定义：
群时延随频率变化，或者群时延对角频率的微分。
光学元件的群时延色散（有时也被称为二阶色散）是指群时延对角频率的微分，或者光谱相位的二阶微分： 

通常采用的单位为fs2或ps2。正值（负值）代表正常（反常）色散(Chromatic dispersion)。例如，1 mm厚的二氧化硅片的群时延色散在800 nm时为+35 fs2（正常色散）而在1500 nm时为-26 fs2（反常色散）。图1给出了另一个例子。 

图1：反射器上放置5微米厚的二氧化硅层的Gires-Tournois干涉仪中的群时延色散随波长的变化。 
群时延色散（GDD）通常是指一些光学元件或者一定长度的介质（如光纤）。单位长度的GDD（单位为s2/m）就是群速度色散（GVD）。

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

Formula symbol: D₂

Units: s²

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.

If two optical pulses travel through an optical element with a frequency-independent group delay dispersion D₂, and their center optical frequencies differ by Δν, their group delay differs by 2π D₂ Δν.

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 D₂ 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 ω₀ is the angular frequency at the center of the spectrum.

Wavelength Instead of Frequency

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:

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_λ. From that, the third-order dispersion can be calculated as follows:

For example, a fiber with zero dispersion slope (wavelength-independent D_λ) 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 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