Slit function

Fig. 40.17. Convolution in the time domain offlf) with h t) carried out as a multiplication in the Fourier domain, (a) A triangular signal (w, = 3 data points) and its FT. (b) A triangular slit function h(t) (wi/, = 5 data points) and its FT. (c) Multiplication of the FT of (a) with that of (b). (d) The inverse FT of (c).

It is instructive to consider a specific example of the method outline above. The triangle fimction (l/l) a (x/l) was discussed in Section 11.1.2. It was pointed out there that it arises in dispersive spectroscopy as the slit function for a monochromator, while in Fourier-transform spectroscopy it is often used as an apodizing function. Its Fourier transform is the function sine2, as shown in Fig. (11-2). The eight points employed to construct the normalized triangle fimction define the matrix... 

Sakai, H. (1962). A Slit Function Correction and an Application to the Study of the Absorption Lines in the H20 Pure Rotation Spectrum. U. S. Armed Services Technical Information Agency Report AD287897. 

We will first consider the error bounds for a spectral density broadened by a Lorentzian slit function, Eq. (15), describing the response to an exponentially damped perturbation. In this case the broadened spectrum,... 

Fig. 1. Error bounds for the nuclear resonance line shape of crystalline CaF2, broadened by a Lorentzian slit function (i.e., the energy absorption by the coupled nuclear spins, due to an exponentially damped harmonic perturbation by a radiofrequency magnetic field).

Another approach to estimating spectral densities, which has the advantage of guaranteeing that the approximate functions are positive, can be based on the error bounds constructed in Section III-A for the spectral density broadened by a Lorentzian slit function. If we had a sufficient number of moments to make the error bounds very precise, then we could reduce the broadening as much as we like, so that the broadened distribution of spectral density becomes as close as we like to the true distribution. In order to estimate these higher moments, we should need to take advantage of some special feature of the distribution. For example, in the case of the harmonic vibrations of a crystalline solid, the distribution of frequencies lies between limits — co,nax and +comax, and is zero outside... 

In many cases, the profile a spectroscopist sees is just the instrumental profile, but not the profile emitted by the source. In the simplest case (geometric optics, matched slits), this is a triangular slit function, but diffraction effects by beam limiting apertures, lens (or mirror) aberrations, poor alignment of the spectroscopic apparatus, etc., do often significantly modify the triangular function, especially if high resolution is employed. 

The second term describes the bound-to-bound contributions, that is the rotovibrational bands of the van der Waals dimer. If the system does not form dimers, this term and the following two terms all vanish. For practical use, the d function in this term should be replaced by an instrumental slit function, or perhaps with some Lorentzian if pressure broadening affects the individual lines (as will often be the case). In any case, the d function is symbolic for the relatively sharp dimer lines that... 

Fig. 6.5. Computed structures due to the hydrogen dimer, in the quadrupole-induced (0223,2023) components near the So(0) line center at 120 K (the temperature of Jupiter s upper atmosphere). Superimposed with the smooth free — free continuum (dashes) are structures arising from bound — free (below 354 cm-1) and free - bound (above 354 cm-1) transitions of the hydrogen pair (dotted). The convolution of the spectrum with a 4.3 cm-1 slit function (approximating the instrumental profile of the Voyager infrared spectrometer) is also shown (heavy line) [282].

I, must be deconvoluted from the laser line shapes the Raman resonance line shapes the detector slit function and the polarization properties of the laser and signal fields. 

Figure 3. Calculated band profiles of Stokes vibrational Raman scattering from Nt at 2000 K assuming a triangular slit function with FWHM = 5.0 cm 1. The bottom curve includes the isotropic part of the Q-branch only. The top curve is a more exact calculation including O- and S-branch scattering, the anisotropic part of the Q-branch and line-strength corrections owing to centrifugal distortion. The base lines have been shifted vertically for clarity.

Figure 1. Relative Stokes vibrational Raman intensity jor nitrogen for a trapezoidal slit function and various center positions...

