Fundamental spectrum

Fig. 2. Overview of pulse shaping results, (a) and (b) depict measured SHG FROG traces before and after adaptive phase correction, respectively. Corresponding retrieved traces are displayed in (c) and (d). (e) Shows fundamental spectrum (shaded contour) measured at the crystal location in the FROG apparatus and the spectrum recovered by the FROG retrieval algorithm (open circles). Dash-dotted curve represents spectral phase prior to adaptive shaping, whereas dashed curve shows the optimized phase, (f) Initial (solid curve) and optimized (shaded contour) temporal intensity profiles. Dashed curve depicts temporal phase of the optimized pulse.

In Sec. 2.4 it was shown that differentiation enables one to eliminate unwanted background caused by light scattering or noise. But sometimes the undisturbed fundamental signal is called for. In that case, the differentiated spectrum must be retransformed, step by step, into the fundamental spectrum by integration (D I method). 

Figure 3-5. Subtraction of spacially delayed signals, a) Fundamental spectrum b) first derivative.

Figure 4-5. Congo Red. Cone. 20 mg L (in water) (slit 2 nm scan speed 2 nm s data interval 2 nm response 20 s). a) Fundamental spectrum b) fourth derivative MD D (S-G 11), corrected.

The derivative spectra of this protein (extracted from bovine pancreas) show the resolution of small irregularities on the flanks of the fundamental spectrum (Fig. 4-7) compare also the RNase spectra in [2], generated by analog differentiation. 

Figure 4-13. Comparison for the true arithmetic mean of 20 independent scans of a MCI2 6 H2O solution in water with the sliding average of 20 scans of the same substance, a) d of the ten averaged fundamental spectrum, b) d of the sliding averaged fundamental spectrum. c) d" of the ten averaged fundamental spectrum, d) d" of the sliding averaged fundamental spectrum.

Figure 4-33. Partitive HODS, la Fundamental spectrum of chymotrypsin cone. 0.250 g L ...

Figure 4-36. Additive HODS. 1 a fundamental spectrum of RNase (cone. 0.250 g L Mn water) lb fourth derivative of la 2a fundamental spectrum of chymotrypsin (cone. 0.125 g L in water) 2b fourth derivative of 2a 3a computed sum of spectra la and 2a 3b computed sum of derivatives lb and 2b 4a mixture of RNase and chymotrypsin (cone. 0.375 g enzymes) 4b fourth derivative of 4a. The d" spectra 3b and 4b are nearly identical this means that the substances show only a small, but noticeable interaction MD D (PP) [10, 29].

It may be that a spectrum of a substance is only known superposed by some background, e.g., background caused by light scattering in a turbid medium or other circumstances. In this case, it is recommended to take a number of derivatives until the background is eliminated. Then, the reverse function of differentiation, integration, must be carried out to restore the undisturbed fundamental spectrum (Sec. 2.6.4.2, [15, 19], and Fig. 4-39). 

In Fig. 4-41, the first to ninth derivative was generated from the flat fundamental spectrum of potassium nitrate. We see in this case that d does not resolve the curve. Even d has a shoulder which is only resolved in the sixth to eight order. 

Figure 4-45. Absorption spectrum of pigment orange 5 (PO 5), adsorbed on a layer of silica with a thickness of 0.2 mm (TLC spot) (High energy, slit -- 5 nm, scan speed 2 nm s response 5 s MD H. a) Fundamental spectrum b) fourth derivative of a) [321.

In this special case, R is equal to R c , the relative directional reflectance (see Eq. (4-14)). The comparison of differentiated absorption spectra with differentiated reflection spectra is then easier, and they are also more clearly arranged [31]. To demonstrate this, in Fig. 4-53 the fundamental spectrum and the fourth derivative of a wheat leaf are given in terms of absorbance and reflectance [57-58]. The shape of the curves are similar, and only the A (Ref) spectrum is slightly shifted to longer wavelengths. 

Figure 4-53. Comparison of absorption and reflection, a) Fundamental spectrum of a wheat leaf in terms of absorbance (2) and reflectance (1) b) fourth derivatives of a) [57, 58].

The full information lies exclusively in the fundamental spectrum. Therefore, the derivative technique cannot give more information, but derivatives, especially higher-order derivatives, make this information more accessible, more visible, and easier to evaluate. 

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