Categories: Laser theory, Optics, Physics.

# Laser rate equations

The Maxwell-Bloch equations (MBEs) give a fundamental description of light-matter interaction for a two-level quantum system for the purposes of laser theory. They govern the electric field $$\vb{E}^{+}$$, the induced polarization $$\vb{P}^{+}$$, and the total population inversion $$D$$:

\begin{aligned} - \mu_0 \pdv[2]{\vb{P}^{+}}{t} &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{n^2}{c^2} \pdv[2]{\vb{E}^{+}}{t} \\ \pdv{\vb{P}^{+}}{t} &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} - \frac{i |g|^2}{\hbar} \vb{E}^{+} D \\ \pdv{D}{t} &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) \end{aligned}

Where $$n$$ is the background medium’s refractive index, $$\omega_0$$ the two-level system’s gap resonance frequency, $$|g| \equiv |\matrixel{e}{\vu{x}}{g}|$$ the transition dipole moment, $$\gamma_\perp$$ and $$\gamma_\parallel$$ empirical decay rates, and $$D_0$$ the equilibrium inversion. Note that $$\vb{E}^{-} = (\vb{E}^{+})^*$$.

Let us make the following ansatz, where $$\vb{E}_0^{+}$$ and $$\vb{P}_0^{+}$$ are slowly-varying envelopes of a plane wave with angular frequency $$\omega \approx \omega_0$$:

\begin{aligned} \vb{E}^{+}(\vb{r}, t) = \frac{1}{2} \vb{E}_0^{+}(\vb{r}, t) \: e^{-i \omega t} \qquad \qquad \vb{P}^{+}(\vb{r}, t) = \frac{1}{2} \vb{P}_0^{+}(\vb{r}, t) \: e^{-i \omega t} \end{aligned}

We insert this into the first MBE, and assume that $$\vb{E}_0^{+}$$ and $$\vb{P}_0^{+}$$ vary so slowly that their second-order derivatives are negligible, i.e. $$\pdv*[2]{\vb{E}_0^{+}\!}{t} \approx 0$$ and $$\pdv*[2]{\vb{P}_0^{+}\!}{t} \approx 0$$, giving:

\begin{aligned} \mu_0 \bigg( i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} \bigg) = \nabla \cross \nabla \cross \vb{E}_0^{+} - \frac{n^2}{c^2} \bigg( i 2 \omega \pdv{\vb{E}_0^{+}}{t} + \omega^2 \vb{E}_0^{+} \bigg) \end{aligned}

To get rid of the double curl, consider the time-independent electromagnetic wave equation, where $$\Omega$$ is an eigenfrequency of the optical cavity in which lasing will occur:

\begin{aligned} \nabla \cross \nabla \cross \vb{E}_0^{+} = \frac{n^2}{c^2} \Omega^2 \vb{E}_0^{+} \end{aligned}

For simplicity, we restrict ourselves to a single-mode laser, where there is only one $$\Omega$$ and $$\vb{E}_0^{+}$$ to care about. Substituting the above equation into the first MBE yields:

\begin{aligned} i 2 \omega \pdv{\vb{P}_0^{+}}{t} + \omega^2 \vb{P}_0^{+} = \varepsilon_0 n^2 \bigg( (\Omega^2 - \omega^2) \vb{E}_0^{+} - i 2 \omega \pdv{\vb{E}_0^{+}}{t} \bigg) \end{aligned}

Where we used $$1 / c^2 = \mu_0 \varepsilon_0$$. Assuming the light is more or less on-resonance $$\omega \approx \Omega$$, we can approximate $$\Omega^2 \!-\! \omega^2 \approx 2 \omega (\Omega \!-\! \omega)$$, so:

\begin{aligned} i 2 \pdv{\vb{P}_0^{+}}{t} + \omega \vb{P}_0^{+} = \varepsilon_0 n^2 \bigg( 2 (\Omega - \omega) \vb{E}_0^{+} - i 2 \pdv{\vb{E}_0^{+}}{t} \bigg) \end{aligned}

Moving on to the second MBE, inserting the ansatz $$\vb{P}^{+} = \vb{P}_0^{+} e^{-i \omega t} / 2$$ leads to:

\begin{aligned} \pdv{\vb{P}_0^{+}}{t} = - \Big( \gamma_\perp + i (\omega_0 - \omega) \Big) \vb{P}_0^{+} - \frac{i |g|^2}{\hbar} \vb{E}_0^{+} D \end{aligned}

