P$.5$ Laser Amplifier Gain and Population Difference. A $15-cm$ long $N{d}^{3+}$ : glass rod used as a laser amplifier has a total small signal gain of $10$ at ${}_0=1\mathrm{.}06m.$ Use the data provided in Table $15\mathrm{.}3-1$ to determine the population difference ions per $c{m}^3)$ required to achieve this gain. $(5$ points$).$

P$.6$ A $0\mathrm{.}1$ m long Nd:YAG laser rod is coated with a $99\%$ reflector on one end and a $95\%$ reflector on the other end. Determine the longitudinal mode spacing. How many modes could operate if the gain bandwidth is a Lorentzian shape with a FWHM of $0\mathrm{.}45$ nm and a peak at $1\mathrm{.}06m?($Assume the laser is mode$-$locked so all possible modes are lasing.$)$ Assume that the maximum small$-$signal gain is $10\%$ at the peak of the emission spectrum and that the cavity losses $($in addition to mirror losses$)$ are $2\%$ per pass. The index of refraction of Nd:YAG at $1\mathrm{.}06m$ is $1\mathrm{.}82.$ Note: Since the transmission losses differ at each mirror, you must consider the total round$-$trip gain and loss. $(10$ Points$)$