Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 46 insertions, 46 deletions

diff --git a/source/know/concept/salt-equation/index.md b/source/know/concept/salt-equation/index.md
index bfd69b9..f5f085d 100644
--- a/source/know/concept/salt-equation/index.md
+++ b/source/know/concept/salt-equation/index.md
@@ -15,11 +15,11 @@ makes it especially appropriate for microscopically small lasers.
 
 Consider the [Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/),
 governing the complex polarization
-vector $\vb{P}^{+}$ and the scalar population inversion $D$ of a set of
+vector $$\vb{P}^{+}$$ and the scalar population inversion $$D$$ of a set of
 active atoms (or quantum dots) embedded in a passive linear background
-material with refractive index $c / v$.
+material with refractive index $$c / v$$.
 The system is affected by a driving [electric field](/know/concept/electric-field/)
-$\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$,
+$$\vb{E}^{+}(t) = \vb{E}_0^{+} e^{-i \omega t}$$,
 such that the set of equations is:
 
 $$\begin{aligned}
@@ -34,13 +34,13 @@ $$\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 $\hbar \omega_0$ is the band gap of the active atoms,
-and $\gamma_\perp$ and $\gamma_\parallel$ are relaxation rates
+Where $$\hbar \omega_0$$ is the band gap of the active atoms,
+and $$\gamma_\perp$$ and $$\gamma_\parallel$$ are relaxation rates
 of the atoms' polarization and population inversion, respectively.
-$D_0$ is the equilibrium inversion, i.e. the value of $D$ if there is no lasing.
-Note that $D_0$ also represents the pump,
-and both $D_0$ and $v$ depend on position $\vb{x}$.
-Finally, the transition dipole matrix elements $\vb{p}_0^{-}$ and $\vb{p}_0^{+}$ are given by:
+$$D_0$$ is the equilibrium inversion, i.e. the value of $$D$$ if there is no lasing.
+Note that $$D_0$$ also represents the pump,
+and both $$D_0$$ and $$v$$ depend on position $$\vb{x}$$.
+Finally, the transition dipole matrix elements $$\vb{p}_0^{-}$$ and $$\vb{p}_0^{+}$$ are given by:
 
 $$\begin{aligned}
     \vb{p}_0^{-}
@@ -51,12 +51,12 @@ $$\begin{aligned}
     = (\vb{p}_0^{-})^*
 \end{aligned}$$
 
-With $q < 0$ the electron charge, $\vu{x}$ the quantum position operator,
-and $\Ket{g}$ and $\Ket{e}$ respectively
+With $$q < 0$$ the electron charge, $$\vu{x}$$ the quantum position operator,
+and $$\Ket{g}$$ and $$\Ket{e}$$ respectively
 the ground state and first excitation of the active atoms.
 
-We start by assuming that the cavity has $N$ quasinormal modes $\Psi_n$,
-each with a corresponding polarization $\vb{p}_n$ of the active matter.
+We start by assuming that the cavity has $$N$$ quasinormal modes $$\Psi_n$$,
+each with a corresponding polarization $$\vb{p}_n$$ of the active matter.
 Note that this ansatz already suggests
 that the interactions between the modes are limited:
 
@@ -80,8 +80,8 @@ $$\begin{aligned}
     + \frac{i}{\hbar} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n \: D
 \end{aligned}$$
 
-With being $\vb{p}_0^{+} \vb{p}_0^{-}$ a dyadic product.
-Isolating the latter equation for $\vb{p}_n$ gives us:
+With being $$\vb{p}_0^{+} \vb{p}_0^{-}$$ a dyadic product.
+Isolating the latter equation for $$\vb{p}_n$$ gives us:
 
 $$\begin{aligned}
     \vb{p}_n
@@ -89,7 +89,7 @@ $$\begin{aligned}
     = \frac{\gamma(\omega_n) D}{\hbar \gamma_\perp} \big(\vb{p}_0^{+} \vb{p}_0^{-}\big) \cdot \Psi_n
 \end{aligned}$$
 
