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8 files changed, 628 insertions, 23 deletions

diff --git a/content/know/category/laser-theory.md b/content/know/category/laser-theory.md
new file mode 100644
index 0000000..5437821
--- /dev/null
+++ b/content/know/category/laser-theory.md
@@ -0,0 +1,9 @@
+---
+title: "Laser theory"
+firstLetter: "L"
+date: 2022-03-16T20:36:17+01:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/concept/coupled-mode-theory/index.pdc b/content/know/concept/coupled-mode-theory/index.pdc
new file mode 100644
index 0000000..5a44d6e
--- /dev/null
+++ b/content/know/concept/coupled-mode-theory/index.pdc
@@ -0,0 +1,233 @@
+---
+title: "Coupled mode theory"
+firstLetter: "C"
+publishDate: 2022-03-31
+categories:
+- Physics
+- Optics
+
+date: 2022-03-12T20:22:21+01:00
+draft: false
+markup: pandoc
+---
+
+# Coupled mode theory
+
+Given an optical resonator (e.g. a photonic crystal cavity),
+consider one of its quasinormal modes
+with frequency $\omega_0$ and decay rate $1 / \tau_0$.
+Its complex amplitude $A$ is governed by:
+
+$$\begin{aligned}
+    \dv{A}{t}
+    &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A
+\end{aligned}$$
+
+We choose to normalize $A$ so that $|A(t)|^2$
+is the total energy inside the resonator at time $t$.
+
+Suppose that $N$ waveguides are now "connected" to this resonator,
+meaning that the resonator mode $A$ and the outgoing waveguide mode $S_\ell^\mathrm{out}$
+overlap sufficiently for $A$ to leak into $S_\ell^\mathrm{out}$ at a rate $1 / \tau_\ell$.
+Conversely, the incoming mode $S_\ell^\mathrm{in}$ brings energy to $A$.
+Therefore, we can write up the following general set of equations:
+
+$$\begin{aligned}
+    \dv{A}{t}
+    &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A
+    - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A + \sum_{\ell = 1}^N \alpha_\ell S_\ell^\mathrm{in}
+    \\
+    S_\ell^\mathrm{out}
+    &= \beta_\ell S_\ell^\mathrm{in} + \gamma_\ell A
+\end{aligned}$$
+
+Where $\alpha_\ell$ and $\gamma_\ell$ are unknown coupling constants,
+and $\beta_\ell$ represents reflection.
+We normalize $S_\ell^\mathrm{in}$
+so that $|S_\ell^\mathrm{in}(t)|^2$ is the power flowing towards $A$ at time $t$,
+and likewise for $S_\ell^\mathrm{out}$.
+
+Note that we have made a subtle approximation here:
+by adding new damping mechanisms,
+we are in fact modifying $\omega_0$;
+see the [harmonic oscillator](/know/concept/harmonic-oscillator/) for a demonstration.
+However, the frequency shift is second-order in the decay rate,
+so by assuming that all $\tau_\ell$ are large,
+we only need to keep the first-order terms, as we did.
+This is called **weak coupling**.
+
+If we also assume that $\tau_0$ is large
+(its effect is already included in $\omega_0$),
+then we can treat the decay mechanisms separately:
+to analyze the decay into a certain waveguide $\ell$,
+it is first-order accurate to neglect all other waveguides and $\tau_0$:
+
+$$\begin{aligned}
+    \dv{A}{t}
+    \approx \bigg( \!-\! i \omega_0 - \frac{1}{\tau_\ell} \bigg) A + \sum_{\ell' = 1}^N \alpha_\ell S_{\ell'}^\mathrm{in}
+\end{aligned}$$
+
+To determine $\gamma_\ell$, we use energy conservation.
