Spectrometer response function

In concluding this section, we point out that the effect of any electrical filter composed of purely linear elements, whether they be passive like resistors, capacitors, and inductors or active like linear amplifiers, can be represented as a convolution. The various other spreading phenomena that are described by convolution in the same domain may therefore be lumped together with the electrical contribution and comprehensively called the spectrometer response function. Even inherent line broadening may be included, provided that the convolution does not appear in an exponent, as in the case of absorption spectra. 

We have shown that the radiant flux spectrum, as recorded by the spectrometer, is given by the convolution of the true radiant flux spectrum (as it would be recorded by a perfect instrument) with the spectrometer response function. In absorption spectroscopy, absorption lines typically appear superimposed upon a spectral background that is determined by the emission spectrum of the source, the spectral response of the detector, and other effects. Because we are interested in the properties of the absorbing molecules, it is necessary to correct for this background, or baseline as it is sometimes called. Furthermore, we shall see that the valuable physical-realizability constraints presented in Chapter 4 are easiest to apply when the data have this form. 

In Eq. (45), TM represents the true transmittance only if instrumental spreading is negligible. In previous sections, however, we learned that instrumental spreading may be described by the convolution product of the flux and the spectrometer response function, here called r(x), that incorporates all the instrumental spreading phenomena ... 

In applying the technique of deconvolution, we take as known the spectrometer response function. It seems reasonable that the more accurately we know this function, the more accurate will be the deconvolved result. Although the nonlinear methods described in Chapter 4 are more tolerant of error, they too require a knowledge of the response function. 

The method has utility where only modest correction is required, such as in the determination of the spectrometer response function when a narrow line has been measured (Chapter 2, Section II.G). 

Fig. 4. Position spectra of silicon and sulphur Ka fluorescence lines, the transitions p3He and pNe used to determine the crystal spectrometer response function, and the Rainier a lines from the hydrogen isotopes. For p3He, the parallel transition (5/ — 4d) is well resolved from the circular transitions (5g — 4f)...

Fig. 5. tyNe(6 — 5) transitions. The line shape is identical with the spectrometer response function because of the small radiative width of 10 meV. A line width of 26 or 550 meV is achieved for a silicon crystal of 95 mm in diameter... 

Many more cross sections in the literature were determined by comparison of Raman intensity of an unknown to that from one of the standards in Table 2.2. Provided the measurement conditions are the same and the spectrometer response function is known, the ratio of the peak areas of two bands adjusted for relative number density will equal the ratio of cross sections. One such procedure is described in Chapter 10, but cross sections for a variety of samples are listed in Table 2.3. These values were not critically evaluated for accuracy and should not be considered as accurate as those in Table 2.3. Nevertheless, they are useful for observing comparative magnitudes and trends. 

Conversely, we may observe an exceedingly narrow spectral line, so that o(x ) is approximated by <5(x ). Now the data i(x) represent the response function. This principle can, in fact, be used to determine the response function of a spectrometer. The laser, for example, is a tempting source of monochromatic radiation for measuring the response function of an optical spectrometer. Coherence effects, however, complicate the issue. We present further detail in Section II of Chapter 2. 

It is possible to estimate the influence of the spectrometer on linewidth measurements by convolving the response function with a mathematical model of the line. The resulting curve may be compared with the observed line and the true width inferred. Hunt et al. (1968) have employed corrections of this type supplied by the author in their determination of carbon monoxide self-broadened linewidths. 

Typically, t(co) is small for co large. A spectrometer suppresses high frequencies. If the data i(x) have appreciable noise content at those frequencies, it is certain that the restored object will show the noise in a more-pronounced way. It is clearly not possible to restore frequencies beyond the band limit Q by this method when such a limit exists. (Optical spectrometers having sine or sine-squared response-function components do indeed band-limit the data.) Furthermore, where the frequencies are strongly suppressed, the signal-to-noise ratio is poor, and T(cu) will amplify mainly the noise, thus producing a noisy and unusable object estimate. 

In Chapter 2, Jansson describes the determination of the system response function for a dispersive spectrometer system. We have made a number of such determinations using very-low-pressure samples of, for example, CO in the 5-fim region. As discussed by Jansson, one records the data and then removes the Doppler profile using deconvolution, yielding the system response function. 

Fig. 26 Fourier transform spectrum of v2 of ammonia. Trace (a) is a section of the infrared absorption spectrum of ammonia recorded on a Digilab Fourier transform spectrometer at a nominal resolution of 0.125 cm-1. In this section of the spectrum near 848 cm-1 the sidelobes of the sine response function partially cancel, but the spectrum exhibits negative absorption and some sidelobes. Trace (b) is the same section of the ammonia spectrum using triangular apodiza-tion to produce a sine-squared transfer function. Trace (c) is the deconvolution of the sine-squared data using a Jansson-type weight constraint.

Figures 5-11 illustrate the restoration process in the presence of a drifting base line. These data are methane absorption lines taken with a four-pass Littrow-type diffraction grating spectrometer. For these data 2048 data points were taken. The impulse response function was approximated by a gaussian. The true width of these lines is approximately 0.02 cm-1.

Figure 1.14 Instrumental response function of an electrostatic energy analyser shown as a function of the spectrometer voltage l/sp. Maximum transmission is achieved at the nominal voltage l/°p, and this maximum value is equal to the luminosity L (left-hand scale) or set to unity (right-hand scale), respectively. For values other than l/°p the response function decreases, and the characteristic fwhm value is indicated. For the relation fU% = °, see...

Figure 5.29 Response function of a photoelectron spectrometer equipped with a large-scale detector to ensure the recording of the whole xenon 4d5/2 photoline with constant efficiency. The accepted range of approximately 5% corresponds to +1.35 eV at a photoelectron kinetic energy of 27 eV, and this value is large compared to the photon bandpass (0.4 eV at 94.5 eV), the contribution from the electron spectrometer (0.22 eV), and the natural linewidth (T = 0.12 eV). From [KKS93].

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