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# Group Velocity Dispersion

Acronym: GVD

Definition: the frequency dependence of the group velocity in a medium, or (quantitatively) the derivative of the inverse group velocity with respect to angular frequency

More general term: chromatic dispersion

German: Gruppengeschwindigkeitsdispersion

Formula symbol: β2

Units: s2/m

Author:

Group velocity dispersion is the phenomenon that the group velocity of light in a transparent medium depends on the optical frequency or wavelength. The term can also be used as a precisely defined quantity, namely the derivative of the inverse group velocity with respect to the angular frequency (or sometimes the wavelength): where k is the frequency-dependent wavenumber. (For waveguides, it is replaced with the phase constant β.)

The group velocity dispersion is the group delay dispersion per unit length. The basic units are s2/m. For example, the group velocity dispersion of fused silica is +35 fs2/mm at 800 nm and −26 fs2/mm at 1500 nm. Somewhere between these wavelengths (at about 1.3 μm), there is the zero-dispersion wavelength.

For optical fibers (e.g. in the context of optical fiber communications), the group velocity dispersion is usually defined as a derivative with respect to wavelength (rather than angular frequency). This can be calculated from the above-mentioned GVD parameter: where c is the vacuum velocity of light.

This quantity is usually specified with units of ps/(nm km) (picoseconds per nanometer wavelength change and kilometer propagation distance). For example, 20 ps/(nm km) at 1550 nm (a typical value for telecom fibers) corresponds to −25 509 fs2/m.

## Conversion of Chromatic Dispersion Values

 Center wavelength: Group velocity dispersion: calc (1 fs2 = 1e-30 s2) Dispersion parameter: calc

Enter input values with units, where appropriate. After you have modified some values, click a "calc" button to recalculate the field left of it.

It is important to realize the different signs of GVD and Dλ, resulting from the fact that longer wavelengths correspond to smaller optical frequencies. In order to avoid confusion, the terms normal and anomalous dispersion can be used instead of positive and negative dispersion. Normal dispersion implies that the group velocity decreases for increasing optical frequency; this is the most common situation.

Depending on the situation, group velocity dispersion can have different important effects:

## Questions and Comments from Users

2020-05-12

I am wondering about the example numbers you are giving here. Specifically the −26 fs2/mm at 1500 nm; by using the calculator, I get 2.25 · 103 fs2/m. Why is there a difference or am I missing something here?

The calculator is not calculating the chromatic dispersion of silica, but only converting chromatic dispersion values given with different units. The value which you obtain just resulted from the conversion of the original value of −1.88 ps/(nm km) to the other units.

2020-06-03

To calculate the GDD of a waveguide, would one just need to multiply GVD with the length of the waveguide?

Exactly.

2021-01-28

In order to get the shortest possible pulse duration from a mode-locked laser, should one minimize the total GDD inside the laser resonator?

When choosing chirped mirrors or GTI mirrors, should the ones with no fluctuation of GDD curve be better?

The shortest possible pulse duration is often achieved when the total group delay dispersion is negative, so that quasi-soliton pulses are formed. However, that depends on the type of laser.

To which extent spectral variations of the GDD matter, depends on the bandwidth of the pulses to be generated. Certainly, the GDD in spectral regions totally outside the pulse spectrum does not matter. On the other hand, it is not necessarily required that the GDD is hardly varying within the pulse bandwidth. A better criterion is that the round-trip phase shift should not very too much within the spectrum. However, it again depends on the type of laser how critical that is.

2021-02-10

What is the difference between GVD and modal dispersion?

Modal dispersion, or more precisely intermodal dispersion, is related to differences between different propagation modes, not within a single mode.

2021-02-23

In order to determine the material dispersion in units of ps/nm km, what is the value of speed of light that we must use to get this unit of material dispersion?

It's essentially the frequency derivative of the group velocity.

2021-07-26

GVD comes from the differentiation of group delay with respect to angular frequency. Hence I think unit should be (s / m) / (rad / s) = s2 / (rad m). But your units are s2/m, as in many textbooks. How did the 'radian' disappear?

Radians are no real dimension; they are defined as a ratio of arc length to radius. Therefore, they are often just omitted. For example, angular frequencies are often specified in units of s−1 rather than rad/s.

2021-09-17

Can I calculate the third order dispersion using the GVD value?

Yes, if you know its frequency derivative, and not just the value at one frequency.

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