Editing Dispersion (optics)

{{Short description|Effect of a material on light}}
[[File:Light dispersion conceptual waves.gif|thumb|In a [[dispersive prism]], material dispersion (a [[wavelength]]-dependent [[refractive index]]) causes different colors to [[Refraction|refract]] at different angles, splitting white light into a [[spectrum]].]]
[[File:Light dispersion of a compact fluorescent lamp seen through an Amici direct-vision prism PNr°0114.jpg|thumb|A [[compact fluorescent lamp]] seen through an [[Amici prism]]]]

'''Dispersion''' is the phenomenon in which the [[phase velocity]] of a [[wave]] depends on its frequency.<ref>{{cite book |last1=Born |first1=Max |author-link=Max Born |last2=Wolf |first2=Emil |title=[[Principles of Optics]]|publisher=[[Cambridge University Press]] |date=October 1999 |location=Cambridge |pages=[https://archive.org/details/principlesofopti0006born/page/n43 14]–24 |isbn=0-521-64222-1}}</ref> Sometimes the term '''chromatic dispersion''' is used to refer to [[optics]] specifically, as opposed to [[wave propagation]] in general.  A medium having this common property may be termed a '''dispersive medium'''.

Although the term is used in the field of optics to describe [[light]] and other [[electromagnetic wave]]s, dispersion in the same sense can apply to any sort of wave motion such as [[acoustic dispersion]] in the case of sound and seismic waves, and in [[gravity wave]]s (ocean waves). Within optics, dispersion is a property of telecommunication signals along [[transmission line]]s (such as [[microwaves]] in [[coaxial cable]]) or the [[Pulse (signal processing)|pulses]] of light in [[optical fiber]].

In optics, one important and familiar consequence of dispersion is the change in the angle of [[refraction]] of different colors of light,<ref>[http://www.proximion.com/about-dispersion Dispersion Compensation]. Retrieved 25-08-2015.</ref> as seen in the spectrum produced by a dispersive [[Prism (optics)|prism]] and in [[chromatic aberration]] of lenses. Design of compound [[achromatic lens]]es, in which chromatic aberration is largely cancelled, uses a quantification of a glass's dispersion given by its [[Abbe number]] ''V'', where ''lower'' Abbe numbers correspond to ''greater'' dispersion over the [[visible spectrum]]. In some applications such as telecommunications, the absolute phase of a wave is often not important but only the propagation of [[wave packet]]s or "pulses"; in that case one is interested only in variations of [[group velocity]] with frequency, so-called [[#Group-velocity_dispersion|group-velocity dispersion]].

All common [[transmission media]] also vary in [[attenuation]] (normalized to transmission length) as a function of frequency, leading to [[attenuation distortion]]; this is not dispersion, although sometimes reflections at closely spaced [[impedance matching|impedance boundaries]] (e.g. crimped segments in a cable) can produce signal distortion which further aggravates inconsistent transit time as observed across signal bandwidth.

== Examples ==
The most familiar example of dispersion is probably a [[rainbow]], in which dispersion causes the spatial separation of a white [[light]] into components of different [[wavelengths]] (different [[color]]s). However, dispersion also has an effect in many other circumstances: for example, [[group-velocity dispersion]] causes [[Pulse (signal processing)|pulses]] to spread in [[optical fiber]]s, degrading signals over long distances; also, a cancellation between group-velocity dispersion and [[Nonlinear system|nonlinear]] effects leads to [[soliton]] waves.

== Material and waveguide dispersion ==
Most often, chromatic dispersion refers to bulk material dispersion, that is,  the change in [[refractive index]] with optical frequency. However, in a [[waveguide]] there is also the phenomenon of ''waveguide dispersion'', in which case a wave's [[phase velocity]] in a structure depends on its frequency simply due to the structure's geometry. More generally, "waveguide" dispersion can occur for waves propagating through any inhomogeneous structure (e.g., a [[photonic crystal]]), whether or not the waves are confined to some region.{{dubious|date=October 2014}} In a waveguide, ''both'' types of dispersion will generally be present, although they are not strictly additive.{{Citation needed|date=October 2014}} For example, in fiber optics the material and waveguide dispersion can effectively cancel each other out to produce a [[zero-dispersion wavelength]], important for fast [[fiber-optic communication]].

