From Fig. 9.4 it can be seen that a device operated at around 100 Hz with a load resistance of 10 MQ will operate on the plateau and maximize the responsivity. Typical values of the thermal and electronic time constants are t =16 ms and RC = 100 ps (R = 107 ohms, C =10 pF).

The main source of noise in pyroelectric bolometers, as in most thermal detectors, is Johnson noise in the equivalent resistance. This is due to thermally excited random motions of the free electrons in a conductor, produced by collisions with the atoms of the lattice. Johnson (1928) showed empirically that the noise power, expressed as the square of the noise voltage Vn, in a resistance R is proportional to R and to the temperature, T. If the noise is independent of frequency, the power also depends directly on the bandwidth Af (Hz). A statistical analysis, due originally to Nyquist (1928), shows that the constant of proportionality is 4 times Boltzmann's constant k. Hence Nyquist's formula

We can now write down an expression for the NEP (W Hz-1/2) of this, the best-performing and most commonly used room-temperature thermal detector, in terms of its physical properties and the operating conditions j2kT/t0R

G* and t are minimized by making the flake very thin (a few x 10 pm). Typically then pA/G*T « 1 pA W-1; R is maximized (108 ohm is about the largest practical value), Af = 1 Hz by definition of NEP (signal power to match noise per unit bandwidth) and NEP is typically « 10-8 to 10-9 W Hz-1/2.

The signal-to-noise ratio (SNR) is often used to describe detector performance in a situation where it is receiving r.m.s. power P

As we have already noted in our definitions of NEP and SNR, the dependence on the bandwidth Af arises because the signal is at a single frequency (the chopping frequency fc) while the noise is present at all frequencies (Fig. 9.5). Many common noise sources, including Johnson noise, are 'white', that is, they have the same mean amplitude at all frequencies. Electronic filtering thus enhances SNR, and we can calculate its expected value for a given set of measurement parameters. In particular, it is useful to replace the obscure quantity Af with

FIG. 9.5. Measurements made in a limited bandwidth centred on the signal frequency have better signal-to-noise ratio than unfiltered measurements, since white noise is present at all frequencies.

the convenient to, so that the signal-to-noise ratio is expressed in quantities that are likely to be well known. To derive the relevant expression, the bandwidth is defined in terms of the power of the signal

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