# Soliton Period

Definition: the distance over which higher-order solitons reproduce their temporal and spectral shape

German: Solitonenperiode

Categories: nonlinear optics, light pulses

Units: m

Formula symbol: <$z_\textrm{s}$>

Author: Dr. Rüdiger Paschotta

The soliton period is defined as the period with which higher-order soliton pulses evolve: after that propagation distance, they reproduce their original temporal and spectral shape.

The soliton period can be calculated according to

$${z_{\rm{s}}} \approx \frac{{\pi \;{{({\tau _{\rm{p}}}/1.76)}^2}}}{{{\rm{2}}\left| {{\beta _2}} \right|}} \approx \frac{{{\tau _{\rm{p}}}^2}}{{{\rm{2}}\left| {{\beta _2}} \right|}}$$where <$\tau_\textrm{p}$> is the pulse duration (full width at half-maximum, FWHM) and <$\beta_2$> is the group delay dispersion of the fiber (in s^{2}/m).

Although fundamental solitons do *not* exhibit a periodic behavior, their soliton period is often calculated because it is related to the propagation distance over which nonlinear phase shifts become substantial. We can use the fundamental soliton equation

in order to rewrite the equation for the soliton period of a fundamental soliton to

$${z_{\rm{s}}} = \frac{\pi }{{2\left| \gamma \right|{P_{\rm{p}}}}}$$This shows that the nonlinear phase shift for the peak of that pulse in a *non-dispersive* fiber would be <$\pi / 2$>. The influence of chromatic dispersion is to obtain only a <$\pi / 4$> phase shift, but this for the whole soliton pulse and not only for its peak.

In situations where solitons are periodically disturbed (e.g. in a soliton mode-locked laser or in an amplified optical fiber communications system), the effect of these disturbances depends strongly on the ratio of the period of the disturbances to the soliton period. If this ratio is well below unity (as is often the case in lasers), the solitons essentially experience just the average values of chromatic dispersion and Kerr nonlinearity, but for larger values of this ratio solitons can become unstable.

See also: solitons, higher-order solitons, adiabatic soliton compression

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