Glossary entry

Population inversion

A non-equilibrium condition in which more atoms (or carriers) occupy an upper energy level than a lower one. A necessary condition for net stimulated emission and laser oscillation.

In thermal equilibrium, populations of two energy levels with separation $E_2 - E_1 = h\nu$ obey Boltzmann statistics:

\frac{N_2}{N_1} \;=\; \frac{g_2}{g_1} \exp\!\left( -\frac{h\nu}{k T} \right),

where $g_i$ are degeneracies. At any positive temperature, $N_2 / g_2 < N_1 / g_1$ , so absorption dominates over stimulated emission and no net optical gain is possible.

Population inversion is the non-equilibrium condition $N_2 / g_2 > N_1 / g_1$ . Under inversion, stimulated emission outpaces absorption, and an optical signal at frequency $\nu$ is amplified rather than attenuated. This is the gain condition required for any laser or optical amplifier to operate.

Achieving inversion requires pumping — an external energy source that selectively populates the upper laser level. Pumping schemes:

Scheme	Examples	Pump source
Optical pumping	Solid-state lasers (Nd:YAG, Ti:Sapphire), EDFA	Lamp, diode laser, or another laser
Electrical pumping	Semiconductor diode lasers (DFB, VCSEL, QCL)	Direct current injection
Gas discharge	HeNe, argon-ion, CO $_2$	Electrical discharge
Chemical pumping	HF/DF chemical lasers (specialized)	Exothermic chemical reaction
Particle beam	Heavy-ion-pumped excimer (research)	Accelerator

Three-level vs four-level systems.

A three-level laser system (lower laser level is the ground state) requires inversion against the ground-state population — at least half of the ground-state atoms must be pumped to the upper level before transparency is reached. Threshold pump power is correspondingly high.

A four-level laser system has a lower laser level that lies above the ground state by several $kT$ , so it remains nearly empty by thermal depopulation. Population inversion is achieved by pumping even a small fraction of atoms to the upper level. Threshold is much lower than for three-level operation.

Most modern solid-state and fiber lasers are four-level or quasi-four-level. Ruby is the classic three-level laser (and the first ever demonstrated). Er-doped fiber operates as a quasi-three-level system at the C-band laser wavelengths, requiring careful population analysis for EDFA gain modeling.

For semiconductor laser diodes, the analog is the carrier-density inversion condition: above threshold, the joint electron–hole density across the active region exceeds the transparency density at the lasing wavelength. Below transparency, the active region absorbs the lasing mode; above transparency, it amplifies. The threshold current corresponds to the carrier density at which modal gain equals total cavity loss.