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Dispersion

Dispersion (optics) 1

Dispersion (optics)This article is about dispersion of waves in optics. For other forms of dispersion, see Dispersion (disambiguation).

In a dispersive prism, material dispersion (awavelength-dependent refractive index) causes

different colors to refract at different angles,splitting white light into a rainbow.

A compact fluorescent lamp seen through anAmici prism

In optics, dispersion is the phenomenon in which the phase velocity ofa wave depends on its frequency, or equivalently when the groupvelocity depends on the frequency. Media having such a property aretermed dispersive media. Dispersion is sometimes called chromaticdispersion to emphasize its wavelength-dependent nature, orgroup-velocity dispersion (GVD) to emphasize the role of the groupvelocity. Dispersion is most often described for light waves, but it mayoccur for any kind of wave that interacts with a medium or passesthrough an inhomogeneous geometry (e.g., a waveguide), such assound waves. A material's dispersion for optical wavelengths ismeasured by its Abbe number, V, with low Abbe numberscorresponding to strong dispersion.

Examples of dispersion

The most familiar example of dispersion is probably a rainbow, inwhich dispersion causes the spatial separation of a white light intocomponents of different wavelengths (different colors). However,dispersion also has an effect in many other circumstances: for example,GVD causes pulses to spread in optical fibers, degrading signals overlong distances; also, a cancellation between group-velocity dispersionand nonlinear effects leads to soliton waves.

Sources of dispersion

There are generally two sources of dispersion: material dispersion and waveguide dispersion. Material dispersioncomes from a frequency-dependent response of a material to waves. For example, material dispersion leads toundesired chromatic aberration in a lens or the separation of colors in a prism. Waveguide dispersion occurs whenthe speed of a wave in a waveguide (such as an optical fiber) depends on its frequency for geometric reasons,independent of any frequency dependence of the materials from which it is constructed. More generally,"waveguide" dispersion can occur for waves propagating through any inhomogeneous structure (e.g., a photoniccrystal), whether or not the waves are confined to some region. In general, both types of dispersion may be present,although they are not strictly additive. Their combination leads to signal degradation in optical fibers used fortelecommunications, because the varying delay in arrival time between different components of a signal "smears out"the signal in time.

Dispersion (optics) 2

Material dispersion in optics

The variation of refractive index vs. vacuum wavelength for various glasses. Thewavelengths of visible light are shaded in yellow.

Influences of selected glass component additions on the mean dispersion of a specificbase glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm (red))[1]

Material dispersion can be a desirableor undesirable effect in opticalapplications. The dispersion of light byglass prisms is used to constructspectrometers and spectroradiometers.Holographic gratings are also used, asthey allow more accurate discriminationof wavelengths. However, in lenses,dispersion causes chromatic aberration,an undesired effect that may degradeimages in microscopes, telescopes andphotographic objectives.

The phase velocity, v, of a wave in agiven uniform medium is given by

where c is the speed of light in avacuum and n is the refractive index ofthe medium.

In general, the refractive index is somefunction of the frequency f of the light,thus n = n(f), or alternatively, withrespect to the wave's wavelength n =n(λ). The wavelength dependence of amaterial's refractive index is usuallyquantified by its Abbe number or itscoefficients in an empirical formulasuch as the Cauchy or Sellmeierequations.

Because of the Kramers–Kronigrelations, the wavelength dependence ofthe real part of the refractive index isrelated to the material absorption,described by the imaginary part of therefractive index (also called theextinction coefficient). In particular, for non-magnetic materials (μ = μ0), the susceptibility that appears in theKramers–Kronig relations is the electric susceptibility .

The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrumby a prism. From Snell's law it can be seen that the angle of refraction of light in a prism depends on the refractiveindex of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the lightis refracted by will also vary with wavelength, causing an angular separation of the colors known as angulardispersion.