== Material dispersion in optics ==
[[File:Mplwp dispersion curves.svg|upright=1.35|thumb|The variation of refractive index vs. vacuum wavelength for various glasses. The wavelengths of visible light are shaded in grey.]]
[[File:Spidergraph Dispersion.GIF|upright=1.45|thumb|Influences of selected glass component additions on the mean dispersion of a specific base glass (''n''<sub>F</sub> valid for ''λ''&nbsp;=&nbsp;486&nbsp;nm (blue), ''n''<sub>C</sub> valid for ''λ''&nbsp;=&nbsp;656&nbsp;nm (red))<ref>[http://glassproperties.com/dispersion/ Calculation of the Mean Dispersion of Glasses].</ref>]]

Material dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used to construct [[spectrometer]]s and [[spectroradiometer]]s. However, in lenses, dispersion causes [[chromatic aberration]], an undesired effect that may degrade images in microscopes, telescopes, and photographic objectives.

The ''[[phase velocity]]'' ''v'' of a wave in a given uniform medium is given by
: <math>v = \frac{c}{n},</math>
where ''c'' is the [[speed of light]] in vacuum, and ''n'' is the [[refractive index]] of the medium.

In general, the refractive index is some function of the frequency ''f'' of the light, thus ''n''&nbsp;=&nbsp;''n''(''f''), or alternatively, with respect to the wave's wavelength ''n''&nbsp;=&nbsp;''n''(''λ''). The wavelength dependence of a material's refractive index is usually quantified by its [[Abbe number]] or its coefficients in an empirical formula such as the [[Cauchy's equation|Cauchy]] or [[Sellmeier equation]]s.

Because of the [[Kramers–Kronig relations]], the wavelength dependence of the real part of the refractive index is related to the material [[absorption (electromagnetic radiation)|absorption]], described by the imaginary part of the refractive index (also called the [[refractive index#Dispersion and absorption|extinction coefficient]]). In particular, for non-magnetic materials ([[permeability (electromagnetism)|''μ'']]&nbsp;=&nbsp;[[magnetic constant|''μ''<sub>0</sub>]]), the [[linear response function|susceptibility]] ''χ'' that appears in the Kramers–Kronig relations is the [[electric susceptibility]] ''χ''<sub>e</sub>&nbsp;=&nbsp;''n''<sup>2</sup>&nbsp;−&nbsp;1.

The most commonly seen consequence of dispersion in optics is the separation of [[electromagnetic spectrum#Visible radiation (light)|white light]] into a [[optical spectrum|color spectrum]] by a [[triangular prism (optics)|prism]]. From [[Snell's law]] it can be seen that the angle of [[refraction]] of light in a prism depends on the refractive index of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the light is refracted by will also vary with wavelength, causing an angular separation of the colors known as ''angular dispersion''.

For visible light, refraction indices ''n'' of most transparent materials (e.g., air, glasses) decrease with increasing wavelength ''λ'':
: <math>1 < n(\lambda_\text{red}) < n(\lambda_\text{yellow}) < n(\lambda_\text{blue}),</math>
or generally,
: <math>\frac{dn}{d \lambda} < 0.</math>

In this case, the medium is said to have ''normal dispersion''. Whereas if the index increases with increasing wavelength (which is typically the case in the ultraviolet<ref>Born, M. and Wolf, E. (1980) "[[Principles of Optics]]", 6th&nbsp;ed., p.&nbsp;93. Pergamon Press.</ref>), the medium is said to have ''anomalous dispersion''.

At the interface of such a material with air or vacuum (index of ~1), Snell's law predicts that light incident at an angle ''θ'' to the [[surface normal|normal]] will be refracted at an angle arcsin({{sfrac|sin ''θ''|''n''}}). Thus, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-known [[rainbow]] pattern.

== Group-velocity dispersion ==
{{main|Group-velocity dispersion}}
[[File:Optical dispersion dynamics.gif|thumb|Time evolution of a short pulse in a hypothetical dispersive medium (''k''&nbsp;=&nbsp;''ω''<sup>2</sup>) showing that the longer-wavelength components travel faster than the shorter-wavelength components (positive GVD), resulting in chirping and pulse broadening]]

Beyond simply describing a change in the phase velocity over wavelength, a more serious consequence of dispersion in many applications is termed  [[group-velocity dispersion]] (GVD). While phase velocity ''v'' is defined as ''v'' = ''c''/''n'', this describes only one frequency component. When different frequency components are combined, as when considering a signal or a pulse, one is often more interested in the [[group velocity]], which describes the speed at which a pulse or information superimposed on a wave (modulation) propagates. In the accompanying animation, it can be seen that the wave itself (orange-brown) travels at a phase velocity much faster than the speed of the ''envelope'' (black), which corresponds to the group velocity. This pulse might be a communications signal, for instance, and its information only travels at the group velocity rate, even though it consists of wavefronts advancing at a faster rate (the phase velocity).

