Page 65 - Introduction to Information Optics

50 1. Entropy Information and Optics

where d is the diameter of A/1, and 8 is the subtended half-angle of lens
aperture. By referring to the well-known resolving power of the microscope, we
have

where 2 sin $ is the numerical aperture. The frequency required for the observa-
tion must satisfy the inequality

ai70)

where c is the speed of light.
By using the lower bound of the preceding equation, the characteristic
diameter (or distance) of the detector can be defined. We assume the detector
maintains at a constant temperature T:

kt \.64kTdsin9 \.64dsm9'
Then the characteristic diameter d can be shown as

Thus for high-frequency observation, (hv » kT), we see that the resolving
distance d is smaller than the characteristic distance d 0; that is,
d«d 0. (1.173)

However, for low-frequency observation, (hv « kT), we have

d»d 0. (1.174)

We stress that d 0 possesses no physical significance except that, at a given
temperature T, for which d 0 provides a distance boundary for low- and high-
frequency observations.
Let us recall the high-frequency observation for which we have
1
/iv > E 0 = -kTln [3 - (i) /*], (1.175)
where E 0 is the threshold energy level for the photodetector.