Kramers-Kronig reflection

The other two types of external reflection microspectroscopy are less well suited to the characterization of tissue samples. In the first type, which is variously called specular reflection, front-surface reflection or Kramers-Kronig reflection, the reflectance... 

The other two types of external reflection microspectroscopy are less well suited to the characterization of tissue samples. In the first type, which is variously called specular reflection, front-surface reflection, or Kramers-Kronig reflection, the reflectance spectra of thick, nonscattering, bulk samples are measured and converted to the wavenumber-dependent optical constants, that is, the refractive index (v) and the absorption index k(v) by the Kramers-Kronig transform, as discussed by Griffiths and de Haseth [10]. As the requirement for thick nonscattering samples is essentially never met for tissue samples, this type of measurement is never used in medical diagnosis but has occasionally been used for the study of polymer blends. 

Dielectric constants of metals, semiconductors and insulators can be detennined from ellipsometry measurements [38, 39]. Since the dielectric constant can vary depending on the way in which a fihn is grown, the measurement of accurate film thicknesses relies on having accurate values of the dielectric constant. One connnon procedure for detennining dielectric constants is by using a Kramers-Kronig analysis of spectroscopic reflectance data [39]. This method suffers from the series-tennination error as well as the difficulty of making corrections for the presence of overlayer contaminants. The ellipsometry method is for the most part free of both these sources of error and thus yields the most accurate values to date [39]. 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

The 1/to4 high frequency limit for R can be useful in determining optical constants from Kramers-Kronig analysis of reflectance data (see Section 2.7). Reflectances at frequencies higher than the greatest far-ultraviolet frequency for which measurements are made can be calculated from (9.15) and used to complete the Kramers-Kronig integral to infinite frequency. 

Figure 6.10 Optical properties of CdS (a) experimental reflectance spectrum of single crystals of CdS (b) refractive index n of CdS obtained from data given in (a) through the Kramers-Kronig relation.

The Pt + Pt intervalence transitions of such chain complexes occur in the regions 25,000-18,200 cm 1, 23,600-14,300 cm 1 and 20,600-7,500 cm 1 for chloro-, bromo-, and iodo-bridged complexes, respectively, the trend Cl > Br > I being the reverse of that of the conductivity of the complexes. The transition wavenumbers may be determined either by Kramers-Kronig analysis of specular reflectance measurements or from plots of the excitation profiles of Raman bands enhanced at or near resonance with the Pt I-PtIV intervalence band. The maxima have been found to be related to the Pt —PtIV chain distance, the smaller the latter the less being the intervalence transition energy (3). 

Fourier transform infrared microscopes are equipped with a reflection capability that can be used under these circumstances. External reflection spectroscopy (ERS) requires a flat, reflective surface, and the results are sensitive to the polarization of the incident beam as well as the angle of incidence. Additionally, the orientations of the electric dipoles in the films are important to the selection rules and the intensities of the reflected beam. In reflectance measurements, the spectra are a function of the dispersion in the refractive index and the spectra obtained are completely different from that obtained through a transmission measurement that is strongly influenced by the absorption index, k. However, a complex refractive index, n + ik can be determined through a well-known mathematical route, namely, the Kramers-Kronig analysis. 

The IR reflectivity measurements were performed on single crystals of 2 0.5 0.3 mm3 in size. A FT-IR Perkin-Elmer 1725X spectrometer equipped with microscope and a helium cryostat was used. Polarized reflectivity spectra (R(ro)) were measured from the conducting plane in two principal directions. Optical conductivity a(co) was obtained by Kramers-Kronig transformation. 

The reflectivity spectra R(E) and the reflectivity-EXAFS Xr(E) = R(E) — Rq(E)]/R()(E) are similar, but not identical, to the absorption spectra and x(E) obtained in transmission mode. R(E) is related to the complex refraction index n(E) = 1 — 8(E) — ifl(E) and P(E) to the absorption coefficient /i(E) by ji fil/An. P and 8 are related to each other by a Kramers-Kronig transformation, p and 8 may be also separated in an oscillatory (A/ , AS) and non-oscillatory part (P0,80) and may be used to calculate Xr- This is, briefly, how the reflectivity EXAFS may be calculated from n(E). which itself can be obtained by experimental transmission EXAFS of standards, or by calculation with the help of commercial programs such as FEFF [109] with the parameters Rj, Nj and a, which characterize the near range order. The fit of the simulated to measured reflectivity yields then a set of appropriate structure parameters. This method of data evaluation has been developed and has been applied to a few oxide covered metal electrodes [110, 111], Fig. 48 depicts a condensed scheme of the necessary procedures for data evaluation. 

Absorption Spectra as Kramers-Kronig-Transformed Reflection Spectra... 

Kramers-Kronig (KK) transformation of the reflection spectra. This provides the optical absorption "(cu) semiexperimentally and allows a thorough analysis of the various relaxation mechanisms creating the absorption lineshape (2.102), (2.111) of an ideal finite crystal in its phonon bath. This method is currently used. However, two major difficulties often obscure the credibility of the results ... 

Figure 2.11. The integration contour in the complex plane allowing one to generalize the Kramers-Kronig relations to the modulus and phase of the reflectivity amplitude.

Figure 3.12. Simulation of the b-polarized (0-0) reflectivity of the anthracene crystal using the bulk reflectivity amplitude derived from a Kramers-Kronig analysis (Section II.C). The total reflectivity is calculated from the scheme of Fig. 3.11 and (3.24)-(3.25) for various values of the nonradiative broadening parameter 7% Comparison with spectra of our best crystals gives the value / ° = 3cm 1 for T = 1.7 K.

Hie evaluation of the data yields Rjy Nj, and Sjy i.e., the near-range order parameters of the material seen from the absorber atom. XAS permits the evaluation of the near-range order in the vicinity of the atoms of various elements of one specimen if the energies of their absorption edges are different enough and thus are well separated within the spectrum. It should be mentioned that XAS in reflection looks similar to XAS in transmission mode, however it is different and the evaluation of measurements requires the comparison with reflectivity data calculated form transmission EXAFS spectra. These evaluation procedures involving Kramers-Kronig transform are described in the literature [i-v]. 

Usually, the absorption coefficient of the conducting crystals is so high that producing a crystal sufficiently thin and suitable for absorption measurements presents a great difficulty. If this is so, the bulk optical constants of a solid may be computed from the normal-incidence reflectivity of that material over an extended range of frequencies, followed by a Kramers-Kronig analysis of the measurements [12,14]. In this method the real, n, and imaginary, k, parts of the complex index of refraction... 

The absorption is very intense, so direct measurement of absorption spectra requires very thin films. When the sample surface is smooth enough, the absorption can be calculated by Kramers-Kronig inversion of the reflectance, which can be done rather accurately for a well-isolated transition [113]. A strongly scattering sample must be studied by attenuated total internal reflectance. 

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