Glossary entry

Stimulated emission

The process by which an incident photon induces a population-inverted atom or molecule to emit a coherent photon identical in frequency, phase, and polarization. The mechanism of all laser amplification.

When an atom or molecule is in an excited state, it can return to a lower energy state by three processes:

Process	Trigger	Output
Spontaneous emission	Vacuum fluctuations	Photon of random direction, phase, and (within natural linewidth) frequency
Stimulated emission	An incident photon	Coherent photon: same frequency, phase, polarization, and direction as the trigger
Non-radiative recombination	Phonons, defects, Auger interaction	Heat (no photon)

The rate equations for a two-level system with population $N_2$ (upper) and $N_1$ (lower):

\frac{dN_2}{dt} \;=\; -\frac{N_2}{\tau_{sp}} - B_{21} N_2 I + B_{12} N_1 I + \text{(pumping)}.

The first term is spontaneous emission with lifetime $\tau_{sp}$ , the second is stimulated emission with rate proportional to incident intensity $I$ , and the third is absorption with rate also proportional to $I$ . Einstein established that $B_{12} = B_{21}$ — absorption and stimulated emission have the same per-atom rate coefficient. The detailed balance gives the relation $A_{21}/B_{21} = 8\pi h\nu^3 / c^3$ for the spontaneous-to-stimulated coefficient ratio in 3D vacuum.

Population inversion ( $N_2 > N_1$ , weighted by degeneracies) is the condition under which stimulated emission exceeds absorption, producing net optical gain:

g \;\propto\; \sigma (N_2 - N_1),

where $\sigma$ is the transition cross-section.

A laser requires:

Population inversion maintained by continuous pumping faster than the spontaneous emission rate
Optical feedback (a cavity) so that emitted photons re-stimulate further emission

Why stimulated emission is coherent. The probability of stimulated emission scales with the photon mode occupation in the field where the emitting atom is located. The emitted photon must be in the same mode (frequency, direction, polarization) as the stimulating one — the process is essentially quantum-mechanical mode amplification. This is in contrast to spontaneous emission, which couples to all vacuum modes with their respective densities of states.

Above lasing threshold, stimulated emission depletes the upper state much faster than spontaneous emission, so the spontaneous emission fraction drops sharply — leading to the characteristic kink in LIV curves at threshold and the dramatic linewidth narrowing as the laser passes through threshold.

The Schawlow–Townes linewidth limit derives from the irreducible contribution of spontaneous emission into the lasing mode — even above threshold, $\sim$ one spontaneous photon per cavity mode per inversion lifetime adds quantum-limited phase noise.