Rv Ro x RiA x R2f x Rsxy x R404 941

Thus, the overall response Rv of the system depends on all three domains, so that, for example, increasing spectral resolution reduces the energy throughput and hence the signal. In general however the variables are not coupled, so that this change should not directly affect the field of view, for example.

System definition begins with a consideration of which properties are fixed by the application and which are free parameters in the optimization of the design. It is also necessary, of course, to take account of purely practical factors such as weight and cost: a design which has been optimized only on the basis of physics is not necessarily the best overall.

9.6.1 Spectral properties

Spectral properties can be placed into three categories: wavelength definition or selection, spectral resolution, and wavelength calibration. The first two are intrinsic properties of the wavelength-selection technique and the optical materials and detectors employed; the last involves the separate provision of one or more reference wavelengths.

Usually, radiation consisting of a wide range of wavelengths will be incident on the entrance aperture. The degree to which each is attenuated in the output from the detector depends on: (i) the wavelength-dependent throughput (transmissiv-ity or reflectivity) of each lens, mirror, and window in the optical chain; (ii) the wavelength-dependent response of the detector itself, and (iii) the performance of the wavelength selective components specifically included for the purpose. The last may include gratings, etalons, and/or blocking filters as appropriate. The wavelength response of the device is the product of the response of all of these.

The resulting spectral bandwidth is generally dominated by a single component such as a filter or grating. The key variable is not just the resolution (AA or Av), defined and discussed below, but also the precise shape of the wavelength-response curve of the system as a whole, which is generally nearly the same as that of the principal spectrally selective component. In the simplest case, the spectral profile of the system is a rectangle or 'boxcar' of width AA. Some filter profiles approximate this behaviour, while grating instruments employing slits give rise in general to a triangular transmission function. For interferometers the

Av

FlG. 9.8. Generalized wavelength responses of; (a) a filter instrument, (b) a grating spectrometer, and (c) a Fourier spectrometer, all with the same spectral resolution AA.

Waven umber

FlG. 9.8. Generalized wavelength responses of; (a) a filter instrument, (b) a grating spectrometer, and (c) a Fourier spectrometer, all with the same spectral resolution AA.

corresponding profile is the Fourier transform of a boxcar profile, which can be shown to be the sinc function sin(x)/x (Fig. 9.8).

The boxcar profile is obviously the ideal, in that it maximizes spectral resolution and throughput simultaneously. The spectral response function R1(A) is simply

and the spectral resolution is defined as AA = A2 -A1. For other profile shapes, it is useful to introduce the concept of equivalent width AW, where AW = J R\dA so that AW is the width of the boxcar profile with the same area under a curve of response versus wavelength. For the boxcar AW and AA are obviously identical; for other shapes the two are interchangeable in the expression for throughput whenever the source radiance is slowly varying across the spectral passband, which is often the case if the source is a continuum emitter such as a blackbody, or when the spectral resolution is high, or both.

In spectroscopy, the key factor in defining spectral resolution is the ability of a scanning instrument to separate the signals from two line sources separated by AA. The features can be considered to be resolved if a spectral scan shows a minimum or dip between the maximum intensity corresponding to each. For spectral profiles of more-or-less regular shape, the resolution is then the width

FlG. 9.9. Features in an intensity scan; (a) fully, (b) well, (c) barely, and (d) not resolved.

of the profile between the half-maximum points. The convention for Fourier transform spectrometers is to take the full width at half-height of the central peak, which can be shown from the properties of the sine function to have a value of 0.6043/L in wave number units, where L is the maximum path difference in cm. In the convention introduced by Rayleigh, the requirement for resolution of two features is for a drop of 17% in the signal midway between the two lines, and this can be adopted for passbands of complex or irregular shape (Fig. 9.9).

9.6.2 Wavelength calibration

Wavelength calibration of a spectrometer involves inserting an emission or absorption source of known properties in place of the target. For best results the spectral features of the reference source should be sharp, strong, stable, numerous and distributed across the operating range of the instrument. Laser diodes, discharge lamps, cells containing gases, and thin plastic films are among the candidates available.

9.6.3 Geometrical optical properties

9.6.3.1 Aperture and field of view The most fundamental geometric optical properties of a system are the optical axis - the direction in which the detector is viewing the target - and the aperture, defined as the area of the entrance pupil measured perpendicular to the axis. The functions R3(x,y) and R4(O,0) specifying the relative response for displacements around the axis, spatially or angularly, which define the field of view, describe the part of the target that is sampled. This can be complex and unevenly weighted, but for the common case of a system with cylindrical symmetry, entrance aperture of area A, and a well-defined field of view of solid angle Q, we expect the number of photons entering the system to be proportional to the product AQ. Many instruments that view external targets have a scanning ability, i.e. can alter the direction of the optical axis by moving internal mirrors, etc. In such cases, an additional term in cos p, where p is the angle between the optical axis and the normal to the surface of the source, enters as a consequence of the definition of radiance. Then we have

The product AO, is sometimes called the etendue or energy grasp of the system. In an ideal system, etendue is conserved throughout; in real systems, the value corresponding to the whole is its lowest value in any stage of the optical train.