+If all $S_{\ell'}^\mathrm{in} = 0$,
+then the energy in $A$ decays as:
+
+$$\begin{aligned}
+    \dv{|A|^2}{t}
+    &= \dv{A}{t} A^* + A \dv{A^*}{t}
+    \\
+    &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_\ell} \bigg) |A|^2
+    + \bigg( i \omega_0 - \frac{1}{\tau_\ell} \bigg) |A|^2
+    \\
+    &= - \frac{2}{\tau_\ell} |A|^2
+\end{aligned}$$
+
+Since all other mechanisms are neglected,
+all this energy must go into $S_\ell^\mathrm{out}$, meaning:
+
+$$\begin{aligned}
+    |S_\ell^\mathrm{out}|^2
+    = - \dv{|A|^2}{t}
+    = \frac{2}{\tau_\ell} |A|^2
+\end{aligned}$$
+
+Taking the square root, we clearly see that $|\gamma_\ell| = \sqrt{2 / \tau_\ell}$.
+Because the phase of $S_\ell^\mathrm{out}$ is arbitrarily defined,
+$\gamma_\ell$ need not be complex, so we choose $\gamma_\ell = \sqrt{2 / \tau_\ell}$.
+
+Next, to find $\alpha_\ell$, we exploit the time-reversal symmetry
+of [Maxwell's equations](/know/concept/maxwells-equations/),
+which govern the light in the resonator and the waveguides.
+In the above calculation of $\gamma_\ell$, $A$ evolved as follows,
+with the lost energy ending up in $S_\ell^\mathrm{out}$:
+
+$$\begin{aligned}
+    A(t)
+    = A e^{-i \omega_0 t - t / \tau_\ell}
+\end{aligned}$$
+
+After reversing time, $A$ evolves like so,
+where we have taken the complex conjugate
+to preserve the meanings of the symbols
+$A$, $S_\ell^\mathrm{out}$, and $S_\ell^\mathrm{in}$:
+$$\begin{aligned}
+    A(t)
+    = A e^{-i \omega_0 t + t / \tau_\ell}
+\end{aligned}$$
+
+We insert this expression for $A(t)$ into its original differential equation, yielding:
+
+$$\begin{aligned}
+    \dv{A}{t}
+    = \bigg( \!-\! i \omega_0 + \frac{1}{\tau_\ell} \bigg) A
+    = \bigg( \!-\! i \omega_0 - \frac{1}{\tau_\ell} \bigg) A + \alpha_\ell S_\ell^\mathrm{in}
+\end{aligned}$$
+
+Isolating this for $A$ leads us to the following power balance equation:
+
+$$\begin{aligned}
+    A
+    = \frac{\alpha_\ell \tau_\ell}{2} S_\ell^\mathrm{in}
+    \qquad \implies \qquad
+    |\alpha_\ell|^2 |S_\ell^\mathrm{in}|^2
+    = \frac{4}{\tau_\ell^2} |A|
+\end{aligned}$$
+
+But thanks to energy conservation,
+all power delivered by $S_\ell^\mathrm{in}$ ends up in $A$, so we know:
+
+$$\begin{aligned}
+    |S_\ell^\mathrm{in}|^2
+    = \dv{|A|^2}{t}
+    = \frac{2}{\tau_\ell} |A|^2
+\end{aligned}$$
+
+To reconcile the two equations above,
+we need $|\alpha_\ell| = \sqrt{2 / \tau_\ell}$.
+Discarding the phase thanks to our choice of $\gamma_\ell$,
+we conclude that $\alpha_\ell = \sqrt{2 / \tau_\ell} = \gamma_\ell$.
+
+Finally, $\beta_\ell$ can also be determined using energy conservation.
+Again using our weak coupling assumption,
+if energy is only entering and leaving $A$ through waveguide $\ell$, we have:
+
+$$\begin{aligned}
+    |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
+    = \dv{|A|^2}{t}
+\end{aligned}$$
+
+Meanwhile, using the differential equation for $A$,
+we find the following relation:
+
+$$\begin{aligned}
+    \dv{|A|^2}{t}
+    &= \dv{A}{t} A^* + A \dv{A^*}{t}
+    \\
+    &= - \frac{2}{\tau_\ell} |A|^2 + \alpha_\ell \Big( S_\ell^\mathrm{in} A^* + (S_\ell^\mathrm{in})^* A \Big)
+\end{aligned}$$
+