Typically, $$\gamma_\perp$$ is much larger than the rate of any other decay process, in which case $$\pdv*{\vb{P}_0^{+}\!}{t}$$ is negligible compared to $$\gamma_\perp \vb{P}_0^{+}$$. Effectively, this means that the polarization $$\vb{P}_0^{+}$$ near-instantly follows the electric field $$\vb{E}^{+}\!$$. Setting $$\pdv*{\vb{P}_0^{+}\!}{t} \approx 0$$, the second MBE becomes:

\begin{aligned} \vb{P}^{+} = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 \!-\! \omega))} \vb{E}^{+} D = \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D \end{aligned}

Where the Lorentzian gain curve $$\gamma(\omega)$$ (which also appears in the SALT equation) represents a laser’s preferred spectrum for amplification, and is defined like so:

\begin{aligned} \gamma(\omega) \equiv \frac{\gamma_\perp}{(\omega - \omega_0) + i \gamma_\perp} \end{aligned}

Note that $$\gamma(\omega)$$ satisfies the following relation, which will be useful to us later:

\begin{aligned} \gamma^*(\omega) - \gamma(\omega) = \frac{\gamma_\perp (i \gamma_\perp + i \gamma_\perp)}{(\omega - \omega_0)^2 + \gamma_\perp^2} = i 2 |\gamma(\omega)|^2 \end{aligned}

Returning to the first MBE with $$\pdv*{\vb{P}_0^{+}\!}{t} \approx 0$$, we substitute the above expression for $$\vb{P}_0^{+}$$:

\begin{aligned} \pdv{\vb{E}_0^{+}}{t} &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{\omega}{2 \varepsilon_0 n^2} \vb{P}_0^{+} \\ &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \vb{E}_0^{+} D \end{aligned}

Next, we insert our ansatz for $$\vb{E}^{+}\!$$ and $$\vb{P}^{+}\!$$ into the third MBE, and rewrite $$\vb{P}_0^{+}$$ as above. Using our identity for $$\gamma(\omega)$$, and the fact that $$\vb{E}_0^{+} \cdot \vb{E}_0^{-} = |\vb{E}|^2$$, we find:

\begin{aligned} \pdv{D}{t} &= \gamma_\parallel (D_0 - D) + \frac{i}{2 \hbar} \Big( \frac{|g|^2 \gamma^*(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{-} D \cdot \vb{E}_0^{+} - \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}_0^{+} D \cdot \vb{E}_0^{-} \Big) \\ &= \gamma_\parallel (D_0 - D) + \frac{i |g|^2}{2 \hbar^2 \gamma_\perp} \Big( \gamma^*(\omega) - \gamma(\omega) \Big) |\vb{E}|^2 D \\ &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D \end{aligned}

This is the prototype of the first laser rate equation. However, in order to have a practical set, we need an equation for $$|\vb{E}|^2$$, which we can obtain using the first MBE:

\begin{aligned} \pdv{|\vb{E}|^2}{t} &= \vb{E}_0^{+} \pdv{\vb{E}_0^{-}}{t} + \vb{E}_0^{-} \pdv{\vb{E}_0^{+}}{t} \\ &= -i (\omega - \Omega^*) |\vb{E}|^2 - i \frac{|g|^2 \omega \gamma^*(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D + i (\omega - \Omega) |\vb{E}|^2 + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} |\vb{E}|^2 D \\ &= i (\Omega^* - \Omega) |\vb{E}|^2 + i \frac{|g|^2 \omega}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \Big(\gamma(\omega) - \gamma^*(\omega)\Big) |\vb{E}|^2 D \\ &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D \end{aligned}

Where $$\Im(\Omega) < 0$$ represents the fact that the laser cavity is leaky. We now have the laser rate equations, although they are still in an unidiomatic form:

\begin{aligned} \boxed{ \begin{aligned} \pdv{|\vb{E}|^2}{t} &= 2 \Im(\Omega) |\vb{E}|^2 + \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 |\vb{E}|^2 D \\ \pdv{D}{t} &= \gamma_\parallel (D_0 - D) - \frac{|g|^2}{\hbar^2 \gamma_\perp} |\gamma(\omega)|^2 |\vb{E}|^2 D \end{aligned} } \end{aligned}

To rewrite this, we replace $$|\vb{E}|^2$$ with the photon number $$N_p$$ as follows, with $$U = \varepsilon_0 n^2 |\vb{E}|^2 / 2$$ being the energy density of the light:

\begin{aligned} N_{p} = \frac{U}{\hbar \omega} = \frac{\varepsilon_0 n^2}{2 \hbar \omega} |\vb{E}|^2 \end{aligned}