-Where we have defined the Lorentzian gain curve $\gamma(\omega_n)$ as follows,
+Where we have defined the Lorentzian gain curve $$\gamma(\omega_n)$$ as follows,
 which represents the laser's preferred frequencies for amplification:
 
 $$\begin{aligned}
@@ -97,10 +97,10 @@ $$\begin{aligned}
     \equiv \frac{\gamma_\perp}{(\omega_n - \omega_0) + i \gamma_\perp}
 \end{aligned}$$
 
-Inserting this expression for $\vb{p}_n$
+Inserting this expression for $$\vb{p}_n$$
 into the first Maxwell-Bloch equation yields
 the prototypical form of the SALT equation,
-where we still need to replace $D$ with known quantities:
+where we still need to replace $$D$$ with known quantities:
 
 $$\begin{aligned}
     0
@@ -108,8 +108,8 @@ $$\begin{aligned}
     - \omega_n^2 \frac{\mu_0 \gamma(\omega_n) D}{\hbar \gamma_\perp} (\vb{p}_0^{+} \vb{p}_0^{-}) \cdot \bigg) \Psi_n 
 \end{aligned}$$
 
-To rewrite $D$, we turn to its (Maxwell-Bloch) equation of motion,
-making the crucial **stationary inversion approximation** $\ipdv{D}{t} = 0$:
+To rewrite $$D$$, we turn to its (Maxwell-Bloch) equation of motion,
+making the crucial **stationary inversion approximation** $$\ipdv{D}{t} = 0$$:
 
 $$\begin{aligned}
     D
@@ -119,8 +119,8 @@ $$\begin{aligned}
 This is the most aggressive approximation we will make:
 it removes all definite phase relations between modes,
 and effectively eliminates time as a variable.
-We insert our ansatz for $\vb{E}^{+}$ and $\vb{P}^{+}$,
-and point out that only excited lasing modes contribute to $D$:
+We insert our ansatz for $$\vb{E}^{+}$$ and $$\vb{P}^{+}$$,
+and point out that only excited lasing modes contribute to $$D$$:
 
 $$\begin{aligned}
     D
@@ -130,9 +130,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Here, we make the [rotating wave approximation](/know/concept/rotating-wave-approximation/)
-to neglect all terms where $\nu \neq \mu$
+to neglect all terms where $$\nu \neq \mu$$
 on the basis that they oscillate too quickly,
-leaving only $\nu = \mu$:
+leaving only $$\nu = \mu$$:
 
 $$\begin{aligned}
     D
@@ -140,8 +140,8 @@ $$\begin{aligned}
     \bigg( \vb{p}_\nu^* \cdot \Psi_\nu - \vb{p}_\nu \cdot \Psi_\nu^* \bigg)
 \end{aligned}$$
 
-Inserting our earlier equation for $\vb{p}_n$
-and using the fact that $\vb{p}_0^{+} = (\vb{p}_0^{-})^*$ leads us to:
+Inserting our earlier equation for $$\vb{p}_n$$
+and using the fact that $$\vb{p}_0^{+} = (\vb{p}_0^{-})^*$$ leads us to:
 
 $$\begin{aligned}
     D
@@ -168,7 +168,7 @@ $$\begin{aligned}
     = i 2 \big|\gamma(\omega_\nu)\big|^2
 \end{aligned}$$
 
-Inserting this into our equation for $D$ gives the following expression:
+Inserting this into our equation for $$D$$ gives the following expression:
 
 $$\begin{aligned}
     D
@@ -176,7 +176,7 @@ $$\begin{aligned}
     \Big|\gamma(\omega_\nu) \vb{p}_0^{-} \cdot \Psi_\nu\Big|^2
 \end{aligned}$$
 
-We then properly isolate this for $D$ to get its final form, namely:
+We then properly isolate this for $$D$$ to get its final form, namely:
 
 $$\begin{aligned}
     D
@@ -197,7 +197,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where we have defined **spatial hole burning** function $h(\vb{x})$ like so,
+Where we have defined **spatial hole burning** function $$h(\vb{x})$$ like so,
 representing the depletion of the supply of charge
 carriers as they are consumed by the active lasing modes:
 