It is possible to calculate the group velocity from the refractive-index curve ''n''(''ω'') or more directly from the wavenumber ''k'' = ''ωn''/''c'', where ''ω'' is the radian frequency ''ω''&nbsp;=&nbsp;2''πf''. Whereas one expression for the phase velocity is ''v''<sub>p</sub>&nbsp;=&nbsp;''ω''/''k'', the group velocity can be expressed using the [[derivative]]: ''v''<sub>g</sub>&nbsp;=&nbsp;''dω''/''dk''. Or in terms of the phase velocity ''v''<sub>p</sub>,

: <math>v_\text{g} = \frac{v_\text{p}}{1 - \dfrac{\omega}{v_\text{p}} \dfrac{dv_\text{p}}{d\omega}}.</math>

When dispersion is present, not only the group velocity is not equal to the phase velocity, but generally it itself varies with wavelength. This is known as group-velocity dispersion and causes a short pulse of light to be broadened, as the different-frequency components within the pulse travel at different velocities. Group-velocity dispersion is quantified as the derivative of the ''reciprocal'' of the group velocity with respect to [[angular frequency]], which results in ''group-velocity dispersion''&nbsp;=&nbsp;''d''<sup>2</sup>''k''/''dω''<sup>2</sup>.

If a light pulse is propagated through a material with positive group-velocity dispersion, then the shorter-wavelength components travel slower than the longer-wavelength components. The pulse therefore becomes ''positively [[chirp]]ed'', or ''up-chirped'', increasing in frequency with time. On the other hand, if a pulse travels through a material with negative group-velocity dispersion, shorter-wavelength components travel faster than the longer ones, and the pulse becomes ''negatively chirped'', or ''down-chirped'', decreasing in frequency with time.

An everyday example of a negatively chirped signal in the acoustic domain is that of an approaching train hitting deformities on a welded track. The sound caused by the train itself is impulsive and travels much faster in the metal tracks than in air, so that the train can be heard well before it arrives. However, from afar it is not heard as causing impulses, but leads to a distinctive descending chirp, amidst reverberation caused by the complexity of the vibrational modes of the track. Group-velocity dispersion can be heard in that the volume of the sounds stays audible for a surprisingly long time, up to several seconds.

== Dispersion control ==
{{unreferenced section|date=March 2023}}
The result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge, rendering the bit-stream unintelligible. This limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fibre at a wavelength where the GVD is zero (e.g., around 1.3–1.5&nbsp;μm in [[silica]] [[fibres]]), so pulses at this wavelength suffer minimal spreading from dispersion. In practice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifies other nonlinear effects (such as [[four-wave mixing]]). Another possible option is to use [[soliton (optics)|soliton]] pulses in the regime of negative dispersion, a form of optical pulse which uses a [[nonlinear optics|nonlinear optical]] effect to self-maintain its shape. Solitons have the practical problem, however, that they require a certain power level to be maintained in the pulse for the nonlinear effect to be of the correct strength. Instead, the solution that is currently used in practice is to perform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion so that the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as [[self-phase modulation]], which interact with dispersion to make it very difficult to undo.

Dispersion control is also important in [[laser]]s that produce [[ultrashort pulse|short pulses]]. The overall dispersion of the [[laser construction|optical resonator]] is a major factor in determining the duration of the pulses emitted by the laser. A pair of [[Prism (optics)|prisms]] can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. [[Diffraction grating]]s can also be used to produce dispersive effects; these are often used in high-power laser amplifier systems. Recently, an alternative to prisms and gratings has been developed: [[chirped mirror]]s. These dielectric mirrors are coated so that different wavelengths have different penetration lengths, and therefore different group delays. The coating layers can be tailored to achieve a net negative dispersion.

== In waveguides ==
[[Waveguide]]s are highly dispersive due to their geometry (rather than just to their material composition). [[Optical fiber]]s are a sort of waveguide for optical frequencies (light) widely used in modern telecommunications systems. The rate at which data can be transported on a single fiber is limited by pulse broadening due to chromatic dispersion among other phenomena.

