Everything about Dispersion Relation totally explained
Dispersion relations describe the ways that wave propagation varies with the
wavelength or
frequency of a
wave. This variation has long explained how white light is dispersed into different colors, thus making
rainbows possible. It turns out, thanks to the wave nature of all traveling objects, that dispersion relations are key to understand how energy and objects are transported from point to point in any medium. This story likely began, however, with interest in the dispersion of waves on water for example by
Pierre-Simon Laplace in
1776.
Important clues to the wide-ranging utility of dispersion relations came from work in the early 20th century by
H. Kramers and
R. Kronig. Their relations take the form of
integrals relating the real and imaginary parts of a property, called the
complex refractive index, of any medium in which waves travel. The
real part of this index describes how waves of different frequency
refract (change speed and hence bend or
disperse) through different angles on entering the medium. The
imaginary part of the index describes how the wave is
absorbed in the medium.
The universality of the concept became apparent with subsequent papers, on the dispersion relation's connection to causality in the
scattering theory of all types of waves and particles. For scattering processes where absorption can be ignored (for example attention focuses on the real refractive index), the term
dispersion relation has also been applied to the dependence of wave
frequency ω on
wave number k, or equivalently through
de Broglie's relations to the dependence of
energy E=ħω on
momentum p=
ħk. From dispersion relations in this form, the refractive index and the wave's "particle" or
group velocity v are obtained by taking the derivative for example
v = dω/dk = dE/dp.
Kramers-Kronig relations and waves
This is an overview of applications for the
Kramers-Kronig integral dispersion relations that connect real and imaginary parts of a medium's index of refraction.
Electron spectroscopy
In
electron energy loss spectroscopy, Kramers-Kronig analysis allows one to calculate the energy dependence of both real and imaginary parts of a specimen's light optical
permittivity, together with other optical properties such as the
absorption coefficient and
reflectivity.
In short, by measuring the number of high energy (for example 200 keV) electrons which lose energy
ΔE over a range of energy losses in traversing a very thin specimen (single scattering approximation), one can calculate the energy dependence of permittivity's imaginary part. The dispersion relations allow one to then calculate the energy dependence of the real part.
This measurement is made with electrons, rather than with light, and can be done with very high spatial resolution! One might thereby, for example, look for ultraviolet (UV) absorption bands in a laboratory specimen of
interstellar dust less than a 100 nm across, for example too small for UV spectroscopy. Although electron spectroscopy has poorer energy resolution than light
spectroscopy, data on properties in visible, ultraviolet and soft x-ray
spectral ranges may be recorded in the same experiment.
Frequency versus wavenumber
As mentioned above, when the focus in a medium is on refraction rather than absorption for example on the real part of the refractive index, it's common to refer to the functional dependence of frequency on wavenumber as the
dispersion relation. For particles, this translates to a knowledge of energy as a function of momentum.
Waves and optics
For
electromagnetic waves, the energy is proportional to the
frequency of the wave and the momentum to the
wavenumber. In this case,
Maxwell's equations tell us that the dispersion relation for vacuum is linear:
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By using the same reasoning, we can infer the speed of those waves:
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