A quantum-dot (QD) laser's active region is an ensemble of self-assembled dots with a spread of sizes. Each dot behaves as a discrete two-level emitter with ground-state occupation n (0 = empty, 1 = fully filled by two spin-paired carriers). The population-inversion factor of a two-level system is (2n − 1): negative below half-filling (net absorption), positive above (net gain), zero at transparency (n = 0.5).
Modal gain: G(n) = Γ · N_dot · g₀ · (2n − 1) / (1 + σ/σ₀)
Threshold: G(n) = α_i + α_m (modal gain = internal + mirror loss)
Carriers: dn/dt = P(1 − n) − n/τ_sp − G(n)·S
Photons: dS/dt = [G(n) − Loss]·S + β·n/τ_sp
- Injection current — sets the pump rate P that fills empty dot states; higher P drives n toward 1 (state-filling saturation, so gain itself saturates — a hallmark of QD lasers).
- Cavity loss — the mirror/outcoupling loss α_m the modal gain must overcome; raising it pushes the threshold current up.
- Dot-size dispersion σ — inhomogeneous broadening from the size distribution of self-assembled dots. The same oscillator strength spread over a wider linewidth lowers the *peak* achievable gain, exactly the 1/(1+σ/σ₀) factor above — this is why tightly size-controlled dots lase at lower current.
- Areal dot density — more dots per unit area raise the maximum achievable modal gain g₀·N_dot, letting the laser reach threshold against a given loss.
The photon field only grows once modal gain exceeds total loss (net gain > 0) — the sharp kink in the light-output-vs-current (L–I) curve that defines the lasing threshold. Below threshold, light comes only from spontaneous emission (the β term); above it, stimulated emission dominates and photon density (and hence output power) rises steeply and starts clamping the carrier occupation.