Glossary entry

Spontaneous emission factor (β)

The fraction of spontaneous emission from a laser's active region that couples into the lasing mode. It sets how abrupt the laser threshold is: small in ordinary edge emitters, where the threshold is sharp, and approaching one in nanocavity lasers, where it smooths out.

Lasers & gainUpdated September 2026

The spontaneous emission factor $β$ is the fraction of all spontaneous emission in a laser's active region that goes into the lasing cavity mode rather than into the many other optical modes available:

β = \frac{R_{s p, mode}}{R_{s p, total}} .

In a large laser, such as a Fabry-Perot edge emitter, the active region radiates into a very large number of modes and $β$ is small, of order $10^{- 5}$ to $10^{- 4}$ . In very small cavities, such as microdisks and photonic-crystal nanocavities, few modes lie within the emission spectrum and $β$ can exceed 0.1, approaching one in the best cavity-QED devices.

β and the threshold

The same factor fixes how sharp the laser threshold is. In the simplest rate-equation model, with all recombination radiative and the gain proportional to the carrier number, the steady-state number of photons in the lasing mode, $S$ , depends on the pump rate normalized to its threshold value, $r = R / R_{th}$ , as

S = \frac{(r - 1) + \sqrt{(r - 1)^{2} + 4 β r}}{2 β},

and the output power is proportional to $S$ . Well below threshold $S \approx r$ ; well above it $S \approx (r - 1) / β$ , the familiar straight line of the L-I curve; at threshold $S = 1 / \sqrt{β}$ . On a logarithmic plot the two branches are separated by a factor of $1 / β$ , so the output rises steeply over a narrow range of pump when $β$ is small:

$r = R / R_{th}$	0.1	0.5	1	2	10
$S$ for $β = 10^{- 4}$	0.11	1.0	100	10,002	90,001
$S$ for $β = 10^{- 2}$	0.11	0.98	10	102	901
$S$ for $β = 1$	0.1	0.5	1	2	10

Between half and twice the threshold pump, the photon number rises ten-thousandfold for $β = 10^{- 4}$ , about a hundredfold for $β = 10^{- 2}$ , and fourfold for $β = 1$ , where the curve is a straight line with no kink at all.

Thresholdless lasers

A laser with $β$ near one puts all of its spontaneous emission into the lasing mode, so its light-current curve has no kink, and such devices are called thresholdless. The name overstates the case in two ways. The character of the light still changes as the pump rises, from the thermal photon statistics of spontaneous emission to the Poissonian statistics of a laser, and measurements of the photon correlation $g^{(2)} (0)$ , which falls from 2 toward 1 across that change, are used to locate the transition when the L-I curve cannot. And the gain medium must still reach transparency before it amplifies, so the lowest useful operating current is set by transparency and by non-radiative recombination rather than by $β$ .

Measuring β

The usual method fits the rate-equation model to a light-current curve measured over several decades of output power and plotted logarithmically; the height of the step between the two branches gives $1 / β$ . The fit depends on how the non-radiative recombination and the gain are modeled, so values from different groups are best compared through their stated models. For nanolasers whose curves show no clear step, photon-correlation measurements are added.

Relation to the Purcell factor

The Purcell factor $F_{P}$ is the enhancement of the emission rate into a cavity mode over the free-space rate. If emission into all other modes proceeds at a fraction $γ$ of the free-space rate, then

β = \frac{F_{P}}{F_{P} + γ},

so a large Purcell factor, which requires a high quality factor, a small mode volume and an emitter placed at the field maximum, is the route to high $β$ . The factor $F_{P}$ describes how fast the emitter radiates into the mode, and $β$ what share of its emission ends up there.

In mass-produced telecom and datacom lasers $β$ is small enough to have no practical effect, and it is not specified. It becomes a design parameter in nanolasers for low-power on-chip interconnects and in single-photon and other quantum light sources.

References: G. Björk and Y. Yamamoto, "Analysis of semiconductor microcavity lasers using rate equations," IEEE Journal of Quantum Electronics 27, 2386 (1991). H. Yokoyama, "Physics and device applications of optical microcavities," Science 256, 66 (1992). O. Painter et al., "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819 (1999). L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).