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diff --git a/source/know/concept/laser-rate-equations/index.md b/source/know/concept/laser-rate-equations/index.md
index 939a1a0..a84d274 100644
--- a/source/know/concept/laser-rate-equations/index.md
+++ b/source/know/concept/laser-rate-equations/index.md
@@ -12,9 +12,9 @@ layout: "concept"
 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 $D$:
+They govern the [electric field](/know/concept/electric-field/) $$\vb{E}^{+}$$,
+the induced polarization $$\vb{P}^{+}$$,
+and the total population inversion $$D$$:
 
 $$\begin{aligned}
     - \mu_0 \pdvn{2}{\vb{P}^{+}}{t}
@@ -28,16 +28,16 @@ $$\begin{aligned}
     &= \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 $\vb{E}^{-} = (\vb{E}^{+})^*$.
+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 $\omega \approx \omega_0$:
+where $$\vb{E}_0^{+}$$ and $$\vb{P}_0^{+}$$ are slowly-varying envelopes
+of a plane wave with angular frequency $$\omega \approx \omega_0$$:
 
 $$\begin{aligned}
     \vb{E}^{+}(\vb{r}, t)
@@ -48,9 +48,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We insert this into the first MBE,
-and assume that $\vb{E}_0^{+}$ and $\vb{P}_0^{+}$
+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 $\ipdvn{2}{\vb{P}_0^{+}\!}{t} \approx 0$,
+i.e. $$\ipdvn{2}{\vb{E}_0^{+}\!}{t} \approx 0$$ and $$\ipdvn{2}{\vb{P}_0^{+}\!}{t} \approx 0$$,
 giving:
 
 $$\begin{aligned}
@@ -62,7 +62,7 @@ $$\begin{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
+where $$\Omega$$ is an eigenfrequency of the optical cavity
 in which lasing will occur:
 
 $$\begin{aligned}
@@ -71,7 +71,7 @@ $$\begin{aligned}
 \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.
+where there is only one $$\Omega$$ and $$\vb{E}_0^{+}$$ to care about.
 Substituting the above equation into the first MBE yields:
 
 $$\begin{aligned}
@@ -79,9 +79,9 @@ $$\begin{aligned}
     = \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:
+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:
 
 $$\begin{aligned}
     i 2 \pdv{\vb{P}_0^{+}}{t} + \omega \vb{P}_0^{+}
@@ -89,18 +89,18 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Moving on to the second MBE,
-inserting the ansatz $\vb{P}^{+} = \vb{P}_0^{+} e^{-i \omega t} / 2$ leads to:
+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:
+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:
 
 $$\begin{aligned}
     \vb{P}^{+}
@@ -108,7 +108,7 @@ $$\begin{aligned}
     = \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D
 \end{aligned}$$
 
-Where the Lorentzian gain curve $\gamma(\omega)$
+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:
@@ -118,7 +118,7 @@ $$\begin{aligned}
     \equiv \frac{\gamma_\perp}{(\omega - \omega_0) + i \gamma_\perp}
 \end{aligned}$$
 
-Note that $\gamma(\omega)$ satisfies the following relation,
+Note that $$\gamma(\omega)$$ satisfies the following relation,
 which will be useful to us later:
 
 $$\begin{aligned}
@@ -127,8 +127,8 @@ $$\begin{aligned}
     = 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 $\vb{P}_0^{+}$:
+Returning to the first MBE with $$\ipdv{\vb{P}_0^{+}}{t} \approx 0$$,
+we substitute the above expression for $$\vb{P}_0^{+}$$:
 
 $$\begin{aligned}
     \pdv{\vb{E}_0^{+}}{t}
@@ -137,10 +137,10 @@ $$\begin{aligned}
     &= 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:
+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:
 
 $$\begin{aligned}
     \pdv{D}{t}
@@ -155,7 +155,7 @@ $$\begin{aligned}
 
 This is the prototype of the first laser rate equation.
 However, in order to have a practical set,
-we need an equation for $|\vb{E}|^2$,
+we need an equation for $$|\vb{E}|^2$$,
 which we can obtain using the first MBE:
 
 $$\begin{aligned}
@@ -171,7 +171,7 @@ $$\begin{aligned}
     &= 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.
+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:
 
@@ -187,8 +187,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-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:
+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:
 
 $$\begin{aligned}
     N_{p}
@@ -196,12 +196,12 @@ $$\begin{aligned}
     = \frac{\varepsilon_0 n^2}{2 \hbar \omega} |\vb{E}|^2
 \end{aligned}$$
 
-Furthermore, consider the definition of the inversion $D$:
+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.
+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:
 
 $$\begin{aligned}
@@ -216,10 +216,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $\gamma_e$ is a redefinition of $\gamma_\parallel$
+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 $R_\mathrm{pump}$
+and the photon loss rate $$\gamma_p$$, the gain $$G$$,
+and the carrier supply rate $$R_\mathrm{pump}$$
 are defined like so:
 
 $$\begin{aligned}
@@ -234,14 +234,14 @@ $$\begin{aligned}
     \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
+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 $t$:
+i.e. where $$N_p$$ and $$N_e$$ are constant in $$t$$:
 
 $$\begin{aligned}
     0
@@ -251,9 +251,9 @@ $$\begin{aligned}
     &= 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:
+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:
 
 $$\begin{aligned}
      N_e
@@ -265,27 +265,27 @@ $$\begin{aligned}
     }
 \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
+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 $R_\mathrm{pump}$,
+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,
+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 $N_p$,
+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.
+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 $N_e$.
+in a more empirical form, which "bookkeeps" the processes affecting $$N_p$$ and $$N_e$$.
 Consider the following example:
 
 $$\begin{aligned}
@@ -301,17 +301,17 @@ $$\begin{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
+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 $N_e$,
+To make matters even worse, many of these decay rates depend on $$N_p$$ or $$N_e$$,
 so solutions can only be obtained numerically.