Page 63 - Introduction to Information Optics

48 1. Entropy Information and Optics

bounded; otherwise, the information cannot be defined, since the information
provided would lead to an infinite amount.
Let us assume a spatial domain A, which corresponds to the total field of
view of an optical system. The spatial domain A is then subdivided into small
subareas AA, which are limited by the resolvable power of the optical system.
Of course, light is necessary for the observation, in which a particle or object
is assumed to be wandering within the spatial domain A. In practice, we look
at each subarea A/4 until we locate the particle; then the accuracy of observation
can be written as

= ai (1.159)

where a is the total number of A^4's within A. To look for the particle we simply
illuminate each A/4 by a beam of light, and each A/1 is assumed to be equipped
with a photodetector able to detect scattered light, if any, from the particle. We
further assume that each of the photodetectors is maintained at a constant
Kelvin temperature T. Let us now investigate each of the succeeding photo-
detectors until a positive reading is obtained; say, from the qth&A, where
q < a. The reading may be caused by thermal fluctuation in the detector, or it
could be a positive reading for which the particle has been found in one of the
A/4's out of the q possibilities. Hence the amount of information obtained by
this sequential observation is

/ = Iog 2 g bits. (1.160)

Since the positive reading was obtained from the absorption of scattered light
by the qth photodetector, the accompanying entropy increase in the qth
detector is

AS ^ -Jcln[l - (|) 1/<r |, q > 1. (1.161)

For a large value of q, the right-hand side of the preceding equation can be
written as

AS ^ fc(ln q + 0.367) > kin 2. (1.162)

Thus
AS-/fcln2^0.367k>0, (1.163)

which is a positive quantity.