Page 146 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 127

and at an oriented distance from the origin
W 40 + W 20 (z)
x (z) =− (4.65)
2
4W + 2
40
in such a way that it is possible to express
1
¯ I(r N , ,z) = RW q (x (z), ) (4.66)
2
( f + z) 2
The main conclusion of all this is that it is possible to obtain the ir-
radiance at any point in image space through the values of the RWT
of a given function q(s, r N , ,z) related to the pupil of the system.
Note, however, that this function depends in general on the particular
coordinates r N , , and z of the observation point. Thus, a different
function RW q (x , ) has to be considered for different points in image
space. This major drawback can be overcome for special sets of points
or trajectories in image space that share the same associated mapped
pupil q(s, r N , ,z).
To describe such trajectories in image space, let us express these
lines in parametric form r N (z), (z). By substituting Jacobi’s identity

+∞
, n
exp(i cos ) = i J n ( ) exp (−in ) , ∈ R (4.67)
n=−∞
where J n (x) represents the Bessel function of the first kind and order
n, into Eq. (4.55), it is straightforward to obtain

+∞
, −2
n
Q(r ,r N (z), (z),z) = i J n r r N (z) Q n (r )
N
N
N
( f + z)
n=−∞
× exp [in (z)] (4.68)
where Q n (r ) stands for the n-order circular harmonic of Q n (r , ),

N N
that is,
2

Q n (r ) = Q(r , ) exp −in d (4.69)
N
N
0
Note that the dependence on the position parameter z in Eq. (4.68)
is established exclusively in the argument of the Bessel functions—
through r N (z)—and the phase exponentials—through (z). Thus, the
only way to strictly cancel this dependence is to consider the trajecto-
ries
r N (z) = K( f + z), (z) = o (4.70)