Stimulated Emission

Optical gain is the engine of the photonics industry. Most conventional lasers use a pumped gain medium inside an optical cavity, where stimulated emission amplifies radiation initially supplied by spontaneous emission or by another seed field. This includes diode lasers and high-power fibre lasers used for applications such as barcode scanning and materials processing. In fibre-optic telecommunications, erbium-doped fibre amplifiers (EDFAs) exploit stimulated emission near 1.55 μm to boost signals across ocean-spanning links without ever converting light to electronics, and semiconductor optical amplifiers do the same on a chip. The same physics reaches into imaging and metrology. Stimulated Emission Depletion (STED) microscopy uses a doughnut-shaped depletion beam to stimulate fluorophores back to the ground state everywhere except a nanometre-scale spot, beating the diffraction limit (2014 Nobel Prize in Chemistry). On cosmic scales, interstellar and circumstellar clouds of OH, H$_2$O, and SiO can be pumped into population inversion by collisions and radiation, producing astrophysical maser emission over enormous distances. The underlying stimulated-emission and absorption processes obey the Einstein reciprocity relation $g_1B_{12}=g_2B_{21}$; for non-degenerate levels this reduces to $B_{21}=B_{12}$.

The cleanest way to feel stimulated emission is to picture a field mode as a bucket already holding $n$ photons. An excited atom coupled to that mode emits at a rate proportional to $(n+1)$: the "$+1$" is spontaneous emission — the atom decaying into the vacuum in a random direction with random phase — while the "$n$" is stimulated emission, driven by the photons already present and delivered straight back into their mode. Absorption from a lower-state atom goes at rate proportional to $n$. When $n$ is large the "$+1$" is negligible and the only thing that matters is whether more atoms are up or down. Stimulated emission is a phase-locked process. The applied field drives the atomic dipole into oscillation, and a coherently oscillating dipole radiates in phase with its driver, adding constructively to the mode. The emitted radiation is therefore added coherently to the existing optical mode, sharing its wavevector, polarization, and phase coherence. This mode-selective amplification is a central reason laser light can remain highly directional and coherent rather than becoming diffuse emission. The first figure shows the clone process against its spontaneous rival; the second is the punchline a laser engineer actually reaches for — the gain coefficient as a function of inversion, with the crossover from absorption to amplification and the universal peak cross section marked on the plot.

The engineering payoff is a single sign. Because absorption and stimulated emission cancel state-for-state, the net effect of a beam passing through the medium is set by the difference $N_2-(g_2/g_1)N_1$. Flip the sign of that difference — invert the populations — and Beer–Lambert absorption becomes exponential amplification. The gain per unit length is the cross section times the inversion, and the cross section itself, on a radiatively broadened line, has a beautifully simple ceiling.

:::svg[Gain coefficient $\gamma(\nu)=\sigma_{\rm peak}[N_2-(g_2/g_1)N_1]$ versus inversion: absorption ($\gamma<0$) below the transparency point $N_2=(g_2/g_1)N_1$, and gain ($\gamma>0$) above it. For a radiatively broadened line at $\lambda=589$ nm, $\sigma_{\rm peak}=\lambda^2/(2\pi)=5.52\times10^{-14}~\mathrm{m^2}$. A representative inversion $\Delta N=1.0\times10^{13}~\mathrm{m^{-3}}$ then gives $\gamma\approx0.55~\mathrm{m^{-1}}$.]

Einstein's phenomenology assigns three elementary processes to a two-level atom of transition frequency $\nu_0=(E_2-E_1)/h$ bathed in radiation of spectral energy density $\rho(\nu)$: absorption at rate $B_{12}\rho N_1$, stimulated emission at rate $B_{21}\rho N_2$, and spontaneous emission at rate $A_{21}N_2$. The stimulated rate per excited atom is the linear-in-field term that makes amplification possible.

where: $W_{21}$ — probability per unit time that the field stimulates an excited atom to emit; $B_{21}$ — Einstein coefficient for stimulated emission; $\rho(\nu_0)$ — spectral energy density at the transition frequency. Because $W_{21}$ is proportional to $\rho$, driving harder means emitting faster — the runaway that lets a mode fill with coherent photons. The single most important structural fact is that absorption and stimulated emission share their coefficient. Requiring the two-level rate equations to reproduce the Planck spectrum in thermal equilibrium (detailed balance) forces this, and quantum mechanics confirms it: aq-qm-absorption-stimulated-emission shows both rates come from the same first-order dipole matrix element $|\langle 2|\mathbf{d}|1\rangle|^2$, with only the initial state swapped.

where: $B_{12}$ — absorption coefficient; $g_1,g_2$ — degeneracies of the lower and upper levels. The equality is why a beam neither grows nor shrinks when the populations are balanced: the two rates cancel state-for-state, and only a population imbalance breaks the tie. A propagating beam grows or decays exponentially. Counting photons added by stimulated emission against those removed by absorption over a slab of thickness $\mathrm{d}z$ gives Beer–Lambert with a signed coefficient — the gain equation of laser physics.

where: $\gamma(\nu)$ — gain coefficient per unit length (negative $\Rightarrow$ absorption); $\sigma(\nu)$ — stimulated-emission cross section; $N_1,N_2$ — lower/upper number densities. Net gain ($\gamma>0$) demands the bracket be positive, i.e. population inversion $N_2/g_2 > N_1/g_1$ — impossible at thermal equilibrium, where Boltzmann guarantees $N_2<N_1$. The cross section links the microscopic $A$ coefficient to the macroscopic gain through the line shape $g(\nu)$ (normalized, $\int g\,\mathrm{d}\nu=1$). On a radiatively broadened line this collapses to a universal value.

where: $\lambda$ — transition wavelength in the medium; $A_{21}=1/\tau$ — spontaneous rate; $g(\nu_0)=2/(\pi\Delta\nu)$ — Lorentzian peak; $\Delta\nu=A_{21}/2\pi$ — natural FWHM. The dipole strength cancels entirely: a radiatively broadened two-level atom presents the same peak cross section $\lambda^2/2\pi$ regardless of how strong or weak its transition is. Finally, the gain does not grow without limit: as the beam intensity approaches the saturation intensity the inversion is bleached and the gain drops.

where: $I_{\text{sat}}$ — saturation intensity; $\tau$ — upper-state lifetime; $\gamma_0$ — small-signal gain. Above $I_{\text{sat}}$ the amplifier extracts energy faster than the pump can replenish the inversion, and the gain clamps — the mechanism that stabilizes a laser's output power.

