# Group Velocity Dispersion

Author: the photonics expert Dr. Rüdiger Paschotta

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

Categories: general optics, fiber optics and waveguides, light pulses

Units: s^{2}/m

Formula symbol: <$\beta_2$>

**Summary:** This authoritative article explains

- how the group velocity dispersion is defined qualitatively and quantitatively,
- how the conversion between wavelength-based and frequency-based definitions of group velocity dispersion works, and
- how the signs of those quantities separate the regimes of normal and anomalous dispersion.

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), called <$\beta_2$>:

where <$k$> is the frequency-dependent wavenumber. (For waveguides, it is replaced with the phase constant <$\beta$>.)

The group velocity dispersion is the group delay dispersion per unit length. The basic SI units are s^{2}/m. For example, the group velocity dispersion of fused silica is +35 fs^{2}/mm at 800 nm and −26 fs^{2}/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:

$${D_\lambda } = \frac{\partial }{{\partial \lambda }}\frac{1}{{{\upsilon _{\rm{g}}}}} = - \frac{{2\pi c}}{{{\lambda ^2}}} \cdot {\beta _2}{\rm{ = }} - \frac{{2\pi c}}{{{\lambda ^2}}} \cdot \frac{{{\partial ^2}k}}{{\partial {\omega ^2}}}$$where <$c$> is the vacuum velocity of light. One can also relate <$D_{\lambda }$> to the second wavelength derivative of the refractive index:

$${D_\lambda } = - \frac{\lambda }{c} \cdot \frac{{{\partial ^2}n}}{{\partial {\lambda ^2}}}$$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 fs^{2}/m.

It is important to realize the different signs of GVD and <$D_{\lambda }$>, 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:

- It is responsible for dispersive temporal broadening or compression of ultrashort pulses.
- In optical fibers, the effect of nonlinearities strongly depends on the group velocity dispersion. For example, there may be spectral broadening (even supercontinuum generation) or compression, depending on the dispersion properties.
- Dispersion is also responsible for the group velocity mismatch of different waves in parametric nonlinear interactions. For example, it can limit the interaction bandwidth in frequency doublers, optical parametric oscillators and amplifiers.

## Case Study: Dispersion Engineering for Telecom Fibers

We explore different ways of optimizing refractive index profile for specific chromatic dispersion properties of telecom fibers, resulting in dispersion-shifted or dispersion-flattened fibers. This also involves automatic optimizations.

## More to Learn

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## Questions and Comments from Users

2020-06-03

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

The author's answer:

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 author's answer:

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?

The author's answer:

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?

The author's answer:

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) = s^{2} / (rad m). But your units are s^{2}/m, as in many textbooks. How did the 'radian' disappear?

The author's answer:

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?

The author's answer:

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

2022-01-01

Can we express the dispersion parameter in units of ps/km unit rather than ps/(nm km)?

The author's answer:

Such units do occur, but for different quantities – in particular, for intermodal dispersion, where differences in group delay result from different modes rather than from differences in wavelength.

2023-01-19

Is group velocity dispersion time-dependent? If not, I am wondering which effect might cause the signal propagation delay to vary over time.

The author's answer:

The group velocity dispersion of the fiber in a fiber-optic link, for example, may vary due to changes of temperature and/or mechanical stress. That also affects propagation delays.

2023-08-26

If I put all the amounts of wavelengths in microns, then the derivation of refractive index with respect to wavelength will also be in 1/micron. If I want to calculate GVD, what units for wavelength and speed of light should I use to obtain amount in fs^{2}/mm?

The author's answer:

The simplest way is always to calculate all numerical quantities in fundamental SI units (e.g., wavelengths in meters), and then convert to the desired units only in the end.

2024-02-05

Does positive dispersion parameter (negative GVD) help with pulse compression?

The author's answer:

That always depends on how the original pulse is chirped. Negative GVD helps for up-chirped pulses.

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2020-05-12

I am wondering about the example numbers you are giving here. Specifically the −26 fs

^{2}/mm at 1500 nm; by using the calculator, I get 2.25 · 10^{3}fs^{2}/m. Why is there a difference or am I missing something here?The author's answer:

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.