= Solution
The physical branch can be plotted without a nonlinear solver by parameterizing
$$
r(x)=\frac{-\ln(1-x)}x,\qquad 0<x<1.
$$
It 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 $x\simeq2(r-1)$. For strong over-pumping the residual inversion fraction is exponentially small, $1-x\simeq e^{-r}$. Thus the curve saturates at the upper bound $\beta\eta=1$.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019-q-switch-extraction.png]
{title=Ideal Q-switched inversion extraction versus over-pumping}
{description=The nonzero pulse branch is plotted using $r=-\ln(1-x)/x$, where $x=\beta\eta$. The threshold and the 95 percent extraction point are marked.}
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