@@ -209,10 +209,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Many authors assume that $\vb{p}_0^- \parallel \Psi_n$,
-so that only its amplitude $|g|^2 \equiv \vb{p}_0^{+} \cdot \vb{p}_0^{-}$ matters.
-In that case, they often non-dimensionalize $D$ and $\Psi_n$
-by dividing out the units $d_c$ and $e_c$:
+Many authors assume that $$\vb{p}_0^- \parallel \Psi_n$$,
+so that only its amplitude $$|g|^2 \equiv \vb{p}_0^{+} \cdot \vb{p}_0^{-}$$ matters.
+In that case, they often non-dimensionalize $$D$$ and $$\Psi_n$$
+by dividing out the units $$d_c$$ and $$e_c$$:
 
 $$\begin{aligned}
     \tilde{\Psi}_n
@@ -228,8 +228,8 @@ $$\begin{aligned}
     \equiv \frac{\varepsilon_0 \hbar \gamma_\perp}{|g|^2}
 \end{aligned}$$
 
-And then the SALT equation and hole burning function $h$ are reduced to the following,
-where the vacuum wavenumber $k_n = \omega_n / c$:
+And then the SALT equation and hole burning function $$h$$ are reduced to the following,
+where the vacuum wavenumber $$k_n = \omega_n / c$$:
 
 $$\begin{aligned}
     0
@@ -249,7 +249,7 @@ $$\begin{aligned}
     = \nabla (\nabla \cdot \Psi) - \nabla^2 \Psi
 \end{aligned}$$
 
-Where $\nabla \cdot \Psi = 0$ thanks to [Gauss' law](/know/concept/maxwells-equations/),
+Where $$\nabla \cdot \Psi = 0$$ thanks to [Gauss' law](/know/concept/maxwells-equations/),
 so we get an even further simplified SALT equation:
 
 $$\begin{aligned}
@@ -258,21 +258,21 @@ $$\begin{aligned}
     + \gamma(c k_n) \frac{\tilde{D}_0}{1 + h} \bigg] \bigg) \tilde{\Psi}_n
 \end{aligned}$$
 
-The challenge is to solve this equation for a given $\varepsilon_r(\vb{x})$ and $D_0(\vb{x})$,
-with the boundary condition that $\Psi_n$ is a plane wave at infinity,
+The challenge is to solve this equation for a given $$\varepsilon_r(\vb{x})$$ and $$D_0(\vb{x})$$,
+with the boundary condition that $$\Psi_n$$ is a plane wave at infinity,
 i.e. that there is light leaving the cavity.
 
-If $k_n$ has a negative imaginary part, then that mode is behaving as an LED.
-Gradually increasing the pump $D_0$ in a chosen region
-causes the $k_n$'s imaginary parts become less negative,
+If $$k_n$$ has a negative imaginary part, then that mode is behaving as an LED.
+Gradually increasing the pump $$D_0$$ in a chosen region
+causes the $$k_n$$'s imaginary parts become less negative,
 until one of them hits the real axis, at which point that mode starts lasing.
-After that, $D_0$ can be increased even further until some other $k_n$ become real.
+After that, $$D_0$$ can be increased even further until some other $$k_n$$ become real.
 
-Below threshold (i.e. before any mode is lasing), the problem is linear in $\Psi_n$,
-but above threshold it is nonlinear, and the amplitude of $\Psi_n$ is adjusted
-such that the corresponding $k_n$ never leaves the real axis.
+Below threshold (i.e. before any mode is lasing), the problem is linear in $$\Psi_n$$,
+but above threshold it is nonlinear, and the amplitude of $$\Psi_n$$ is adjusted
+such that the corresponding $$k_n$$ never leaves the real axis.
 When any mode is lasing, hole burning makes it harder for other modes to activate,
-since it effectively reduces the pump $D_0$.
+since it effectively reduces the pump $$D_0$$.
 
 
 ## References