Page 54 - Photodetection and Measurement - Maximizing Performance in Optical Systems

Fundamental Noise Basics and Calculations

Fundamental Noise Basics and Calculations 47

matic and perfectly polarized laser source is incident on a photodetector, the
actual instant of generation of individual photoelectrons is a stochastic process.
For a power of P s watts, the mean number of photoevents detected per second
is P s/h = K m, where = c/l is the optical frequency, l is the wavelength, and c
is the velocity of light in vacuum. The probability P(r) of measuring r photo-
events per second is given by:

r
Pr () = K m e - K m (3.3)
r!

This is the Poisson distribution, which has a variance equal to its mean
. However, for situations where the mean
K m. The standard deviation is K m
detection rate K m of photoelectrons is large, the Poisson distribution becomes
identical to the Gaussian distribution. This is why in practice noise current
appears to be distributed as a Gaussian function about the mean current, as in
Fig. 3.1.
To obtain the full variation in photoevents detected per unit time, the above
Poisson distribution for photodetection needs to be multiplied by the stochas-
tic distribution of the incident radiation photons. This depends on whether the
incident optical field is highly coherent, like that from a well-controlled laser,
or is thermal radiation such as from a hot filament, gas discharge tube, or LED.
These complex issues are treated by the subject of photon statistics and are not
addressed further here. The interested reader is directed to the excellent book
by Goodman and to many other papers on the subject.
For the purposes of this deliberately practical treatment, we can assume that
the total photodetection process also exhibits Poisson (or Gaussian) statistics,
with variations in photoevents and photoelectrons per unit time exhibiting a
variance equal to the mean. This variance in the photocurrent gives rise to the
fundamental noise contribution we call shot noise.
The basic results of photon statistics can be applied to the electrons making
up the photocurrent from our detector. For example, a photocurrent I p is made
-1
up of a stream of electrons with a mean arrival rate of I p/qs . In a measure-
ment time of T i seconds, on average I p T i /q electrons will be counted. However,
as the electrons follow Poisson statistics, this number can be determined only
to within IT q/ . Converting back to current, the uncertainty with which the
pi
current can be determined is given by:

q IT ∫ qI p (3.4)
pi
T i q T i

Writing the effective bandwidth of the measurement as B = 1/(2T i) Hz we obtain
for the uncertainty 2qI B , in agreement with the current noise expression
p
given above.

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