Page 185 - Phase Space Optics Fundamentals and Applications

166 Chapter Five

5.2 The Product-Space Representation and
Product Spectrum Representation
In early studies on the use of coherent optical systems, Leith and
his colleagues 5−7 recognized that several one-dimensional signals can
be processed in parallel by using suitable two-dimensional masks.
As is depicted in Fig. 5.1, one can implement optically a spectrum
analyzer of several one-dimensional signals, if one is able to generate
a suitable two-dimensional mask. Here, we consider the following
two-dimensional, complex amplitude transmittance
y y

p(x, y) = u x + u ∗ x − (5.1)
2 2
In this chapter, the complex amplitude transmittance in Eq. (5.1) is
8
referred to (see Wood and Barry ) as the product-space representation
of the signal u(x). For a rectangular window, u(x) = rect (x/X), the
product-space representation is

y y
x + 2 x − 2
p(x, y) = rect rect
X X
y

x
= rect rect 2 (5.2)
X −|y| X
Product spectrum representation
n
P(m,n)

Product space representation m
y
p(x, y)

x
f

f

f
f

FIGURE 5.1 Optical setup for mapping the product space representation
into the product spectrum representation.