Consider a teleservices provider based in Jaipur, has been assigned a task to deliver data to subscribers at Pilani Town. The company decides to connect central office at Pilani using a step$-$index SMF and deliver data to subscribers at last mile by employing a step$-$index MMF$,$ both operating at $1550$ nm $.$ The source at Jaipur is a laser emitting a power of $2$ mW and FWHM of $3$ nm while a LED is employed at Pilani office with a power output of $10$ mW and FWHM of $25$ nm $.$ The SMF with $\backslash (8\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ diameter has a core index of $1\mathrm{.}485$ with a $\backslash (\backslash$Delta$=0\mathrm{.}003\backslash )$ while MMF has a diameter of $\backslash (12\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ with core index of $1\mathrm{.}480$ and $\backslash (\backslash$Delta$=0\mathrm{.}002\backslash ).$ Assume a distance of $225$ km from Jaipur to Pilani and all subscribers are located within a radius of $10$ km around the office at Pilani. The SMF roll has a maximum span of $50$ km with an attenuation coefficient of $\backslash (0\mathrm{.}22\backslash$mathrm$\{$~dB$\}/\backslash$mathrm$\{$km$\}\backslash ).$ The MMF exhibit an attenuation coefficient of $\backslash (3\backslash$mathrm$\{$~dB$\}/\backslash$mathrm$\{$km$\}\backslash ).$

Determine:

$[1]$ The numerical aperture and acceptance angle for both SMF and MMF$.[5]$

$[2]$ The combined $($multipath $+$ material$)$ dispersion for the MMF$.$ Which one dominates?

$[3]$ The total number of modes propagating in the MMF and their effective indexes and phase velocities.

$[4]$ The power carried in each mode observed in $(3)$ and confined in core of the MMF$.$

$[5]$ The percent reduction in the diameter of SMF when required total dispersion, $\backslash ($ D$=0\backslash )$ and $\backslash ($ D$\_\{\backslash$mathrm$\{$m$\}\}\backslash )$ for SMF is $\backslash (10\backslash$mathrm$\{$ps$\}/\backslash$mathrm$\{$nm$\}.\backslash$mathrm$\{$km$\}\backslash ).$

$[6]$ The mode field diameter, WP and associated joint loss $($in dB$)$ for SMF$.$ Assume lateral misalignment of $\backslash (0\mathrm{.}2\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash ).$

$[7]$ The splicing losses due to extrinsic factors in SMF to increase span in order to cover $225$ km $,$ consider the same lateral misalignment as mentioned in $(6)$ to be the only reason for splice loss.

$[8]$ The power $($in nW$)$ received at Pilani office and power $($in mW $)$ received by a subscriber at Pilani. Assume the signal is regenerated at the office and each connector at an end add to a loss of $0\mathrm{.}5$ dB $.$