Group velocity dispersion (${\beta }_2=d^{2}k/d{\omega }^2$)

Variation of inverse group velocity with angular frequency; for a uniform medium of length $L$, the accumulated group delay dispersion is $L{d}^{2}k/d{\omega }^2$.

A pulse's frequency components accumulate a differential group delay $\Delta t=L|{\partial }_f(1/v_g)|\Delta f$. If that spread approaches the pulse repetition period $T_r$, neighbouring symbols overlap. The resulting length estimate depends on bandwidth, initial pulse width, chirp and the receiver's allowable intersymbol interference. For $n(f)=n_0+\tau (f-f_0)$, the group index is $n_0+\tau (2f-f_0)$ and ${\partial }_f(1/v_g)=2\tau /c$: differentiating the phase index alone loses a factor of two.

A digital repeater can recover the symbol sequence and generate new pulses with restored amplitude, shape and timing. This resets waveform degradation between spans, provided decisions remain reliable. An amplifier alone increases the broadened pulse amplitude and also adds noise; it does not undo group velocity dispersion. optical dispersion compensation can instead reverse an accumulated frequency-dependent delay within its design bandwidth.

The distance over which second-order group velocity dispersion substantially changes a pulse of characteristic duration $T_0$. Width conventions and initial chirp affect numerical broadening factors; the displayed definition is the standard envelope-equation scale.