+By isolating both of the above relations for $\dv*{|A|^2}{t}$
+and equating them, we arrive at:
+
+$$\begin{aligned}
+    |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
+    &= - \frac{2}{\tau_\ell} |A|^2 + \alpha_\ell \Big( S_\ell^\mathrm{in} A^* + (S_\ell^\mathrm{in})^* A \Big)
+\end{aligned}$$
+
+We insert the definition of $\gamma_\ell$ and $\beta_\ell$,
+namely $\gamma_\ell A = S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in}$,
+and use $\alpha_\ell = \gamma_\ell$:
+
+$$\begin{aligned}
+    |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
+    &= - \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big) \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big)
+    \\
+    &\quad\; + S_\ell^\mathrm{in} \Big( (S_\ell^\mathrm{out})^* - \beta_\ell^* (S_\ell^\mathrm{in})^* \Big)
+    + (S_\ell^\mathrm{in})^* \Big( S_\ell^\mathrm{out} - \beta_\ell S_\ell^\mathrm{in} \Big)
+    \\
+    &= - |\beta_\ell|^2 |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
+    + \beta_\ell S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + \beta_\ell^* (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out}
+    \\
+    &\quad\; + S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* - \beta_\ell^* |S_\ell^\mathrm{in}|^2
+    + (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out} - \beta_\ell |S_\ell^\mathrm{in}|^2
+    \\
+    &= - (|\beta_\ell|^2 + \beta_\ell + \beta_\ell^*) |S_\ell^\mathrm{in}|^2
+    + (1 - \beta_\ell) S_\ell^\mathrm{in} (S_\ell^\mathrm{out})^* + (1 - \beta_\ell^*) (S_\ell^\mathrm{in})^* S_\ell^\mathrm{out}
+\end{aligned}$$
+
+This equation is only satisfied if $\beta_\ell = -1$.
+Combined with $\alpha_\ell = \gamma_\ell = \sqrt{2 / \tau_\ell}$,
+the **coupled-mode equations** take the following form:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \dv{A}{t}
+            &= - i \omega_0 A - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A
+            + \sum_{\ell = 1}^N \sqrt{\frac{2}{\tau_\ell}} S_\ell^\mathrm{in}
+            \\
+            S_\ell^\mathrm{out}
+            &= - S_\ell^\mathrm{in} + \sqrt{\frac{2}{\tau_\ell}} A
+        \end{aligned}
+    }
+\end{aligned}$$
+
+By connecting multiple resonators with waveguides,
+optical networks can be created,
+whose dynamics are described by these equations.
+
+The coupled-mode equations are extremely general,
+since we have only used weak coupling,
+conservation of energy, and time-reversal symmetry.
+Even if the decay rates are quite large,
+coupled mode theory still tends to give qualitatively correct answers.
+
+
+
+## References
+1.  H.A. Haus,
+    *Waves and fields in optoelectronics*,
+    1984, Prentice-Hall.
+2.  J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade,
+    *Photonic crystals: molding the flow of light*,
+    2nd edition, Princeton.
+
diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc
index b56af77..d50235f 100644
--- a/content/know/concept/einstein-coefficients/index.pdc
+++ b/content/know/concept/einstein-coefficients/index.pdc
@@ -5,9 +5,9 @@ publishDate: 2021-07-11
 categories:
 - Physics
 - Optics
-- Electromagnetism
 - Quantum mechanics
 - Two-level system
+- Laser theory
 
