Page 173 - Phase Space Optics Fundamentals and Applications

154 Chapter Four

under study for = o . Thus, by using the results in Sec. 4.2.1, it is
straightforward to obtain

⎛ 2 ⎞
l f f m o
1 − − − f − m o f + m o l +
f Z( )
⎜ Z( ) ⎟
M( ) = ⎝ ⎠ (4.102)
1
m o
f
where the following restriction applies to the fixed desired output
plane
f f 2
l = R o + f − − (4.103)
m o Z o
where m o =−d /f is the magnification obtained at the fixed image

o
plane (for = o ). The relationship between the RWTs of the input
object and the output Fresnel pattern for each spectral channel now
can be established by application of Eqs. (4.27) and (4.28). In particular,
by setting = 0wefind

I (x; ) ∝ RW t x ( ), ( )

o
2
f x

tan ( ) = R o + − 1 , x = cos ( ) (4.104)

Z o o m o
Therefore, for the polychromatic description of the output diffraction
pattern we have to sum values of the RWT of the transmittance of the
object in a region in the Radon domain whose size in both dimensions
is given by

x
=| max − min |, x = (cos max − cos ) (4.105)

min
m o
where

max = max{ ( )| ∈ [ 1 , 2 ]}, min = min{ ( )| ∈ [ 1 , 2 ]}
(4.106)
The specific values of these limits, which define the extension of the
integration region in Radon space in the polychromatic case, depend
on the particular values of the geometric design parameters f and
Z o of the imaging system. We now try to find a case that minimizes
the chromatic blur in the output pattern. It is worth mentioning that

exact achromatization of the pattern is achieved only when ( ) =
( o ) ∀ ∈ [ 1 , 2 ], which cannot be fulfilled in practice, as can be

seen from Eq. (4.104). However, a first-order approximation to that
ideal correction can be achieved by imposing a stationary behavior
for ( ) around = o . Mathematically, we impose

d ( )
= 0 (4.107)
d
o