In
opticsOptics is the branch of physics which studies the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
,
dispersion is the phenomenon in which the
phase velocityThe phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
of a wave depends on its frequency, or alternatively when the
group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
depends on the frequency.
Media having such a property are termed
dispersive media. Dispersion is sometimes called chromatic dispersion
to emphasize its wavelength-dependent nature, or group-velocity dispersion
(GVD) to emphasize the role of the group velocity.
The most familiar example of dispersion is probably a
rainbowA rainbow is an optical and meteorological phenomenon that causes a spectrum of light to appear in the sky when the Sun shines onto droplets of moisture in the Earth's atmosphere...
, in which dispersion causes the spatial separation of a white light into components of different wavelengths (different
colorColor or colour is the visual perceptual property corresponding in humans to the categories called red, yellow, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...
s). However, dispersion also has an effect in many other circumstances: for example, GVD causes
pulsesIn signal processing, the term pulse has the following meanings:#A rapid, transient change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value....
to spread in
optical fiberAn optical fiber is a glass or plastic fiber that carries light along its length. Fiber optics is the overlap of applied science and engineering concerned with the design and application of optical fibers...
s, degrading signals over long distances; also, a cancellation between group-velocity dispersion and nonlinear effects leads to
solitonIn mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. "Dispersive effects" refer to dispersion relations between the frequency...
waves. Dispersion is most often described for
lightLight is electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye ....
waves, but it may occur for any kind of wave that interacts with a medium or passes through an inhomogeneous geometry (e.g. a
waveguideA waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguide for each type of wave...
), such as
soundSound is a travelling wave which is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.- Perception of sound...
waves.
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 aberrationIn optics, chromatic aberration is the failure of a lens to focus all colors to the same point. It occurs because lenses have a different refractive index for different wavelengths of light...
in a
lensA lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens is a lens consisting of a single optical element...
or the separation of colors in a
prismIn optics, a dispersive prism is a type of optical prism, normally having the shape of a geometrical triangular prism. It is the most widely-known type of optical prism, although perhaps not the most common in actual use. Triangular prisms are used to disperse light, that is to break light up into...
. 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 crystalPhotonic crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons...
), 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 fiberAn optical fiber is a glass or plastic fiber that carries light along its length. Fiber optics is the overlap of applied science and engineering concerned with the design and application of optical fibers...
s for
telecommunicationTelecommunication is transmission over a distance for the purpose of communication. In earlier times, this may have involved the use of smoke signals, drums, semaphore, flags or heliograph. In modern times, telecommunication typically involves the use of electronic devices such as the telephone,...
s, because the varying delay in arrival time between different components of a signal "smears out" the signal in time.
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
spectrometerA spectrometer is an instrument used to measure properties of light over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify materials. The variable measured is most often the light's intensity but could also, for instance, be the polarization...
s and
spectroradiometerSpectroradiometers are designed to measure the spectral power distributions of illuminants. They operate almost like spectrophotometers in the visible region...
s. Holographic gratings are also used, as they allow more accurate discrimination of wavelengths. However, in lenses, dispersion causes
chromatic aberrationIn optics, chromatic aberration is the failure of a lens to focus all colors to the same point. It occurs because lenses have a different refractive index for different wavelengths of light...
, an undesired effect that may degrade images in microscopes, telescopes and photographic objectives.
The
phase velocityThe phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
, v
, of a wave in a given uniform medium is given by
where c
is the speed of lightIn physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
in a vacuum and n
is the refractive indexThe refractive index of a medium is a measure of how much the speed of light is reduced inside the medium. For example, typical soda-lime glass has a refractive index close to 1.5, which means that in glass, light travels at 1 / 1.5 = 2/3 the speed of light in a vacuum...
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 dependence of a material's refractive index is usually quantified by an empirical formula, the
CauchyCauchy's equation is an empirical relationship between the refractive index and wavelength of light for a particular transparent material. It is named for the mathematician Augustin Louis Cauchy, who defined it in 1836.-The equation:...
or
Sellmeier equationThe Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium.It was first proposed in 1871 by W...
s.
Because of the Kramers–Kronig relations, the wavelength dependence of the real part of the refractive index is related to the material
absorptionIn physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy, for example, to heat. The absorption of light during wave propagation...
, described by the imaginary part of the refractive index (also called the extinction coefficient). In particular, for non-magnetic materials (
μIn electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. Magnetic permeability is typically represented by the Greek letter μ. The term was coined in September, 1885 by Oliver Heaviside...
= μ
0), the
susceptibilityA linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist...
that appears in the Kramers–Kronig relations is the
electric susceptibilityThe electric susceptibility χe of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of...
.
