Page 64 - Introduction to Information Optics

1.7. Accuracy and Reliability Observation 49

For the case of simultaneous observations, we have simultaneously observed
7 positive readings from the a photodetectors. The amount of information
obtained is therefore

/ = log 2 - bits, (1.164)

where y < a. Again, any reading could be due to thermal fluctuation. Since
there are y detectors absorbing the scattered light and the observations are
made on all the a photodetectors, the overall amount of entropy increase in
the y photodetectors is
1/a
AS^ -y/cln[l -i ]. (1.165)

For a » 1, the preceding equation can be approximated by

AS^yfc(lna + 0.367), (1.166)

which increases with respect to y and a. Thus

AS - J/cln2 ^ fc[lny + (}> - l)lna + 0.367y] > 0. (1.167)

It is interesting to note that, if it takes only one of the a photodetectors to
provide a positive reading (y = 1), then Eq. (1.167) is essentially identical to
Eq. (1.163). However, if y » 1, then the amount of information obtained from
the simultaneous observations of Eq. (1.164) is somewhat less than that
obtained from the sequential observations of Eq. (1.160), for q » 1, and the
amount of entropy increase for the simultaneous observations is also greater.
Since it is assumed that only one particle is wandering in the spatial domain
A, for y = 1, Eq. (1.167) yields the smallest trade-off of entropy and information.
At the same time, for a large number of photodetectors (a » 1), Eq. (1.166) is
the asymptotic approximation used in high-accuracy observation (hv » kT).
Remember that any other arrangement of the photodetectors may result in
higher entropy. For example, if all the photodetectors are arranged to receive
light directly, rather than from scattered light, it can be seen that a much higher
cost of entropy will be paid.
We now turn our attention to high-accuracy observation. Note that, if AA
becomes very small, then higher-frequency illumination (a shorter wavelength)
is necessary for the observation. As illustrated in Fig. 1.16, this observation
cannot be efficient unless the wavelength of the light source is shorter than

1.64dsin0, (1.168)