In general, for a waveguide mode with an [[angular frequency]] ''ω''(''β'') at a [[propagation constant]] ''β'' (so that the electromagnetic fields in the propagation direction ''z'' oscillate proportional to ''e''<sup>''i''(''βz''−''ωt'')</sup>), the group-velocity dispersion parameter ''D'' is defined as<ref>Ramaswami, Rajiv and Sivarajan, Kumar N. (1998) ''Optical Networks: A Practical Perspective''. Academic Press: London.</ref>
: <math>D = -\frac{2\pi c}{\lambda^2} \frac{d^2 \beta}{d\omega^2} = \frac{2\pi c}{v_g^2 \lambda^2} \frac{dv_g}{d\omega},</math>

where ''λ''&nbsp;=&nbsp;2{{pi}}''c''/''ω'' is the vacuum wavelength, and ''v''<sub>g</sub>&nbsp;=&nbsp;''dω''/''dβ'' is the group velocity. This formula generalizes the one in the previous section for homogeneous media and includes both waveguide dispersion and material dispersion. The reason for defining the dispersion in this way is that |''D''| is the (asymptotic) temporal pulse spreading Δ''t'' per unit bandwidth
Δ''λ'' per unit distance travelled, commonly reported in [[picosecond|ps]]/([[nanometre|nm]]⋅[[kilometre|km]]) for optical fibers.

In the case of [[multi-mode optical fiber]]s, so-called [[modal dispersion]] will also lead to pulse  broadening. Even in [[single-mode fiber]]s, pulse broadening can occur as a result of [[polarization mode dispersion]] (since there are still two polarization modes). These are ''not'' examples of chromatic dispersion, as they are not dependent on the wavelength or [[Spectral linewidth|bandwidth]] of the pulses propagated.

== Higher-order dispersion over broad bandwidths ==
When a broad range of frequencies (a broad bandwidth) is present in a single wavepacket, such as in an [[ultrashort pulse]] or a [[chirp]]ed pulse or other forms of [[spread spectrum]] transmission, it may not be accurate to approximate the dispersion by a constant over the entire bandwidth, and more complex calculations are required to compute effects such as pulse spreading.

In particular, the dispersion parameter ''D'' defined above is obtained from only one derivative of the group velocity. Higher derivatives are known as ''higher-order dispersion''.<ref>[http://www.rp-photonics.com/chromatic_dispersion.html Chromatic Dispersion], ''Encyclopedia of Laser Physics and Technology'' (Wiley, 2008).</ref><ref>{{Cite journal|last1=Mai|first1=Wending|last2=Campbell|first2=Sawyer D.|last3=Whiting|first3=Eric B.|last4=Kang|first4=Lei|last5=Werner|first5=Pingjuan L.|last6=Chen|first6=Yifan|author7-link=Douglas Werner|last7=Werner|first7=Douglas H.|date=2020-10-01|title=Prismatic discontinuous Galerkin time domain method with an integrated generalized dispersion model for efficient optical metasurface analysis|journal=[[Optical Materials Express]]|language=EN|volume=10|issue=10|pages=2542–2559|doi=10.1364/OME.399414|bibcode=2020OMExp..10.2542M|issn=2159-3930|doi-access=free}}</ref> These terms are simply a [[Taylor series]] expansion of the [[dispersion relation]] ''β''(''ω'') of the medium or waveguide around some particular frequency. Their effects can be computed via numerical evaluation of [[Fourier transform]]s of the waveform, via integration of higher-order [[slowly varying envelope approximation]]s, by a [[split-step method]] (which can use the exact dispersion relation rather than a Taylor series), or by direct simulation of the full [[Maxwell's equations]] rather than an approximate envelope equation.

== Spatial dispersion ==
{{Main|Spatial dispersion}}

In electromagnetics and optics, the term ''dispersion'' generally refers to aforementioned temporal or frequency dispersion. Spatial dispersion refers to the non-local response of the medium to the space; this can be reworded as the wavevector dependence of the permittivity. For an exemplary [[Anisotropy|anisotropic]] medium, the spatial relation between [[Electric field|electric]] and [[electric displacement field]] can be expressed as a [[convolution]]:<ref name="landau"/>

: <math>D_i(t, r) = E_i(t, r) + \int_0^\infty \int f_{ik}(\tau; r, r') E_k(t - \tau, r') \,dV'\,d\tau,</math>

