(Solved): 3. (20\%) (a) For graded index fiber with refractive index n1 for the core center and n0 as ...

3. (20\%) (a) For graded index fiber with refractive index $$n_1$$ for the core center and $$n_0$$ as the cladding (using quasi-plane wave condition) with polynomial dependent with order $$\mathrm{p}$$, $$n^2(r)=n_1^2\left[1?2{\left(\frac{r}{a}\right)}^p?\right],r?a,$$ (i) derive the total mode number $$\mathrm{M}$$ allowed for given $$\mathrm{V}$$ number $$M?\frac{p}{p+2}{n}_1^2{k}_o^2{a}^2?=\frac{p}{p+2}\frac{V^2}{2}$$ (ii) derive the $${\mathrm{q}}^{\text{th }}$$ mode group velocity $$v_q?c_1\left[1?\frac{p?2}{p+2}{\left(\frac{q}{M}\right)}^{p/(p+2)}?\right].$$ (b) Explain the advantage of using $$p=2$$ for graded index fiber from the point of view (i) mode number (ii) modal dispersion?