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# Third-order Dispersion

Acronym: TOD

Definition: chromatic dispersion related to a third-order dependence of the phase change on the frequency offset

German: Dispersion dritter Ordnung

Units: s3

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Third-order dispersion results from the frequency dependence of the group delay dispersion. In the Taylor expansion of the spectral phase versus angular frequency offset (see the article on chromatic dispersion), it is related to the third-order term. It can be written as

$$k''' = \frac{{\partial k''}}{{\partial \omega }} = \frac{{{\partial ^3}\varphi }}{{\partial {\omega ^3}}}$$

which is also the derivative of the group delay per unit length with respect to angular frequency.

The corresponding change in the spectral phase within a propagation length <$L$> is

$$\Delta \varphi (\omega ) = \frac{1}{6}\frac{{{\partial ^3}k}}{{\partial {\omega ^3}}}{\left( {\omega - {\omega _0}} \right)^3}L$$

The third-order dispersion of an optical element is usually specified in units of fs3, whereas the units of <$k'''$> are fs3/m.

It is also possible to specify TOD with respect to the vacuum wavelength rather than the optical frequency, which leads to units of fs/nm2, for example. It can be calculated as follows:

$${D_{3\lambda }} = {\left( {\frac{{2\pi c}}{{{\lambda ^2}}}} \right)^2} \cdot \frac{{{\partial ^3}{T_{\rm{g}}}}}{{\partial {\omega ^2}}} = {\left( {\frac{{2\pi c}}{{{\lambda ^2}}}} \right)^2} \cdot \frac{{{\partial ^3}\varphi }}{{\partial {\omega ^3}}}$$

In practice, one often has the dispersion parameter <$D_{\lambda }$> and its wavelength derivative, called the dispersion slope, and can calculate the TOD from those as:

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

In mode-locked lasers for pulse durations below roughly 30 fs, it is necessary to provide dispersion compensation not only for the average group delay dispersion (second-order dispersion), but also for the third-order dispersion and possibly for even higher orders.

In many cases, the investigation of the effect of third-order dispersion requires numerical pulse propagation modeling.

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