dispersion (optics) - wikipedia, the free encyclopedia

7/28/2019 Dispersion (Optics) - Wikipedia, The Free Encyclopedia

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In a dispersive prism, material

dispersion (a wavelength-dependent

refractive index) causes different

colorsto refract at different angles,

splitting white light into a rainbow.

A compact fluorescent lampseen

through an Amici prism

The variation of refractive index vs. vacuum wavelength

for various glasses.The wavelengths of visiblelight areshaded in red.

Influences of selected glass component additions on the

mean dispersion of a specific base glass (nF

valid for =

486 nm(blue), nC

valid for = 656 nm(red))[2]

ispersion (optics)m Wikipedia, the free encyclopedia

optics, dispersion is the phenomenon in which the phase velocityof a wave depends on its frequency,[1] or alternatively when the group velocity

ends on the frequency. Media having such a propertyare termed dispersive media. Dispersion is sometimes called chromatic dispersion to

phasize its wavelength-dependent nature, or group-velocity dispersion (GVD) to emphasize the role of the group velocity. Dispersion is most

en described for light waves, but it may occur for any kind of wave that interacts with a medium or passes through an inhomogeneous geometry

., a waveguide), such as sound waves. A material's dispersion is measured by its Abbe number, V, with low Abbe numbers corresponding to

ng dispersion.

Contents

1 Examples of dispersion2 Sources of dispersion

3 Material dispersion in optics

4 Group and phase velocity

5 Dispersion in waveguides

6 Higher-order dispersion over broad bandwidths

7 Dispersion in gemology

8 Dispersion in imaging

9 Dispersion in pulsar timing

10 See also

11 References

12 External links

xamples of dispersion

e most familiar example of dispersion is probably a rainbow, in which dispersion causes the spatial separation of a white light into componentsdifferent wavelengths (different colors). However, dispersion also has an effect in many other circumstances: for example, GVD causes pulses to spread in optical fibers, degrading

nals over long distances; also, a cancellation between group-velocity dispersion and nonlinear effects leads to soliton waves.

ources of dispersion

ere are generally two sources of dispersion: material dispersion and waveguide dispersion. Material dispersion comes from a frequency-dependent response of a material to waves. For

mple, material dispersion leads to undesired chromatic aberration in a lens or the separation of colors in a prism. Waveguide dispersion occurs when the speed of a wave in a

veguide (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

erally, "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.

eneral, both types of dispersion may be present, although they are not strictly additive. Their combination leads to signal degradation in optical fibers for telecommunications, because

varying delay in arrival time between different components of a signal "smears out" the signal in time.

aterial dispersion in optics

terial dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used tostruct spectrometers and spectroradiometers. Holographic gratings are also used, as they allow more accurate discrimination of

velengths. However, in lenses, dispersion causes chromatic aberration, an undesired effect that may degrade images in

roscopes, telescopes and photographic objectives.

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

ere c is the speed of light in a vacuum and n is the refractive index of the medium.

general, the refractive index is some function of the frequency fof the light, thus n = n(f), or alternatively, with respect to the

ve's wavelength n = n(). The wavelength dependence of a material's refractive index is usually quantified by its Abbe number or

coefficients in an empirical formula such as the Cauchy or Sellmeier equations.

ause of the KramersKronig relations, the wavelength dependence of the real part of the refractive index is related to the

erial absorption, described by the imaginary part of the refractive index (also called the extinction coefficient). In particular, for-magnetic materials ( =

0), the susceptibility that appears in the KramersKronig relations is the electric susceptibility

.

e most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. From

ll'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

refractive index varies with wavelength, it follows that the angle that the light is refracted by will also vary with wavelength,

sing an angular separation of the colors known as angular dispersion.

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

alternatively:

his case, the medium is said to have normal dispersion. Whereas, if the index increases with increasing wavelength (which is

ically the case for X-rays), the medium is said to have anomalous dispersion.

he interface of such a material with air or vacuum (index of ~1), Snell's law predicts that light incident at an angle to the normal will be refracted at an angle arcsin(sin()/n). Thus,

e light, with a higher refractive index, will be bent more stronglythan red light, resulting in the well-known rainbow pattern.