For visible light, refraction indices n of most transparent materials (e.g., air, glasses) decrease with increasingwavelength λ:

Dispersion (optics) 3

or alternatively:

In this case, the medium is said to have normal dispersion. Whereas, if the index increases with increasingwavelength (which is typically the caseWikipedia:Citation needed for X-rays), the medium is said to haveanomalous dispersion.At the interface of such a material with air or vacuum (index of ~1), Snell's law predicts that light incident at anangle θ to the normal will be refracted at an angle arcsin(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 and phase velocityAnother consequence of dispersion manifests itself as a temporal effect. The formula v = c / n calculates the phasevelocity of a wave; this is the velocity at which the phase of any one frequency component of the wave willpropagate. This is not the same as the group velocity of the wave, that is the rate at which changes in amplitude(known as the envelope of the wave) will propagate. For a homogeneous medium, the group velocity vg is related tothe phase velocity v by (here λ is the wavelength in vacuum, not in the medium):

The group velocity vg is often thought of as the velocity at which energy or information is conveyed along the wave.In most cases this is true, and the group velocity can be thought of as the signal velocity of the waveform. In someunusual circumstances, called cases of anomalous dispersion, the rate of change of the index of refraction withrespect to the wavelength changes sign (becoming positive), in which case it is possible for the group velocity toexceed the speed of light (vg > c). Anomalous dispersion occurs, for instance, where the wavelength of the light isclose to an absorption resonance of the medium. When the dispersion is anomalous, however, group velocity is nolonger an indicator of signal velocity. Instead, a signal travels at the speed of the wavefront, which is c irrespectiveof the index of refraction.[2] Recently, it has become possible to create gases in which the group velocity is not onlylarger than the speed of light, but even negative. In these cases, a pulse can appear to exit a medium before it enters.Even in these cases, however, a signal travels at, or less than, the speed of light, as demonstrated by Stenner, et al.The group velocity itself is usually a function of the wave's frequency. This results in group velocity dispersion(GVD), which causes a short pulse of light to spread in time as a result of different frequency components of thepulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter (again, thisformula is for a uniform medium only):

If D is less than zero, the medium is said to have positive dispersion. If D is greater than zero, the medium hasnegative dispersion. If a light pulse is propagated through a normally dispersive medium, the result is the higherfrequency components travel slower than the lower frequency components. The pulse therefore becomes positivelychirped, or up-chirped, increasing in frequency with time. Conversely, if a pulse travels through an anomalouslydispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negativelychirped, or down-chirped, decreasing in frequency with time.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

Dispersion (optics) 4

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 μm in silica fibres), so pulses at this wavelength suffer minimal spreading from dispersion—inpractice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifiesother nonlinear effects (such as four wave mixing). Another possible option is to use soliton pulses in the regime ofanomalous dispersion, a form of optical pulse which uses a nonlinear optical effect to self-maintain itsshape—solitons have the practical problem, however, that they require a certain power level to be maintained in thepulse for the nonlinear effect to be of the correct strength. Instead, the solution that is currently used in practice is toperform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion sothat the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as self-phasemodulation, which interact with dispersion to make it very difficult to undo.Dispersion control is also important in lasers that produce short pulses. The overall dispersion of the opticalresonator is a major factor in determining the duration of the pulses emitted by the laser. A pair of prisms can bearranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the lasermedium. Diffraction gratings can also be used to produce dispersive effects; these are often used in high-power laseramplifier systems. Recently, an alternative to prisms and gratings has been developed: chirped mirrors. Thesedielectric mirrors are coated so that different wavelengths have different penetration lengths, and therefore differentgroup delays. The coating layers can be tailored to achieve a net negative dispersion.

Dispersion in waveguidesOptical fibers, which are used in telecommunications, are among the most abundant types of waveguides. Dispersionin these fibers is one of the limiting factors that determine how much data can be transported on a single fiber.The transverse modes for waves confined laterally within a waveguide generally have different speeds (and fieldpatterns) depending upon their frequency (that is, on the relative size of the wave, the wavelength) compared to thesize of the waveguide.In general, for a waveguide mode with an angular frequency ω(β) at a propagation constant β (so that theelectromagnetic fields in the propagation direction (z) oscillate proportional to ), the group-velocitydispersion parameter D is defined as:[3]

where is the vacuum wavelength and is the group velocity. This formula generalizesthe one in the previous section for homogeneous media, and includes both waveguide dispersion and materialdispersion. The reason for defining the dispersion in this way is that |D| is the (asymptotic) temporal pulse spreading

per unit bandwidth per unit distance travelled, commonly reported in ps / nm km for optical fibers.A similar effect due to a somewhat different phenomenon is modal dispersion, caused by a waveguide havingmultiple modes at a given frequency, each with a different speed. A special case of this is polarization modedispersion (PMD), which comes from a superposition of two modes that travel at different speeds due to randomimperfections that break the symmetry of the waveguide. Modal dispersion can also be used to generate large,tunable group delay dispersion in a compact footprint using chromo-modal dispersion.[4]

