Page 148 - Introduction to Information Optics

2.8. Processing with Incoherent Light ! 33

then the corresponding Fourier transform would be

rv
,
r
F(p,q) = Fi x
in which we see that F(p,q) is smeared into rainbow colors at the Fourier
domain. Thus, a high temporal coherence Fourier spectrum within a narrow-
spectral-band filter can be obtained, as given by

(2.132)
Po

Since the spectral content of the input object is dispersed in rainbow colors, it
is possible to synthesize a set of narrow-spectral-band filters to accommodate
the dispersion, as illustrated in Fig. 2.45a.
On the other hand, if the spatial filtering is a 1 -D operation, it is possible
to construct a fan-shaped spatial filter to cover the entire smeared Fourier
spectrum, as illustrated in Fig. 2.45b. Thus, we see that a high degree of
temporally coherent filtering can be carried out by a simple white light source.
Needless to say, the (broadband) spatial filters can be synthesized by com-
puter-generated techniques.
In the preceding we have shown that spatial and temporal coherence can be
exploited by spatial encoding and spectral dispersion of an incoherent source.
We have shown that complex amplitude processing can be carried out with
either a set of 2-D narrow-spectral-band filters or with a 1-D fan-shaped
broadband filter.
Let us first consider that a set of narrow-spectral-band filters is being used,
as given by

7
H n(p n,q n,i n), forn = 1 2, . . . , ,\ ,
where (p n, q n) represents the angular frequency coordinates and /L n is the center
wavelength of the narrow-width filter. It can then be shown that the output
light intensity would be the incoherent superposition of the filtered signals, as
given by

2
/(*, y) =£ £ A^|/(.x, y- A n) * h(x, y; A n)\ , (2.1 33)

where * denotes the convolution operation, f(x, y,A n) represents the input
signal illuminated by A n , A/1,, is the narrow spectral width of the narrow-