Kramers-Kronig rule

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 CPA method has important properties apart from its relative simplicity. It is analytic in z,158 and thus respects the elementary physical constraints causality, Kramers-Kronig relations, sum rules, positive definite spectrum, etc. What is more, it is universal this method describes the virtual-crystal limit A W, the isolated-impurity limit cA - 0 or cA - 1, and the isolated-molecule limit W- 0, with the correct contribution of each molecular level.122 Indeed, the CPA may be derived159 from this last limit, as well as from that in the locator formalism.122... 

Sum rules can be obtained, not only for absorption or emission, but for many other optical functions, e.g. dielectric permeability, refractive index, rotatory power and ellipticity or circular dichroism. In fact, any quantities in physics which are related by a Kramers-Kronig transformation4 provide... 

However, the degree of modification we can exert on the properties of optical phenomena by introducing a complex medium appears to be limited. These limits, called conservation or sum rules, seem to have fundamental physical reasons. For instance, the Eamett-Loudon sum rule [1] places a restriction on the modification of spontaneous emission rate regardless of the means used and is derived from the general causality-related laws such as the Kramers-Kronig relation. 

If the transfer function H is in accordance with the causality rule, the components R and X are no longer independent of each other. Causality in the meaning of system theory forces couplings between the real and imaginary part, which are known as Kramers-Kronig relations (KKT) or Hilbert relations (HT), for details see Section 3.1.2.9 (The use of Kramers-Kronig Transforms). 

Here, = k T, 0 is the momentum transfer and hco the energy transfer, hk and hk are the momenta of the incoming and outgoing neutrons, and m is the bare electron mass. The Kramers-Kronig relations (White 1970) yield the sum rule... 

From the Kramers-Kronig transformation and from the quantum-theoretical description, sum rules for the rotational strength analogous to the Kuhn-Reichert sum rule for the absorption can be derived. This means that, for the sum over the rotational strength of all transitions of the CD spectrum of a molecule, one obtains... 

One such link between semiempirical theory and experiment that appeared about that time was the development of calculational methods for optical rotatory dispersion. Moffitt s theoretical work with Kronig—Kramers transforms coupled with Djerassi s experimental data on steroids gave rise to rules for the prediction of the sign of optical rotation. Computer calculations with semiempirical methods played a role. i Wavefunctions of at least an approximate sort were needed for the dipole and dipole velocity matrix elements of the theory. 

KKTs are tools brought to network theory by the work of Kramers (1926) and Kronig (1929) on X-ray optics. Just as the reciprocity theorem, they are purely mathematical rules of general validity in any passive, linear, reciprocal network of a minimum phase shift type. By minimum-phase networks, we mean ladder networks that do not have poles in the right half plane of the Wessel diagram. A ladder network is of minimum phase type a bridge where signal can come from more than one ladder is not necessarily of the minimum-phase type. The transforms are only possible when the functions are finite-valued at all frequencies. With impedance Z = R- -jX the transforms are ... 

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