where the [[Integral transform|kernel]] <math>f_{ik}</math> is dielectric response (susceptibility); its indices make it in general a [[tensor]] to account for the anisotropy of the medium. Spatial dispersion is negligible in most macroscopic cases, where the scale of variation of <math>E_k(t - \tau, r')</math> is much larger than atomic dimensions, because the dielectric kernel dies out at macroscopic distances. Nevertheless, it can result in non-negligible macroscopic effects, particularly in conducting media such as [[metal]]s, [[electrolyte]]s and [[Plasma (physics)|plasmas]]. Spatial dispersion also plays role in [[Optical rotation|optical activity]] and [[Doppler broadening]],<ref name="landau">{{cite book
 |first1=L. D. |last1=Landau |author1-link= Lev Landau
 |first2= E. M. |last2= Lifshitz |author2-link= Evgeny Lifshitz
 |first3= L. P. |last3= Pitaevskii |author3-link= Lev Pitaevskii
 |year=1984
 |title=Electrodynamics of Continuous Media
 |edition=2nd |volume= 8
 |publisher=[[Butterworth-Heinemann]]
 |isbn=978-0-7506-2634-7
 |title-link=Course of Theoretical Physics}}</ref> as well as in the theory of [[metamaterial]]s.<ref>{{cite journal |last1=Demetriadou |first1=A. |last2=Pendry |first2= J. B. |author-link2=John Pendry |date=1 July 2008 |title= Taming spatial dispersion in wire metamaterial|journal=[[Journal of Physics: Condensed Matter]] |volume=20 |issue=29 |pages=295222 |doi=10.1088/0953-8984/20/29/295222 |bibcode=2008JPCM...20C5222D |s2cid=120249447 }}</ref>

