$(25$ Points$)$ A pulsed laser beam machine is used to drill microholes on zirconium $($vaporization energy $42\mathrm{.}5\frac{J}{m}{m}^3).$ This machine generates laser pulses with $1\mathrm{.}5\frac{J}{?}$ pulse energy and $0\mathrm{.}001$ s pulse duration. The beam divergence of the laser is $0\mathrm{.}002$ rad $,$ and its radiation is focused with a $30$ mm focal length lens. Suppose the conversion efficiency of input energy to thermal energy is $0\mathrm{.}5\%,$ which corresponds to material removal.

a$.$ What will be the smallest diameter of the hole drilled using the laser?

b$.$ How many pulses are needed to drill a $1$ mm deep hole with the smallest diameter?

c$.$ What is the difference between percussive and trepanning laser drilling?

d$.$ How can this laser be used to drill a hole of $5$ mm diameter and $1$ mm depth?

e$.$ How many pulses will be required to drill this larger hole?