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 \pdvn{2}{\vb{P}^{+}}{t} &= \nabla \cross \nabla \cross \vb{E}^{+} + \frac{n^2}{c^2} \pdvn{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. $\ipdvn{2}{\vb{E}_0^{+}\!}{t} \approx 0$ and $\ipdvn{2}{\vb{P}_0^{+}\!}{t} \approx 0$, 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 $\ipdv{\vb{P}_0^{+}\!}{t}$ 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 $\ipdv{\vb{P}_0^{+}\!}{t} \approx 0$, 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 the 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 $\ipdv{\vb{P}_0^{+}}{t} \approx 0$, 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 the aforementioned 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 \Imag(\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 $\Imag(\Omega) < 0$ 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 \Imag(\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. 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 \Imag(\Omega) = \frac{Q}{\Real(\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.