Page 53 - Photodetection and Measurement - Maximizing Performance in Optical Systems

Fundamental Noise Basics and Calculations

46 Chapter Three

1.4 1.4
1.2 1.2
1 1
Current 0.8 Current 0.8

0.6
0.6
0.4 0.4
0.2 0.2
0 0
0 250 500 750 1000 0 25 50 75 100
Time (Sample No.) Likelihood
Figure 3.1 Multiple measurements of a photocurrent should provide a Gaussian (actually Poisson)
histogram, with a variance equal to its mean.

measured value is the nominal current, here I mean =1.0, but there will always be
some fuzziness.
A constant photocurrent (I p) exhibits a current noise power spectral density
2
(i n ) given by:
i n = 2 qIB or i n = 2 qIB (3.1)
2
p
p
where q is the charge on the electron (1.602·10 -19 C) and B is the measure-
ment bandwidth in hertz. This is called the Schottky formula. Note that the
current noise power spectral density is proportional to the measurement
bandwidth, at least up to some high frequency limit defined by the structure
of the photodiode, such as the reciprocal of the transit time of electrons
through the detector junction. The spectral density of shot noise is also “white,”
meaning that the power per unit bandwidth is independent of frequency. The
current noise power spectral density can be interpreted as the variance of the
current I p , assuming the distribution of current values looks Gaussian, as in
Fig. 3.1. Actually the probability P(I) of a given current cannot be Gaussian, of
the form:

2
-
PI () ª e -( II mean ) variance (3.2)
because this would give a finite probability of our measuring negative currents,
which is not physical. The true distribution comes from the analysis of photon
statistics.

3.3 Photon Statistics

An alternative description of shot noise in photodetection can be obtained from
analysis of photon statistics. When a constant intensity, perfectly monochro-

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