Page 172 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 153

for the diffracted pattern, and, therefore, the final superposition of all
the spectral components of the incoming light produces a chromatic
blur of the monochromatic result. To analyze this effect, we again use
the RWT approach to describe the spectral components of this Fresnel
pattern. Since the above dispersion volume is transformed, by means
of the imaging system, into a different image volume, it is interesting
to derive the geometric conditions that provide a minimum value for
its axial elongation in the image space. Equivalently, the same RWT
analysis will be performed at the output of the system to analyze the
chromatic blur at the output plane for the reference wavelength o .
For the sake of simplicity, we consider a one-dimensional ampli-
tude transmittance t(x) for the diffracting object. Let us now apply
our approach to calculate the irradiance free-space diffraction pattern
under issue through the RWT of the object mask. If we recall the result
in Eq. (4.36) for z = R o , we obtain that each spectral component of this
Fresnel pattern is given by

I o (x; ) ∝ RW t (x ( ), ( )) , tan ( ) =− R o ,
x
x ( ) = sin ( ) (4.100)
R o
In this equation the chromatic blur is considered through the spec-
tral variation of the coordinates in the RWT for any given transverse
position x in the diffraction pattern. Thus, for a fixed observation
position there is a region in the Radon space that contains all the
points needed to compute the polychromatic irradiance. If we define
i = arctan ( i R o ), for i = 1, 2, the width of this region in both Radon-
space directions can be estimated as

x sin 1 sin 2
=| 1 − 2 |, x = − (4.101)

R o 1 2
Note that the smaller this region is, the less is the effect of the chro-
matic blur affecting the irradiance at the specified observation point.
To achieve an achromatization of the selected diffraction pattern, this
region has to be reduced in the output plane of the optical setup.
Let us now fix our attention on the effect of the imaging system on
the polychromatic diffraction pattern under issue. Again, we use the
RWT approach to achieve this description by simply noting that the
system behaves as an abcd device that links the object plane and the se-
lected output plane. The transformation matrix M achr can be obtained
as a sequence of elemental transformations (see Fig. 4.29), namely, free
propagation at a distance l, propagation through the achromatic lens,
free propagation to the focal plane of that element, passage through

the zone plate, and, finally free propagation at a distance d . The out-
o
put plane is selected as the image plane of the diffraction pattern