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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,^{ [1] }^{ [2] }

**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.

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

The **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.

- Applications
- Derivation
- Alternate derivation
- Group delay dispersion
- See also
- External links
- References

where and are angular frequencies, and the group velocity is defined as . The units of group velocity dispersion are [time]^{2}/[distance], often expressed in fs^{2}/mm.

Equivalently, group velocity dispersion can be defined in terms of the medium-dependent wave vector according to

or in terms of the refractive index according to

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

Group velocity dispersion is most commonly used to estimate the amount of chirp that will be imposed on a pulse of light after passing through a material of interest. The relevant expression is given by

A **chirp** is a signal in which the frequency increases (*up-chirp*) or decreases (*down-chirp*) with time. In some sources, the term *chirp* is used interchangeably with **sweep signal**. It is commonly used in sonar, radar, and laser, but has other applications, such as in spread-spectrum communications.

A simple illustration of how GVD can be used to determine pulse chirp can be seen by looking at the effect of a transform-limited pulse of duration passing through a planar medium of thickness *d*. Before passing through the medium, the phase offsets of all frequencies are aligned in time, and the pulse can be described as a function of time according to the expression

A **bandwidth-limited pulse** is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. Bandwidth-limited pulses have a constant phase across all frequencies making up the pulse. Optical pulses of this type can be generated by mode-locked lasers.

or equivalently, as a function of frequency according to the expression

(the parameters *A* and *B* are normalization constants). Passing through the medium results in a frequency-dependent phase accumulation , such that the post-medium pulse can be described by

In general, the refractive index , and therefore the wave vector , can be an arbitrary function of , making it difficult to analytically perform the inverse Fourier transform back into the time domain. However, if the bandwidth of the pulse is narrow relative to the curvature of , then good approximations of the impact of the refractive index can be obtained by replacing with its Taylor expansion centered about :

Truncating this expression and inserting it into the post-medium frequency-domain expression results in a post-medium time-domain expression of

- .

On balance, the pulse will have lengthened to an intensity standard deviation value of

In statistics, the **standard deviation** is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

thus validating the initial expression. Note that for a transform-limited pulse σ_{t}σ_{t} = 1/2, which makes it appropriate to identify 1/(2σ_{t}) as the bandwidth.

An alternate derivation of the relationship between pulse chirp and GVD, which more immediately illustrates the reason why GVD can be defined by the derivative of inverse group velocity, can be outlined as follows. Consider two transform-limited pulses of carrier frequencies and , which are initially overlapping in time. After passing through the medium, these two pulses will exhibit a time delay between their respective pulse-envelope centers, given by

The expression can be approximated as a Taylor expansion, giving

or,

From here it is possible to imagine scaling this expression up two pulses to infinitely many. The frequency difference must be replaced by the bandwidth, and the time delay evolves into the induced chirp.

A closely related yet independent quantity is the **group delay dispersion** (**GDD**), defined such that group velocity dispersion is the group delay dispersion per unit length. GDD is commonly used as a parameter in characterizing layered mirrors, where the group velocity dispersion is not particularly-well defined, yet the chirp induced after bouncing off the mirror can be well-characterized. The units of group delay dispersion are [time]^{2}, often expressed in fs^{2}.

The group delay dispersion (GDD) of an optical element is the derivative of the group delay with respect to angular frequency, and also the second derivative of the optical phase. . It is a measure of the chromatic dispersion of the element. GDD is related to the total dispersion parameter as

- Online refractive index database: .
- RP Photonics Encyclopedia: .
- Commercial Optical Dispersion Measurement with White Light Interferometry

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

In physics, a **wave** is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals.

The **active laser medium** is the source of optical gain within a laser. The gain results from the stimulated emission of electronic or molecular transitions to a lower energy state from a higher energy state previously populated by a pump source.

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.

The **Drude model** of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials. The model, which is an application of kinetic theory, assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like a pinball machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.

In 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.

In optics, an **ultrashort pulse** of light is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.

In fluid dynamics, **dispersion** of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

**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.

A **perfectly matched layer** (**PML**) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods. The key property of a PML that distinguishes it from an ordinary absorbing material is that it is designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML to strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior.

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:

The **McCumber relation** is a relationship between the effective cross-sections of absorption and emission of light in the physics of solid-state lasers. It is named after Dean McCumber, who proposed the relationship in 1964.

The **omega equation** is of great importance in meteorology and atmospheric physics. It is a partial differential equation for the vertical velocity, , which is defined as the Lagrangian rate of change of pressure with time. Mathematically, , where represents a material derivative. It is valid for large scale flows under the conditions of quasi-geostrophy and hydrostatic balance. In fact, one may consider the vertical velocity that results from solving the omega equation as that which is needed to maintain quasi-geostrophy and hydrostasy.

The equation reads:

**Precursors** are characteristic wave patterns caused by dispersion of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of electromagnetic radiation as well as in fluid surface waves and seismic waves.

A **Sverdrup wave** is a wave in the ocean, which is affected by gravity and Earth's rotation.

In the fields of nonlinear optics and fluid dynamics, **modulational instability** or **sideband instability** is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.

The **Vanna–Volga method** is a mathematical tool used in finance. It is a technique for pricing first-generation exotic options in foreign exchange market (FX) derivatives.

In the physics of continuous media, **spatial dispersion** is a phenomenon where material parameters such as permittivity or conductivity have dependence on wavevector. Normally, such a dependence is assumed to be absent for simplicity, however spatial dispersion exists to varying degrees in all materials.

- ↑ Boyd, Robert. W (2007).
*Nonlinear Optics*(3rd ed.). Elsevier. - ↑ Paschotta, Dr. Rüdiger. "Encyclopedia of Laser Physics and Technology - group velocity dispersion".
*www.rp-photonics.com*. Retrieved 2016-05-15.

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