Page 136 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 117

3. Magnifier. If a uniform scale factor m is applied to the input
function, the associated M matrix is given by

1 0
m
M m = (4.38)
0 m
In this case, the RWT is transformed according to the law
1

RW g (x , ) ∝ RW f (x , ), tan = 2 tan ,

m
mx
x = sin (4.39)

sin
The vertical and horizontal projections are given here simply by
the following formulas.

2 x 0
RW g (x 0 , 0) = |g(x 0 )| ∝ RW f , 0
m
(4.40)

RW g x /2 , ∝ RW f mx /2 ,
2 2
4.2.2 Optical Implementation of the RWT:
The Radon-Wigner Display
Like any other phase-space function, the RWT also enables an optical
implementation that is desirable for applications in the analysis and
processing of optical signals. The correct field identification requires
a large number of Wigner distribution projections, which raises the
necessity to design flexible optical setups to obtain them. The rela-
tionship between the RWT and the FrFT, expressed mathematically
by Eq. (4.19), suggests that the optical computation of the RWT is pos-
sible directly from the input function, omitting the passage through
its WDF. In fact, the RWT for a given projection angle is simply the in-
tensity registered at the output plane of a given FrFT transformer. For
one-dimensional signals, the RWT for all possible projection angles
simultaneously displays a continuous representation of the FrFT of a
signal as a function of the fractional Fourier order p, and it is known
as the Radon-Wigner display (RWD). This representation, proposed by
Wood and Barry for its application to the detection and classification
1
of linear FM components, has found several applications in optics as
we will see later in this chapter.
Different and simple optical setups have been suggested to im-
plement the FrFT, and most have been the basis for designing other
systemstoobtaintheRWD.Thefirstonedescribedintheliterature,de-
signed to obtain the RWD of one-dimensional signals, was proposed
4
by Mendlovic et al. It is based on Lohmann’s bulk optics systems
5
for obtaining the FrFT. In this method, the one-dimensional input