 Dispersion-limited digital-fibre link length

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.