Page 158 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 139

both the space and spatial frequency domains, could also be evaluated
from the RWD.
To derive the formal relationship between the PSF (and the OTF)
and the RWT, let us consider the monochromatic wave field, with
wavelength , generated by an optical imaging system characterized
by a one-dimensional pupil function t(x), when a unit amplitude point
source is located at the object plane. In the neighborhood of the image
plane, located at z = 0, the field amplitude distribution can be written,
according to the Fresnel scalar approximation, as
+∞

−i 2 i 2

U (x, z) = t(x ) exp x exp (x − x) dx
f ( f + z)
−∞
(4.80)
where f is the distance from the pupil to the image plane. The trans-
formation of t(x) to obtain the field U (x, z) is given by a two-step
sequence of elementary abcd transforms, namely, a spherical wave-
front illumination (with focus at = f ) and a free-space propagation
(for a distance f + z). Considering the results presented in Sec. 4.2.1,
the abcd matrix associated with this transform can be found to be

1 0 1 − ( f + z) 1 − ( f + z)
M = = (4.81)
1 1 z
f 1 0 1 f − f
and, therefore, the equivalent relationships to that given by Eq. (4.80)
in terms of the corresponding RWTs can be expressed as [see Eq. (4.25)]

RW U(x,z) (x , ) ∝ RW t (x , )

(4.82)
2
f tan − f ( f + z) x

tan = , x = sin

tan − z sin + ( f + z) cos
In particular, the value = 0 provides the irradiance distribution at
the considered observation point, as stated in Sec. 4.2.1. This function
is the PSF of the imaging system, as a function of the distance z to the
image plane. Thus,
2
RW U(x,z) (x 0 , 0) = |U(x 0 ,z)| = I (x 0 ,z) ∝ RW t (x , )

0

0
(4.83)
f ( f + z) x 0

tan = , x = sin

0 0
z 0 ( f + z)
For the optical axis (x 0 = 0) the PSF can be expressed as

f ( f + z)
2
RW U(x,z) (0, 0) = |U(0,z)| = I (0,z) ∝ RW t 0, arctan
z
(4.84)