== In gemology ==
{| class="wikitable sortable collapsible collapsed" style="float: right; text-align: center;"
|+ Dispersion values of minerals<ref name="b1">{{cite book |last=Schumann |first=Walter |year=2009 |title=Gemstones of the World |edition=4th newly revised & expanded |publisher=Sterling Publishing Company |isbn=978-1-4027-6829-3 |pages=41–42 |url=https://books.google.com/books?id=V9PqVxpxeiEC&pg=PA42 |access-date=31 December 2011}}</ref>
! Mineral name !! {{nobr|{{mvar|n}}{{sub|{{sc|B}} }} − {{mvar|n}}{{sub|{{sc|G}} }} }} !! {{nobr|{{mvar|n}}{{sub|{{sc|C}} }} − {{mvar|n}}{{sub|{{sc|F}} }} }}
|-
| [[Hematite]] || 0.500 || —
|-
| [[Cinnabar]] (HgS) || 0.40 || —
|-
| synth. [[Rutile]] || 0.330 || 0.190
|-
| [[Rutile]] (TiO<sub>2</sub>) || 0.280 || 0.120–0.180
|-
| [[Anatase]] (TiO<sub>2</sub>) || 0.213–0.259 || —
|-
| [[Wulfenite]] || 0.203 || 0.133
|-
| [[Vanadinite]] || 0.202 || —
|-
| [[Fabulite]] || 0.190 || 0.109
|-
| [[Sphalerite]] (ZnS) || 0.156 || 0.088
|-
| [[Sulfur]] (S) || 0.155 || —
|-
| [[Stibiotantalite]] || 0.146 || —
|-
| [[Goethite]] (FeO(OH)) || 0.14 || —
|-
| [[Brookite]] (TiO<sub>2</sub>) || 0.131 || 0.12–1.80
|-
| [[Linobate]] || 0.13 || 0.075
|-
| [[Zincite]] (ZnO) || 0.127 || —
|-
| synth. [[Moissanite]] (SiC) || 0.104 || —
|-
| [[Cassiterite]] (SnO<sub>2</sub>) || 0.071 || 0.035
|-
| [[Zirconia]] (ZrO<sub>2</sub>) || 0.060 || 0.035
|-
| [[Powellite]] (CaMoO<sub>4</sub>) || 0.058 || —
|-
| [[Andradite]] || 0.057 || —
|-
| [[Demantoid]] || 0.057 || 0.034
|-
| [[Cerussite]] || 0.055 || 0.033–0.050
|-
| [[Titanite]] || 0.051 || 0.019–0.038
|-
| [[Benitoite]] || 0.046 || 0.026
|-
| [[Anglesite]] || 0.044 || 0.025
|-
| [[Diamond]] (C) || 0.044 || 0.025
|-
| synth. [[Cassiterite]] (SnO<sub>2</sub>) || 0.041 || —
|-
| [[Flint glass]] || 0.041 || —
|-
| [[Hyacinth (mineral)|Hyacinth]] || 0.039 || —
|-
| [[Jargoon]] || 0.039 || —
|-
| [[Starlite]] || 0.039 || —
|-
| [[Scheelite]] || 0.038 || 0.026
|-
| [[Zircon]] (ZrSiO<sub>4</sub>) || 0.039 || 0.022
|-
| [[Gadolinium gallium garnet|GGG]] || 0.038 || 0.022
|-
| [[Dioptase]] || 0.036 || 0.021
|-
| [[Whe Vinay wellite]] || 0.034 || —
|-
| [[Gypsum]] || 0.033 || 0.008
|-
| [[Alabaster]] || 0.033 || —
|-
| [[Epidote]] || 0.03 || 0.012–0.027
|-
| [[Tanzanite]] || 0.030 || 0.011
|-
| [[Thulite]] || 0.03 || 0.011
|-
| [[Zoisite]] || 0.03 || —
|-
| [[Yttrium aluminium garnet|YAG]] || 0.028 || 0.015
|-
| [[Spessartine]] || 0.027 || 0.015
|-
| [[Uvarovite]] || 0.027 || 0.014–0.021
|-
| [[Almandine]] || 0.027 || 0.013–0.016
|-
| [[Hessonite]] || 0.027 || 0.013–0.015
|-
| [[Willemite]] || 0.027 || —
|-
| [[Pleonaste]] || 0.026 || —
|-
| [[Rhodolite]] || 0.026 || —
|-
| [[Boracite]] || 0.024 || 0.012
|-
| [[Cryolite]] || 0.024 || —
|-
| [[Staurolite]] || 0.023 || 0.012–0.013
|-
| [[Pyrope]] || 0.022 || 0.013–0.016
|-
| [[Diaspore]] || 0.02 || —
|-
| [[Grossular]] || 0.020 || 0.012
|-
| [[Hemimorphite]] || 0.020 || 0.013
|-
| [[Kyanite]] || 0.020 || 0.011
|-
| [[Peridot]] || 0.020 || 0.012–0.013
|-
| [[Spinel]] || 0.020 || 0.011
|-
| [[Vesuvianite]] || 0.019–0.025 || 0.014
|-
| [[Gahnite]] || 0.019–0.021 || —
|-
| [[Clinozoisite]] || 0.019 || 0.011–0.014
|-
| [[Labradorite]] || 0.019 || 0.010
|-
| [[Axinite]] || 0.018–0.020 || 0.011
|-
| [[Diopside]] || 0.018–0.020 || 0.01
|-
| [[Ekanite]] || 0.018 || 0.012
|-
| [[Corundum]] (Al<sub>2</sub>O<sub>3</sub>) || 0.018 || 0.011
|-
| synth. [[Corundum]] || 0.018 || 0.011
|-
| [[Ruby]] (Al<sub>2</sub>O<sub>3</sub>) || 0.018 || 0.011
|-
| [[Sapphire]] (Al<sub>2</sub>O<sub>3</sub>) || 0.018 || 0.011
|-
| [[Kornerupine]] || 0.018 || 0.010
|-
| [[Sinhalite]] || 0.018 || 0.010
|-
| [[Sodalite]] || 0.018 || 0.009
|-
| [[Rhodizite]] || 0.018 || —
|-
| [[Hiddenite]] || 0.017 || 0.010
|-
| [[Kunzite]] || 0.017 || 0.010
|-
| [[Spodumene]] || 0.017 || 0.010
|-
| [[Tourmaline]] || 0.017 || 0.009–0.011
|-
| [[Cordierite]] || 0.017 || 0.009
|-
| [[Danburite]] || 0.017 || 0.009
|-
| [[Herderite]] || 0.017 || 0.008–0.009
|-
| [[Rubellite]] || 0.017 || 0.008–0.009
|-
| [[Achroite]] || 0.017 || —
|-
| [[Dravite]] || 0.017 || —
|-
| [[Elbaite]] || 0.017 || —
|-
| [[Indicolite]] || 0.017 || —
|-
| [[Liddicoatite]] || 0.017 || —
|-
| [[Scapolite]] || 0.017 || —
|-
| [[Schorl]] || 0.017 || —
|-
| [[Verdelite]] || 0.017 || —
|-
| [[Andalusite]] || 0.016 || 0.009
|-
| [[Baryte]] (BaSO<sub>4</sub>) || 0.016 || 0.009
|-
| [[Euclase]] || 0.016 || 0.009
|-
| [[Datolite]] || 0.016 || —
|-
| [[Alexandrite]] || 0.015 || 0.011
|-
| [[Chrysoberyl]] || 0.015 || 0.011
|-
| [[Rhodochrosite]] || 0.015 || 0.010–0.020
|-
| [[Sillimanite]] || 0.015 || 0.009–0.012
|-
| [[Hambergite]] ||  0.015 || 0.009–0.010
|-
| [[Pyroxmangite]] || 0.015 || —
|-
| synth. [[Scheelite]] || 0.015 || —
|-
| [[Smithsonite]] || 0.014–0.031 || 0.008–0.017
|-
| [[Amblygonite]] || 0.014–0.015 || 0.008
|-
| [[Aquamarine (gemstone)|Aquamarine]] || 0.014 || 0.009–0.013
|-
| [[Beryl]] || 0.014 || 0.009–0.013
|-
| [[Emerald]] || 0.014 || 0.009–0.013
|-
| [[Heliodor]] || 0.014 || 0.009–0.013
|-
| [[Morganite]] || 0.014 || 0.009–0.013
|-
| [[Brazilianite]] || 0.014 || 0.008
|-
| [[Celestine (mineral)|Celestine]] || 0.014 || 0.008
|-
| [[Topaz]] || 0.014 || 0.008
|-
| [[Goshenite]] || 0.014 || —
|-
| [[Apatite]] || 0.013 || 0.008–0.010
|-
| [[Aventurine]] || 0.013 || 0.008
|-
| [[Amethyst]] (SiO<sub>2</sub>) || 0.013 || 0.008
|-
| [[Citrine quartz]] || 0.013 || 0.008
|-
| [[Prasiolite]] || 0.013 || 0.008
|-
| [[Quartz]] (SiO<sub>2</sub>) || 0.013 || 0.008
|-
| Rose quartz (SiO<sub>2</sub>) || 0.013 || 0.008
|-
| Smoky quartz (SiO<sub>2</sub>) || 0.013 || 0.008
|-
| [[Anhydrite]] || 0.013 || —
|-
| [[Dolomite (mineral)|Dolomite]] || 0.013 || —
|-
| [[Morion (mineral)|Morion]] || 0.013 || —
|-
| [[Feldspar]] || 0.012 || 0.008
|-
| [[Moonstone (gemstone)|Moonstone]] || 0.012 || 0.008
|-
| [[Orthoclase]] || 0.012 || 0.008
|-
| [[Pollucite]] || 0.012 || 0.007
|-
| [[Albite]] || 0.012 || —
|-
| [[Bytownite]] || 0.012 || —
|-
| [[Emerald#Synthetic emerald|synth. Emerald]] || 0.012 || —
|-
| [[Magnesite]] (MgCO<sub>3</sub>) || 0.012 || —
|-
| [[Sanidine]] || 0.012 || —
|-
| [[Sunstone]] || 0.012 || —
|-
| synth. [[Alexandrite]] || 0.011 || —
|-
| synth. [[Sapphire]] (Al<sub>2</sub>O<sub>3</sub>) || 0.011 || —
|-
| [[Phosphophyllite]] || 0.010–0.011 || —
|-
| [[Phenakite]] || 0.01 || 0.009
|-
| [[Cancrinite]] || 0.010 || 0.008–0.009
|-
| [[Leucite]] || 0.010 || 0.008
|-
| [[Enstatite]] || 0.010 || —
|-
| [[Obsidian]] || 0.010 || —
|-
| [[Anorthite]] || 0.009–0.010 || —
|-
| [[Actinolite]] || 0.009 || —
|-
| [[Jeremejevite]] || 0.009 || —
|-
| [[Nepheline]] || 0.008–0.009 || —
|-
| [[Apophyllite]] || 0.008 || —
|-
| [[Hauyne]] || 0.008 || —
|-
| [[Natrolite]] || 0.008 || —
|-
| synth. [[Quartz]] (SiO<sub>2</sub>) || 0.008 || —
|-
| [[Aragonite]] || 0.007–0.012 || —
|-
| [[Augelite]] || 0.007 || —
|-
| [[Beryllonite]] || 0.010 || 0.007
|-
| [[Strontianite]] || 0.008–0.028 || —
|-
| [[Calcite]] (CaCO<sub>3</sub>) || 0.008–0.017 || 0.013–0.014
|-
| [[Fluorite]] (CaF<sub>2</sub>) || 0.007 || 0.004
|-
| [[Tremolite]] || 0.006–0.007 || —
|}

