# Group Delay Dispersion

Acronym: GDD

Definition: the frequency dependency of the group delay, or (quantitatively) the corresponding derivative with respect to angular frequency

Alternative term: second-order dispersion

German: Gruppenverzögerungsdispersion

Categories: general optics, light pulses

Formula symbol: *D*_{2}

Units: s^{2}

How to cite the article; suggest additional literature

Author: Dr. Rüdiger Paschotta

The group delay dispersion (also sometimes called *second-order dispersion*) of an optical element is a quantitative measure for chromatic dispersion.
It is defined as the derivative of the group delay, or the second derivative of the change in spectral phase, with respect to the angular optical frequency:

That derivative is always evaluated at a certain angular optical frequency – for example, at the center frequency of a laser pulse when considering the impact of chromatic dispersion on that pulse. If the group delay dispersion is independent of optical frequency, we have pure second-order dispersion and no higher-dispersion. Otherwise, third-order and other higher-order dispersion may be calculated via frequency derivatives of group delay dispersion.

The fundamental unit of group delay dispersion is s^{2} (seconds squared), but in practice it is usually specified in units of fs^{2} or ps^{2}.
Positive (negative) values correspond to normal (anomalous) chromatic dispersion.
For example, the group delay dispersion of a 1-mm thick silica plate is +35 fs^{2} at 800 nm (normal dispersion) or −26 fs^{2} at 1500 nm (anomalous dispersion).
Another example is given in Figure 1.

If an optical element has only second order dispersion, i.e., a frequency-independent *D*_{2} value, and no higher-order dispersion, its effect on an optical pulse or signal can be described as a change of the spectral phase:

where ω_{0} is the angular frequency at the center of the spectrum.

Note that the group delay dispersion (GDD) always refers to some optical element or to some given length of a medium (e.g. an optical fiber).
The GDD per unit length (in units of s^{2}/m) is the *group velocity dispersion* (GVD).

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### Bibliography

[1] | K. Naganuma et al., “Group-delay measurement using the Fourier transform of an interferometric cross correlation generated by white light”, Opt. Lett. 15 (7), 393 (1990), doi:10.1364/OL.15.000393 |

[2] | A. P. Kovacs et al., “Group-delay measurement on laser mirrors by spectrally resolved white-light interferometry”, Opt. Lett. 20 (7), 788 (1995), doi:10.1364/OL.20.000788 |

[3] | S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry”, J. Opt. Soc. Am. B 13 (6), 1120 (1996), doi:10.1364/JOSAB.13.001120 |

[4] | A. Gosteva et al., “Noise-related resolution limit of dispersion measurements with white-light interferometers”, J. Opt. Soc. Am. B 22 (9), 1868 (2005), doi:10.1364/JOSAB.22.001868 |

[5] | T. V. Amotchkina et al., “Measurement of group delay of dispersive mirrors with white-light interferometer”, Appl. Opt. 48 (5), 949 (2009), doi:10.1364/AO.48.000949 |

See also: chromatic dispersion, group velocity dispersion, group delay

and other articles in the categories general optics, light pulses

2020-11-13

Is it possible to estimate the range of GDD values expected for generic broadband dielectric mirrors?

Answer from the author:

That depends on the mirror design and refractive index contrast. For a given design, this could be easily calculated with suitable thin-film coating software such as RP Coating.