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+---
+title: "Laser rate equations"
+firstLetter: "L"
+publishDate: 2022-03-16
+categories:
+- Physics
+- Optics
+- Laser theory
+
+date: 2022-03-12T20:23:42+01:00
+draft: false
+markup: pandoc
+---
+
+# Laser rate equations
+
+The [Maxwell-Bloch equations](/know/concept/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](/know/concept/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](/know/concept/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](/know/concept/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](/know/concept/einstein-coefficients/),
+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.