 Solution

Packet and group index. A wave packet superposes nearby wavevectors to form a localized envelope. The phase velocity is $v_p=\omega /k$; for a narrow packet with weak distortion the envelope velocity is $v_g=d\omega /dk$. Since $k=2\pi fn(f)/c$,For the stipulated linear index, $n_g=1.5+\tau (2f-f_0)$, not simply $n(f)$.

bandwidth and speed spread. A pulse lasting $t_p$ has Fourier bandwidth of order $1/t_p$, so $\Delta f\sim 10^{10}\mathrm{Hz}$; an exact width coefficient depends on pulse shape and width convention. At the carrier,Across a full bandwidth $\Delta f$,Higher-frequency components have the lower group velocity. For a band symmetric about $f_0$, the endpoint velocities are approximately $v_g(f_0)\pm 370\mathrm{m}{\mathrm{s}}^{-1}$.

dispersion-limited digital-fibre link length. Differential group delay accumulates asThe pulse spacing is $t_r=10^{-9}\mathrm{s}$. Pulse overlap and intersymbol interference become serious when the dispersive spread approaches that spacing, givingA conservative full-width estimate $t_p+\Delta t_{\mathrm{disp}}\lesssim t_r$ gives about 45 km; this refinement is not a universal sharp boundary because pulse shape, chirp and detection tolerance are unspecified. Attenuation and modal or waveguide dispersion could impose further limits, but are not quantified by the supplied index law.

digital optical pulse regeneration. Each stage must recover and regenerate the data with fresh, short pulses and correct timing, as well as restore signal level. Amplifying the already broadened waveform alone does not remove accumulated dispersion; explicit dispersion compensation is an alternative to full regeneration.