$6.$ The $45-$degree prism shown is to be used to deflect light by $90$ degrees, so the facet should provide total internal reflection.

a$)$ For normal incidence light, what is the minimum value of the index of refraction $\backslash ($ n $\backslash )$ such that the prism functions as intended?

b$)$ What is the fraction of transmitted power at normal incidence $($the ratio of the intensity in air propagating down after deflection to the intensity incoming from the left$)$ as a function of the index $\backslash ($ n $\backslash )?$ What is the value of the transmitted fraction for the specific value of $\backslash ($ n $\backslash )$ found in part $($a$)?$ You may assume the dielectric material has $\backslash (\backslash$mu$=\backslash$mu$\_\{0\}\backslash ).$

c$)$ If it is required that any rays within an acceptance angle of $\backslash (\backslash$pm $15^\{\backslash$circ$\}(\backslash$pm $\backslash$pi $/12)\backslash )-$ as measured inside the prism $-$ should also be deflected by total internal reflection at the $45-$degree facet, what is the minimum value of index $\backslash ($ n $\backslash )$ needed? What is the acceptance angle in air $\backslash (\backslash$theta$\_\{$a$\}\backslash )$ that corresponds to $\backslash (15^\{\backslash$circ$\}\backslash )$ inside the prism? You may use $\backslash (\backslash$sin $(\backslash$pi $/12)=0\mathrm{.}25\backslash ).$