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# 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

Units: s2

Formula symbol: <$D_2$>

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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:

$${D_2}(\omega ) = \frac{{\partial {T_{\rm{g}}}}}{{\partial \omega }} = \frac{{{\partial ^2}\varphi }}{{\partial {\omega ^2}}}$$

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.

If two optical pulses travel through an optical element with a frequency-independent group delay dispersion <$D_2$>, and their center optical frequencies differ by <$\Delta \nu$>, their group delay differs by <$2\pi \: D_2 \: \Delta\nu$>.

The fundamental unit of group delay dispersion is s2 (seconds squared), but in practice it is usually specified in units of fs2 or ps2. (Note that 1 ps = 1000 fs, thus 1 ps2 = 1,000,000 fs2.) Positive (negative) values correspond to normal (anomalous) chromatic dispersion. For example, the group delay dispersion of a 1-mm thick silica plate is +35 fs2 at 800 nm (normal dispersion) or −26 fs2 at 1500 nm (anomalous dispersion). Another example is given in Figure 1. Figure 1: Wavelength-dependent group delay dispersion of a Gires–Tournois interferometer made of a 5-μm thick silica layer on a high reflector.

## Spectral Phase and Group Delay

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:

$$\Delta \varphi (\omega ) = \frac{{{D_2}}}{2}{\left( {\omega - {\omega _0}} \right)^2}$$

where <$\omega_0$> is the angular frequency at the center of the spectrum.

An alternative way of specifying group delay dispersion is referring to the vacuum wavelength instead of the angular optical frequency. That leads to a value in units of ps/nm (picoseconds per nanometer), for example. It can be calculated from the GDD as defined above:

$$D_{2\lambda} = - \frac{{2\pi c}}{{{\lambda ^2}}} \cdot {D_2} = - \frac{{2\pi c}}{{{\lambda ^2}}} \cdot \frac{{{\partial ^2}\varphi }}{{\partial {\omega ^2}}}$$

Note the different signs of both quantities: higher optical frequencies are associated with shorter wavelengths.

Higher-order dispersion is often specified in the form of the dispersion slope, i.e., the wavelength derivative of <$D_{\lambda }$>. From that, the third-order dispersion can be calculated as follows:

$$D_{3\lambda} = - \frac{{{\lambda ^2}}}{{2\pi c}}\frac{\partial }{{\partial \lambda }}\left( { - \frac{{{\lambda ^2}}}{{2\pi c}}D_{2\lambda }} \right) = {\left( {\frac{{{\lambda ^2}}}{{2\pi c}}} \right)^2}\frac{{\partial D_{2\lambda }}}{{\partial \lambda }} + \frac{{{\lambda ^3}}}{{2{\pi ^2}{c^2}}}D_{2\lambda }$$

For example, a fiber with zero dispersion slope (wavelength-independent <$D_{\lambda }$>) would generally have some non-zero third-order dispersion.

## Relation to Group Velocity Dispersion

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 s2/m) is the group velocity dispersion (GVD).

## Measurement of Group Delay Dispersion

There are various methods for measuring the GDD of an optical element. In the case of optical fibers, one may use the pulse delay technique, based on measuring the difference in propagation time (group delay) for light pulses with different center wavelengths. There are also methods based on interferometry. For details, see the article on chromatic dispersion.

### Bibliography

  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); https://doi.org/10.1364/OL.15.000393  A. P. Kovacs et al., “Group-delay measurement on laser mirrors by spectrally resolved white-light interferometry”, Opt. Lett. 20 (7), 788 (1995); https://doi.org/10.1364/OL.20.000788  S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry”, J. Opt. Soc. Am. B 13 (6), 1120 (1996); https://doi.org/10.1364/JOSAB.13.001120  A. Gosteva et al., “Noise-related resolution limit of dispersion measurements with white-light interferometers”, J. Opt. Soc. Am. B 22 (9), 1868 (2005); https://doi.org/10.1364/JOSAB.22.001868  T. V. Amotchkina et al., “Measurement of group delay of dispersive mirrors with white-light interferometer”, Appl. Opt. 48 (5), 949 (2009); https://doi.org/10.1364/AO.48.000949

## Questions and Comments from Users

2020-11-13

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

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.

2021-02-28

What is “group delay matching”?

That will probably depend on the context. For example, it may mean the matching of the arm lengths in an interferometer such that pulses going through the two arms can interfere with each other when getting recombined. Another context would be a synchronously pumped optical parametric oscillator, where the group delay of one resonator round-trip needs to be matched to the spacing of the pump pulses.

2021-03-03

How do pulses propagate when there is zero GDD but non-zero TOD?

That also leads to a kind of pulse broadening, but more complicated than for GDD only. Such things can be most conveniently studied with suitable software, e.g. RP Fiber Power.

2023-09-01

It seems most suppliers have extensive catalogs of low-GDD mirrors, but not low-GDD lenses. Is there a reason why GDD is less of a problem in the case of lenses vs mirrors?

Mirrors have various design parameters (layer thickness parameters) which can be optimized concerning GDD, but that is not the case for lenses. By the way, for lenses it is not even straight-forward to define the GDD, as the output is distorted by chromatic aberrations.

2023-10-05

For Gaussian pulses a constant GDD leads to a linear frequency chirp. Is this generally the case also for other pulse shapes? Would for instance a sech2-shaped pulse also exactly have a linear chirp for constant GDD? Share this with your friends and colleagues, e.g. via social media:   These sharing buttons are implemented in a privacy-friendly way!