In a prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a rainbow. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium Refraction is the change in direction of a Wave due to a change in its Speed. A rainbow is an optical and meteorological phenomenon that causes a spectrum of Light to appear in the Sky when the Sun

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. The phase velocity (or phase speed) of a Wave is the rate at which the phase of the wave propagates in space [1] 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 colors). A rainbow is an optical and meteorological phenomenon that causes a spectrum of Light to appear in the Sky when the Sun In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. However, dispersion also has an impact in many other circumstances: for example, it causes pulses to spread in optical fibers, degrading signals over long distances; also, a cancellation between dispersion and nonlinear effects leads to soliton waves. In 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 An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length This article describes the use of the term nonlinearity in mathematics In Mathematics and Physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse that maintains its shape while it travels at constant speed 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. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 g. a waveguide), such as sound waves. A waveguide is a structure which guides waves such as Electromagnetic waves Light, or Sound waves Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies 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 or the separation of colors in a prism. In Optics, chromatic aberration is caused by a lens having a different Refractive index for different Wavelengths of Light A lens is an optical device with perfect or approximate Axial symmetry which transmits and refracts Light, converging or diverging In Geometry, a triangular prism or three-sided prism is a type of prism; it is a Polyhedron made of a triangular base a translated 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. An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length 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. Photonic crystals are periodic Optical (nanostructures that are designed to affect the motion of Photons in a similar way that periodicity of a Semiconductor In general, 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 the varying delay in arrival time between different components of a signal "smears out" the signal in time. An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length

## Material dispersion in optics

The variation of refractive index vs. wavelength for various glasses. The wavelengths of visible light are shaded in red.
Influences of selected glass component additions on the mean dispersion of a specific base glass (nF valid for λ = 486 nm (blue), nC valid for λ = 656 nm (red))[2]

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. A spectrometer is an Optical instrument used to measure properties of Light over a specific portion of the Electromagnetic spectrum, typically used Spectroradiometers are designed to measure the Spectral power distributions of Illuminants They operate almost like Spectrophotometers in the visible Holographic gratings are also used, as they allow more accurate discrimination of wavelengths. Holography (from the Greek, ὅλος - hólos whole + γραφή - grafē writing drawing is a technique that allows the However, in lenses, dispersion causes chromatic aberration, an undesired effect that may degrade images in microscopes, telescopes and photographic objectives. In Optics, chromatic aberration is caused by a lens having a different Refractive index for different Wavelengths of Light

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

$v = \frac{c}{n}$

where c is the speed of light in a vacuum and n is the refractive index of the medium. The phase velocity (or phase speed) of a Wave is the rate at which the phase of the wave propagates in space The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside 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). In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. The wavelength dependency of a material's refractive index is usually quantified by an empirical formula, the Cauchy or Sellmeier equations. Cauchy's equation is an Empirical relationship between the Refractive index n and Wavelength of light λ for a particular transparent In Optics, the Sellmeier equation is an Empirical relationship between Refractive index n and Wavelength λ for a

The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. In Optics, a dispersive prism is a type of optical prism, normally having the shape of a geometrical triangular 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. In Optics and Physics, Snell's law (also known as Descartes' law or the law of refraction) is a formula used to describe the relationship Refraction is the change in direction of a Wave due to a change in its Speed. 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:

$1 < n(\lambda_{\rm red}) < n(\lambda_{\rm yellow}) < n(\lambda_{\rm blue})\ ,$

or alternatively:

$\frac{{\rm d}n}{{\rm d}\lambda} < 0,$

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 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. A rainbow is an optical and meteorological phenomenon that causes a spectrum of Light to appear in the Sky when the Sun

## 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 velocity at which the phase of any one frequency component of the wave will propagate. In Physics, velocity is defined as the rate of change of Position. The 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 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 of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system The group velocity vg is related to the phase velocity by, for a homogeneous medium (here λ is the wavelength in vacuum, not in the medium):

$v_g = c \left( n - \lambda \frac{dn}{d\lambda} \right)^{-1}.$

The group velocity vg 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. The signal velocity is the speed at which a Wave carries information In some unusual circumstances, where the wavelength of the light is close to an absorption resonance of the medium, it is possible for the group velocity to exceed the speed of light (vg > c), leading to the conclusion that superluminal (faster than light) communication is possible. In Physics, absorption of electromagnetic radiation is the process by which the Energy of a Photon is taken up by matter typically the electrons of an 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):

$D = - \frac{\lambda}{c} \, \frac{d^2 n}{d \lambda^2}.$

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. Frequency is a measure of the number of occurrences of a repeating event per unit Time. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. A chirp is a signal in which the Frequency increases ('up-chirp' or decreases ('down-chirp' 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 chirped, or down-chirped, decreasing in frequency with time. A chirp is a signal in which the Frequency increases ('up-chirp' or decreases ('down-chirp' 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. An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length 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). The Chemical compound silicon dioxide, also known as silica or silox (from the Latin " Silex " is an Oxide Fiber or fibre is a class of Materials that are continuous filaments or are in discrete elongated pieces similar to lengths of thread. Four-wave mixing is an Intermodulation distortion in Optical systems similar to the Third-order intercept point in electrical systems Another possible option is to use soliton pulses in the regime of anomalous 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. In 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 Nonlinear optics (NLO is the branch of Optics that describes the behaviour of Light in nonlinear media, that is media in which the dielectric polarization 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. Self-phase modulation (SPM is a nonlinear optical effect of Light - Matter interaction

