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Third-order Dispersion

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

Acronym: TOD

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

Category: article belongs to category physical foundations physical foundations

Units: s³

Page views in 12 months: 1939

DOI: 10.61835/gbz Cite the article: BibTex plain text HTML Link to this page! LinkedIn

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.

More to Learn

Dispersion

Group delay dispersion

Dispersion compensation

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