Dispersion (optics)

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In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. [1]

Optics The branch of physics that studies light

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

Phase velocity rate at which the phase of the wave propagates in space

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity 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. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

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Media having this common property may be termed dispersive media. Sometimes the term chromatic dispersion is used for specificity.

Although the term is used in the field of optics to describe light and other electromagnetic waves, dispersion in the same sense can apply to any sort of wave motion such as acoustic dispersion in the case of sound and seismic waves, in gravity waves (ocean waves), and for telecommunication signals along transmission lines (such as coaxial cable) or optical fiber.

Light electromagnetic radiation in or near visible spectrum

Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to visible light, which is the visible spectrum that is visible to the human eye and is responsible for the sense of sight. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), or 4.00 × 10−7 to 7.00 × 10−7 m, between the infrared and the ultraviolet. This wavelength means a frequency range of roughly 430–750 terahertz (THz).

Acoustic dispersion is the phenomenon of a sound wave separating into its component frequencies as it passes through a material. The phase velocity of the sound wave is viewed as a function of frequency. Hence, separation of component frequencies is measured by the rate of change in phase velocities as the radiated waves pass through a given medium.

Gravity wave Wave in or at the interface between fluids where gravity is the main equilibrium force

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the ocean, which gives rise to wind waves.

In a dispersive prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a spectrum. Prism rainbow schema.png
In a dispersive prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a spectrum.
A compact fluorescent lamp seen through an Amici prism. Light dispersion of a compact fluorescent lamp seen through an Amici direct-vision prism PNrdeg0114.jpg
A compact fluorescent lamp seen through an Amici prism.

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 number V, 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.

Refraction refraction of light

In physics, refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

Prism transparent optical element with flat, polished surfaces that refract light

In optics, a prism is a transparent optical element with flat, polished surfaces that refract light. At least two of the flat surfaces must have an angle between them. 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 "prism" usually refers to this type. Some types of optical prism are not in fact in the shape of geometric prisms. Prisms can be made from any material that is transparent to the wavelengths for which they are designed. Typical materials include glass, plastic, and fluorite.

Chromatic aberration Failure of a lens to focus all colors on the same point

In optics, chromatic aberration is a failure of a lens to focus all colors to the same point. It is caused by dispersion: the refractive index of the lens elements varies with the wavelength of light. The refractive index of most transparent materials decreases with increasing wavelength. Since the focal length of a lens depends on the refractive index, this variation in refractive index affects focusing. Chromatic aberration manifests itself as "fringes" of color along boundaries that separate dark and bright parts of the image.

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 (GVD) 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.

Rainbow meteorological phenomenon

A rainbow is a meteorological phenomenon that is caused by reflection, refraction and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky. It takes the form of a multicoloured circular arc. Rainbows caused by sunlight always appear in the section of sky directly opposite the sun.

Color Characteristic of human visual perception

Color, or colour, is the characteristic of human visual perception described through color categories, with names such as red, orange, yellow, green, blue, or purple. This perception of color derives from the stimulation of cone cells in the human eye by electromagnetic radiation in the visible spectrum. Color categories and physical specifications of color are associated with objects through the wavelength of the light that is reflected from them. This reflection is governed by the object's physical properties such as light absorption, emission spectra, etc.

In optics, group velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium will affect the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the inverse of group velocity of light in a material with respect to angular frequency,

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 ] 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.

Refractive index dimensionless number that describes how fast light propagates through the material

In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light travels through the material. It is defined as

Waveguide structure that guides waves, typically electromagnetic waves

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

Photonic crystal a periodic optical nanostructure that affects the motion of photons in much the same way that ionic lattices affect electrons in solids

A photonic crystal is a periodic optical nanostructure that affects the motion of photons in much the same way that ionic lattices affect electrons in solids. Photonic crystals occur in nature in the form of structural coloration and animal reflectors, and, in different forms, promise to be useful in a range of applications.

