Page 152 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 133

FIGURE 4.13 RDW showing the evolution of the field diffracted by a double
slit. (a) Experimental result. (b) Cross section of the RWD showing the
intensity profile near the Fraunhofer region. For comparison purposes, the
theoretical sinc envelope of the Young fringes is also shown by the dotted
line.

the experimental RWD for values of y/h close to 1 is also represented
in Fig. 4.13.
Another classic example is diffraction by periodic objects. Here,
self-imaging phenomena, such as the Talbot effect, are interesting and
stimulating and usually attract the students’ attention. As illustrated
earlier in Fig. 4.6, which shows the diffraction patterns of a Ronchi
grating, several self-imaging planes can be identified. It can be clearly
seen that, due to the finite extent of the grating at the input, the number
of Talbot images is limited by the so-called walk-off effect. Self imag-
ing phenomena are discussed in more detail in Chap. 9 by Markus
Testorf.
In addition to its use as an educational tool for displaying diffrac-
tion patterns, the RWD has been used to investigate diffraction by a
variety of different interesting structures, including fractal diffraction
screens. In fact, the properties of diffraction patterns produced by frac-
tal objects and their potential applications have attracted the attention
of several researchers during recent years because many natural phe-
nomena and physical structures, such as phase transition, turbulence,
or optical textures, can be analyzed and described by assuming fractal
symmetry. Most research has been devoted to the study of diffraction
25
patterns obtained from fractal objects in the Fraunhofer region, yet it
is in the Fresnel region where interesting features appear. For instance,
Fresnel diffraction of a Cantor set 26 shows an irradiance distribution
along the optical axis having a periodicity that depends on the level of
the set. Furthermore, the intensity distributions at transverse planes