Furthermore, consider the definition of the inversion $$D$$: because a photon emission annihilates an electron-hole pair, it reduces $$D$$ by $$2$$. Since lasing is only possible for $$D > 0$$, we can replace $$D$$ with the conduction band’s electron population $$N_e$$, which is reduced by $$1$$ whenever a photon is emitted. The laser rate equations then take the following standard form:

\begin{aligned} \boxed{ \begin{aligned} \pdv{N_p}{t} &= - \gamma_p N_p + G N_p N_e \\ \pdv{N_e}{t} &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e \end{aligned} } \end{aligned}

Where $$\gamma_e$$ is a redefinition of $$\gamma_\parallel$$ depending on the electron decay processes, and the photon loss rate $$\gamma_p$$, the gain $$G$$, and the carrier supply rate $$R_\mathrm{pump}$$ are defined like so:

\begin{aligned} \gamma_p = - 2 \Im(\Omega) = \frac{Q}{\Re(\Omega)} \qquad \quad G \equiv \frac{|g|^2 \omega}{\hbar \varepsilon_0 \gamma_\perp n^2} |\gamma(\omega)|^2 \qquad \quad R_\mathrm{pump} \equiv \gamma_\parallel D_0 \end{aligned}

With $$Q$$ being the cavity mode’s quality factor. The nonlinear coupling term $$G N_p N_e$$ represents stimulated emission, which is the key to lasing.

To understand the behaviour of a laser, consider these equations in a steady state, i.e. where $$N_p$$ and $$N_e$$ are constant in $$t$$:

\begin{aligned} 0 &= - \gamma_p N_p + G N_p N_e \\ 0 &= R_\mathrm{pump} - \gamma_e N_e - G N_p N_e \end{aligned}

In addition to the trivial solution $$N_p = 0$$, we can also have $$N_p > 0$$. Isolating $$N_p$$’s equation for $$N_e$$ and inserting that into $$N_e$$’s equation, we find:

\begin{aligned} N_e = \frac{\gamma_p}{G} \qquad \implies \qquad \boxed{ N_p = \frac{1}{\gamma_p} \bigg( R_\mathrm{pump} - \frac{\gamma_e \gamma_p}{G} \bigg) } \end{aligned}

The quantity $$R_\mathrm{thr} \equiv \gamma_e \gamma_p / G$$ is called the lasing threshold: if $$R_\mathrm{pump} \ge R_\mathrm{thr}$$, the laser is active, meaning that $$N_p$$ is big enough to cause a “chain reaction” of stimulated emission that consumes all surplus carriers to maintain a steady state.

The point is that $$N_e$$ is independent of the electron supply $$R_\mathrm{pump}$$, because all additional electrons are almost immediately annihilated by stimulated emission. Consequently $$N_p$$ increases linearly as $$R_\mathrm{pump}$$ is raised, at a much steeper slope than would be possible below threshold. The output of the cavity is proportional to $$N_p$$, so the brightness is also linear.

Unfortunately, by deriving the laser rate equations from the MBEs, we lost some interesting and important effects, most notably spontaneous emission, which is needed for $$N_p$$ to grow if $$R_\mathrm{pump}$$ is below threshold.

For this reason, the laser rate equations are typically presented in a more empirical form, which “bookkeeps” the processes affecting $$N_p$$ and $$N_e$$. Consider the following example:

\begin{aligned} \boxed{ \begin{aligned} \pdv{N_p}{t} &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e \\ \pdv{N_e}{t} &= R_\mathrm{pump} + \gamma_\mathrm{abs} N_p - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e - G_\mathrm{stim} N_p N_e \end{aligned} } \end{aligned}

Where $$\gamma_\mathrm{out}$$ represents the cavity’s usable output, $$\gamma_\mathrm{abs}$$ the medium’s absorption, $$\gamma_\mathrm{loss}$$ scattering losses, $$\gamma_\mathrm{spon}$$ spontaneous emission, $$\gamma_\mathrm{n.r.}$$ non-radiative electron-hole recombination, and $$\gamma_\mathrm{leak}$$ the fact that some carriers leak away before they can be used for emission.

Unsurprisingly, this form is much harder to analyze, but more accurately describes the dynamics inside a laser. To make matters even worse, many of these decay rates depend on $$N_p$$ or $$N_e$$, so solutions can only be obtained numerically.

## References

1. D. Meschede, Optics, light and lasers, Wiley.
2. L.A. Coldren, S.W. Corzine, M.L. Mašanović, Diode lasers and photonic integrated circuits, 2nd edition, Wiley.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.