 date: 2021-07-11T18:22:14+02:00
 draft: false
@@ -275,7 +275,7 @@ $$\begin{aligned}
     \boxed{
         B_{21} = B_{12} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2}
         \qquad
-        A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon \hbar c^3}
+        A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon_0 \hbar c^3}
     }
 \end{aligned}$$
 
@@ -332,7 +332,7 @@ $$\begin{aligned}
     \boxed{
         B_{21} = B_{12} = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2}
         \qquad
-        A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon \hbar c^3}
+        A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon_0 \hbar c^3}
     }
 \end{aligned}$$
 
diff --git a/content/know/concept/fabry-perot-cavity/index.pdc b/content/know/concept/fabry-perot-cavity/index.pdc
index 50b7c62..e4195d0 100644
--- a/content/know/concept/fabry-perot-cavity/index.pdc
+++ b/content/know/concept/fabry-perot-cavity/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-09-18
 categories:
 - Physics
 - Optics
+- Laser theory
 
 date: 2021-09-18T00:42:59+02:00
 draft: false
diff --git a/content/know/concept/laser-rate-equations/index.pdc b/content/know/concept/laser-rate-equations/index.pdc
new file mode 100644
index 0000000..d087035
--- /dev/null
+++ b/content/know/concept/laser-rate-equations/index.pdc
@@ -0,0 +1,330 @@
+---
+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.
diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc
index e3a3680..2e0cdd9 100644
--- a/content/know/concept/maxwell-bloch-equations/index.pdc
+++ b/content/know/concept/maxwell-bloch-equations/index.pdc
@@ -7,6 +7,7 @@ categories:
 - Quantum mechanics
 - Two-level system
 - Electromagnetism
+- Laser theory
 
 date: 2021-09-09T21:17:52+02:00
 draft: false
@@ -34,10 +35,10 @@ $\hat{H}_1$ is given by:
 $$\begin{aligned}
     \hat{H}_1(t)
     = - \hat{\vb{p}} \cdot \vb{E}(t)
-    \qquad \quad
+    \qquad \qquad
     \vu{p}
     \equiv q \vu{x}
-    \qquad \quad
+    \qquad \qquad
     \vb{E}(t)
     = \vb{E}_0 \cos\!(\omega t)
 \end{aligned}$$
@@ -72,7 +73,7 @@ Similarly, we define the transition dipole moment $\vb{p}_0^{-}$:
 $$\begin{aligned}
     \vb{p}_0^{-}
     \equiv q \matrixel{e}{\vu{x}}{g}
-    \qquad \quad
+    \qquad \qquad
     \vb{p}_0^{+}
     \equiv (\vb{p}_0^{-})^*
     = q \matrixel{g}{\vu{x}}{e}
@@ -194,7 +195,7 @@ both decay with rate $\gamma_\perp$:
 $$\begin{aligned}
     \Big( \dv{\rho_{eg}}{t} \Big)_{\perp}
     = - \gamma_\perp \rho_{eg}
-    \qquad \quad
+    \qquad \qquad
     \Big( \dv{\rho_{ge}}{t} \Big)_{\perp}
     = - \gamma_\perp \rho_{ge}
 \end{aligned}$$
@@ -295,7 +296,7 @@ towards an equilbrium $d_0$:
 $$\begin{aligned}
     2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
     = \gamma_\parallel (d_0 - d)
-    \qquad \quad
+    \qquad \qquad
     d_0
     \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e}
 \end{aligned}$$
@@ -367,37 +368,66 @@ Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$
 together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields:
 
 $$\begin{aligned}
+    \nabla \cross \big( \nabla \cross \vb{E} \big)
+    = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t}
+\end{aligned}$$
+
+Where $\sigma$ is the active material's conductivity, if any;
+almost all authors assume $\sigma = 0$.
+
+Recall that we are describing the dynamics of a two-level system.
+In reality, such a system (e.g. a quantum dot)
+is suspended in a passive background medium,
+which reacts with a polarization $\vb{P}_\mathrm{med}$
+to the electric field $\vb{E}$.
+If the medium is linear, i.e. $\vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E}$,
+then:
+
+$$\begin{aligned}
+    \mu_0 \pdv[2]{\vb{P}}{t}
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}_\mathrm{med}}{t}
+    \\
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \pdv[2]{t} \Big( \varepsilon_0 \vb{E} + \varepsilon_0 \chi \vb{E} \Big)
+    \\
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t}
+\end{aligned}$$
+
+Where $\varepsilon_r \equiv 1 + \chi_e$ is the medium's relative permittivity.
+The speed of light $c^2 = 1 / (\mu_0 \varepsilon_0)$,
+and the refractive index $n^2 = \mu_r \varepsilon_r$,
+where $\mu_r = 1$ due to our assumption that $\vb{M} = 0$, so:
+
+$$\begin{aligned}
     \boxed{
-        \nabla \cross \big( \nabla \cross \vb{E} \big)
-        = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t}
+        \mu_0 \pdv[2]{\vb{P}}{t}
+        = - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}}{t}
     }
 \end{aligned}$$
 
