Exercise $2$: $(7$ points$)$

A Nd:YLF laser is pumped by an input power of $P_p.$

$($a$)$ Show that the relation between the threshold of pump power $P_{th}$ and the minimum threshold of pump power could be expressed by the equation:

$P_{th}=\frac{}{{}_i+\frac{{}_1}{2}}{P}_{mth}$

$($b$)$ Assuming $P_p=7W,R_1=100\%,$ the internal losses is ${}_i=3\%$ and $P_{mth}=350mW,$ calculate the optimum transmission $T_2=1-R_2$ of the output mirror.