= Solution
A forward-biased <p-n junction> injects <electrons> from its <n-type semiconductor> and <electron holes> from its <p-type semiconductor> into an active region. A <direct band gap> allows radiative transitions without a compensating <phonon>, so <stimulated emission> can compete effectively with nonradiative processes. At thermal equilibrium there is one <Fermi level>; forward injection instead establishes separate electron and valence-electron <quasi-Fermi levels> $F_c$ and $F_v$. Sufficient injection makes conduction states more occupied than the corresponding valence states, producing <optical gain>.
Two parallel cleaved facets supply an <optical resonator>. At normal incidence their approximate intensity <reflectance> follows the <Fresnel equations>:
$$
R=\left(\frac{n-1}{n+1}\right)^2=0.3195.
$$
A resonant <normal mode> grows when its round-trip amplification exceeds its losses. For length $L$, material <optical gain> $g$, overlap factor $\Gamma$, internal intensity loss $\alpha_i$ and facet reflectances $R_1,R_2$,
$$
R_1R_2e^{2(\Gamma g-\alpha_i)L}\geq1,
\qquad \boxed{\Gamma g\geq\alpha_i+\frac{1}{2L}\ln\frac1{R_1R_2}.}
$$
The pump supplies the emitted <energy>; the facets provide feedback and let some output escape. Positive <optical gain> alone does not establish <laser threshold>.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2015-injected-diode-and-guided-mode-cutoff.png]
{title=Injected <laser diode> levels and the first additional guided spatial mode}
{description=Original schematic: the left panel uses stipulated energy units to distinguish equilibrium and injected <quasi-Fermi levels>, with an allowed downward optical transition. The right panel plots $J_0(u)$, whose first zero fixes the weak-guidance spatial-mode cutoff. The separate panels do not represent a shared spatial coordinate.}
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