Page 57 - Photodetection and Measurement - Maximizing Performance in Optical Systems

Fundamental Noise Basics and Calculations

50 Chapter Three

rail built with vacuum tubes or high voltage transistors. All in all, it’s surprisingly
difficult to demonstrate shot noise in this simple way with a scope; for more convincing
demonstrations we will need to design receivers more appropriate than the scope’s
front-end amplifier. Designing to see noise is just as difficult as designing to minimize
noise. We’ll return to this experiment later.

Nevertheless, this experiment points out some of the tricks of design for low
noise. The greater the optical power detected and the longer the measurement
period, the greater the noise but the less is the relative noise. Hence, other
things being equal, we should usually strive in instrument designs for the
highest detected power and measure with as small a bandwidth as possible for
as long as possible. It also shows that we need to be very careful not to take
anything at face value, especially not an innocent looking resistor, even if it is
just inside our expensive new scope.

3.5 Thermal (Johnson) Noise
The second fundamental source of noise, thermal noise, was investigated by
Johnson and Nyquist in the 1920s. It is present in all resistors at a tempera-
ture above absolute zero and is characterized by internal current fluctuations
and fluctuations in voltage across their open circuit terminals, even when no
external current is flowing. If connected into an external circuit, these will also
cause external current fluctuations. Although the warm resistor acts as a little
generator, it is not possible to extract power from it. Anything connected to it
deposits as much power into the resistor as is extracted from it. Analogous to
the treatment of shot noise currents, a resistor of RW will show a noise power
spectral density given in voltage or current by:
e n = 4 kTBR (in units of V ) or e n = 4 kTR (in V Hz ) (3.7)
2
2
2
2
or i nt = 4 kT B R (in units of A ) or i nt = 4 kT R (in A Hz ) (3.7a)
where k is Boltzmann’s constant (1.381 ¥ 10 -23 J/K) and T is the absolute tem-
perature in K (ª300K at room temp.).
Equation 3.7 is the Johnson or Nyquist formula. As with shot noise, this
thermal noise power is proportional to the measurement bandwidth. In pho-
toreceiver design it is usually more convenient to calculate and measure volt-
ages and currents rather than powers, so we can rewrite Eq. 3.7 in convenient
engineering units, either as a voltage source in series with a (noiseless) resis-
tor R or as a current generator in parallel with R:

e n = 4 kTR = 4 R(kW ) nV Hz (at room temp. ) (3.8)
i nt = 4 kT R = 4 R(kW ) pA Hz (at room temp. ) (3.8a)

A few calculated values will give a feel for the voltage magnitudes (Table 3.2):

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