Dispersion (optics) 5

Higher-order dispersion over broad bandwidthsWhen a broad range of frequencies (a broad bandwidth) is present in a single wavepacket, such as in an ultrashortpulse or a chirped pulse or other forms of spread spectrum transmission, it may not be accurate to approximate thedispersion by a constant over the entire bandwidth, and more complex calculations are required to compute effectssuch 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.[5] These terms are simply a Taylor series expansion of thedispersion relation of the medium or waveguide around some particular frequency. Their effects can becomputed via numerical evaluation of Fourier transforms of the waveform, via integration of higher-order slowlyvarying envelope approximations, by a split-step method (which can use the exact dispersion relation rather than aTaylor series), or by direct simulation of the full Maxwell's equations rather than an approximate envelope equation.

Dispersion in gemology

Dispersion values of minerals

Name B–G C–F

Cinnabar (HgS) 0.40 –

Synth. rutile 0.330 0.190

Rutile (TiO2) 0.280 0.120–0.180

Anatase (TiO2) 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 (TiO2) 0.131 0.12–1.80

Zincite (ZnO) 0.127 –

Linobate 0.13 0.075

Synth. moissanite (SiC) 0.104 –

Cassiterite (SnO2) 0.071 0.035

Zirconia (ZrO2) 0.060 0.035

Powellite (CaMoO4) 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

Dispersion (optics) 6

Flint glass 0.041 –

Hyacinth 0.039 –

Jargoon 0.039 –

Starlite 0.039 –

Zircon (ZrSiO4) 0.039 0.022

GGG 0.038 0.022

Scheelite 0.038 0.026

Dioptase 0.036 0.021

Whewellite 0.034 –

Alabaster 0.033 –

Gypsum 0.033 0.008

Epidote 0.03 0.012–0.027

Achroite 0.017 –

Cordierite 0.017 0.009

Danburite 0.017 0.009

Dravite 0.017 –

Elbaite 0.017 –

Herderite 0.017 0.008–0.009

Hiddenite 0.017 0.010

Indicolite 0.017 –

Liddicoatite 0.017 –

Kunzite 0.017 0.010

Rubellite 0.017 0.008–0.009

Schorl 0.017 –

Scapolite 0.017 –

Spodumene 0.017 0.010

Tourmaline 0.017 0.009–0.011

Verdelite 0.017 –

Andalusite 0.016 0.009

Baryte (BaSO4) 0.016 0.009

Euclase 0.016 0.009

Alexandrite 0.015 0.011

Chrysoberyl 0.015 0.011

Hambergite 0.015 0.009–0.010

Phenakite 0.01 0.009

Rhodochrosite 0.015 0.010–0.020

Sillimanite 0.015 0.009–0.012

Smithsonite 0.014–0.031 0.008–0.017

Amblygonite 0.014–0.015 0.008

Dispersion (optics) 7

Aquamarine 0.014 0.009–0.013

Beryl 0.014 0.009–0.013

Brazilianite 0.014 0.008

Celestine 0.014 0.008

Goshenite 0.014 –

Heliodor 0.014 0.009–0.013

Morganite 0.014 0.009–0.013

Pyroxmangite 0.015 –

Synth. scheelite 0.015 –

Dolomite 0.013 –

Magnesite (MgCO3) 0.012 –

Synth. emerald 0.012 –

Synth. alexandrite 0.011 –

Synth. sapphire (Al2O3) 0.011 –

Phosphophyllite 0.010–0.011 –

Enstatite 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 (SiO2) 0.008 –

Aragonite 0.007–0.012 –

Augelite 0.007 –

Tanzanite 0.030 0.011

Thulite 0.03 0.011

Zoisite 0.03 –

YAG 0.028 0.015

Almandine 0.027 0.013–0.016

Hessonite 0.027 0.013–0.015

Spessartine 0.027 0.015

Uvarovite 0.027 0.014–0.021

Willemite 0.027 –

Pleonaste 0.026 –

Rhodolite 0.026 –

Boracite 0.024 0.012

Cryolite 0.024 –

Dispersion (optics) 8

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

Clinozoisite 0.019 0.011–0.014

Labradorite 0.019 0.010

Axinite 0.018–0.020 0.011

Ekanite 0.018 0.012

Kornerupine 0.018 0.010

Corundum (Al2O3) 0.018 0.011

Rhodizite 0.018 –

Ruby (Al2O3) 0.018 0.011

Sapphire (Al2O3) 0.018 0.011

Sinhalite 0.018 0.010

Sodalite 0.018 0.009

Synth. corundum 0.018 0.011

Diopside 0.018–0.020 0.01

Emerald 0.014 0.009–0.013

Topaz 0.014 0.008

Amethyst (SiO2) 0.013 0.008

Anhydrite 0.013 –

Apatite 0.013 0.010

Apatite 0.013 0.008

