Design a single$-$mode step$-$index fiber $($with V $=2\mathrm{.}2)$ for operation at $850$ nm with a fused silica core $($n $=1\mathrm{.}458)$

$($a$)$ Find the cladding index n$2$ if the diameter of the fiber is $8\mathrm{.}0$ um$.$

$($b$)$ Calculate the cutoff wavelength lamda$\_$c$.($Note: the range for single$-$mode,$2\mathrm{.}05<$ V$<2\mathrm{.}405).$

$($c$)$ It is possible to make single$-$mode graded$-$index fiber. What would be the critical values of the V$-$parameter $($Vco$)$ for a parabolic$-$index $($g $=2)$ fiber and a triangular$-$ index $($g $=1)$ fiber? $($Hint: Vco $=2\mathrm{.}405[(1+(2/$g$)]^(1/2)\}$

$($d$)$ Estimate the coupling efficiency n$,$ if numerical aperture of this fiber NA $=0\mathrm{.}25.$