Spontaneous emission factor (β)

The spontaneous emission factor $\beta$ is the fraction of all spontaneous emission events in a laser's active region that emit a photon into the lasing cavity mode, rather than into any of the many other available optical modes:

$$\beta \;=\; \frac{R_{sp,\text{mode}}}{R_{sp,\text{total}}}.$$

For a large laser with many cavity modes (e.g., a long Fabry-Perot edge-emitter), $\beta$ is small ($\sim 10^{-5} - 10^{-4}$); the active region's spontaneous emission spreads across thousands of available modes, and only a tiny fraction couples into the specific mode that lases.

For a very small cavity (VCSEL, photonic-crystal laser, nanocavity), the mode density is suppressed and $\beta$ can approach unity — every spontaneously emitted photon couples into the cavity mode.

Why $\beta$ matters. The lasing transition (the L-I curve "knee" at threshold) is a continuous statistical transition rather than an abrupt phase change. The sharpness of this transition is controlled by $\beta$:

• $\beta \ll 1$: sharp threshold; below threshold, the laser is essentially off; above threshold, light output rises linearly with current
• $\beta \sim 0.1 - 1$: smooth threshold; below threshold, significant spontaneous emission is in the lasing mode; the lasing transition is gradual
• $\beta \rightarrow 1$ ("thresholdless laser"): no clear threshold; output rises smoothly with input across all currents

Mathematical form of the L-I curve. The output power $P$ versus injection current $I$ for a single-mode laser with general $\beta$:

$$P(I) \;\propto\; \frac{1}{2} \left[ \beta I + \sqrt{(\beta I)^2 + 4 \beta I_\text{th,th}} \right] - \frac{1}{2} I_\text{th,th},$$

approaching the standard $P \propto (I - I_\text{th})$ form for $\beta \to 0$ and approaching the smooth curve $P \propto I$ for $\beta \to 1$.

Material implications.

Laser type	Typical $\beta$
Edge-emitting Fabry-Perot laser (long cavity)	$10^{-5} - 10^{-4}$
DFB laser	$10^{-4} - 10^{-3}$
VCSEL (oxide-confined, $\sim 5 \times 5$ μm aperture)	$10^{-3} - 10^{-2}$
Microdisk laser ($< 5$ μm diameter)	$0.01 - 0.1$
Photonic-crystal cavity laser	$0.1 - 0.7$
Nanowire laser	$0.1 - 0.5$
Single-quantum-dot cavity QED	$0.5 - 1$

Why $\beta = 1$ matters (thresholdless lasers). A laser with $\beta$ near unity dissipates all of its electrical input into the lasing mode (rather than into thousands of useless modes). This in principle:

• Drops the threshold current to the radiative-recombination floor (every photon goes into the mode)
• Eliminates the lasing-threshold-nonlinearity that makes lasers difficult to control near turn-on
• Enables ultra-low-power on-chip optical interconnects (theoretical sub-μW operating power)

In practice, even $\beta \approx 1$ lasers still need to reach a population inversion before they exhibit gain rather than absorption, so the practical lower limit of operating current is still set by transparency current, not by $\beta$.

Extraction. Below threshold:

$$R_{sp,\text{mode}} \;=\; \beta \, R_{sp,\text{total}} \;=\; \beta \, \frac{N}{\tau_r},$$

where $N$ is carrier density and $\tau_r$ is radiative lifetime. Measure the integrated spontaneous emission collected from the mode (via OSA or photodetector after spectral filter on the lasing mode position only), divide by the calculated total radiative recombination. Standard extraction has $\pm 30$% uncertainty due to challenges in calibrating the mode-vs-non-mode coupling.

Comparison to Purcell factor. The Purcell factor $F_P$ is the enhancement of the local spontaneous emission rate at a specific position relative to free-space rate. $\beta$ and $F_P$ are related but distinct: $F_P$ relates to the speed of emission, $\beta$ to the directionality of emission. A high-$F_P$ cavity at a high-Q mode usually has high $\beta$, but the relationship depends on the spatial overlap between the emitter and the mode.

Significance for telecom/datacom. For mass-market lasers, $\beta$ is so small that it does not affect operating performance — threshold is sharp, L-I curves are linear, and $\beta$ is not separately specified. For emerging applications (extremely low-power on-chip interconnects, quantum light sources), $\beta$ becomes a central design parameter.

References: Yokoyama et al., Strong coupling regime for vacuum Rabi splitting, Phys. Rev. Lett. 1995 (the canonical β factor measurement); Coldren, Corzine, Mašanović, Diode Lasers, Ch. 5 (rate equation treatment); Painter et al., Two-Dimensional Photonic Band-Gap Defect Mode Laser, Science 1999 (high-β photonic crystal laser demonstration).