9.6.3.2 Beam modulation The geometric properties of an optical system include not only the direction and field of view, but also factors such as modulation of the beam, which can be deliberate (chopping) or undesired (vibration due to external disturbances). In the most common form of beneficial modulation, mechanical chopping, the beam is interrupted by a vibrating or rotating blade in order to convert the output from the detector from a d.c. to an a.c. signal, for ease of amplification. If the modulator has a period very much shorter than the thermal time constants of the sources of the background, chopping enables the output due to the background including its low-frequency fluctuations to be rejected. In a real instrument there will be sources of additional low-frequency fluctuations in the detector and the analogue part of the signal-processing subsystem. The use of modulation at a sufficiently high frequency also overcomes these effects as far as their influence on measurement of the mean signal flux is concerned.

We can express the same idea in frequency-domain language as follows. The noise power spectrum of the background is flat at high frequencies and rises at low frequencies. In d.c. systems (i.e. without modulators) the signal-amplitude information captured occupies a bandwidth (its baseband) extending from zero Hz (d.c.) up to a value of the order of the reciprocal of the response time of the signal processing electronics. A modulator shifts the signal information (but not the background) from its baseband to a higher frequency where the excess low-frequency noise has fallen below the white level, i.e. the unavoidable fluctuations. The wanted information (the amplitude of the signal flux) becomes the modulation on a carrier at the modulator frequency. The amplitude information is recovered by demodulation in the signal-processing electronics. By this means the majority of the drifts in the optical system, the detector and the analogue part of the signal-processing subsystem (except gain drifts) can be avoided.

Usually, the modulator has no spectral dependence of significance to the system but in the special case where this is a feature of the design, it is called a se

lective modulator. Selective modulation can be used to distinguish not only the scene radiation from the background but also different components of the scene emission, for example those from a particular atmospheric gas. Physically, a nonselective modulator is usually either a metal disc with sectors cut out rotating about its axis or an oscillating metal shutter, hence the name 'chopper' for these mechanisms. The surfaces that actually interrupt the beam may be either black on the side facing the optical system or reflecting and angled so that the field of view of the system is directed intermittently to a stationary black reference source. The stability of the temperature of the moving parts is less important if the second choice is made. Selective modulation can be achieved using interrupting surfaces that have spectrally dependent transmissions or reflectivities. Selective modulation restricted to the minute spectral intervals occupied by the absorption lines of a gas can be obtained by using a modulator consisting of a cell containing the gas whose pressure is caused to oscillate. By this means the emission from a particular gas in the scene can alone be detected in the presence of other emissions.

Chopping is used to 'label' the incoming beam and reduce the sensitivity of the system to fluctuations in the background flux. Even when an instrument is cooled to the point where the background photon noise is negligible, there may still be a large background signal. In this situation, slow drifts in the background signal level, due for example to small changes in the temperature of the instrument housing, can upset the radiometric calibration of the device. The solution in this case is, obviously, simply to cool the instrument further; however, extensive cooling is awkward to provide in practice and is employed only when absolutely essential. An alternative approach is to chop the incoming signal beam against a cold reference target, to modulate the signal but not the background. The a.c. component of the detector output is amplified by a phase-sensitive amplifier with a band-pass filter tuned to the frequency of the mechanical chopper.

A disadvantage of chopping is that it reduces the throughput of the system by at least a factor of 2, since half of the incoming energy is rejected. This can be important in systems where the source is weak and signal is at a premium. In fact, for the most common kind of chopper, a rotating toothed wheel, the blades take a finite time to cover and uncover the beam and the net effect is a reduction of a factor between 2 and 4. The latter value applies when the chopper segment is the same size as the beam, so the total obscuration (and total passage) of the flux is only momentary, resulting in a triangular waveform at the detector. An intermediate value of 2a/2 applies if the modulation is sinusoidal.

9.6.3.3 Ray tracing The geometrical properties of a system may be determined by ray tracing, graphically for simple systems or using one of the many advanced computer packages for complex instruments. For infra-red systems, where visible alignment is usually not possible and where unwanted sources and reflections may be dispersed through the instrument, ray tracing is an essential part of the design process. The procedure is to launch a number of rays, some parallel to the optical axis and some skewed to it, each defined by their direction cosines X, Y, and Z and the co-ordinates x, y, and z of the point where they intercept the entrance aperture. At each optical surface, the direction of reflection is calculated geometrically and of transmission using Snell's law for each ray. They are then traced to their point of impact with the next surface, and the process repeated, until they reach the detector (or not, as the case may be). Many thousands of rays may be traced in this way until all of the aperture has been mapped out in space and in incident angle to the required detail. It is often most instructive to launch the rays backwards through the system, starting at the detector; a uniform package of rays ends up forming a pattern in object space, the distortions in which show the departure of the field of view from the specification. Minute adjustments can then be made in the positions or properties of each component, often automatically by the program, until the result is acceptable within some criterion specified by the experimenter.

