Group Delay Dispersion
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.
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 Dλ. From that, the third-order dispersion can be calculated as follows:
For example, a fiber with zero dispersion slope (wavelength-independent Dλ) 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.
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