In
optics and in
wave propagation in general, dispersion is the phenomenon in which the
phase velocity of a
wave depends on its frequency;[1] sometimes the term chromatic dispersion is used for specificity to optics in particular.
A medium having this common property may be termed a dispersive medium (plural dispersive media).
In optics, one important and familiar consequence of dispersion is the change in the angle of
refraction of different colors of light,[2] as seen in the spectrum produced by a dispersive
prism and in
chromatic aberration of lenses. Design of compound
achromatic lenses, in which chromatic aberration is largely cancelled, uses a quantification of a glass's dispersion given by its
Abbe numberV, 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 packets or "pulses"; in that case one is interested only in variations of
group velocity with frequency, so-called
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 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
colors). However, dispersion also has an effect in many other circumstances: for example,
group-velocity dispersion causes
pulses to spread in
optical fibers, degrading signals over long distances; also, a cancellation between group-velocity dispersion and
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 –
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 a
zero-dispersion wavelength, important for fast
fiber-optic communication.
Material dispersion in optics
Material dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used to construct
spectrometers and
spectroradiometers. However, in lenses, dispersion causes
chromatic aberration, an undesired effect that may degrade images in microscopes, telescopes, and photographic objectives.
The phase velocityv of a wave in a given uniform medium is given by
In general, the refractive index is some function of the frequency f of the light, thus n = n(f), or alternatively, with respect to the wave's wavelength n = 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 or
Sellmeier equations.
Because of the
Kramers–Kronig relations, the wavelength dependence of the real part of the refractive index is related to the material
absorption, described by the imaginary part of the refractive index (also called the
extinction coefficient). In particular, for non-magnetic materials (
μ =
μ0), the
susceptibilityχ that appears in the Kramers–Kronig relations is the
electric susceptibilityχe = n2 − 1.
The most commonly seen consequence of dispersion in optics is the separation of
white light into a
color spectrum by a
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:
or generally,
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[4]), 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
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.
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 = 2πf. Whereas one expression for the phase velocity is vp = ω/k, the group velocity can be expressed using the
derivative: vg = dω/dk. Or in terms of the phase velocity vp,
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 = 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 becomes positively
chirped, 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.
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 in
silicafibres), 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 pulses in the regime of negative dispersion, a form of optical pulse which uses a
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
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 the 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 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.
In waveguides
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 an
angular frequencyω(β) at a
propagation constantβ (so that the electromagnetic fields in the propagation direction z oscillate proportional to ei(βz−ωt)), the group-velocity dispersion parameter D is defined as[5]
where λ = 2πc/ω is the vacuum wavelength, and vg = 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 bandwidth
Δλ per unit distance travelled, commonly reported in
ps/(
nm⋅
km) for optical fibers.
When 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, 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.[6][7] These terms are simply a
Taylor series expansion of the
dispersion relationβ(ω) of the medium or waveguide around some particular frequency. Their effects can be computed via numerical evaluation 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 relation rather than a Taylor series), or by direct simulation of the full
Maxwell's equations rather than an approximate envelope equation.
Generalized formulation of the high orders of dispersion – Lah-Laguerre optics
The description of the chromatic dispersion in a perturbative manner through Taylor coefficients is advantageous for optimization problems where the dispersion from several different systems needs to be balanced. For example, in chirp pulse laser amplifiers, the pulses are first stretched in time by a stretcher to avoid optical damage. Then in the amplification process, the pulses accumulate inevitably linear and nonlinear phase passing through materials. And lastly, the pulses are compressed in various types of compressors. To cancel any residual higher orders in the accumulated phase, usually individual orders are measured and balanced. However, for uniform systems, such perturbative description is often not needed (i.e., propagation in waveguides).
The dispersion orders have been generalized in a computationally friendly manner, in the form of Lah-Laguerre type transforms.[8][9]
The dispersion orders are defined by the Taylor expansion of the phase or the wavevector.
The dispersion relations for the wavector and the phase can be expressed as:
,
The derivatives of any differentiable function in the wavelength or the frequency space is specified through a Lah transform as:
The matrix elements of the transformation are the Lah coefficients:
Written for the GDD the above expression states that a constant with wavelength GGD, will have zero higher orders. The higher orders evaluated from the GDD are:
Substituting equation (2) expressed for the refractive index or optical path into equation (1) results in closed-form expressions for the dispersion orders. In general, the order dispersion POD is a Laguerre type transform of negative order two:
The matrix elements of the transforms are the unsigned Laguerre coefficients of order minus 2, and are given as:
The first ten dispersion orders, explicitly written for the wavevector, are:
The group refractive index is defined as: .
Explicitly, written for the phase , the first ten dispersion orders can be expressed as a function of wavelength using the Lah transforms (equation (2)) as:
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
anisotropic medium, the spatial relation between
electric and
electric displacement field can be expressed as a
convolution:[10]
where the
kernel 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 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
metals,
electrolytes and
plasmas. Spatial dispersion also plays role in
optical activity and
Doppler broadening,[10] as well as in the theory of
metamaterials.[11]
In the
technical terminology of
gemology, 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 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.[12][13]
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
achromats, 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
Pulsars 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, 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 ν is
and the dispersion measure (DM) is the column density of free electrons (
total electron content) – i.e. the
number density of electrons ne integrated along the path traveled by the photon from the pulsar to the Earth – and is given by
with units of
parsecs per cubic centimetre (1 pc/cm3 = 30.857×1021 m−2).[15]
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 Δt between a high-frequency νhi and a low-frequency νlo 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.