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Dispersion values of minerals[8]

Name BG CF

Cinnabar (HgS) 0.40

Synth. rutile 0.330 0.190

roup and phase velocity

other consequence of dispersion manifests itself as a temporal effect. The formula v = c /n calculates the phase velocity of a wave; this is the velocity at which the phase of any one

quency component of the wave will propagate. 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

ve) will propagate. For a homogeneous medium, the group velocity vg

is related to the phase velocityby (here is the wavelength in vacuum, not in the medium):

e group velocity vg

is often thought of as the velocityat which energy or information is conveyed along the wave. In most cases this is true, and the group velocity can be thought of as

signal velocity of the waveform. In some unusual circumstances, called cases of anomalous dispersion, the rate of change of the index of refraction with respect to the wavelength

nges sign, in which case it is possible for the group velocity to exceed the speed of light (vg

> c). Anomalous dispersion occurs, for instance, where the wavelength of the light is close

n absorption resonance of the medium. When the dispersion is anomalous, however, group velocityis no longer an indicator of signal velocity. Instead, a signal travels at the speed of

wavefront, which is c irrespective of the index of refraction.[3]

Recently, it has become possible to create gases in which the group velocity is not only larger than the speed of light, butn negative. In these cases, a pulse can appear to exit a medium before it enters.[4] Even in these cases, however, a signal travels at, or less than, the speed of light, as demonstrated by

nner, et al.[5]

e group velocityitself 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

erent frequency components of the pulse travelling at different velocities. GVD is often quantified as the group delay dispersion parameter(again, this formula is for a uniform

dium only):

D is less than zero, the medium is said to have positive dispersion. IfD is greater than zero, the medium has negative dispersion. If a light pulse is propagated through a normally

persive medium, the result is the higher frequency components travel slower than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped,

reasing 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

omes negatively chirped, or down-chirped, decreasing in frequency with time.

e result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systemsed 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

gth 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

., around 1.31.5 m in silica fibres), so pulses at this wavelength suffer minimal spreading from dispersionin practice, however, this approach causes more problems than it solves

ause zero GVD unacceptably amplifies other nonlinear effects (such as four wave mixing). Another possible option is to use soliton pulses in the regime of anomalous dispersion, a

m of optical pulse which uses a nonlinear optical effect to self-maintain its shapesolitons have the practical problem, however, that they require a certain power level to be maintained

he 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

er 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

ract with dispersion to make it very difficult to undo.

persion control is also important in lasers that produce short pulses. The overall dispersion of the optical resonator is a major factor in determining the duration of the pulses emitted by

laser. A pair of prisms can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. Diffraction gratings can

o 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 mirrors.

ese 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

ative dispersion.

spersion in waveguidesical fibers, which are used in telecommunications, are among the most abundant types of waveguides. Dispersion in these fibers is one of the limiting factors that determine how much

a can be transported on a single fiber.

e transverse modes for waves confined laterally within a waveguide generally have different speeds (and field patterns) depending upon their frequency (that is, on the relative size of

wave, the wavelength) compared to the size of the waveguide.

general, for a waveguide mode with an angular frequency () at a propagation constant (so that the electromagnetic fields in the propagation direction (z) oscillate proportional to

), the group-velocity dispersion parameter D is defined as:[6]

ere is the vacuumwavelength and is the group velocity. This formula generalizes the one in the previous section for homogeneous media, and includes

h waveguide dispersion and material dispersion. The reason for defining the dispersion in this way is that |D| is the (asymptotic) temporal pulse spreading per unit bandwidth

unit distance travelled, commonly reported in ps / nm km for optical fibers.

imilar effect due to a somewhat different phenomenon is modal dispersion, caused by a waveguide having multiple modes at a given frequency, each with a different speed. A special

e of this is polarization mode dispersion (PMD), which comes from a superposition of two modes that travel at different speeds due to random imperfections that break the symmetry of

waveguide.

igher-order dispersion over broad bandwidths

en a broad range of frequencies (a broad bandwidth) is present in a single wavepacket, such as in an ultrashort pulse or a chirped pulse or other forms of spread spectrum transmission,

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.

articular, the dispersion parameter D defined above is obtained from only one derivative of the group velocity. Higher derivatives are known as higher-order dispersion.[7] These

ms are simply a Taylor series expansion of the dispersion relation of the medium or waveguide around some particular frequency. Their effects can be computed via numerical

luation of Fourier transforms of the waveform, via integration of higher-order slowly varying envelope approximations, by a split-step method (which can use the exact dispersion

tion rather than a Taylor series), or by direct simulation of the full Maxwell's equations rather than an approximate envelope equation.

spersion in gemology

he technical terminology of gemology, dispersion is the difference in the refractive index of a material at the B and G (686.7 nm

