Page 147 - Introduction to Information Optics

132 2. Signal Processing with Optics

length of the extended source. Thus, we see that the spatial coherence at the
input plane and the source-encoding intensity transmittance form a Fourier
transform pair, as given by

y( Xl) = :F[F(x 2 - x' 2)], and F(x 2 - xV) = ..^~^(x,)], (2,128)

5
where J " denotes the Fourier transform operation. In other words, if a •specific
spatial coherence requirement is needed for certain information processing, a
source encoding can be performed. The source-encoding }'(xj) can consist of
apertures of different shapes or slits, but it should be a positive real function
that satisfies the following physically realizable constraint:

0^}<x,)s$l. (2/1.29)

For the exploitation of temporal coherence, note that the Fourier spectrum
is linearly proportional to the wavelength of the light source. It is apparently
not capable of (or is inefficient at) using a broadband source for complex
amplitude processing. To do so, a narrow-spectral-band (i.e., temporally
coherent) source is needed. In other words, the spectral spread of the input
object should be confined within a small fraction fringe spacing of the spatial
filter, which is given by

(2.130)
271
where d is the fringe spacing of the spatial filter, p m is the upper angular spatial
frequency content of the input object, / is the focal length of the transform lens,
and AA is the spectral bandwidth of the light source. In order to have a higher
temporal coherence requirement, the spectral width of the light source should
satisfy the following constraint:

A/, 71
— «7— , (2.131)
A hp m
where 1 is the center wavelength of the light source, 2h is the size of the input
object transparency, and 2h = (4f)/d.
There are ways to exploit the temporal coherence content from a broadband
source. One of the simplest methods is by dispersing the Fourier spectrum,
which can be obtained by placing a sampling grating at the input domain. For
example, if the input object is sampled by a phase grating as given by

f(x 2)T(x 2) = /(x 2)exp(j> 0x 2),