What follows is a chapter from my final year report on ‘Simulating SPIRE using IDL’. When researching the project I found it difficult to find some basic introduction to bolometer theory, so I decided to put my version online. You can also see the chapter in full context by downloading my fourth report report: Simulating SPIRE using IDL.

• Basic Principles
• Time Constant
• Responsivity
• Time Response of a Bolometer
• Noise Equivalent Power
  • Photon Shot Noise and Wave Noise
  • Photon Noise Limited NEP
• Photon Detector Efficiency
• Other Sources of Noise
• Minimising Noise
• Addition of Noise Terms
• Overall Noise and NEP

Basic Principles

A bolometer is a device which detects incoming radiation by producing a change in electrical resistance proportional to the amount of radiation received.  Incoming radiation is absorbed by the bolometer which causes an increase in its temperature, which in turn causes a change in its electrical resistance.

The essential features of a bolometer are as follows:

Figure 3.1 - Diagram of a Bolometric Detector

The bolometer itself comprises of an absorber material linked to a heat sink of fixed temperature.  Incoming electromagnetic (EM) radiation is absorbed by the material increasing the kinetic energy of free electrons.  The collisions of free electrons with atoms in the material cause lattice vibrations which are observed as a change in temperature.

Typical materials for the thermometer are semiconductors such as doped germanium.  The resistance for such a material changes significantly for a small change in temperature and can be characterised by the equation,

(3.1)

where is a constant called the resistance parameter (Ohms), is the resistance (Ohms), is the temperature of the resistor, and (K) is the material band gap temperature. The value of is called the material parameter and is given the symbol . The temperature coefficient of resistance is defined by,

(3.2)

The operations of a bolometer detector are illustrated in Figure 3.1. The bolometer at temperature is linked to a heat sink of fixed temperature by a thermal conductance . A dc bias current flows through the bolometer generating a voltage . Changes in the incoming radiation power give rise to changes in the resistance , and therefore in the output voltage . The typical bolometer biasing and readout circuit is pictured in Figure 3.2.

Figure 3.2 - Bolometer biasing and readout circuit

The current which flows through the resistor causes a dissipation of power into the absorber material. In addition, the amount of radiant energy absorbed by the absorber is denoted by . The total power dissipated in the bolometer is therefore given by ,

(3.3)

Under steady state conditions the energy absorbed by the absorber will be removed to the heatsink by the thermal link, this is given by the following relationship,

(3.4)

The dc voltage-current (VI) curve for the bolometer is defined by the equations,

(3.5)

In practise the bolometer is biased by a battery of voltage V₀ and load resistance R_L. The resistance of the load resistor is normally designed to be much higher than the resistance of the bolometer over its entire operating range. This is to keep the current passing through the bolometer at a stable level so that the power dissipated in the bolometer by the resistance thermometer stays somewhat constant.

The operating point of the bolometer is then given at the intersection of the load curve and the load line, determined by the equation,

(3.6)

A graph showing a typical VI curve and load line from results of a simulation program that I produced in IDL is given in Figure 3.3.

Figure 3.3 - IDL output of several VI curves with a load line over plotted

As can be seen from Figure 3.3, the resistance of the bolometer is incredibly high at small currents. The bolometer resistance begins to decrease and eventually levels off at higher currents because additional power is dissipated into the absorber material. If radiation is incident on the detector, the power dissipated in the absorber will also increase. This has the effect of squashing the VI curve as shown in the diagram above. The electrical power dissipated in the absorber material, as derived in [8], is given by,

(3.7)

where represents a fractional increase in the temperature of the absorber where represents the absorber material being at a temperature of . is the static thermal conductance of the thermal link at the ³He refrigerator temperature (WK^-1), which is given by the following power law,

(3.8)

where is the static thermal conductance at 300mK (WK^-1), and and is called the thermal conductivity index. The term is called the loading parameter and is given by the equation,

(3.9)

It is the loading parameter which causes the squashing effect of the VI curve when there is incident EM radiation on the detector.

Time Constant

As with most physical systems a bolometer does not respond instantly to an instant change in its inputs. In many cases the response of a detector to a step change in the input is an exponential change in the output. This is analogous to the charging and discharging of a capacitor in an RC circuit.