Material dispersion in optics

The variation of refractive index vs. vacuum wavelength for various glasses. The wavelengths of visible light are shaded in grey. Mplwp dispersion curves.svg
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 for l = 486 nm (blue), nC valid for l = 656 nm (red)) Spidergraph Dispersion.GIF
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))

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. Holographic gratings are also used, as they allow more accurate discrimination of wavelengths. However, in lenses, dispersion causes chromatic aberration, an undesired effect that may degrade images in microscopes, telescopes, and photographic objectives.

Spectrometer instrument used to measure properties of light

A spectrometer is a scientific instrument used to separate and measure spectral components of a physical phenomenon. Spectrometer is a broad term often used to describe instruments that measure a continuous variable of a phenomenon where the spectral components are somehow mixed. In visible light a spectrometer can separate white light and measure individual narrow bands of color, called a spectrum. A mass spectrometer measures the spectrum of the masses of the atoms or molecules present in a gas. The first spectrometers were used to split light into an array of separate colors. Spectrometers were developed in early studies of physics, astronomy, and chemistry. The capability of spectroscopy to determine chemical composition drove its advancement and continues to be one of its primary uses. Spectrometers are used in astronomy to analyze the chemical composition of stars and planets, and spectrometers gather data on the origin of the universe.

A spectroradiometer is a light measurement tool that is able to measure both the wavelength and amplitude of the light emitted from a light source. Spectrometers discriminate the wavelength based on the position the light hits at the detector array allowing the full spectrum to be obtained with a single acquisition. Most spectrometers have a base measurement of counts which is the un-calibrated reading and is thus impacted by the sensitivity of the detector to each wavelength. By applying a calibration, the spectrometer is then able to provide measurements of spectral irradiance, spectral radiance and/or spectral flux. This data is also then used with built in or PC software and numerous algorithms to provide readings or Irradiance (W/cm2), Illuminance, Radiance (W/sr), Luminance (cd), Flux, Chromaticity, Color Temperature, Peak and Dominant Wavelength. Some more complex spectrometer software packages also allow calculation of PAR µmol/m²/s, Metamerism, and candela calculations based on distance and include features like 2- and 20-degree observer, baseline overlay comparisons, transmission and reflectance.

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

where c is the speed of light in a vacuum and n is the refractive index of the medium.

In general, the refractive index is some function of the frequency f of the light, thus n = n(f), or 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 susceptibility χ that appears in the Kramers–Kronig relations is the electric susceptibility χe = 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 alternatively:

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.

Group velocity dispersion

Time evolution of a short pulse in a hypothetical dispersive medium (k=w^2) showing that the longer wavelength components travel faster than the high frequency ones (positive GVD), resulting in chirping and pulse broadening. Optical dispersion dynamics.gif
Time evolution of a short pulse in a hypothetical dispersive medium (k=w^2) showing that the longer wavelength components travel faster than the high frequency ones (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 termed group velocity dispersion. While phase velocity v is defined as v = c/n, this describes only one frequency component. When different frequency components are combined together, 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 which is 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 v=ω/k, the group velocity can be expressed using the derivative: vg=dω/dk. Or in terms of the phase velocity vp,

For a more complete explanation of the significance of group velocity and mathematics surrounding it, the reader may consult the article: group velocity.

When dispersion is present, not only will the group velocity not be equal to the phase velocity, but generally will itself vary with wavelength. This is known as group velocity dispersion (GVD) and causes a short pulse of light to be broadend, as the different frequency components within the pulse travel at different velocities. GVD is quantified as the derivative of the reciprocal of the group velocity with respect to radian frequency which results in GVD=d2k/dω2. GVD is also often quantified using the group velocity dispersion parameter:

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, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped , or down-chirped, decreasing in frequency with time.

Dispersion control

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 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 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 frequency ω(β) at a propagation constant β (so that the electromagnetic fields in the propagation direction z oscillate proportional to ei(βzωt)), the group-velocity dispersion parameter D is defined as: [5]

where λ = 2πc/ω is the vacuum wavelength and vg = / 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.