430.8 nm) or C and F (653.3 nm and 486.1 nm) Fraunhofer wavelengths, and is meant to express the degree to which a prism cut

m the gemstone shows "fire", or color. Dispersion is a material property. Fire depends on the dispersion, the cut angles, the

hting environment, the refractive index, and the viewer.[8]


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Rutile (TiO2) 0.280 0.1200.180

Anatase (TiO2) 0.2130.259

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 0.121.80

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 0.0330.050

Titanite 0.051 0.0190.038

Benitoite 0.046 0.026

Anglesite 0.044 0.025

Diamond (C) 0.044 0.025

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 0.0120.027

Achroite 0.017

Cordierite 0.017 0.009

Danburite 0.017 0.009

Dravite 0.017

Elbaite 0.017

Herderite 0.017 0.0080.009

Hiddenite 0.017 0.010

Indicolite 0.017

Liddicoatite 0.017

Kunzite 0.017 0.010

Rubellite 0.017 0.0080.009

Schorl 0.017

Scapolite 0.017

Spodumene 0.017 0.010

Tourmaline 0.017 0 .0090.011

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 .0090.010Phenakite 0.01 0.009

Rhodochrosite 0.015 0.0100.020

Sillimanite 0.015 0.0090.012

Smithsonite 0.0140.031 0.0080.017

Amblygonite 0.0140.015 0.008

Aquamarine 0.014 0.0090.013

Beryl 0.014 0.0090.013

spersion in imaging

photographic and microscopic lenses, dispersion causes chromatic aberration, which causes the different colors in the image not to

rlap properly. Various techniques have been developed to counteract this, such as the use of achromats, multielement lenses with

sses of different dispersion. They are constructed in such a way that the chromatic aberrations of the different parts cancel out.

spersion in pulsar timing

sars are spinning neutron stars that emit pulses at very regular intervals ranging from milliseconds to seconds. Astronomers

eve that the pulses are emitted simultaneously over a wide range of frequencies. However, as observed on Earth, the components

ach pulse emitted at higher radio frequencies arrive before those emitted at lower frequencies. This dispersion occurs because of

ionized component of the interstellar medium, mainly the free electrons, which makes the group velocity frequency dependent.

e extra delay added at a frequency is

ere the dispersion constant is given by

,

the dispersion measure DMis the column density of electrons i.e. the number density of electrons (electrons/cm3)

grated along the path traveled by the photon from the pulsar to the Earth and is given by

h units of parsecs per cubic centimetre (1pc/cm3 = 30.8571021 m2).[9]

pically for astronomical observations, this delay cannot be measured directly, since the emission time is unknown. What can be

asured is the difference in arrival times at two different frequencies. The delay between a high frequency and a low

quency component of a pulse will be

writing the above equation in terms ofDMallows one to determine the DMby measuring pulse arrival times at multiple

quencies. This in turn can be used to study the interstellar medium, as well as allow for observations of pulsars at different

quencies to be combined.

e also

Dispersion relation

Sellmeier equationCauchy's equation

Abbe number

KramersKronig relations

Group delay

Calculation of glass properties incl.

dispersionLinear response function

GreenKubo relations

Fluctuation theorem

Multiple-prism dispersion theory

Ultrashort pulse

Intramodal dispersion

eferences

1. ^ Born,Max; Wolf, Emil (October 1999). Principles of Optics.

Cambridge: Cambridge University Press. pp. 1424.

ISBN 0-521-64222-1.

2. ^ Calculation of the Mean Dispersion of Glasses (http://

glassproperties.com/dispersion/)

3. ^ Brillouin,Lon.Wave Propagationand GroupVelocity.(Academic

Press: San Diego,1960). See esp.Ch. 2 by A.Sommerfeld.

4. ^ Wang, L.J., Kuzmich,A., and Dogariu, A. (2000). "Gain-assisted

superluminal light propagation".Nature 406 (6793): 277.

Bibcode:2000Natur .406..277W (http://adsabs.harvard.edu/abs/2000Natur.406..277W).doi:10.1038/35018520 (http://

dx.doi.org/10.1038%2F35018520).

5. ^ Stenner, M. D., Gauthier,D. J., and Neifeld,M. A.(2003). "The

speed of information in a 'fast-light'optical medium".Nature 425

(6959) : 6958. Bibcode:2003Natur .425..695S (http://

adsabs.harvard.edu/abs/2003Natur.425..695S). doi:10.1038/

nature02016 (http://dx.doi.org/10.1038%2Fnature02016).

