Page 144 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 125

Thus we arrive at a compact form for the irradiance at a point in the
image space

1
¯ I(r N , ,z) = 2 2
( f + z)
2
1 4 2
i2 W 40 r N i2 W 20 (z) r N

× Q(r ,r N , ,z) exp exp r dr
N
N
N

0
(4.56)
By using the mapping transformation
1
r 2 = s + , Q(r ,r N , ,z) = q(s, r N , ,z) (4.57)

N N
2
we finally obtain
1
¯ I(r N , ,z) =
2
( f + z) 2
2
0.5 2

i2 W 40 s i2 [W 40 + W 20 (z)] s
× q(s, r N , ,z) exp exp ds

−0.5
(4.58)
Note that in this expression all the dependence on the observation
coordinates is concentrated in the mapped pupil q(s, r N , ,z) and the
defocus coefficient W 20 (z). If we expand the modulus square in this
equation, we find
1
¯ I(r N , ,z) =
2
( f + z) 2
0.5
0.5 2 2
i2 W 40 (s − s )
∗

× q(s, r N , ,z)q (s ,r N , ,z) exp

−0.5 −0.5

i2 [W 40 + W 20 (z) ] (s − s )

× exp ds ds (4.59)

which by using the change of variables
s + s
t = , u = s − s (4.60)
2