Characterizations of the exponential function

The refraction of a light in a prism is due to dispersion.

In physical sciences and electrical engineering, dispersion relations describe the effect of dispersion from a medium on the properties of a wave traveling within that medium. A dispersion relation connects different properties of the wave such as its energy, frequency, wavelength and wavenumber. From these relations the phase velocity and group velocity of the wave have convenient expressions which can thereby determine a refractive index of the medium. More general to the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media.

In the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase velocity and group velocity.

Dispersion

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space. The speed of a plane wave, v, is a function of the wave's wavelength $\lambda$:

$v=v(\lambda ).\,$

The wave's speed, wavelength, and frequency, f, are related by the dispersion relation

$v(\lambda )=\lambda \ f(\lambda ).\,$

Dispersion relations are more commonly expressed in terms of the angular frequency $\omega =2\pi f$ and wavenumber $k=2\pi /\lambda$. Rewriting the relation above in these variables gives

$\omega (k)=v(k)\ k.\,$

The use of ω(k) to describe the dispersion relation is standard because both the phase velocity ω/k and the group velocity dω/dk have convenient representations via this function.

The elementary sorts of waves being considered can be described by

$A(x,t)=A_{0}e^{2\pi i{\frac {x-vt}{\lambda }}}=A_{0}e^{i(kx-\omega t)},$

where

A is the amplitude of the wave,

A₀ = A(0,0),

x is a position along the wave's direction of travel, and

t is the time at which the wave is described.

Plane waves in vacuum

Plane waves in vacuum are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.

Electromagnetic waves

For electromagnetic waves in vacuum, the angular frequency is proportional to the wavenumber:

$\omega =ck.\,$

This is a linear dispersion relation. In this case, phase velocity and group velocity are the same:

$v={\frac {\omega }{k}}={\frac {\partial \omega }{\partial k}}=c;$

they are given by c, the speed of light in vacuum, a frequency-independent constant.

De Broglie dispersion relations

The free-space dispersion plot of kinetic energy versus momentum, for many objects of everyday life.

Total energy, momentum, and mass of particles are connected through the relativistic relation

$E^{2}=(mc^{2})^{2}+(pc)^{2}\,$  

which in the ultrarelativistic limit is essentially

$E=pc\,$

and in the nonrelativistic limit is

$E=mc^{2}+{\frac {p^{2}}{2m}}$

where in both equations $m$ is the rest mass. In the nonrelativistic limit, $mc^{2}$ is essentially constant and $p^{2}/(2m)$ is the familiar kinetic energy in terms of momentum $p=mv$.

The transition from ultrarelativistic to nonrelativistic behaviour shows up as a slope change from p to p² as shown in the log-log dispersion plot of E vs. p.

Elementary particles, atomic nuclei, atoms, and even molecules behave in some context as matter waves. According to the de Broglie relations, their kinetic energy E can be expressed as a frequency ω, and their momentum p as a wavenumber k, using the reduced Planck constant ħ:

$E=\hbar \omega ,\quad p=\hbar k.$

Accordingly, angular frequency and wavenumber are connected through a dispersion relation, which in the nonrelativistic limit reads

$\omega ={\frac {\hbar k^{2}}{2m}}.$

Animation: phase and group velocity of electrons

This animation portrays the de Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4 Ångstroms in width. The momentum per unit mass (proper velocity) of the middle electron is lightspeed so that its group velocity is 0.707 c. The top electron has twice the momentum, while the bottom electron has half. Note that wavelength and phase velocity decrease as the group velocity increases, until the wave packet and its phase maxima move together near the speed of light and only wavelength noticeably decreases past that. Both transverse and longitudinal coherence widths (packet sizes) of such high energy electrons in the lab may be orders of magnitude larger than the ones shown here.

Frequency versus wavenumber

As mentioned above, when the focus in a medium is on refraction rather than absorption—that is, on the real part of the refractive index—it is common to refer to the functional dependence of angular frequency on wavenumber as the dispersion relation. For particles, this translates to a knowledge of energy as a function of momentum.

Waves and optics

Template:See The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction, or by using light in a non-uniform medium such as a waveguide. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e., be dispersed. In these materials, ${\frac {\partial \omega }{\partial k}}$ is known as the group velocity and corresponds to the speed at which the peak propagates, a value different from the phase velocity.

Deep water waves

Frequency dispersion of surface gravity waves on deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red dot traverses the figure in the time it takes the green dot to traverse half.

The dispersion relation for deep water waves is often written as

$\omega ={\sqrt {gk}},$

where g is the acceleration due to gravity. Deep water, in this respect, is commonly denoted as the case where the water depth is larger than half the wavelength. In this case the phase velocity is

$v_{p}={\frac {\omega }{k}}={\sqrt {\frac {g}{k}}}$

and the group velocity is

$v_{g}={\frac {d{\omega }}{dk}}={\frac {1}{2}}v_{p}.$

Waves on a string

Two-frequency beats of a non-dispersive transverse wave. Since the wave is non-dispersive, phase (red) and group (green) velocities are equal.

For an ideal string, the dispersion relation can be written as

$\omega =k{\sqrt {\frac {T}{\mu }}}$

where T is the tension force in the string and μ is the string's mass per unit length. As for the case of electromagnetic waves in a vacuum, ideal strings are thus a non-dispersive medium i.e., the phase and group velocities are equal and independent (to first order) of vibration frequency.

For a nonideal string, where stiffness is taken into account, the dispersion relation is written as

$\omega ^{2}={\frac {T}{\mu }}k^{2}+\alpha k^{4}$

where "$\alpha$" is a constant that depends on the string.

Solid state

In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals means that many levels of energy are possible for a given momentum and that some energies might not be available at any momentum. The collection of all possible energies and momenta is known as the band structure of a material. Properties of the band structure define whether the material is an insulator, semiconductor or conductor.

Phonons

Phonons are to sound waves in a solid what photons are to light: They are the quanta that carry it. The dispersion relation of phonons is also important and non-trivial. Most systems will show two separate bands on which phonons live. Phonons on the band that cross the origin are known as acoustic phonons, the others as optical phonons.

Electron optics

With high energy (e.g., 200 keV) electrons in a transmission electron microscope, the energy dependence of higher order Laue zone (HOLZ) lines in convergent beam electron diffraction (CBED) patterns allows one, in effect, to directly image cross-sections of a crystal's three-dimensional dispersion surface. This dynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.

History

Isaac Newton studied refraction in prisms. He failed, however, to recognize the material dependence of the dispersion relation. Had he done so, he would almost certainly have invented the achromatic lens.

Dispersion of waves on water was studied by Pierre-Simon Laplace in 1776.

The universality of the Kramers-Kronig relations (1926/27) became apparent with subsequent papers on the dispersion relation's connection to causality in the scattering theory of all types of waves and particles.

See also

• Ellipsometry
• Ultrashort pulse

References

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External links

• Poster on CBED simulations to help visualize dispersion surfaces, by Andrey Chuvilin and Ute Kaiser