Rate equations

The rate equations are the standard model of laser dynamics: one equation balances the supply and loss of carriers (or of excited atoms), the other balances the generation and loss of photons in the lasing mode, and stimulated emission couples them. They average over the mode profile and leave out the optical phase; with a phase equation, Langevin noise terms and the linewidth enhancement factor they also give intensity noise, chirp and linewidth. The threshold, slope, resonance frequency and switching delay quoted for a diode laser are usually obtained from these two equations.

The diode-laser equations

In the form used by Coldren, Corzine and Mašanović, with $N$ the carrier density in the active volume $V$ and $S$ the photon density in the mode volume $V/\Gamma$:

$$\frac{dN}{dt} = \frac{\eta_i I}{qV} - \frac{N}{\tau_n} - v_g\,g(N)\,S$$ $$\frac{dS}{dt} = \Gamma v_g\,g(N)\,S + \Gamma\beta_{sp}\frac{N}{\tau_r} - \frac{S}{\tau_p}$$

Here $\eta_i$ is the injection efficiency, $\tau_n$ the carrier lifetime, $v_g$ the group velocity, $g(N)$ the material gain, $\Gamma$ the confinement factor, $\beta_{sp}$ the spontaneous emission factor, $\tau_r$ the radiative lifetime and $\tau_p$ the photon lifetime. The gain is often written with a compression factor, $g(N)/(1+\varepsilon S)$, and with a logarithmic or linear dependence on $N$.

Steady state

Setting both derivatives to zero and neglecting spontaneous emission into the mode gives two regimes. Below threshold $S \approx 0$ and $N = \eta_i I \tau_n/(qV)$. Above threshold the photon equation requires $\Gamma v_g g(N) = 1/\tau_p$, which is the threshold condition, so $N$ clamps at $N_\text{th}$ and

$$I_\text{th} = \frac{qVN_\text{th}}{\eta_i \tau_n}.$$

The extra current goes into photons, and the output power is

$$P = \eta_i \frac{\alpha_m}{\alpha_i + \alpha_m}\frac{h\nu}{q}(I - I_\text{th}).$$

For a 300 µm × 2 µm ridge with 40 nm of quantum wells ($V$ = 2.4 × 10⁻¹¹ cm³), $\Gamma$ = 0.05, $\alpha_i$ = 10 cm⁻¹, cleaved facets ($\alpha_m$ = 40.1 cm⁻¹), $\eta_i$ = 0.8 and $\tau_n$ = 2 ns, the logarithmic gain model with $g_0$ = 1500 cm⁻¹ and $N_\text{tr}$ = 1.5 × 10¹⁸ cm⁻³ gives $N_\text{th}$ = 2.93 × 10¹⁸ cm⁻³ and $I_\text{th}$ = 7.0 mA. The differential efficiency is 0.64 and $h\nu/q$ at 1550 nm is 0.80 V, so the slope is 0.51 W/A from both facets, or 0.26 W/A per facet. At 20 mA above threshold the laser emits 10 mW in total, and the photon density in the mode is 5 × 10¹⁴ cm⁻³. Treating $\tau_n$ as constant is a simplification; with Auger and bimolecular recombination it shortens as $N$ rises, and the threshold current is computed with the value at $N_\text{th}$.

Small-signal response

Linearizing about a bias point above threshold gives a damped second-order system. Its natural frequency is

$$f_R = \frac{1}{2\pi}\sqrt{\frac{v_g\,a\,S}{\tau_p}},$$

where $a = dg/dN$ is the differential gain. For the example laser with $a$ = 5 × 10⁻¹⁶ cm² and $\tau_p$ = 2.4 ps, $f_R$ = 4.7 GHz at 10 mW. The damping rate grows as $K f_R^2 + \gamma_0$, mostly through gain compression; the relaxation oscillations entry covers the response function, the damping and the bandwidth limits that follow.

Large-signal behavior: turn-on delay

When a laser biased at $I_b$ below threshold is switched to $I$ above it, the carriers must first build up to $N_\text{th}$, during which there is almost no light. With a constant lifetime the delay is

$$t_d = \tau_n \ln\frac{I - I_b}{I - I_\text{th}}.$$

From zero bias ($I_b$ = 0) to twice threshold, $t_d = \tau_n \ln 2$ = 1.39 ns for $\tau_n$ = 2 ns; to three times threshold it is 0.81 ns. Prebiasing at 0.9 $I_\text{th}$ and switching to 2 $I_\text{th}$ cuts it to 0.19 ns. Because the delay depends on the preceding bit pattern it causes timing jitter, which is why directly modulated lasers are biased at or above threshold. Once the photons build up, the carrier density overshoots and the output rings at $f_R$; short single pulses obtained this way are the basis of gain switching.

Solid-state and other lasers

For a four-level laser the same structure applies with the upper-level population density $N$ and the intracavity photon density $\phi$:

$$\frac{dN}{dt} = R_p - \frac{N}{\tau} - \frac{c}{n}\sigma N\phi$$ $$\frac{d\phi}{dt} = \frac{c}{n}\sigma N\phi - \frac{\phi}{\tau_c}$$

with $R_p$ the pump rate, $\tau$ the upper-state lifetime, $\sigma$ the emission cross section and $\tau_c$ the cavity lifetime (filling factors are omitted here). Because $\tau$ is hundreds of microseconds in Nd:YAG, the relaxation oscillations are at tens of kilohertz, and the same equations describe Q-switching when $\tau_c$ is switched from short to long.

Common questions

What are the laser rate equations?

Two coupled balance equations, for the carrier (or inversion) density and the photon density, linked by stimulated emission.

What do the rate equations predict above threshold?

The carrier density clamps at its threshold value, and the output power rises linearly with current at a slope set by the injection efficiency and the ratio of mirror loss to total loss.

How is turn-on delay calculated?

From the time the carrier density takes to rise from its initial value to threshold, $t_d = \tau_n \ln[(I - I_b)/(I - I_\text{th})]$; for a laser switched from zero to twice threshold with a 2 ns lifetime, 1.39 ns.

References: L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993). A. E. Siegman, Lasers (University Science Books, 1986). O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010).