In the [[technical terminology]] of [[gemology]], ''dispersion'' is the difference in the refractive index of a material at the B and G (686.7&nbsp;[[Nanometre|nm]] and 430.8&nbsp;nm) or C and F (656.3&nbsp;nm and 486.1&nbsp;nm) [[Fraunhofer lines|Fraunhofer wavelengths]], and is meant to express the degree to which a prism cut from the [[gemstone]] demonstrates "fire".  Fire is a colloquial term used by gemologists to describe a gemstone's dispersive nature or lack thereof. Dispersion is a material property. The amount of fire demonstrated by a given gemstone is a function of the gemstone's facet angles, the polish quality, the lighting environment, the material's refractive index, the saturation of color, and the orientation of the viewer relative to the gemstone.<ref name="b1" /><ref>{{cite web |title=What is gemstone dispersion? |website=International Gem Society (GemSociety.org) |url=http://www.gemsociety.org/article/gemstone-dispersion/ |access-date=2015-03-09}}</ref>

== In imaging ==
In photographic and microscopic lenses, dispersion causes [[chromatic aberration]], which causes the different colors in the image not to overlap properly. Various techniques have been developed to counteract this, such as the use of [[achromat]]s, multielement lenses with glasses of different dispersion. They are constructed in such a way that the chromatic aberrations of the different parts cancel out.