The most commonly seen consequence of dispersion in optics is the separation of
white lightWhite Light may refer to:* Light with the color white*White Light/White Heat , The Velvet Underground's second album.*White Light , a 1971 album*White Light a 1980 novel by Rudy Rucker...
into a color spectrum by a prism. From
Snell's lawIn optics and physics, Snell's law , is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...
it can be seen that the angle of
refractionRefraction is the change in direction of a wave due to a change in its velocity. This is most commonly observed when a wave passes from one medium to another...
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, most transparent materials (e.g. glasses) have:
or alternatively:
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 normalA surface normal, or simply normal, to a flat surface is a vector which is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to...
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 rainbowA rainbow is an optical and meteorological phenomenon that causes a spectrum of light to appear in the sky when the Sun shines onto droplets of moisture in the Earth's atmosphere...
pattern.
Group and phase velocity
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 velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
at which the phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
of any one frequency component of the wave will propagate. This is not the same as the group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
of the wave, which is the rate that changes in amplitudeAmplitude is the magnitude of change in the oscillating variable, with each oscillation, within an oscillating system. For instance, sound waves are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
(known as the envelope
of the wave) will propagate. The group velocity v
g is related to the phase velocity by, for a homogeneous medium (here λ is the wavelength in vacuum, not in the medium):
The group velocity v
g 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 velocityThe signal velocity is the speed at which a wave carries information. It describes how quickly a message can be communicated between two separated parties...
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 changes sign, in which case it is possible for the group velocity to exceed the speed of light (v
g > c
). Anomalous dispersion occurs, for instance, where the wavelength of the light is close to an absorption resonance of the medium. When the dispersion is anomalous, however, group velocity is no longer an indicator of signal velocity. Instead, a signal travels at the speed of the wavefront, which is c irrespective of the index of refraction. Recently, it has become possible to create gases in which the group velocity is not only larger 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 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):
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
chirpA chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
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
chirpA chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
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—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
solitonIn optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:...
pulses in the regime of anomalous dispersion, a form of optical pulse which uses a
nonlinear opticalNonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light. This nonlinearity is typically only observed at very high light intensities such as...
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 modulationSelf-phase modulation is a nonlinear optical effect of light-matter interaction.An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect...
, which interact with dispersion to make it very difficult to undo.
Dispersion control is also important in
laserA laser is a device that emits light through a process called stimulated emission. Laser light is usually spatially coherent, which means that the light either is emitted in a narrow, low-divergence beam, or can be converted into one with the help of optical components such as lenses...
s that produce
short pulsesIn optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is on the order of the femtosecond . Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators...
. The overall dispersion of the
optical resonatorA laser is constructed from three principal parts:*An energy source ,*A gain medium or laser medium, and*Two or more mirrors that form an optical resonator.-Pump source:...
is a major factor in determining the duration of the pulses emitted by the laser. A pair of
prismIn optics, a prism is a transparent optical element with flat, polished surfaces that refract light. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use...
s can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium.
DiffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
gratingA grating is any regularly spaced collection of essentially identical, parallel, elongated elements. Gratings usually consist of a single set of elongated elements, but can consist of two sets, in which case the second set is usually perpendicular to the first...
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 mirrorA chirped mirror is a dielectric mirror with chirped spaces—spaces of varying depth designed to reflect varying wavelengths of lights—between the dielectric layers ....
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 modeA transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
s for waves confined laterally within a
waveguideA waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguide for each type of wave...
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 frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
ω(β) at a
propagation constantThe propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...
β (so that the electromagnetic fields in the propagation direction z
oscillate proportional to ), the group-velocity dispersion parameter D
is defined as:
where is the vacuum wavelength and 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 per unit bandwidth
per unit distance travelled, commonly reported in
psA picosecond is 10-12 of a second. That is one trillionth, or one millionth of one millionth of a second, or 0.000 000 000 001 seconds. The name is formed by the SI prefix pico and the SI unit second. It is abbreviated as ps....
/ nm km for optical fibers.
A similar effect due to a somewhat different phenomenon is
modal dispersionModal dispersion is a distortion mechanism occurring in multimode fibers and other waveguides, in which the signal is spread in time because the propagation velocity of the optical signal is not the same for all modes...
, caused by a waveguide having multiple modes at a given frequency, each with a different speed. A special case of this is
polarization mode dispersionPolarization mode dispersion is a form of modal dispersion where two different polarizations of light in a waveguide, which normally travel at the same speed, travel at different speeds due to random imperfections and asymmetries, causing random spreading of optical pulses...
(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.
Higher-order dispersion over broad bandwidths
When a broad range of frequencies (a broad bandwidth) is present in a single wavepacket, such as in an
ultrashort pulseIn optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is on the order of the femtosecond . Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators...
or a
chirpA chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
ed pulse or other forms of
spread spectrumSpread-spectrum techniques are methods by which electromagnetic energy generated in a particular bandwidth is deliberately spread in the frequency domain, resulting in a signal with a wider bandwidth...
