Solution:

λ = 850 nm

σ = 45 nm

pulse broadening due to material dispersion is given by,

$\sigma_m=\sigma L M$

$ \text { Material dispersion constant } D_{\text {mat }}=\frac{-\lambda}{c} \cdot \frac{d^2 n}{d \lambda^2} $

For LED sources operating at 850 nm,

$ \left|\lambda^2 \frac{d^2 n}{d \lambda^2}\right|=0.025 $

$ \begin{aligned} & \mathrm{M}=\frac{1}{\mathrm{c} \lambda}\left|\lambda^2 \frac{\mathrm{d}^2 \mathrm{n}}{\mathrm{d}\\\lambda^2}\right|=\frac{1}{\left(3 \times 10^5\right)(850)} \times 0.025 \\\\ & \mathrm{M}=9.8 \mathrm{ps} / \mathrm{nm} / \mathrm{km} \\\\ & \sigma_{\mathrm{m}}=\mathbf{4 4 1} \mathrm{ns} / \mathbf{k m}\\ \end{aligned} $