Dispersion control is also important in lasers that produce short pulses. A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. In Optics, an ultrashort pulse of light is an Electromagnetic pulse whose time duration is on the order of the femtosecond (10^{-15} second The overall dispersion of the optical resonator is a major factor in determining the duration of the pulses emitted by the laser. A laser is constructed from three principal parts An energy source (usually referred to as the pump or pump source) A 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. In Optics, a prism is a transparent optical element with flat polished surfaces that refract Light. Diffraction gratings can also be used to produce dispersive effects; these are often used in high-power laser amplifier systems. Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle A grating is any regularly spaced collection of essentially identical Parallel, elongated elements Recently, an alternative to prisms and gratings has been developed: chirped mirrors. A chirped mirror is a Dielectric mirror with chirped spaces—spaces of varying depth designed to reflect varying wavelengths of lights—between the dielectric layers 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. An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length Dispersion in these fibers is one of the limiting factors that determine how much data can be transported on a single fiber.

The 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 the wave, the wavelength) compared to the size of the waveguide. A waveguide is a structure which guides waves such as Electromagnetic waves Light, or Sound waves In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency.

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 ei(βz − ωt)), the group-velocity dispersion parameter D is defined as:[3]

$D = -\frac{2\pi c}{\lambda^2} \frac{d^2 \beta}{d\omega^2} = \frac{2\pi c}{v_g^2 \lambda^2} \frac{dv_g}{d\omega}$

where λ = 2πc / ω is the vacuum wavelength and vg = dω / dβ is the group velocity. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency The 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 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. To help compare Orders of magnitude of different Times this page lists times between 10&minus12 seconds and 10&minus11 seconds (1 Pico A nanometre ( American spelling: nanometer, symbol nm) ( Greek: νάνος nanos dwarf; μετρώ metrό count) is a The kilometre ( American spelling: kilometer) symbol km is a unit of Length in the Metric system, equal to one thousand

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. Modal dispersion is a distortion mechanism occurring in Multimode fibers and other Waveguides in which the signal is spread in time because the propagation 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. Polarization mode dispersion (PMD is a form of Modal dispersion where two different Polarizations of light in a waveguide which normally travel at the same speed

## 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 wavelengths of 686. Technical terminology is the specialized Vocabulary of a field Gemology ( gemmology outside the United States) is the Science, Art and Profession of identifying and evaluating Gemstones In Physics and Optics, the Fraunhofer lines are a set of Spectral lines named for the German physicist Joseph von Fraunhofer ( 1787 In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. 7 nm and 430. A nanometre ( American spelling: nanometer, symbol nm) ( Greek: νάνος nanos dwarf; μετρώ metrό count) is a 8 nm and is meant to express the degree to which a prism cut from the gemstone shows "fire", or color. A gemstone or gem, also called a precious or semi-precious stone, is a piece of attractive Mineral, which &mdash when cut and polished &mdash 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. In Optics, chromatic aberration is caused by a lens having a different Refractive index for different Wavelengths of Light

## Dispersion in pulsar timing

Pulsars are spinning neutron stars that emit pulses at very regular intervals ranging from milliseconds to seconds. Pulsars are highly magnetized rotating Neutron stars that emit a beam of Electromagnetic radiation in the form of radio waves 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 ν is

$D = 4.15 ms (\frac{\nu}{GHz})^{-2} \times (\frac{DM}{cm^{-3} pc})$

where the dispersion measure DM is

$DM = \int_0^d{n_e\;dl}$

is the integrated free electron column density ne out to the pulsar at a distance d[4].

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 ΔT between a high frequency νhi and a low frequency νlo component of a pulse will be

$\Delta T = 4.15 ms [(\frac{\nu_{lo}}{GHz})^{-2} - (\frac{\nu_{hi}}{GHz})^{-2} ] \times (\frac{DM}{cm^{-3} pc})$

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 relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave. In Optics, the Sellmeier equation is an Empirical relationship between Refractive index n and Wavelength λ for a Cauchy's equation is an Empirical relationship between the Refractive index n and Wavelength of light λ for a particular transparent In Physics and Optics, the Abbe number, also known as the V-number or constringence of a transparent material is a measure The Kramers–Kronig relations are mathematical properties connecting the real and imaginary parts of any complex function which is analytic Group delay is a measure of the transit time of a signal through a Device under test (DUT versus frequency The calculation of glass properties ( glass modeling) is used to predict Glass properties of interest or glass behavior under certain conditions (e dispersion
• Linear response function
• Green-Kubo relations
• Fluctuation theorem

## References

1. ^ Born, Max (October 1999). A linear response function describes the input-output relationshipof a signal transducer such as a radio turning electromagnetic waves into musicor a neuron turning synaptic input into Green–Kubo relations give exact mathematical expression for transport coefficients in terms of integrals of time correlation functions The fluctuation theorem (FT is a theorem from Statistical mechanics dealing with the relative probability that the Entropy of a system which is currently away from Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum Principle of Optics. Cambridge: Cambridge University Press, pp. Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 14-24. ISBN 0521642221.
2. ^ Calculation of the Mean Dispersion of Glasses
3. ^ Rajiv Ramaswami and Kumar N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press: London 1998).
4. ^ 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.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9184602499008179, "perplexity_flag": "middle"}