Population inversion

Population inversion is the state in which an upper energy level holds more population per state than a lower level, $N_2\mathrm{/}{g}_2>N_1\mathrm{/}{g}_1$. Under this condition stimulated emission outpaces absorption, and a beam at the transition frequency is amplified; this is the operating condition of conventional lasers and of doped-fiber and semiconductor optical amplifiers. It never occurs in thermal equilibrium; it has to be produced by pumping. The inversion density a laser needs is often small: a 1 cm Nd:YAG rod with a 95% output coupler reaches threshold at an inversion of about $9\times {10}^{16}$ cm⁻³, under 0.1% of the neodymium ions in a 1 at.% crystal.

Why equilibrium forbids it. In thermal equilibrium two levels separated by $h\nu$ obey Boltzmann statistics:

$$\frac{N_2}{N_1}=\frac{g_2}{g_1}\exp \text{\hspace{0.17em}⁣}(-\frac{h\nu }{kT})\mathrm{.}$$

At any positive temperature $N_2\mathrm{/}{g}_2<N_1\mathrm{/}{g}_1$, so absorption wins and no net optical gain is possible. At 300 K the Boltzmann factor is $3.7\times {10}^{-14}$ for a 1550 nm transition ($h\nu =0.80$ eV) and $1.1\times {10}^{-2}$ for the 10.6 μm CO₂ line ($h\nu =0.117$ eV). Formally, inversion corresponds to a negative temperature for that pair of levels; the host crystal or gas keeps an ordinary positive temperature.

Gain and threshold. The gain coefficient is

$$g=\sigma (N_2-\frac{g_2}{g_1}{N}_1)=\sigma \mathrm{\Delta }N\mathrm{.}$$

Inversion makes $g$ positive; the laser reaches threshold when the round-trip gain equals the round-trip loss. For a rod of length $L$ between mirrors of reflectivity $R_1$ and $R_2$, ignoring internal loss,

$$g_{\text{th}}=\frac{1}{2L}\ln \frac{1}{R_1{R}_2}\mathrm{.}$$

Worked example. Take Nd:YAG with $\sigma =2.8\times {10}^{-19}$ cm², $L=1$ cm, $R_1=1$, and $R_2=0.95$. Then $g_{\text{th}}=0.0256$ cm⁻¹ and $\mathrm{\Delta }N=g_{\text{th}}\mathrm{/}\sigma =9.2\times {10}^{16}$ cm⁻³. A 1 at.% Nd:YAG crystal holds $1.38\times {10}^{20}$ Nd ions per cm³, so the required inversion is 0.066% of the ions.

Pumping schemes. Inversion requires an external energy source that populates the upper laser level selectively:

A closed two-level system cannot be inverted by optical pumping in steady state: as the pump intensity rises, stimulated emission by the pump itself drives the populations toward equality, and the medium becomes transparent without gaining. Practical gain media therefore use three or four levels.

Three-level and four-level systems. In a three-level system the lower laser level is the ground state. With equal degeneracies, at least half of all the ions must be pumped out of the ground state to reach transparency, so threshold pump power is high. Ruby, the first laser demonstrated (Maiman, 1960), is the classic example.

In a four-level system the lower laser level lies well above the ground state and empties quickly. In Nd:YAG it sits about 2110 cm⁻¹ up; with $kT=208.5$ cm⁻¹ at 300 K the Boltzmann factor is about $4\times {10}^{-5}$, so the level is nearly empty and a small upper-state population already gives inversion, as in the worked example.

Most modern solid-state and fiber lasers are four-level or quasi-four-level. Er-doped silica operates as a quasi-three-level system at C-band wavelengths: the lower level is the ground-state manifold, and the emission and absorption cross-sections differ across the band. Whether a given wavelength sees gain depends on the fraction of ions inverted, which is why EDFA gain modeling tracks the population along the fiber and why a weakly pumped erbium fiber absorbs at 1530 nm while still amplifying longer wavelengths.

Semiconductors. In a laser diode the inversion condition is stated with quasi-Fermi levels: gain at photon energy $h\nu$ requires

$$E_g<h\nu <E_{Fc}-E_{Fv},$$

the Bernard–Duraffourg condition. The carrier density at which the separation first reaches the band gap is the transparency density, of order ${10}^{18}$ cm⁻³ in III–V active regions; the corresponding injection current is the transparency current. Below transparency the active region absorbs the lasing mode; above it, the region amplifies. The threshold current corresponds to the carrier density at which modal gain equals the total cavity loss.

How inversion is observed. Populations are rarely counted directly. In practice inversion is inferred from a probe beam that is amplified instead of absorbed, from bleaching of the absorption spectrum under pump, from the fluorescence intensity of the upper level, or in diodes from the gain spectrum measured below threshold. In fiber amplifiers the transparency pump power at a signal wavelength is a routine measurement.

Pitfalls. Inversion is necessary but not sufficient for lasing; the gain must also exceed all cavity losses. Comparing raw populations instead of $N\mathrm{/}g$ gives the wrong condition when degeneracies differ. In quasi-three-level media, inversion is wavelength-dependent, and a medium can amplify at one wavelength and absorb at another at the same time.

Common questions

Why is population inversion needed for a laser?

Absorption and stimulated emission are driven by the same photons with the same per-emitter coefficient (for equal degeneracies). Net amplification therefore requires more emitters in the upper state than the lower one.

Why can't a two-level system be inverted?

Optical pumping of a two-level system stimulates emission as fast as it excites. At high pump intensity the populations approach equality, and the medium becomes transparent. A third level that the upper laser level fills from, or a lower laser level that empties quickly, breaks this symmetry.

References: Siegman, Lasers (University Science Books, 1986); Svelto, Principles of Lasers (5th ed., 2010); Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019); Coldren, Corzine & Mašanović, Diode Lasers and Photonic Integrated Circuits (2nd ed., 2012); Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994).