In the case of multi-mode optical fibers, so-called modal dispersion will also lead to pulse broadening. Even in single-mode fibers, pulse broadening can occur as a result of polarization mode dispersion (since there are still two polarization modes). These are not examples of chromatic dispersion as they are not dependent on the wavelength or bandwidth of the pulses propagated.

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 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] These terms are simply a Taylor series expansion of the dispersion relation β(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.

In gemology

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. [7] [8]

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

where the dispersion constant kDM is given by

[9]

and the dispersion measure (DM) is the column density of free electrons (total electron content) — i.e. the number density of electrons ne (electrons/cm3) 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). [10]

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 for observations of pulsars at different frequencies to be combined.

See also

Related Research Articles

Wavelength spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

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.

Wavenumber spatial frequency of a wave

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by John Kerr, a Scottish physicist.

Dispersion relation Relation of wavelength/wavenumber as a function of a waves frequency

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

Optical ring resonators

An optical ring resonator is a set of waveguides in which at least one is a closed loop coupled to some sort of light input and output. The concepts behind optical ring resonators are the same as those behind whispering galleries except that they use light and obey the properties behind constructive interference and total internal reflection. When light of the resonant wavelength is passed through the loop from input waveguide, it builds up in intensity over multiple round-trips due to constructive interference and is output to the output bus waveguide which serves as a detector waveguide. Because only a select few wavelengths will be at resonance within the loop, the optical ring resonator functions as a filter. Additionally, as implied earlier, two or more ring waveguides can be coupled to each other to form an add/drop optical filter.

Self-phase modulation (SPM) 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. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.

Fiber Bragg grating

A fiber Bragg grating (FBG) is a type of distributed Bragg reflector constructed in a short segment of optical fiber that reflects particular wavelengths of light and transmits all others. This is achieved by creating a periodic variation in the refractive index of the fiber core, which generates a wavelength-specific dielectric mirror. A fiber Bragg grating can therefore be used as an inline optical filter to block certain wavelengths, or as a wavelength-specific reflector.

Grating-eliminated no-nonsense observation of ultrafast incident laser light e-fields (GRENOUILLE) is an ultrashort pulse measurement technique based on frequency-resolved optical gating (FROG). The acronym was chosen because of the technique's relationship to FROG; grenouille is French for frog.

Slow light is the propagation of an optical pulse or other modulation of an optical carrier at a very low group velocity. Slow light occurs when a propagating pulse is substantially slowed down by the interaction with the medium in which the propagation takes place.

Acousto-optics

Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound through an ultrasonic grating.

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 and linear effects in the medium. There are two main kinds of solitons:

References

  1. Born, Max; Wolf, Emil (October 1999). Principles of Optics. Cambridge: Cambridge University Press. pp. 14–24. ISBN   0-521-64222-1.
  2. Dispersion Compensation Retrieved 25-08-2015.
  3. Calculation of the Mean Dispersion of Glasses
  4. Born, M. and Wolf, E. (1980) "Principles of Optics, 6th ed." pg. 93. Pergamon Press.
  5. Ramaswami, Rajiv and Sivarajan, Kumar N. (1998) Optical Networks: A Practical Perspective. Academic Press: London.
  6. Chromatic Dispersion, Encyclopedia of Laser Physics and Technology (Wiley, 2008).
  7. 1 2 Schumann, Walter (2009). Gemstones of the World: Newly Revised & Expanded Fourth Edition. Sterling Publishing Company, Inc. pp. 41–2. ISBN   978-1-4027-6829-3 . Retrieved 31 December 2011.
  8. What is Gemstone Dispersion? by International Gem Society (IGS). Retrieved 03-09-2015
  9. Single-Dish Radio Astronomy: Techniques and Applications, ASP Conference Proceedings, Vol. 278. Edited by Snezana Stanimirovic, Daniel Altschuler, Paul Goldsmith, and Chris Salter. ISBN   1-58381-120-6. San Francisco: Astronomical Society of the Pacific, 2002, p. 251-269
  10. 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.