The important figure of merit for a bolometer is the sensitivity (S), detectivity (D), noise equivalent power (NEP) and time constant (τ ). The NEP is a measure of the sensitivity of a bolometer, and is defined as the power absorbed that produces a signal to noise of unity at the output. NEP can also be defined as the minimum incident radiation power necessary to obtain a signal to noise of unity.
Radiation incident on a solid state device will cause the absorption of photons and the creation of lattice vibrations or phonons. This will act to increase the resistivity of a solid state device. Two parameters are convenient to define before stating an expression for a bolometer’s votage signal. The first is the heat conductance of G defined as dP/dT where P is the radiative power lost or absorbed by the device and T is temperature. The second parameter is the temperature coefficient of resistance, α, where α = (1/R)dR/dT is the fractional change in resistance per degree Kelvin. We now assume the device is driven by a constant current I, and is enclosed in a chamber at temperature T0. Since the current through the resistor heats it slightly, we define the temperature of the detector element as
T and assume (T-T0) << T0. The signal voltage becomes It follows that bolometer sensitivity, S=VS/Ps, is given by
G R S = Iα
(2-3)
For high sensitivity, one requires a large electrical resistance, a large current, a large temperature coefficient of resistance and a small heat conductance. In order to obtain a large temperature coefficient of resistance is accomplished by choosing the proper material. The bolometer’s resistance is controlled by maintaining a small film thickness and a large length to width ratio. The latter requirement, however, conflicts the need for resistance uniformity across an array of bolometers for infrared video imaging.
In such cases, it may be necessary to sacrifice the benefits of a large resistance.
Minimizing the heat conduction is probably the most interesting aspect of bolometer design. The large thermal conductance provided by a bulk substrate will draw heat from the temperature sensing element too fast and lower the sensitivity. A large thermal conductance, however, may be desirable in the construction of very high speed bolometer if a reduction in sensitivity is tolerable. By quickly drawing the excess heat from the bolometer resistance element, fast changes in temperature or infrared images can be detected.
To quantify the speed of a bolometer, consider a detector element with heat capacity C defined as C = dE/dT where E is the total internal energy of the element. The bolometer time constant is then τ = C/G in analogy to a single time constant circuit. It should be noted that optimizing the sensitivity by lowering G and optimizing the speed by increasing G cannot be done without simultaneously lowering the heat capacity.
Intuitively, the need for a small C by considering that a smaller change in the bolometer internal energy is needed to create an appreciable change in temperature. Therefore a
small incident energy can be used to cause a change in the voltage signal.
Equation (2-3) can be further modified to include the effect of illumination chopping frequency, ω, as the following:
A factor, η, has also added to account for incomplete optical absorption by the absorbing layer. The factor is called the coefficient of absorptance.
In terms of noise figure for a bolometer, the three dominant noise factors must be considered. All solids at a finite temperature suffer from temperature fluctuations and Johnson noise. The third noise source arises from the electrical noise in the amplifier electronics. The mean-square fluctuation in the voltage signal per unit of bandwidth is obtained by summing the square of the two independent factors. The first factor is the amplifier noise and the second factor arises from the temperature fluctuations in the bolometer resistive element.
2
Where k is Boltzmann’s constant and TN is the effective input noise temperature of the amplifier. Because the body has a conductance G and capacity C with time constant τ, we know from basic network theory that the power spectrum of such a body is of the form 2 2 2
where K is a constant of proportionality found by considering the entire temperature fluctuation over all frequencies. Besides, from Boltzmann’s relationship, the probability of the resistive element has energy Ei and can be expressed
p(Ei)=Ae−Ei/kT with
∑
− /= 1
From the definition of heat capacity,
2 2 2 12 2 Combining the above with equation (2-6), we have,
∞
∫
+ = Solving for K yields the noise spectral density,) For frequencies much less than 1/τ, the final temperature fluctuation of the sensor, per unit of bandwidth, is For a body in thermal equilibrium, one-half of the fluctuation is due to emission of radiation while the other half is due to absorption. It follows that the mean-square temperature fluctuation per unit of bandwidth is
4 (1 ) The resultant voltage fluctuation is
4 (1 )
The NEP is defined as the minimum incident radiation power, PS, necessary to produce S/N = 1. From the latter equation we therefore have the noise equivalent power in units of watts per Hz1/2,
The second term indicated an increasing NEP with bias current. The third term is the Johnson noise and indicates a decreasing NEP with bias current, associated with the increased signal response. It can be written in terms of the sensitivity, using equation (2-2),
The specific detectivity, D*, is defined as
NEP BW
D A( )
*= (2-20)
Where A is the device area and BW is the bandwidth of the noise power. The detectivity D* is less sensitive to the detector area A.