Figure 2. Intensity ratio of anti-Stokes to Stokes vibrational Raman scattering for a trapezoidal slit function. Center position of Stokes bandpass at 6072 A.

A polymer membrane employed in water aeration device is a typical example of macroporous membranes, i.e., membranes with sub-millimeter holes. A rubber film with numbers of slits functions as a thin layer that separates the gas phase and the liquid phase, which allows the transfer of the gas into the liquid phase in a controlled manner. Such a film is appropriately termed as aeration membrane. 

Assuming the true absorptions to have Lorentzian form and assuming a triangular slit function, Ramsay 121, 122) investigated the effect of finite resolving power upon these band shapes for the vibrations of a variety of compounds. This approach reproduced satisfactorily the observed band profiles. However, he did not obtain a simple relationship between true and apparent integrated intensities. Ramsay found that bands are best characterized by their apparent peak intensities [loge(J o/r) J and their apparent half-intensity band widths These quantities are related to... 

Again assuming both a Lorentzian form for the true band shape, and a triangular slit function, a series of apparent band shapes may be calculated for a fixed value of the ratio and various values of the true peak... 

In all three methods, the assumptions of a Lorentzian band profile and of a triangular slit function were made. However, since Methods II and III involve measurement over the complete experimental curve, whereas Method I uses only three points of this curve, the latter is the most sensitive to the first assumption. Method II depends upon fairly small corrections related to the band shape and the slit function. Method HI is almost insensitive to the form of the slit function, but is much more strongly dependent upon the assumed band shape. Consequently, for partially overlapping bands, and in general, Methods II and HI are to be preferred, although Method III has speed in its favor. 

The procedures used to calibrate the Raman shift axis with neon lines can be fairly complex, due to subtleties introduced by the use of CCDs (5, 7-10). The image of the entrance slit usually covers more than one CCD pixel along the wavelength axis, and the Raman bandshape might be distributed over several pixels as well. So the observed Raman peak is a convolution of the slit function, the pixel width, and the line shape. These issues can become important when <1 cm accuracy is required or when comparing spectra from different insturements (7,9). 

The slit function is approximately triangular. Various instrument-related factors combine to produce the shape shown in Figure 25-7. 

Function (e) was convolved with the receiving slit function representing the rectangular function. Figure 6.14 shows axial instrumental functions. The results are consistent with our model. 

In addition to the effects of the laser widths on the CARS signal it is necessary to include the detector response of the spectrometer and other instrument factors. This is usually achieved by a further convolution of the calculated CARS signal with an appropriate Voigt [49] slit function. 

Whiting, E. E. "An Empirical Approximation to the Voigt Slit Function." Journal of Quantitative Spectroscopy and Radiative Transfer 8 (1968) 1379. 

Fig. 4.13. Gaussian shape of an absorbance band (p) and triangular slit function (s). (a) The light of measurement is veiy broad-band, therefore the absorption of the sample only covers a small part of the offered spectrum of light (b) the offered light is absorbed proportional to the concentration of the sample for this specific ratio of slit function and natural band width. Therefore the somewhat polychromatic light is affected by various different absorption coefficients.

When a spectrum is measured on a dispersive instrument, the true spectrum is convolved with the instrumental line shape (ILS) of the monochromator, which is the triangular slit function. The situation with the FT technique is equivalent, except that the true spectrum is convolved with the (sinx)/x function (no apodization) or with the FT of an appropriate apodization function. Hence, FT instruments offer a free choice of ILS according to the apodization selected and thus make it possible to optimise the sampling condition for a particular application. 

More recently, sensitivity quite comparable to that afforded by the line source method has been obtained by timed flash photographic spectroscopy, by which the rate of appearance of OH in the shock-wave decomposition of HgO vapour was reinvestigated. Values of A over the instrumental slit function, averaged for several lines, are included in Fig. 2.3. 

---