Dispersion (optics) 172333 223099121 2008-07-02T15:50:53Z ClueBot 4928500 Reverting possible vandalism by [[Special:Contributions/86.7.148.232|86.7.148.232]] to version by BOTarate. False positive? [[User:ClueBot/FalsePositives|Report it]]. Thanks, [[User:ClueBot]]. (430808) (Bot) [[Image:Prism rainbow schema.png|frame|right|In a prism, material dispersion (a [[wavelength]]-dependent [[refractive index]]) causes different colors to [[refraction|refract]] at different angles, splitting white light into a [[rainbow]].]] In [[optics]], '''dispersion''' is the phenomenon in which the [[phase velocity]] of a wave depends on its frequency.<ref>{{cite book | last = Born | first = Max | authorlink = Max Born | last2 = Wolf | first2 = Emil | title = Principle of Optics | publisher = [[Cambridge University Press]] | date = October 1999 | location = Cambridge | pages = pp. 14-24 | isbn = 0521642221}}</ref> Media having such a property are termed ''dispersive media''. 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 impact in many other circumstances: for example, it causes [[Pulse (signal processing)|pulses]] to spread in [[optical fiber]]s, degrading signals over long distances; also, a cancellation between dispersion and [[nonlinear]] effects leads to [[soliton]] waves. Dispersion is most often described for [[light]] waves, but it may occur for any kind of wave that interacts with a medium or passes through an inhomogeneous geometry (e.g. a [[waveguide]]), such as [[sound]] waves. Dispersion is sometimes called '''''chromatic'' dispersion''' to emphasize its wavelength-dependent nature. There 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 example, material dispersion leads to undesired [[chromatic aberration]] in a [[lens (optics)|lens]] or the separation of colors in a [[triangular prism|prism]]. Waveguide dispersion occurs when the 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 [[photonic crystal]]), 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 fiber]]s for [[telecommunication]]s, because the varying delay in arrival time between different components of a signal "smears out" the signal in time. ==Material dispersion in optics== [[Image:Dispersion-curve.png|right|thumb|320px|The variation of refractive index vs. wavelength for various glasses. The wavelengths of visible light are shaded in red.]] [[Image:Spidergraph Dispersion.GIF|320px|thumb|Influences of selected glass component additions on the mean dispersion of a specific base glass (n<sub>F</sub> valid for λ = 486 nm (blue), n<sub>C</sub> valid for λ = 656 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. [[Holographic]] gratings are also used, as they allow more accurate discrimination of wavelengths. 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 a 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'' = ''n''(f), or alternately, with respect to the wave's [[wavelength]] ''n'' = ''n''(λ). The wavelength dependency of a material's refractive index is usually quantified by an empirical formula, the [[Cauchy's equation|Cauchy]] or [[Sellmeier equation]]s. The most commonly seen consequence of dispersion in optics is the separation of [[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 will also vary with wavelength, causing an angular separation of the colors known as ''angular dispersion''. For visible light, most transparent materials (e.g. glasses) have: :<math>1 < n(\lambda_{\rm red}) < n(\lambda_{\rm yellow}) < n(\lambda_{\rm blue})\ ,</math> or alternatively: :<math>\frac{{\rm d}n}{{\rm d}\lambda} < 0,</math> that is, refractive index ''n'' decreases with increasing wavelength λ. In this case, the medium is said to have ''normal dispersion''. Whereas, if the index increases with increasing wavelength the medium has ''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( 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 velocity ==<!-- This section is linked from [[Speed of sound]] --> Another consequence of dispersion manifests itself as a temporal effect. The formula above, ''v'' = ''c'' / ''n'' calculates the ''phase velocity'' of a wave; this is the [[velocity]] at which the ''[[phase (waves)|phase]]'' of any one frequency component of the wave will propagate. This is not the same as the ''[[group velocity]]'' of the wave, which is the rate that changes in [[amplitude]] (known as the ''envelope'' of the wave) will propagate. The group velocity ''v''<sub>g</sub> is related to the phase velocity by, for a homogeneous medium (here <math>\lambda</math> is the wavelength in vacuum, not in the medium): :<math>v_g = c \left( n - \lambda \frac{dn}{d\lambda} \right)^{-1}.</math> The group velocity ''v''<sub>g</sub> 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 some unusual circumstances, where the wavelength of the light is close to an [[absorption (optics)|absorption]] resonance of the medium, it is possible for the group velocity to exceed the speed of light (''v''<sub>g</sub> &gt; ''c''), leading to the conclusion that superluminal (faster than light) communication is possible. In practice, in such situations the distortion and absorption of the wave is such that the value of the group velocity essentially becomes meaningless, and does not represent the true signal velocity of the wave, which stays less than ''c''. 