Aventurine 0.013 0.008

Citrine 0.013 0.008

Morion 0.013 –

Prasiolite 0.013 0.008

Quartz (SiO2) 0.013 0.008

Smoky quartz (SiO2) 0.013 0.008

Rose quartz (SiO2) 0.013 0.008

Albite 0.012 –

Bytownite 0.012 –

Feldspar 0.012 0.008

Moonstone 0.012 0.008

Dispersion (optics) 9

Orthoclase 0.012 0.008

Pollucite 0.012 0.007

Sanidine 0.012 –

Sunstone 0.012 –

Beryllonite 0.010 0.007

Cancrinite 0.010 0.008–0.009

Leucite 0.010 0.008

Obsidian 0.010 –

Strontianite 0.008–0.028 –

Calcite (CaCO3) 0.008–0.017 0.013–0.014

Fluorite (CaF2) 0.007 0.004

Hematite 0.500 –

Synth. cassiterite (SnO2) 0.041 –

Gahnite 0.019–0.021 –

Datolite 0.016 –

Tremolite 0.006–0.007 –

In the technical terminology of gemology, dispersion is the difference in the refractive index of a material at the Band G (686.7 nm and 430.8 nm) or C and F (656.3 nm and 486.1 nm) Fraunhofer wavelengths, and is meant toexpress the degree to which a prism cut from the gemstone shows "fire", or color. Dispersion is a material property.Fire depends on the dispersion, the cut angles, the lighting environment, the refractive index, and the viewer.

Dispersion in imagingIn photographic and microscopic lenses, dispersion causes chromatic aberration, which causes the different colors inthe image not to overlap properly. Various techniques have been developed to counteract this, such as the use ofachromats, multielement lenses with glasses of different dispersion. They are constructed in such a way that thechromatic aberrations of the different parts cancel out.

Dispersion in pulsar timingPulsars are spinning neutron stars that emit pulses at very regular intervals ranging from milliseconds to seconds.Astronomers believe that the pulses are emitted simultaneously over a wide range of frequencies. However, asobserved on Earth, the components of each pulse emitted at higher radio frequencies arrive before those emitted atlower frequencies. This dispersion occurs because of the ionized component of the interstellar medium, mainly thefree electrons, which make the group velocity frequency dependent. The extra delay added at a frequency is

where the dispersion constant is given by

,

and the dispersion measure DM is the column density of electrons — i.e. the number density of electrons (electrons/cm3) integrated along the path traveled by the photon from the pulsar to the Earth — and is given by

Dispersion (optics) 10

with units of parsecs per cubic centimetre (1pc/cm3 = 30.857×1021 m−2).[6]

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 between a highfrequency and a low frequency component of a pulse will be

Re-writing the above equation in terms of DM allows one to determine the DM by measuring pulse arrival times atmultiple frequencies. This in turn can be used to study the interstellar medium, as well as allow for observations ofpulsars at different frequencies to be combined.

References[1] Calculation of the Mean Dispersion of Glasses (http:/ / glassproperties. com/ dispersion/ )[2][2] Brillouin, Léon. Wave Propagation and Group Velocity. (Academic Press: San Diego, 1960). See esp. Ch. 2 by A. Sommerfeld.[3] Rajiv Ramaswami and Kumar N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press: London 1998).[4][4] E.D. Diebold et al., "Giant tunable optical dispersion using chromo-modal excitation of a multimode waveguide," Optics Express 19 (24)

2011[5] Chromatic Dispersion (http:/ / www. rp-photonics. com/ chromatic_dispersion. html), Encyclopedia of Laser Physics and Technology (Wiley,

2008).[6] Lorimer, D.R., and Kramer, M., Handbook of Pulsar Astronomy, vol. 4 of Cambridge Observing Handbooks for Research Astronomers,

(Cambridge University Press, Cambridge, U.K.; New York, U.S.A, 2005), 1st edition.

