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The Radon-Wigner Transform 157

Amplitude transmittance 1.0 Amplitude transmittance 1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.0 0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r/a r/a
(a) (b)
FIGURE 4.32 (a) Amplitude transmittance of a desired pupil function.
(b) Phase-space tomographic reconstruction of the same pupil.

To illustrate the method, we numerically simulated the synthesis
of an annular apodizer represented in Fig. 4.32. It has been shown
that its main features are to increase the focal depth and to reduce
the influence of SA. From this function we numerically determined
first the W q (x, ) function, using the WDF definition, and thereby
0,0
the projected distributions defined by the RWT, obtaining the axial
irradiance distribution for different values of SA. In this case, we used
1024 values for both W 40 / and W 20 / , ranging from −16 to +16. We
treated these distributions as if they represented the desired axial be-
havior for a variable SA, and we reconstructed the WDF by using
a standard filtered backprojection algorithm for the inverse Radon
transform. From the reconstructed WDF we obtained the synthesized
pupil function p(x) by performing the discrete one-dimensional in-
verse FT of W 0,0(x, ). The result is shown in Fig. 4.32b. As can be
q
seen, the amplitude transmittance of the synthesized pupil function
closely resembles the original apodizer in Fig. 4.32a.

4.4.3 Signal Processing through RWT
Throughout this chapter we have discussed the RWT as a mathemat-
ical tool that allows us to develop novel and interesting applications
in optics. Among several mathematical operations that can be opti-
cally implemented, correlation is one of the most important because it
can be used for different applications, such as pattern recognition and
object localization. Optical correlation can be performed in coherent
systems by use of the fact that the counterpart of this operation in the
Fourier domain is simply the product of both signals. To implement
this operation, several optical architectures were developed, such as
the classic VanderLugt and joint transform correlators. 56,57 Because