# Group velocity dispersion

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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.

## Contents

${\displaystyle {\textrm {GVD}}(\omega _{0})\equiv {\frac {\partial }{\partial \omega }}\left({\frac {1}{v_{g}(\omega )}}\right)_{\omega =\omega _{0}},}$

where ${\displaystyle \omega }$ and ${\displaystyle \omega _{0}}$ are angular frequencies, and the group velocity ${\displaystyle v_{g}(\omega )}$ is defined as ${\displaystyle v_{g}(\omega )\equiv \partial \omega /\partial k}$. The units of group velocity dispersion are [time]2/[distance], often expressed in fs2/mm.

Equivalently, group velocity dispersion can be defined in terms of the medium-dependent wave vector ${\displaystyle k(\omega )}$ according to

${\displaystyle {\textrm {GVD}}(\omega _{0})\equiv \left({\frac {\partial ^{2}k}{\partial \omega ^{2}}}\right)_{\omega =\omega _{0}},}$

or in terms of the refractive index ${\displaystyle n(\omega )}$ 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

${\displaystyle {\textrm {GVD}}(\omega _{0})\equiv {\frac {2}{c}}\left({\frac {\partial n}{\partial \omega }}\right)_{\omega =\omega _{0}}+{\frac {\omega _{0}}{c}}\left({\frac {\partial ^{2}n}{\partial \omega ^{2}}}\right)_{\omega =\omega _{0}}.}$

## Applications

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.

${\displaystyle {\textrm {chirp}}=({\textrm {material}}\,\,{\textrm {thickness}})\,\times \,{\textrm {GVD}}(\omega _{0})\,\times \,({\textrm {bandwidth}}).}$

### Derivation

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 ${\displaystyle \sigma }$ 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.

${\displaystyle E(t)=Ae^{-{\frac {t^{2}}{4\sigma ^{2}}}}e^{-i\omega _{0}t},}$

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

${\displaystyle E(\omega )=Be^{-{\frac {(w-w_{0})^{2}}{4(1/2\sigma )^{2}}}}}$

(the parameters A and B are normalization constants). Passing through the medium results in a frequency-dependent phase accumulation ${\displaystyle \Delta \phi (\omega )=k(\omega )d}$ , such that the post-medium pulse can be described by

${\displaystyle E(\omega )=Be^{-{\frac {(w-w_{0})^{2}}{4(1/2\sigma )^{2}}}}e^{ik(\omega )d}.}$

In general, the refractive index ${\displaystyle n(\omega )}$, and therefore the wave vector ${\displaystyle k(\omega )=n(\omega )\omega /c}$, can be an arbitrary function of ${\displaystyle \omega }$, 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 ${\displaystyle n}$, then good approximations of the impact of the refractive index can be obtained by replacing ${\displaystyle k(\omega )}$ with its Taylor expansion centered about ${\displaystyle \omega _{0}}$:

${\displaystyle {\frac {n(\omega )\omega }{c}}=\underbrace {\frac {n(\omega _{0})\omega _{0}}{c}} _{k(\omega _{0})}\quad +\quad \underbrace {\left[{\frac {n(\omega _{0})+n'(\omega _{0})\omega _{0}}{c}}\right]} _{k'(\omega _{0})}(\omega -\omega _{0})\quad +\quad {\frac {1}{2}}\underbrace {\left[{\frac {2n'(\omega _{0})+n''(\omega _{0})\omega _{0}}{c}}\right]} _{\textrm {GVD}}(\omega -\omega _{0})^{2}\quad +\quad ...}$

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

${\displaystyle E_{post}(t)=A_{post}\exp \left[-{\frac {(t-k'(\omega _{0})d)^{2}}{4(\sigma ^{2}-i\,{\textrm {GVD}}\,d/2)}}\right]e^{i[k(\omega _{0})d-\omega _{0}t]}}$.

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.

${\displaystyle \sigma _{post}={\sqrt {\sigma ^{2}+\left[d\,\times \,{\textrm {GVD}}(\omega _{0})\,\times \,\left({\frac {1}{2\sigma }}\right)\right]^{2}}}}$

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

### Alternate derivation

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 ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$, 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

${\displaystyle \Delta T=d\left({\frac {1}{v_{g}(\omega _{2})}}-{\frac {1}{v_{g}(\omega _{1})}}\right).}$

The expression can be approximated as a Taylor expansion, giving

${\displaystyle \Delta T=d\left({\frac {1}{v_{g}(\omega _{1})}}+{\frac {\partial }{\partial \omega }}\left({\frac {1}{v_{g}(\omega ')}}\right)_{\omega '=\omega _{1}}(\omega _{2}-\omega _{1})-{\frac {1}{v_{g}(\omega _{1})}}\right),}$

or,

${\displaystyle \Delta T=d\,\times \,{\textrm {GVD}}(\omega _{1})\,\times \,(\omega _{2}-\omega _{1}).}$

From here it is possible to imagine scaling this expression up two pulses to infinitely many. The frequency difference ${\displaystyle \omega _{2}-\omega _{1}}$ must be replaced by the bandwidth, and the time delay ${\displaystyle \Delta T}$ evolves into the induced chirp.

## Group delay dispersion

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 fs2.

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. ${\displaystyle D_{2}(\omega )=-{\frac {\partial T_{g}}{d\omega }}={\frac {d^{2}\phi }{d\omega ^{2}}}}$. It is a measure of the chromatic dispersion of the element. GDD is related to the total dispersion parameter ${\displaystyle D_{tot}}$ as

${\displaystyle D_{2}(\omega )=-{\frac {2\pi c}{\lambda ^{2}}}D_{tot}}$