-Where $\sigma$ is the medium's conductivity, if any;
-many authors assume $\sigma = 0$.
-It is trivial to show that $\vb{E}$ and $\vb{P}$
-can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$.
-
+$\vb{E}$ and $\vb{P}$ can trivially be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$.
 It is also simple to convert $\vb{p}^{+}$ and $d$
-into the macroscopic polarization $\vb{P}^{+}$ and total inversion $D$
-by summing over the atoms:
+into the macroscopic $\vb{P}^{+}$ and total $D$
+by summing over all two-level systems in the medium:
 
 $$\begin{aligned}
     \vb{P}^{+}(\vb{x}, t)
-    &= \sum_{n} \vb{p}^{+}_n \: \delta(\vb{x} - \vb{x}_n)
+    &= \sum_{\nu} \vb{p}^{+}_\nu \: \delta(\vb{x} - \vb{x}_\nu)
     \\
     D(\vb{x}, t)
-    &= \sum_{n} d_n \: \delta(\vb{x} - \vb{x}_n)
+    &= \sum_{\nu} d_\nu \: \delta(\vb{x} - \vb{x}_\nu)
 \end{aligned}$$
 
 We thus arrive at the **Maxwell-Bloch equations**,
-which are relevant for laser theory:
+which are the foundation of laser theory:
 
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \mu_0 \pdv[2]{\vb{P}^{+}}{t}
-            &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}^{+}}{t}
+            &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}^{+}}{t}
             \\
             \pdv{\vb{P}^{+}}{t}
             &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}
diff --git a/content/know/concept/multi-photon-absorption/index.pdc b/content/know/concept/multi-photon-absorption/index.pdc
index b208cfe..a5f4ad7 100644
--- a/content/know/concept/multi-photon-absorption/index.pdc
+++ b/content/know/concept/multi-photon-absorption/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "M"
 publishDate: 2022-01-30
 categories:
 - Physics
+- Optics
 - Quantum mechanics
 - Nonlinear optics
 - Perturbation
diff --git a/content/know/concept/salt-equation/index.pdc b/content/know/concept/salt-equation/index.pdc
index 2f2917b..6383469 100644
--- a/content/know/concept/salt-equation/index.pdc
+++ b/content/know/concept/salt-equation/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2022-02-07
 categories:
 - Physics
 - Optics
+- Laser theory
 
 date: 2022-01-20T22:01:48+01:00
 draft: false
@@ -66,10 +67,10 @@ that the interactions between the modes are limited:
 
 $$\begin{aligned}
     \vb{E}^{+}(\vb{x}, t)
-    = \sum_{n = 1}^N \Psi_n(\vb{x}) e^{- i \omega_n t}
+    = \sum_{n = 1}^N \Psi_n(\vb{x}) \: e^{- i \omega_n t}
     \qquad \qquad
     \vb{P}^{+}(\vb{x}, t)
-    = \sum_{n = 1}^N \vb{p}_n(\vb{x}) e^{- i \omega_n t}
+    = \sum_{n = 1}^N \vb{p}_n(\vb{x}) \: e^{- i \omega_n t}
 \end{aligned}$$
 
 Using the modes' linear independence to treat each term of the summation individually,