Stimulated emission

Stimulated emission is the process in which a photon of energy $h\nu$ matching a transition drives an excited emitter down to its lower level, and the emitter releases a second photon into the same mode as the first. The two photons share frequency, phase, polarization, and direction, so the field is amplified coherently. The strength of a transition is usually quoted as a stimulated-emission cross-section $\sigma$: about $2.8\times {10}^{-19}$ cm² for the 1064 nm line of Nd:YAG, with an upper-state lifetime near 230 μs. Einstein introduced the process in 1916–1917 to make Planck's radiation law consistent with detailed balance.

The three radiative and non-radiative channels. An emitter in an excited state can return to a lower state in three ways:

Rate equations. For an upper level with population density $N_2$, a lower level $N_1$, pump rate $R_p$, lifetime $\tau$, and photon flux $\varphi =I\mathrm{/}h\nu$:

$$\begin{array}{rl}{\displaystyle \frac{d{N}_2}{dt}}& {\displaystyle =R_p-\frac{N_2}{\tau }}\\ {\displaystyle }& {\displaystyle -\sigma \varphi (N_2-\frac{g_2}{g_1}{N}_1)}\end{array}$$

The $N_2\mathrm{/}\tau$ term contains spontaneous emission and any non-radiative decay. The last term is stimulated emission minus absorption; both scale with the same flux, and the degeneracies $g_i$ fix their relative weight.

Einstein relations. With the $B$ coefficients defined against spectral energy density, detailed balance with blackbody radiation requires

$$\begin{array}{rl}{\displaystyle g_1{B}_{12}}& {\displaystyle =g_2{B}_{21},}\\ {\displaystyle \frac{A_{21}}{B_{21}}}& {\displaystyle =\frac{8\pi h{\nu }^3{n}^3}{c^3},}\end{array}$$

where $n$ is the refractive index of the medium. For equal degeneracies, absorption and stimulated emission have the same per-emitter coefficient. The cross-section follows from the spontaneous rate and the normalized lineshape $g(\nu )$ (the Füchtbauer–Ladenburg relation):

$$\sigma (\nu )=\frac{{\lambda }^2{A}_{21}}{8\pi n^2}g(\nu )\mathrm{.}$$

Net gain. The gain coefficient is

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

which is positive only under population inversion. This is the microscopic origin of optical gain.

Worked numbers. In thermal equilibrium the ratio of stimulated to spontaneous emission rates equals the mean photon occupation per mode, $1\mathrm{/}(e^{h\nu \mathrm{/}kT}-1)$. At 300 K this is $3.7\times {10}^{-14}$ at 1550 nm ($h\nu =0.80$ eV) and $1.1\times {10}^{-2}$ at 10.6 μm ($h\nu =0.117$ eV). Stimulated emission is negligible in thermal light at optical frequencies; a laser cavity raises the occupation of one mode by many orders of magnitude. For a four-level medium the gain saturates at $I_{\text{sat}}=h\nu \mathrm{/}(\sigma \tau )$. With the Nd:YAG values above and $h\nu =1.165$ eV, $I_{\text{sat}}\approx 2.9$ kW/cm² and the saturation fluence $h\nu \mathrm{/}\sigma \approx 0.67$ J/cm².

How it is measured. Gain is measured directly by passing a weak probe through the pumped medium and comparing output with input (small-signal gain), or in a semiconductor laser below threshold by the Hakki–Paoli method, which extracts gain from the modulation depth of the Fabry–Pérot fringes in the emission spectrum. Cross-sections are usually derived from the measured fluorescence spectrum and lifetime through the relation above, or from absorption spectra through McCumber theory for rare-earth ions such as erbium.

Why the emitted photon is coherent. The stimulated rate into a given mode is proportional to the number of photons already in that mode, and the emitted photon is added to that mode. Spontaneous emission couples to all vacuum modes according to their density of states, which is why it is broadband and isotropic in free space.

Role in lasers. A laser needs inversion sustained by pumping and optical feedback from a cavity. At threshold, stimulated emission into the lasing mode begins to dominate the upper-state decay; the output slope changes abruptly, producing the kink in LIV curves, and the linewidth narrows. Above threshold the gain clamps at the cavity loss and further pump power goes into stimulated emission. Spontaneous emission into the lasing mode persists and sets the quantum-limited Schawlow–Townes linewidth and a floor on phase noise.

Role in amplifiers. In an erbium-doped fiber amplifier or a semiconductor optical amplifier, the signal is amplified by stimulated emission while spontaneous emission is amplified along with it as amplified spontaneous emission. With full inversion and high gain the noise figure approaches 3 dB, a quantum limit for phase-insensitive amplification. Strong signals deplete the inversion, which is gain saturation.

Pitfalls. Degeneracy factors are often dropped, which is harmless only when $g_1=g_2$. The $A\mathrm{/}B$ ratio depends on whether $B$ is defined against energy density or intensity; the two conventions differ by a factor $c\mathrm{/}n$. In quasi-three-level media such as Er:silica or Yb:silica the emission and absorption cross-sections differ at a given wavelength, so a single $\sigma$ is not enough.

Common questions

What is the difference between spontaneous and stimulated emission?

Spontaneous emission occurs without an incident photon, at a rate $A_{21}=1\mathrm{/}{\tau }_{sp}$, into a random mode. Stimulated emission requires an incident photon and adds a photon to that photon's mode. Light-emitting diodes and electroluminescence in general rely on the first; lasers and amplifiers on the second.

Does stimulated emission happen without population inversion?

Yes. It occurs whenever excited emitters meet resonant photons. Without inversion, absorption by the lower-level population outweighs it and the beam is attenuated. Inversion is the condition for net gain.

Is the stimulated photon an exact copy?

It enters the same mode, so it has the same frequency, wavevector, and polarization, and it adds in phase with the driving field. Amplification is still accompanied by spontaneous emission into that mode, which is why no amplifier can copy a quantum state noiselessly.

References: Einstein, "Zur Quantentheorie der Strahlung," Physikalische Zeitschrift 18, 121 (1917); Siegman, Lasers (University Science Books, 1986), Ch. 2–3; Svelto, Principles of Lasers (5th ed., 2010), Ch. 2; Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019).