9.6.4 Radiometric properties

9.6.4.1 Signal-to-nolse ratio and throughput In addition to the other factors discussed above, the responsivity of the system depends on an efficiency factor e, usually called the throughput. Throughput is defined simply as the probability that a photon entering the system within the spectral pass-band and field of view will reach the detector. It can be estimated by calculating the product of the transmission and reflection losses of each component in the system, and determined experimentally by comparing the measured to the calculated signal from a standard source.

Given knowledge of the throughput and the geometrical and spectral factors determining the energy flux onto the detector, the signal at the output of the system can then be calculated. In practice, the important factor is not the size of the signal but the ratio of this to the noise on the output from the detector. The dominant source of this noise is usually the detector itself or fluctuations in the radiation field, signal and background, falling on the detector.

9.6.4.2 Response time Along with signal-to-noise ratio, the radiometric performance is defined by the response time of the system. Where signal levels are high relative to the noise, the important factor determining the repeat time between successive measurements is the time constant of the detector. A system with time constant t requires an integration time of 5t to achieve an output equal to 99.3% of the input. More often, the controlling factor is the dwell time required to achieve the required signal-to-noise ratio by integrating the signal electronically. This can be achieved with a network of time constant RC, which smoothes the signal and the noise so that, if the latter is random, an improvement in signal-to-noise ratio proportional to %/RC is achieved. Further improvements

0 Radiance /

FlG. 9.10. Response of a linear system to input radiance.

can be achieved, and the time required for the measurement of a weak source usefully reduced, by using phase sensitive detection and other active techniques.

9.6.4.3 Signal processing and linearity Depending on the type of detector used, each component of the background and signal photon currents will give rise to a mean output from the detector that is proportional to either the mean number of photons incident in the measuring time or to their energy. The signal-processing subsystems that follow the detector consist of an analogue part followed by a digital part, and these are designed to provide an output count that is proportional to the current or voltage output from the detector, i.e. to provide linear signal processing. Then the entire system is linear from end to end. Considerable care must be applied in practice to the selection of detectors and their operating ranges, and in the design of signal-processing electronics, to ensure that the output from the system is in fact a linear function of the incoming intensity. If this is not the case, the relationship between signal and output can never be determined accurately since, even if the transfer function is measured, it cannot be assumed not to change, particularly if the source of the non-linearity is a detector being operated at a point in its range that is approaching saturation.

9.6.4.4 Intensity calibration The usual principle is to ensure linearity of the device and then to use observations of two standard sources of known intensity to fix the constants of the straight-line response function (Fig. 9.10). Absolute radiometry is impossible without radiometric calibration, since, while it is possible to build a radiometer in which there is a high degree of confidence that at any instant the output will be linearly related to the scene radiance, calculating the position of this straight-line transfer characteristic to the accuracy desired for the intended measurements will almost always be impossible given the amount of detailed knowledge required about the components and their physical state. The usefulness of a radiometric model of the system is limited to detailed verification of the radiometric performance of an instrument during its final tests. In operation, the transfer function (i.e. the straight line that describes output voltage in terms of target radiance) can be established by allowing the instrument to view two sources of known brightness, ideally at each end of the range of target brightnesses, which provide standard radiance levels. The only accurate and reliable reference sources in the infra-red are blackbodies of various designs. Radiance calibration of the system can be performed by pointing its optical system at blackbodies at different temperatures before or after viewing the scene. This can be achieved by moving the entire optical system or steering its field of view with a mirror. The latter method avoids the need to move the (possibly massive) optical system, but requires that changes in the effective emissivity of the mirror, its temperature, and the amount of scattered radiation entering the optical system from the surroundings remain sufficiently small as the three views are selected. For instruments in orbit, which can view cold space, an ideal zero-radiance background is available and this is generally used for the lower calibration point. How long the calibration established in this way remains useful depends on how rapidly changes in the optics transmission, detector response, and the gain and offset of the analogue part of the signal-processing subsystem take place due to temperature changes, aging, power-supply variations, etc.

