Page 299 - Phase Space Optics Fundamentals and Applications

280 Chapter Nine

Closer study, however, reveals a powerful representation of optical
signals and systems adding unprecedented insight and intuition to
well-known optical phenomena, which in turn forms the basis for
new system designs and applications.
While not all effects in classical optics find a preferable interpreta-
tion in phase space, the self-imaging phenomenon can be regarded as
a poster child for promoting phase-space optics as a true alternative
to Fourier optics.
In this chapter, we revisit the self-imaging effect and its closest rel-
atives, including the Talbot effect, the fractional Talbot effect, and the
Lau effect. All essential relationships are derived from simple dia-
grams of the associated phase space. We show that much of the math-
ematics involving the WDF can be avoided once a small set of relation-
ships has been established. Compared to the Fourier optics treatment,
we will then barely need any mathematical instrument, except basic
algebra and geometry. This is only possible if the mathematical tools
of phase-space optics are not applied in a mechanistic way, but are
customized to each situation. Thus, the study of self-imaging may not
provide a foolproof recipe. The chapter rather is intended as a teaser to
illustrate the beauty of phase-space optics. While intellectual pleasure
is guaranteed, the phase-space interpretation of self-imaging may also
have the potential of pointing toward new effects and applications not
immediately obvious from a Fourier optics perspective.

9.2 Phase-Space Optics Minimum Tool Kit
Phase-space optics represents N-dimensional signals in a 2N-
dimensional configuration space. Since we want to use diagrams not
merely for illustration, but also to obtain quantitative results, we re-
strictourdiscussiontoone-dimensionalopticalsignals.Aphase-space
distribution suitable to represent the one-dimensional complex am-
plitude distribution u(x) is the WDF 1–3

x x
∞

W(x, ) = u x + u ∗ x − exp(−i2 x ) dx (9.1)
2 2
−∞
The signal function enters the transform twice which results in a bilin-
ear transformation. The properties of the WDF are highly symmetric
with respect to the two conjugate variables x and . This is reflected
by the alternative definition

∞

W(x, ) = ˜ u + ˜ u ∗ − exp(i2 x) d (9.2)
2 2
−∞