Page 153 - Phase Space Optics Fundamentals and Applications

134 Chapter Four

FIGURE 4.14 RWD as a display of all diffraction patterns generated by a
Cantor grating of level 3.
show a partial self-similar behavior that is increased when moving to-
ward the Fraunhofer region. For this reason, it is useful to represent the
evolution of the complex amplitude of one-dimensional fractals prop-
agating through free space represented on a two-dimensional display,
especially if such a display can be obtained experimentally. In this case
one axis represents the transversal coordinate, and the other is a func-
tion of the axial coordinate. In fact, according to the analysis carried
out in Ref. 27, the evolution of the diffraction patterns allows one to
determine the main characteristic parameters of the fractal. Therefore,
one of the most important applications of the RWD has been in this
field. 28 The RWD obtained for a triadic Cantor grating developed up
to level 3 is shown in Fig. 4.14. Moreover, this result can be favor-
ably compared with the results obtained with other displays. 27 The
magnification provided by the lens L in the experimental setup (see
Fig. 4.4) enables the RWD representation to provide an optimum sam-
pling of the diffracted field. Near the object, where the diffraction pat-
terns change rapidly, the mapping of the propagation distance pro-
vides a fine sampling, whereas the sampling is coarse in the far field
where the variation of the diffraction patterns with the axial distance
is slow. We note that sampling is the subject of Chap. 10.

4.3.2 Inverting RWT: Phase-Space Tomographic
Reconstruction of Optical Fields
The WDF is an elegant and graphical way to describe the propagation
of optical fields through linear systems. Since the WDF of a complex
field distribution contains all the necessary information to retrieve the
field itself, 29,30 many of the methods to obtain the WDF (and the AF)