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. These terms are simply a
Taylor seriesIn mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named after the English mathematician Brook Taylor...
expansion of the dispersion relation of the medium or waveguide around some particular frequency. Their effects can be computed via numerical evaluation of
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
s of the waveform, via integration of higher-order
slowly varying envelope approximationIn physics, the slowly varying envelope approximation is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a period or wavelength...
s, by a
split-step methodIn numerical analysis, the split-step method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the...
(which can use the exact dispersion relation rather than a Taylor series), or by direct simulation of the full
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
rather than an approximate envelope equation.
Dispersion in gemology
In the
technical terminologyTechnical terminology is the specialized vocabulary of a field, the nomenclature. These terms have specific definitions within the field, which is not necessarily the same as their meaning in common use...
of
gemologyGemology or gemmology is the science, art and profession of identifying and evaluating gemstones. It is considered a geoscience and a branch of mineralogy. Some jewelers are academically trained gemologists and are qualified to identify and evaluate gems.- Background :With further post-graduate...
, dispersion is the difference in the refractive index of a material at the B and G Fraunhofer wavelengths of 686.7
nmA nanometre is a unit of length in the metric system, equal to one billionth of a meter....
and 430.8 nm and is meant to express the degree to which a prism cut from the
gemstoneA gemstone or gem is a piece of attractive mineral, which—when cut and polished—is used to make jewelry or other adornments...
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 aberrationIn optics, chromatic aberration is the failure of a lens to focus all colors to the same point. It occurs because lenses have a different refractive index for different wavelengths of light...
, distorting the image, and various techniques have been developed to counteract it such as the use of multielement lenses with glasses with different dispersion characteristics: the net effect is to recombine (at least approximately) all colors.
Dispersion in pulsar timing
PulsarPulsars are highly magnetized, rotating neutron stars that emit a beam of electromagnetic radiation. The observed periods of their pulses range from 1.4 milliseconds to 8.5 seconds. The radiation can only be observed when the beam of emission is pointing towards the Earth. This is called the...
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 mediumIn astronomy, the interstellar medium is the gas and dust that pervade interstellar space: the matter that exists between the star systems within a galaxy. It fills interstellar space and blends smoothly into the surrounding intergalactic space...
, which makes 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 free electron column density (total electron contentTotal electron content is an important descriptive quantity for the ionosphere of the Earth. TEC is the total number of electrons present along a path between two points, with units of electrons per square meter, where 1016 electrons/m² = 1 TEC unit .TEC is significant in determining...
) integrated along the path traveled by the photon from the pulsar to the Earth, and is given by
with units of parsecThe parsec is a unit of length, equal to just under 31 million million kilometres , or about 3.26 light-years. The parsec measurement unit is used in astronomy and is defined as the length of the adjacent side of an imaginary right triangle in space...
s per cubic centimetreA cubic centimetre or cubic centimeter is a commonly used unit of volume extending the derived SI-unit cubic metre and corresponds to the volume of a cube measuring 1×1×1 cm...
(1pc/cc = 30.857×1021 m−2).
Typically for astronometric 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 high frequency 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 at multiple frequencies. This in turn can be used to study the interstellar medium, as well as allow for observations of pulsars at different frequencies to be combined.
See also
- Dispersion relation
Dispersion relations describe the ways that wave propagation varies with the wavelength or frequency of a wave. This variation has long explained how white light is dispersed into different colors, thus making rainbows possible...
- Sellmeier equation
The Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium.It was first proposed in 1871 by W...
- Cauchy's equation
Cauchy's equation is an empirical relationship between the refractive index and wavelength of light for a particular transparent material. It is named for the mathematician Augustin Louis Cauchy, who defined it in 1836.-The equation:...
- Abbe number
In physics and optics, the Abbe number, also known as the V-number or constringence of a transparent material, is a measure of the material's dispersion in relation to the refractive index...
- Kramers–Kronig relation
- Group delay
Group delay is a measure of the transit time of a signal through a device under test , versus frequency. Group delay is a useful measure of phase distortion, and is calculated by differentiating the insertion phase response of the DUT versus frequency. Another way to say this is that group delay is...
- Calculation of glass properties
The calculation of glass properties is used to predict glass properties of interest or glass behavior under certain conditions without experimental investigation, based on past data and experience, with the intention to save time, material, financial, and environmental resources, or to gain...
incl. dispersion
- Linear response function
A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist...
- Green-Kubo relations
Green–Kubo relations give exact mathematical expression for transport coefficients in terms of integrals of time correlation functions.-Thermal and mechanical transport processes:...
- Fluctuation theorem
The fluctuation theorem which originated from statistical mechanics deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium will increase or decrease over a given amount of time...
External links