Page 157 - Phase Space Optics Fundamentals and Applications

138 Chapter Four

of this effect, a very good qualitative agreement can be observed be-
tween the results obtained with the theoretical and experimental data.
The asymmetry in Fig. 4.17 is a consequence of the noise in Fig. 4.16a,
reflecting also the asymmetry on the spatial coordinate in this figure.
In Fig. 4.18 the typical arrow-shaped WDF of a chirp function can be
observed in both cases. The slope in the arrowhead that character-
izes the chirp rate of the signal is the same for the theoretical and the
experimental results.
Several extensions of the proposed method are straightforward. On
one hand, a similar implementation proposed here for the WDF can be
easily derived for the AF, by virtue of Eq. (4.22). Note also that it is easy
to extend this method to obtain two-dimensional samples of the four-
dimensional WDF of a complex two-dimensional signal by use of a
line scanning system. Moreover, since complex optical wave fields can
be reconstructed from the WDF provided the inversion formulas, this
approach can be used as a phase retrieval method that is an alterna-
tive to the conventional interferometric or iterative-algorithm-based
36
techniques. In fact, as demonstrated, phase retrieval is possible with
intensity measurements at two close FrFT domains. This approach,
however, requires some a priori knowledge of the signal bandwidth.
In our method, a continuous set of FrFTs is available simultaneously,
and this redundancy should avoid any previous hypothesis about the
input signal.

4.3.3 Merit Functions of Imaging Systems
in Terms of the RWT
4.3.3.1 Axial Point-Spread Function (PSF) and Optical
Transfer Function (OTF)
There are several criteria for analyzing the performance of an opti-
cal imaging system for aberrations and/or focus errors in which the
on-axis image intensity, or axial point-spread function (PSF), is the
relevant quantity. Among them we mention: 37 Rayleigh’s criterion,
Marechal’s treatment of tolerance, and the Strehl ratio (SR). As Hop-
38
kins suggested, the analysis of Marechal can be reformulated to give
a tolerance criterion based on the behavior of the optical transfer func-
tion (OTF) (spatial frequency information) instead of the PSF (space
information). Phase-space functions were also employed to evaluate
some merit functions and quality parameters. 39–41 This point of view
equally emphasizes both the spatial and the spectral information con-
tents of the diffracted wave fields that propagate in the optical imaging
systems. Particularly, since the information content stored in the FrFT
of an input signal changes from purely spatial to purely spectral as
p varies from p = 0to p = 1, that is, in the domain of the RWT, it
is expected that the imaging properties of a given optical system, in