== Pulsar emissions ==

[[Pulsar]]s are spinning neutron stars that emit [[pulse]]s at very regular intervals ranging from milliseconds to seconds. Astronomers believe that the pulses are emitted simultaneously over a wide range of frequencies. However, as observed on Earth, the components of each pulse emitted at higher radio frequencies arrive before those emitted at lower frequencies. This dispersion occurs because of the ionized component of the [[interstellar medium]], mainly the free electrons, which make the group velocity frequency-dependent. The extra delay added at a frequency {{mvar|ν}} is
: <math>t = k_\text{DM} \cdot \left(\frac{\text{DM}}{\nu^2}\right),</math>
where the dispersion constant ''k''<sub>DM</sub> is given by<ref>"Single-Dish Radio Astronomy: Techniques and Applications", ASP Conference Proceedings, vol.&nbsp;278. Edited by Snezana Stanimirovic, [[Daniel R. Altschuler|Daniel Altschuler]], Paul Goldsmith, and Chris Salter. {{ISBN|1-58381-120-6}}. San Francisco: Astronomical Society of the Pacific, 2002, p.&nbsp;251–269.</ref>
: <math>k_\text{DM} = \frac{e^2}{2 \pi m_\text{e} c} \approx 4.149~\text{GHz}^2\,\text{pc}^{-1}\,\text{cm}^3\,\text{ms},</math>
and the '''dispersion measure''' (DM) is the column density of free electrons ([[total electron content]]){{snd}} i.e. the [[number density]] of electrons ''n''<sub>e</sub> integrated along the path traveled by the photon from the pulsar to the Earth{{snd}} and is given by
: <math>\text{DM} = \int_0^d n_e\,dl</math>
with units of [[parsec]]s per cubic centimetre (1&nbsp;pc/cm<sup>3</sup> = 30.857{{e|21}}&nbsp;m<sup>−2</sup>).<ref>Lorimer, D. R., and Kramer, M., ''Handbook of Pulsar Astronomy'', vol.&nbsp;4 of Cambridge Observing Handbooks for Research Astronomers ([[Cambridge University Press]], Cambridge, U.K.; New York, U.S.A, 2005), 1st edition.</ref>

Typically for astronomical observations, this delay cannot be measured directly, since the emission time is unknown. What ''can'' be measured is the difference in arrival times at two different frequencies. The delay Δ''t'' between a high-frequency {{mvar|ν}}<sub>hi</sub> and a low-frequency {{mvar|ν}}<sub>lo</sub> component of a pulse will be
: <math>\Delta t = k_\text{DM} \cdot \text{DM} \cdot \left( \frac{1}{\nu_\text{lo}^2} - \frac{1}{\nu_\text{hi}^2} \right).</math>

Rewriting the above equation in terms of Δ''t'' allows one to determine the DM by measuring pulse arrival times at multiple frequencies. This in turn can be used to study the interstellar medium, as well as allow observations of pulsars at different frequencies to be combined.

== See also ==
{{div col|colwidth=22em}}
* [[Calculation of glass properties]] incl. dispersion
* [[Cauchy's equation]]
* [[Dispersion relation]]
* [[Fast radio burst]] (astronomy)
* [[Fluctuation theorem]]
* [[Green–Kubo relations]]
* [[Group delay]]
* [[Intramodal dispersion]]
* [[Kramers–Kronig relations]]
* [[Linear response function]]
* [[Multiple-prism dispersion theory]]
* [[Sellmeier equation]]
* [[Ultrashort pulse]]
* [[Virtually imaged phased array]]
{{div col end}}

== References ==
{{Reflist}}

== External links ==
{{Commons}}
* [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page Dispersive Wiki] {{Webarchive|url=https://web.archive.org/web/20170723091053/http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page |date=2017-07-23 }} – discussing the mathematical aspects of dispersion.
* [http://www.rp-photonics.com/dispersion.html Dispersion] – Encyclopedia of Laser Physics and Technology
* [http://qed.wikina.org/dispersion/ Animations demonstrating optical dispersion] by QED
* [http://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_2/chromatic_dispersion/ Interactive webdemo for chromatic dispersion] Institute of Telecommunications, University of Stuttgart

{{Glass science}}

{{Authority control}}

[[Category:Glass physics]]
[[Category:Optical phenomena]]