Page 151 - Phase Space Optics Fundamentals and Applications

132 Chapter Four

have to be considered, the greater the resultant savings in computation
time.
As a final remark on this subject, we want to point out that this
approach can also be applied to other trajectories of interest in im-
age space. For instance, short paths parallel to the optical axis in the
neighborhood of the focal plane 17 or straight lines crossing the focal
point can be considered. 22

4.3.1.2 Parallel Optical Display of Diffraction Patterns
In Sec. 4.2.2 we mentioned that the mathematical relationship be-
tween Fresnel diffraction and the FrFT is given by Eq. (4.41). This
means that the RWD is itself a continuous display of the evolution
of diffraction patterns of one-dimensional objects, and this property
is extremely useful from a pedagogical point of view. In fact, calcu-
lations of Fresnel and Fraunhofer diffraction patterns of uniformly
illuminated one-dimensional apertures are standard topics in under-
graduate optics courses. These theoretical predictions are calculated
analytically for some typical apertures, or, more frequently, they are
computed numerically. The evolution of these diffraction patterns un-
der propagation is often represented in a two-dimensional display of
gray levels in which one axis represents the transverse coordinate—
pattern profile—and the other axis is related to the axial coordinate—
evolution parameter. 23 This kind of representation illustrates, e.g.,
how the geometrical shadow of the object transforms into the Fraun-
hofer diffraction pattern as it propagates, and that the Fraunhofer
diffraction simply is a limiting case of Fresnel diffraction. 24 In addi-
tion to the qualitative physical insight that the RWD provides about
diffraction, it can provide a quantitative measurement of a variety of
terms. These include the precise location y s and the relative magnifica-
tion M s of each diffraction pattern. These two terms are quantitatively
defined in terms of the maximum
h and minimum
0 powers of the
varifocal lens L of the system represented in Fig. 4.5, i.e.,
h
y s = , M s = 1 + (4.76)
2

+ l (
h −
0 ) l (
h −
0 )
2
where
is the axial coordinate at which the corresponding diffraction
pattern is localized under parallel illumination and h is the extent of
the so-called progression zone of the varifocal lens. Figure 4.13 illus-
trates the experimental results registered by a CCD camera using a
double slit as an input object. It can be seen that the RWD is a nice
representation of the evolution by propagation of the interference phe-
nomena. In fact, the Fraunhofer region of the diffracted field clearly
shows the characteristic Young fringes modulated by a sinc function.
To compare the theoretical and experimental results, a cross section of