External links• Dispersive Wiki (http:/ / wiki. math. toronto. edu/ DispersiveWiki/ index. php/ Main_Page) – discussing the

mathematical aspects of dispersion.• Dispersion (http:/ / www. rp-photonics. com/ dispersion. html) – Encyclopedia of Laser Physics and Technology• Animations demonstrating optical dispersion (http:/ / qed. wikina. org/ dispersion/ ) by QED• Interactive webdemo for chromatic dispersion (http:/ / webdemo. inue. uni-stuttgart. de/ webdemos/ 02_lectures/

uebertragungstechnik_2/ chromatic_dispersion/ ) Institute of Telecommunications, University of Stuttgart

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Article Sources and ContributorsDispersion (optics)  Source: http://en.wikipedia.org/w/index.php?oldid=606387550  Contributors: Aarchiba, Acather96, Adoniscik, Afluegel, Alejo2083, Avoided, AxelBoldt, BJJOYCE87,BenFrantzDale, Bernael, Betacommand, Bhadani, Brews ohare, Catslash, Charles Matthews, Cheolsoo, Colliand, Colonies Chris, Coneslayer, Corrigendas, Crowsnest, D-Kuru, David 4d, DavidLegrand, David Shay, Diannaa, DrBob, Dude fri13, EdH, Ediebold, Emijrp, Escape Orbit, Fgnievinski, Flying Oaf, Fred Bauder, Fred12121212, Germendax, Gfutia, Gisling, Glenn, Goudzovski,Hankwang, Headbomb, Incnis Mrsi, Interferometrist, Iridescent, Isarra (HG), J S Lundeen, J04n, Jasper, Jatosado, Jim1138, JimVC3, Jimp, Jitse Niesen, Johnpseudo, Kanags, Keegan, Khazar2,Kku, KuduIO, Laurascudder, Lestrade, LiborX, Looxix, LouScheffer, LucasVB, MK8, Materialscientist, Mejor Los Indios, Michael Hardy, Mikko Paananen, Mild Bill Hiccup, Mpfiz,Mpolyanskiy, NathanielEP, NawlinWiki, Nbarth, Ndrasen, NewEnglandYankee, Nk, Noegenesis, Paul Pot, Pedrose, Persiana, Pfalstad, Pgabolde, Pgan002, Philbull, Pratyya Ghosh, PwnZ0r4t3r,Rabarberski, Reatlas, RedWolf, Rjwilmsi, Robert P. O'Shea, Saintrain, Salsa Shark, Saperaud, ScAvenger, Scarian, Siroxo, Speed8ump, Spigget, Srleffler, Steven Zhang, Stevenj,SunCountryGuy01, Sverdrup, Swpb, Tanaats, Tarquin, Teles, The Thing That Should Not Be, Thewikiguru1, Thrapper, Torsionalmetric, Ulflund, UnHoly, Vasiľ, Vilisvir, Waxigloo, XXAV3NG3R Xx, Yurik, ZoeB, Zzyzx11, Кстати, 208 anonymous edits

Image Sources, Licenses and ContributorsFile:Prism rainbow schema.png  Source: http://en.wikipedia.org/w/index.php?title=File:Prism_rainbow_schema.png  License: GNU Free Documentation License  Contributors: Adoniscik,HenkvD, Joanjoc, Ranveig, Saperaud, Sevela.p, Suidroot, Teebeutel, Ustas, Ævar Arnfjörð Bjarmason, 1 anonymous editsFile:Light dispersion of a compact fluorescent lamp seen through an Amici direct-vision prism PNr°0114.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:Light_dispersion_of_a_compact_fluorescent_lamp_seen_through_an_Amici_direct-vision_prism_PNr°0114.jpg  License: unknown  Contributors:D-KuruFile:Dispersion-curve.png  Source: http://en.wikipedia.org/w/index.php?title=File:Dispersion-curve.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors:Original uploader was DrBob at en.wikipediaFile:Spidergraph Dispersion.GIF  Source: http://en.wikipedia.org/w/index.php?title=File:Spidergraph_Dispersion.GIF  License: Public Domain  Contributors: Afluegel, BenFrantzDale

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