A bolometer has a single energy reservoir in the heat capacitance of the absorber. Therefore a bolometer can be modelled using a single first order differential equation; this also means that a bolometer does not suffer from memory effects. The response can therefore be characterised by a time constant which for a bolometer is given by the equation,

(3.10)

where is the absorber heat capacity (JK^-1). The static thermal conductance is related to a the value of at the ³He refrigerator temperature by,

(3.11)

When radiation is incident on the detector it increases its temperature by a small amount, this affects the absorbers heat capacity and the thermal links conductance. The absorber heat capacity at the increased temperature is related to a known value at 300mK by,

(3.12)

where is the heat capacity index. Looking back at equation (3.2) and using the terms defined above, the temperature coefficient of resistance can be rewritten as,

(3.13)

where is the power law index from resistance-temperature relation in equation (3.1). We see that is negative for a semiconductor bolometer. This leads to a value of which is smaller than that described by equation (3.10). This is due to electrothermal feedback which is described in [9]. We define a new term which is the value of with a correction for the electrothermal feedback,

(3.14)

This new value allows us to define a value of which again contains a correction for electrothermal feedback,

(3.15)

As the thermal resistor is biased by a voltage the electrical power dissipated into the absorber can be given by . An increase in the incident EM signal will increase the temperature of the thermal resistor and therefore also increase its resistance; this in turn will cause a decrease in the dissipated power. If the resistor is acting in the steep part of its curve then the total power dissipated in the absorber will remain constant, as will its temperature. This system is referred to as having negative electrothermal feedback. This has the advantage of reducing the time constant to of the thermal time constant as given in (3.10).

Responsivity

Responsivity is defined as the change of output voltage for a change in the incident power, which in a bolometer is equivalent to a change in temperature. The voltage responsivity of the bolometer is defined as,

(3.16)

and varies as a function of the operating point. If the signal on the detector is modulated the modulation frequency must be low enough so that the detector can respond to the change in power. It is shown that,

(3.17)

where frequency of modulation. The zero frequency (dc) responsivity can be evaluated directly from the load curve using the expression,

(3.18)

where is the zero frequency dynamic impedance (Ohms) of the bolometer at the operating point. can be shown to be given by,

(3.19)

Time Response of a Bolometer

For the majority of bolometers the shape of the VI curve is dominated by a background power level. When a small additional signal is applied to the bolometer the departure from the VI curve can be assumed to be negligible. This is known as the small signal approximation. In the small signal limit i.e. where source background the change in bolometer voltage due to a change in incident radiation power can be given by,

(3.20)

The change in the output voltage does not occur instantly and by comparing the bolometer with an RC circuit the response can be modelled by either of the two following equations,

(3.21)

When plotted these equations have the following form (where and ),

Figure 3.4 - Plots of V against Time for both Positive and Negative V

When large signals are considered, the departure from the VI curve is no longer negligible. Therefore the change in the output voltage due to a change in the incident power can not be calculated by applying responsivity it is now given by the change in the operating point voltage of the bolometer. In moving from the initial to the final VI curve, the time constant of the system varies as a function of the operating point. Therefore the bolometer is no longer a single time constant device and can not be modelled by applying the simple RC circuit response equations.

Figure 3.5 - VI curves for a large signal change

Noise Equivalent Power

Of great importance to any bolometer is the Noise Equivalent Power or NEP. The NEP is the root mean squared signal strength required to equal the root mean square of the detector noise. The best signal-to-noise ratio achievable by a bolometer is given by the equation,

(3.22)

In general, NEP has the units of .

Photon Shot Noise and Wave Noise

If we consider the particle picture of light and realise that light will arrive at the detector in a random or uncorrelated way, we can define photon shot noise. Photon shot noise is justifiable at high frequencies (where the photon picture of light is most suitable) but at lower frequencies the wave picture of light is more appropriate and therefore we define another term wave noise.

By application of Bose-Einstein statistics and assuming that the background for the detection is in the form of a blackbody we find that the root mean squared fluctuations in the number of photons arriving in time , in frequency interval V is given by,

(3.23)

where , = emissivity of the background, and = overall transmission efficiency between the background and the detector. The additional term takes account for the wave noise.