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 the pulse travelling at different velocities. GVD is often quantified as the ''group delay dispersion parameter'' (again, this formula is for a uniform medium only): :<math>D = - \frac{\lambda}{c} \, \frac{d^2 n}{d \lambda^2}. </math> If ''D'' is less than zero, the medium is said to have ''positive dispersion''. If ''D'' is greater than zero, the medium has ''negative dispersion''.If a light pulse is propagated through a normally dispersive medium, the result is the higher [[frequency]] components travel slower than the lower frequency components. The pulse therefore becomes ''positively [[chirp]]ed'', or ''up-chirped'', increasing in frequency with time. Conversely, if a pulse travels through an anomalously dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes ''negatively [[chirp]]ed'', 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 together, 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 μm in [[silica]] [[fibres]]), so pulses at this wavelength suffer minimal spreading from dispersion&mdash;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 anomalous dispersion, a form of optical pulse which uses a [[nonlinear optics|nonlinear optical]] effect to self-maintain its shape&mdash;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)|prism]]s 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. == Dispersion in waveguides == [[Optical 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 data can be transported on a single fiber. The [[transverse mode]]s 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 the wave, the [[wavelength]]) compared to the size of the waveguide. 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 <math>e^{i(\beta z - \omega t)}</math>), the group-velocity [[dispersion parameter]] ''D'' is defined as:<ref>Rajiv Ramaswami and Kumar N. Sivarajan, ''Optical Networks: A Practical Perspective'' (Academic Press: London 1998).</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 <math>\lambda = 2\pi c/\omega</math> is the vacuum wavelength and <math>v_g = d\omega/d\beta</math> 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 <math>\Delta t</math> per unit bandwidth <math>\Delta\lambda</math> per unit distance travelled, commonly reported in [[picosecond|ps]]&nbsp;/&nbsp;[[nanometer|nm]]&nbsp;[[kilometer|km]] for optical fibers. A similar 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 case 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 the waveguide. == Dispersion in gemology == In the [[technical terminology]] of [[gemology]], ''dispersion'' is the difference in the refractive index of a material at the B and G [[fraunhofer line|Fraunhofer]] [[wavelength]]s of 686.7 [[Nanometre|nm]] and 430.8 nm and is meant to express 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 imaging == In photographic and microscopic lenses, dispersion causes [[chromatic aberration]], distorting the image, and various techniques have been developed to counteract it. == Dispersion in pulsar timing == [[Pulsar]]s are spinning neutron stars that emit pulses at very regular intervals ranging from milliseconds to seconds. It is believed 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 ionised component of the [[interstellar medium]], which makes the group velocity frequency dependent. The extra delay added at frequency <math>\nu</math> is :<math>D = 4.15 ms (\frac{\nu}{GHz})^{-2} \times (\frac{DM}{cm^{-3} pc})</math> where the dispersion measure ''DM'' is :<math>DM = \int_0^d{n_e\;dl}</math> is the integrated free electron column density <math>n_e</math> out to the pulsar at a distance ''d''<ref>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.</ref>. Of course, 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 <math>\Delta T</math> between a high frequency <math>\nu_{hi}</math> and a low frequency <math>\nu_{lo}</math> component of a pulse will be :<math>\Delta T = 4.15 ms [(\frac{\nu_{lo}}{GHz})^{-2} - (\frac{\nu_{hi}}{GHz})^{-2} ] \times (\frac{DM}{cm^{-3} pc})</math> and so ''DM'' is normally computed from measurements at two different frequencies. This allows computation of the absolute delay at any frequency, which is used when combining many different pulsar observations into an integrated timing solution. == See also == *[[Dispersion relation]] *[[Sellmeier equation]] *[[Cauchy's equation]] *[[Abbe number]] *[[Kramers–Kronig relation]] *[[Group delay]] *[[Calculation of glass properties]] incl. dispersion *[[Linear response function]] *[[Green-Kubo relations]] *[[Fluctuation theorem]] ==References== <!-- ---------------------------------------------------------- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for a discussion of different citation methods and how to generate footnotes using the<ref>, </ref> and <reference /> tags ----------------------------------------------------------- --> <div class="references-small"> <references /> </div> == External links == {{commons|Dispersion|Dispersion (optics)}} * [http://ioannis.virtualcomposer2000.com/spectroscope/characteristics.html Optical Characteristics of the SF10 Crystal Prism] * [http://ioannis.virtualcomposer2000.com/spectroscope/deviationangle.html Deviation Angle for a Prism] * [http://tosio.math.toronto.edu/wiki/index.php/Main_Page Dispersive Wiki] - discussing the mathematical aspects of dispersion. *[http://www.rp-photonics.com/dispersion.html Dispersion] - Encyclopedia of Laser Physics and Technology {{Glass science}} [[Category:Optics]] [[Category:Glass engineering and science]] [[af:Dispersie (optika)]] [[ar:تشتيت ضوء]] [[bg:Дисперсия (оптика)]] [[cs:Disperze (světlo)]] [[da:Optisk dispersion]] [[de:Dispersion (elektromagnetische Wellen)]] [[et:Dispersioon (füüsika)]] [[es:Dispersión refractiva]] [[eo:Varianco (optiko)]] [[fr:Dispersion]] [[it:Dispersione ottica]] [[he:נפיצה]] [[lt:Šviesos dispersija]] [[hu:Diszperzió]] [[nl:Dispersie (natuurkunde)]] [[ja:分散 (光学)]] [[pl:Rozszczepienie (fizyka)]] [[pt:Dispersão (óptica)]] [[ro:Grad de dispersie]] [[ru:Дисперсия света]] [[fi:Dispersio]] [[sv:Dispersion]] [[uk:Дисперсія світла]] [[zh:光的色散]]