
in closed form, so it is necessary to use a numerical root-search proce-
dure. For the example considered here, this procedure is straightfor-
ward.However,formorecomplicatedopticalsystems,thisroot-search
procedure can be computationally demanding, as each iteration in-
volves tracing a ray across the system. Also notice that, as written,
Eq. (8.108) is not suitable for the evaluation of a field generated by a
point source, since X (z 0 , ) = 0. This problem can be solved by sub-

√
stituting H(z 0 , )X (z 0 , )A 0 [X(z 0 , ),z 0 ] = U 0 , as this substitution

would give the asymptotic estimation of a circular wave in free space.
The momentum-based estimate for two-dimensional propagation is
given by

k H(z 0 , )P (z 0 , )P (z, )

r
U( ) ≈ B 0 [P(z 0 , ),z 0 ]
2 H(z, )
× exp(ik{L(z, ) + [x − X(z, )]P(z, )}) d (8.109)
For calculating the field due to a point source, we use B 0 [P(z 0 , ),z 0 ] =
√
iU 0 /H(z 0 , ) (where the phase factor is inserted so that this estimate
is in phase with the other two).
Figure 8.7 shows (a) some of the rays and (b) a segment of the PSC
for x s = 3, x 0 = 0.8, and z = 10. Notice that these rays present
both a position caustic (at x ≈ 0.55) and a momentum caustic (at
x ≈ 0.78). The intensity estimates at z = 10 for kn 0 / = 1000 are