Dispersion is the phenomenon in which thephase velocity of awave depends on its frequency.[1] Sometimes the termchromatic dispersion is used to refer tooptics specifically, as opposed towave propagation in general. A medium having this common property may be termed adispersive medium.
In optics, one important and familiar consequence of dispersion is the change in the angle ofrefraction of different colors of light,[2] as seen in the spectrum produced by a dispersiveprism and inchromatic aberration of lenses. Design of compoundachromatic lenses, in which chromatic aberration is largely cancelled, uses a quantification of a glass's dispersion given by itsAbbe numberV, wherelower Abbe numbers correspond togreater dispersion over thevisible spectrum. In some applications such as telecommunications, the absolute phase of a wave is often not important but only the propagation ofwave packets or "pulses"; in that case one is interested only in variations ofgroup velocity with frequency, so-calledgroup-velocity dispersion.
All commontransmission media also vary inattenuation (normalized to transmission length) as a function of frequency, leading toattenuation distortion; this is not dispersion, although sometimes reflections at closely spacedimpedance boundaries (e.g. crimped segments in a cable) can produce signal distortion which further aggravates inconsistent transit time as observed across signal bandwidth.
The most familiar example of dispersion is probably arainbow, in which dispersion causes the spatial separation of a whitelight into components of differentwavelengths (differentcolors). However, dispersion also has an effect in many other circumstances: for example,group-velocity dispersion causespulses to spread inoptical fibers, degrading signals over long distances; also, a cancellation between group-velocity dispersion andnonlinear effects leads tosoliton waves.
Most often, chromatic dispersion refers to bulk material dispersion, that is, the change inrefractive index with optical frequency. However, in awaveguide there is also the phenomenon ofwaveguide dispersion, in which case a wave'sphase 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., aphotonic crystal), whether or not the waves are confined to some region.[dubious –discuss] In a waveguide,both types of dispersion will generally be present, although they are not strictly additive.[citation needed] For example, in fiber optics the material and waveguide dispersion can effectively cancel each other out to produce azero-dispersion wavelength, important for fastfiber-optic communication.
The variation of refractive index vs. vacuum wavelength for various glasses. The wavelengths of visible light are shaded in grey.Influences of selected glass component additions on the mean dispersion of a specific base glass (nF valid forλ = 486 nm (blue),nC valid forλ = 656 nm (red))[3]
Material dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used to constructspectrometers andspectroradiometers. However, in lenses, dispersion causeschromatic aberration, an undesired effect that may degrade images in microscopes, telescopes, and photographic objectives.
Thephase velocityv of a wave in a given uniform medium is given by
In general, the refractive index is some function of the frequencyf of the light, thusn = n(f), or alternatively, with respect to the wave's wavelengthn = n(λ). The wavelength dependence of a material's refractive index is usually quantified by itsAbbe number or its coefficients in an empirical formula such as theCauchy orSellmeier equations.
Because of theKramers–Kronig relations, the wavelength dependence of the real part of the refractive index is related to the materialabsorption, described by the imaginary part of the refractive index (also called theextinction coefficient). In particular, for non-magnetic materials (μ = μ0), thesusceptibilityχ that appears in the Kramers–Kronig relations is theelectric susceptibilityχe = n2 − 1.
The most commonly seen consequence of dispersion in optics is the separation ofwhite light into acolor spectrum by aprism. FromSnell's law it can be seen that the angle ofrefraction 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 asangular dispersion.
For visible light, refraction indicesn of most transparent materials (e.g., air, glasses) decrease with increasing wavelengthλ:
or generally,
In this case, the medium is said to havenormal dispersion. Whereas if the index increases with increasing wavelength (which is typically the case in the ultraviolet[4]), 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 an angleθ to thenormal will be refracted at an angle arcsin(sinθ/n). Thus, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-knownrainbow pattern.
Time evolution of a short pulse in a hypothetical dispersive medium (k = ω2) 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 termedgroup-velocity dispersion (GVD). While phase velocityv is defined asv =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 thegroup 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 theenvelope (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 curven(ω) or more directly from the wavenumberk =ωn/c, whereω is the radian frequencyω = 2πf. Whereas one expression for the phase velocity isvp = ω/k, the group velocity can be expressed using thederivative:vg = dω/dk. Or in terms of the phase velocityvp,
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 thereciprocal of the group velocity with respect toangular frequency, which results ingroup-velocity dispersion = d2k/dω2.