Photon Noise Limited NEP

In the best case, the detector and subsequent components will add a negligible amount of additional noise to the signal in addition to the photon shot noise. Therefore, the photon noise limits the sensitivity of the bolometer measurement, this ultimate limit is called the photon noise limited NEP, . This is given by the equation,

(3.24)

Photon Detector Efficiency

In practise it is not possible to obtain the photon noise limited S/N as this assumes that a perfect detector is used. Real detectors differ in operation in the fact that,

• a real detector may not respond to every photon

• the detector and its electronics produce additional noise

Two parameters are defined in order to take into account these to deficiencies in the detection system; these are the Responsive Quantum Efficiency and the Detective Quantum Efficiency.

Responsive Quantum Efficiency (RQE or )

The RQE or accounts for the imperfect absorption of photons and is defined as the fraction of incident photons which contribute to the signal, obviously .

Detective Quantum Efficiency (DQE)

(3.25)

The DQE is the ratio of the actual sensitivity to maximum achievable in principle. The parameter takes both the absorption efficiency and any extra noise generated in the detector into account. This parameter can therefore be used to compare different types of detector with each other.

In practise the bias voltage is chosen in order to obtain the peak DQE for each detector. In the case of SPIRE, it is groups of detectors that share a common bias voltage which can be adjusted to obtain an optimal DQE for the group.

Other Sources of Noise

Johnson Noise

Within any piece of any conducting material the electrons have random thermal motions because the material has a finite temperature. A bolometric detector and its components are - or can be considered to be - a resistor with an electrical contact at each end. If there is no electrical potential across the contacts the voltage in the resistor will fluctuate randomly about zero volts, this is because positive and negative fluctuations are equally probable. The noise power within the component is however proportional to the fluctuation voltage squared i.e. it is always positive. This is called the Johnson or Nyquist Noise.

The Johnson noise NEP, , is,

(3.26)

The frequency spectrum of Johnson noise is flat i.e. it is frequency independent. This can be seen from the above equation where there is no frequency dependence. Noise with a flat spectrum is called white noise.

Phonon Noise

So far we have considered noise created from photons and electrons, we now consider the flow of heat into the heat sink as quantised in the form of phonons (quantised lattice vibrations). This leads to random fluctuations in the temperature of the bolometer. The phonon noise NEP, , is,

(3.27)

Temperature Noise

Temperature noise is caused by the fact that the heat sink is not at a constant temperature and varies slightly over time. The temperature noise NEP, is,

(3.28)

where is the spectral intensity of fluctuation in the temperature of the heat sink (K²Hz^-1).

1/f Noise

This source of noise is very important in practical applications, although the causes are often not very well understood. For most devices, large levels of noise are found at low frequencies.

Figure 3.6 - 1/f Noise

Minimising Noise

Noise affects results in a degrading fashion and therefore we employ several techniques (in order to reduce its effects.

• Make the post detection bandwidth as small as possible

• Try to avoid measuring signals (or frequency band) that coincide with discrete frequency interference sources

• Ensure that the signal frequency (or frequency band) is high enough not to be affected by significant amounts of noise.

Due to noise it is not possible to observe a source for long periods of continuous observation; this would involve working at very low frequencies where the noise would be significant. One technique used to avoid noise is to modulate the signal with a frequency which is high enough that is no longer significant. The modulation frequency can not, however, be so high that the detectors frequency response results in a loss in signal. A further advantage to the modulation technique is that it can be used to subtract the background from a signal by switching between the source signal and a background signal; this is known as ‘chopping’ in FIR/sub-mm observations.

Addition of Noise Terms

The total noise in a system will be the combination of all the individual noise sources present as described previously. We assume that all of the noise sources are uncorrelated i.e. the value of one is not dependant on any other. As they are uncorrelated adding them normally would not take into account the phases of the noise, therefore we take the root mean square of the noise (rms value) as sometimes the noise sources may cancel each other.

(3.29)

where is the noise voltage spectral density (VHz^-1/2) from each of the noise contributions.

Overall Noise and NEP

It is possible to define the NEP in the following way, the signal power which gives a S/N of 1 in an integration time of 0.5 seconds. If we let = detector responsivity (VW^-1), = electromagnetic power incident on the detector (W) and, = total noise voltage spectral density, the signal voltage can be written,

(3.30)

The noise voltage will then be given by,

(3.31)

By the definition of the NEP, if then the signal voltage . Therefore we obtain an equation for the NEP in terms of the Noise voltage spectral density and the responsivity,

(3.32)

The units of NEP are normally given as WHz^-1/2, the Hz^-1/2 terms refers to the post detection bandwidth or the inverse of the integration time.

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