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

The physical branch can be plotted without a nonlinear solver by parameterizingIt begins at $(r,x)=(1,0)$ and increases monotonically to $x=1$ as $r\to \infty$. Expanding near threshold gives $r=1+x/2+O(x^2)$, so $x\simeq 2(r-1)$. For strong over-pumping the residual inversion fraction is exponentially small, $1-x\simeq e^{-r}$. Thus the curve saturates at the upper bound $\beta \eta =1$.