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 becomespositivelychirped, orup-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 becomesnegatively chirped, ordown-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.
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 μm insilicafibres), 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 asfour-wave mixing). Another possible option is to usesoliton pulses in the regime of negative dispersion, a form of optical pulse which uses anonlinear 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 asself-phase modulation, which interact with dispersion to make it very difficult to undo.
Dispersion control is also important inlasers that produceshort pulses. The overall dispersion of theoptical resonator is a major factor in determining the duration of the pulses emitted by the laser. A pair ofprisms 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 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 mirrors. 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.
Waveguides are highly dispersive due to their geometry (rather than just to their material composition).Optical fibers 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 anangular frequencyω(β) at apropagation constantβ (so that the electromagnetic fields in the propagation directionz oscillate proportional toei(βz−ωt)), the group-velocity dispersion parameterD is defined as[5]
whereλ = 2πc/ω is the vacuum wavelength, andvg = 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 Δt per unit bandwidthΔλ per unit distance travelled, commonly reported inps/(nm⋅km) for optical fibers.
When a broad range of frequencies (a broad bandwidth) is present in a single wavepacket, such as in anultrashort pulse or achirped pulse or other forms ofspread 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 parameterD defined above is obtained from only one derivative of the group velocity. Higher derivatives are known ashigher-order dispersion.[6][7] These terms are simply aTaylor series expansion of thedispersion relationβ(ω) of the medium or waveguide around some particular frequency. Their effects can be computed via numerical evaluation ofFourier transforms of the waveform, via integration of higher-orderslowly varying envelope approximations, by asplit-step method (which can use the exact dispersion relation rather than a Taylor series), or by direct simulation of the fullMaxwell's equations rather than an approximate envelope equation.
In electromagnetics and optics, the termdispersion 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 exemplaryanisotropic medium, the spatial relation betweenelectric andelectric displacement field can be expressed as aconvolution:[8]
where thekernel is dielectric response (susceptibility); its indices make it in general atensor to account for the anisotropy of the medium. Spatial dispersion is negligible in most macroscopic cases, where the scale of variation of 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 asmetals,electrolytes andplasmas. Spatial dispersion also plays role inoptical activity andDoppler broadening,[8] as well as in the theory ofmetamaterials.[9]
In thetechnical terminology ofgemology,dispersion is the difference in the refractive index of a material at the B and G (686.7 nm and 430.8 nm) or C and F (656.3 nm and 486.1 nm)Fraunhofer wavelengths, and is meant to express the degree to which a prism cut from thegemstone 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.[10][11]
In photographic and microscopic lenses, dispersion causeschromatic 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 ofachromats, multielement lenses with glasses of different dispersion. They are constructed in such a way that the chromatic aberrations of the different parts cancel out.
Pulsars are spinning neutron stars that emitpulses 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 theinterstellar medium, mainly the free electrons, which make the group velocity frequency-dependent. The extra delay added at a frequencyν is
and thedispersion measure (DM) is the column density of free electrons (total electron content) – i.e. thenumber density of electronsne integrated along the path traveled by the photon from the pulsar to the Earth – and is given by
with units ofparsecs per cubic centimetre (1 pc/cm3 = 30.857×1021 m−2).[13]
Typically for astronomical observations, this delay cannot be measured directly, since the emission time is unknown. Whatcan be measured is the difference in arrival times at two different frequencies. The delay Δt between a high-frequencyνhi and a low-frequencyνlo component of a pulse will be
Rewriting the above equation in terms of Δt 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.
^"Single-Dish Radio Astronomy: Techniques and Applications", ASP Conference Proceedings, vol. 278. Edited by Snezana Stanimirovic,Daniel Altschuler, Paul Goldsmith, and Chris Salter.ISBN1-58381-120-6. San Francisco: Astronomical Society of the Pacific, 2002, p. 251–269.
^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.
