Page 141 - Phase Space Optics Fundamentals and Applications

122 Chapter Four

simulations and the experimental results obtained using a Ronchi
grating as input object.
Interestingly, in Fig. 4.6 the values of p that correspond to the self-
images, both positive and negative, can be clearly identified. The op-
tical setup designed for the experimental implementation of the RWD
was successfully adapted to several applications, as we show later in
this chapter.
In searching for an RWD with an exact scale factor for all the frac-
10
tional orders, this approach also inspired another proposal in which
a bent structure for the detector was suggested. The result is an exact,
but unfortunately impractical, setup to obtain the RWD. This draw-
back was partially overcome in other configurations derived by the
same authors using the abcd matrix formalism. There, the free propa-
gation distances are designed to be fixed or to vary linearly with the
transverse coordinate, 11 so the input plane and/or the output plane
should be tilted instead of bent, resulting in a more realistic configu-
ration, provided that the tilt angles are measured very precisely.

4.3 Analysis of Optical Signals and Systems
by Means of the RWT

4.3.1 Analysis of Diffraction Phenomena
4.3.1.1 Computation of Irradiance Distribution
along Different Paths in Image Space
Determination of the irradiance at a given point in the image space
of an imaging system is a classic problem in optics. The conventional
techniques carry out a finite partition of the pupil of the system to
sum all these contributions at the observation point. 12–16 This time-
consuming procedure needs to be completely repeated for each ob-
servation point, or if the aberration state of the system changes. In this
section we present a useful technique, based on the use of the RWT
of a mapped version of the pupil of the system, for a much more effi-
cient analysis of the irradiance in the image space of imaging systems.
This technique has been successfully applied to the analysis of dif-
ferent optical systems with circular 17 as well as square, 18 elliptical, 19
triangular, 19 and even fractal pupils. 20 The method has also been ap-
plied to the study of multifaceted imaging devices. 21
Let us consider a general imaging system, characterized by an exit
pupil function with generalized amplitude transmittance P(x). The
distance from this pupil to the Gaussian imaging plane is denoted
by f . Note that the function P(x) includes any arbitrary amplitude
variation p(x) and any phase aberration that the imaging system may
suffer from.