9.6.4.5 Reducing the background Ideally, the background is made negligible by cooling the whole optical system sufficiently. Placing a modulator in front of the optical system is a useful alternative method of reducing the effects of the background where cooling the system is impractical, although of course this does not reduce the fundamental fluctuations in the background. In many practical designs the fore optics are warm and the detector is cooled to enhance its performance. It is often worth increasing the amount of the system that is cooled to include the narrowest band-limiting filter and the system stops (which are usually placed just in front of the detector) as this can reduce the background and its fundamental fluctuations considerably. The ideal is to cool the entire instrument, when this is practical; see the example of the tropospheric emission spectrometer in Chapter 10. An additional advantage is gained by doing this, in that drifts in the background signal no longer require calibrating out.

9.7 Radiometric performance

9.7.1 Signal-to-noise ratio

The performance of a measurement system is characterized by the signal-to-noise ratio at its output. In a well-designed infra-red system, this is usually controlled by the noise in the detector, other sources of noise such as vibration, thermal or amplifier noise having been rendered negligible by careful design and

Wavelength (jim)

FlG. 9.11. Photon emittance versus wavelength for various temperatures.

Wavelength (jim)

FlG. 9.11. Photon emittance versus wavelength for various temperatures.

implementation. We distinguish between intrinsic detector noise, such as that due to lattice vibrations in the detector element, or the statistics of current flow through the detector, and extrinsic noise, produced by fluctuations in the photon flux from the background, and ultimately in that from the source itself. The latter generally dominates in the most sensitive observing systems, where thermal effects and the background flux of radiation have both been reduced by cooling parts, or all, of the instrument.

Thus, in the case where the limit on the signal-to-noise ratio attainable in a measurement of the spectral mean intensity of a photon stream is set by the fluctuations in the stream itself, we distinguish two cases, the ideal one in which the entire stream being detected originates in the scene being viewed and the background limited case in which the dominant stream is generated by the measuring instrument rather than the scene. In less-favourable circumstances still the limit may be set by excess noise generated in the detector, the detector or technology limit. While many different configurations are possible, as a general rule moderately cryogenic systems (typically those cooled with liquid nitrogen or with mechanical refrigerators to around 80 K) can reach background limited performance at wavelengths up to about 10 pm or so, while at longer wavelengths they employ cooling with liquid helium or advanced refrigerators to much lower temperatures of a few K. Uncooled devices are generally technology limited at all wavelengths.

Several types of photon detectors, for example extrinsic germanium and silicon, are background limited down to quite low levels of background photon flux (photons cm~2 s_1). The amount by which the background is reduced by cooling depends not only on the temperature but is also a function of wavelength, as shown in Fig. 9.11. At some point, reducing the temperature produces no further improvement as photon noise due to the source itself takes over.

For example, the current noise in for a photovoltaic detector can be written in = yjVNb 2 A/ e2 + VNS 2 A/ e2 + , (9.45)

where Nb and Ns are the total numbers of photons falling on the detector from the background and source, respectively, i0 represents the sum of all sources of noise other than photon noise (1 /f noise, Johnson noise, etc.) and the other symbols have the same meaning as before. The signal-to-noise ratio is

For a sufficiently bright source, the second term in the denominator will dominate the third and for some sufficiently low background this expression reduces to

and the signal-to-noise ratio is proportional to the square root of the signal strength.

On the other hand, for a very weak signal and a low background situation, like that in most astronomical observations, or an uncooled system limited by technology noise, the third term in eqn (9.45) dominates and

Now, the signal-to-noise ratio is directly proportional to the signal. This is also the case for an inadequately cooled system where Ns ^ Nb

In designing a system for a particular application, it clearly is necessary to evaluate each term first for a particular configuration, and to decide what cooling is necessary. Providing the cooling is often difficult or expensive and can introduce other practical problems, hence the designer wants to keep it to a minimum.

Warm cai. larget

Warm cai. larget

FlG. 9.12. Schematic diagram of a simple radiometer.

9.7.2 A generalized radiometer

Consider the simplest kind of radiometer in which the dominant fluctuation in the electrical signal from the detector is produced by processes occurring within the detector, not by fluctuations in the incoming photon stream. This is often the case in real systems where the source is bright and achieving the ultimate sensitivity is unnecessary, for instance net-flux radiometers. The diagram in Fig. 9.12 represents the simplest concept of a radiometer - some light-gathering optics, a detector and its amplifier, and a readout device, in this case a voltmeter. There is also some kind of wavelength filter - if this is not provided explicitly, then the spectral bandpass will be dominated by the wavelength dependence of the components in the optical chain, including the detector, and the complex wavelength-dependent transmission of the intervening atmosphere if the device is in air.

The system is assumed to observe an extended source, that is, one that fills the field of view of the instrument, which is determined by its optical design, and specified by the solid angle 0 within which radiation is accepted at the aperture of area A. If G is the gain of the amplifier and V the reading in volts at the output then the energy falling on the detector of responsivity Rv is V/GRv W. If the target is approximately a blackbody of temperature T, using the Stefan-Boltzmann law with constant a, the signal power S reaching the detector is

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