The competition between stimulated and spontaneous emission is governed by the photon occupation of the driven mode, and the sign of the gain by the inversion:

Example 1 — the peak gain cross section and gain of an inverted vapour (confidence builder). Estimate the peak stimulated-emission cross section of the Na D transition ($\lambda=589$ nm, radiatively broadened) and the small-signal gain produced by a modest inversion $\Delta N = 1.0\times10^{13}\text{ m}^{-3}$.

Step 1. The radiatively broadened peak cross section is dipole-independent: $\sigma_{\text{peak}}=\lambda^2/2\pi=(589\times10^{-9})^2/(2\pi)=\mathbf{5.52\times10^{-14}\ m^2}$. Note this is $\sim5\times10^{-10}\text{ cm}^2$, roughly $10^{5}$ times the geometric cross section of the atom.

Step 2. The small-signal gain coefficient is $\gamma_0=\sigma_{\text{peak}}\,\Delta N=(5.52\times10^{-14})(1.0\times10^{13})=\mathbf{0.55\ m^{-1}}$. Over a 30 cm gain tube the single-pass gain is $e^{\gamma_0 L}=e^{0.55\times0.30}=e^{0.166}\approx\mathbf{1.18}$ — an 18% amplification per pass, comparable to a low-gain He–Ne discharge.

Step 3. Because the gain is only $18\%$ per pass, a laser cavity can recycle the light through the gain medium many times. The exact number of passes depends on the cavity length and mirror losses, while lasing begins when the round-trip gain exceeds the round-trip loss. Thus, even a low-gain medium can lase if the cavity provides sufficiently low losses.

Example 2 — why thermal equilibrium can never amplify (exam level). Show quantitatively that a two-level Na vapour in thermal equilibrium at 300 K cannot produce gain, and estimate how large a stimulating field would be needed for stimulated emission to even match spontaneous emission.

Step 1. In equilibrium the Boltzmann ratio is $N_2/N_1=e^{-h\nu_0/k_BT}$. For Na D, $h\nu_0=2.10$ eV and $k_BT=0.0259$ eV at 300 K, so $h\nu_0/k_BT=81.1$ and $N_2/N_1=e^{-81}\approx\mathbf{6\times10^{-36}}$. The upper state is essentially empty, so $N_2-N_1<0$ and $\gamma<0$: the vapour absorbs, it cannot amplify.

Step 2. The ratio of stimulated to spontaneous emission equals the mean photon number per mode, $\bar n=1/(e^{h\nu_0/k_BT}-1)=e^{-81.1}\approx6\times10^{-36}$. Thermal radiation at optical frequencies supplies far less than one photon per mode, so stimulated emission from a blackbody is utterly negligible at room temperature.

Step 3. To make stimulated emission comparable to spontaneous emission in a given optical mode, the mean photon occupation must reach $\bar n\sim1$, corresponding to a radiation field whose stimulated-emission rate is comparable to the spontaneous rate. To obtain net optical gain, however, the medium must also be driven out of thermal equilibrium to create population inversion. The required field intensity depends on the mode and bandwidth, so $\bar n\sim1$ is the cleanest mode-independent statement here.

Optical gain is what stimulated emission looks like on the macroscopic scale: a negative absorption coefficient, a beam that grows instead of shrinks. Its history is the story of turning Einstein's 1917 prediction into a device. Rudolf Ladenburg's discharge-tube experiments in the late 1920s found “negative dispersion” in excited neon, providing important experimental evidence for the stimulated-emission contribution predicted by Einstein's theory and showing that an excited medium could modify light propagation in the direction associated with gain. But nobody could yet invert a population strongly enough to see net gain. The breakthrough waited for Charles Townes, who in 1954 built the ammonia maser by physically sorting inverted molecules into a microwave cavity, and for the 1958 Schawlow–Townes proposal to extend the idea to optical frequencies. In 1960 Theodore Maiman fired the first laser, a flash-pumped ruby rod, and optical gain became an everyday tool.

References. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 & 5 · Foot, Atomic Physics, Ch. 1 & 7 · Saleh & Teich, Fundamentals of Photonics, 2nd ed., Ch. 13–14 · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 2 · Svelto, Principles of Lasers, Ch. 2 · Einstein, Phys. Z. 18, 121 (1917). The Planck-matching argument yields $A_{21}/B_{21}=8\pi h\nu^3/c^3$ and $B_{12}=B_{21}$. The dipole expression is $B_{21}=\pi|\mathbf d_{21}|^2/(3\varepsilon_0\hbar^2)$. The radiatively broadened peak cross section $\sigma_{\text{peak}}=\lambda^2/2\pi$ evaluates to $5.52\times10^{-14}\text{ m}^2$ at 589 nm.

Builds on: Einstein A and B Coefficients · Resonant Absorption · Electric Dipole Approximation. Leads to: Detailed Balance and the Planck Spectrum · Spontaneous Emission · Population Inversion and Laser Gain.