A step index fiber has a core refractive index of $1\mathrm{.}5$ and a core diameter of $50m.$

The fiber is jointed with a lateral misalignment between the core axes of $5m.$

Estimate the insertion loss at the joint due to the lateral misalignment assuming a

uniform distribution of power between all guided modes when:

$($a$)$ there is a small air gap at the joint;

$($b$)$ the joint is considered index matched.

Give me the complete solution in detail