PMID 14562097 (//www.ncbi.nlm.nih.gov/pubmed/14562097).

6. ^ Rajiv Ramaswami and Kumar N.Sivarajan, Optical Networks: A

Practical Perspective (Academic Press: London 1998).

7. ^ Chromatic Dispersion (http://www.rp-photonics.com/chromatic_

dispersion.html),Encyclopedia of Laser Physics and Technology

(Wiley, 2008).

8. ^a b

Walter Schumann(2009). Gemstones of the World: Newly

Revised & Expanded Fourth Edition (http://books.google.com/

books?id=V9PqVxpxeiEC&pg=PA42). Sterling Publishing

Company,Inc. pp. 412. ISBN 978-1-4027-6829-3.Retrieved 31December 2011.

9. ^ Lorimer,D.R.,and Kramer, M.,Handbook of Pulsar Astronomy,

vol. 4 of Cambridge Observing Handbooks for Research

Astronomers,(Cambridge UniversityPress, Cambridge,U.K.; New

York, U.S.A, 2005),1st edition.

xternal links

Optical Characteristics of the SF10 Crystal Prism (http://ioannis.virtualcomposer2000.com/spectroscope/characteristics.html)

Deviation Angle for a Prism (http://ioannis.virtualcomposer2000.com/spectroscope/deviationangle.html)

Dispersive Wiki (http://tosio.math.toronto.edu/wiki/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

rieved from "http://en.wikipedia.org/w/index.php?title=Dispersion_(optics)&oldid=554671250"

egories: Optics Glass physics

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4/5

Brazilianite 0.014 0.008

Celestine 0.014 0.008

Goshenite 0.014

Heliodor 0.014 0.0090.013

Morganite 0.014 0.0090.013

Pyroxmangite 0.015

Synth. scheelite 0.015

Dolomite 0.013

Magnesite (MgCO3) 0.012

Synth. emerald 0.012

Synth. alexandrite 0.011

Synth. sapphire (Al2O

3) 0.011

Phosphophyllite 0.0100.011

Enstatite 0.010

Anorthite 0.0090.010

Actinolite 0.009

Jeremejevite 0.009

Nepheline 0.0080.009

Apophyllite 0.008

Hauyne 0.008

Natrolite 0.008

Synth. quartz (SiO2) 0.008

Aragonite 0.0070.012

Augelite 0.007

Tanzanite 0.030 0.011

Thulite 0.03 0.011

Zoisite 0.03

YAG 0.028 0.015

Almandine 0.027 0.0130.016

Hessonite 0.027 0.0130.015

Spessartine 0.027 0.015

Uvarovite 0.027 0.0140.021

Willemite 0.027

Pleonaste 0.026

Rhodolite 0.026

Boracite 0.024 0.012

Cryolite 0.024

Staurolite 0.023 0.0120.013

Pyrope 0.022 0.0130.016

Diaspore 0.02

Grossular 0.020 0.012

Hemimorphite 0.020 0.013

Kyanite 0.020 0.011

Peridote 0.020 0.0120.013

Spinel 0.020 0.011

Vesuvianite 0.0190.025 0.014

Clinozoisite 0.019 0 .0110.014

Labradorite 0.019 0.010

Axinite 0.0180.020 0.011

Ekanite 0.018 0.012

Kornerupine 0.018 0.010

Corundum (Al2O

3) 0.018 0.011

Rhodizite 0.018

Ruby (Al2O

3) 0.018 0.011

Sapphire (Al2O

3) 0.018 0.011

Sinhalite 0.018 0.010

Sodalite 0.018 0.009

Synth. corundum 0.018 0.011

Diopside 0.0180.020 0.01

Emerald 0.014 0.0090.013

Topaz 0.014 0.008

Amethyst (SiO2) 0.013 0.008

Anhydrite 0.013

Apatite 0.013 0.010


5/5

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 0.008Moonstone 0.012 0.008

Orthoclase 0.012 0.008

Pollucite 0.012 0.007

Sanidine 0.012

Sunstone 0.012

Beryllonite 0.010 0.007

Cancrinite 0.010 0.0080.009

Leucite 0.010 0.008

Obsidian 0.010

Strontianite 0.0080.028

Calcite (CaCO3) 0.0080.017 0.0130.014

Fluorite (CaF2) 0.007 0.004

Hematite 0.500

Synth. cassiterite (SnO2) 0.041

Gahnite 0.0190.021

Datolite 0.016

Tremolite 0.0060.007

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