Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 1 / c / ii

At the end of the pulse, the photon density returns to negligible size. Put $u=N_f^{\ast }/N_i^{\ast }$ and $r=N_i^{\ast }/N_{\mathrm{th}}^{\ast }$. The result of i givesUsing $x=\beta \eta =1-u$, the nonzero-pulse solution obeysFor $r>1$, select $0<x<1$, corresponding to depletion below threshold. The root $x=0$ is also an algebraic solution for every $r$ but represents no macroscopic pulse. A positive nonzero root exists only above threshold.