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

1 files changed, 23 insertions, 23 deletions

diff --git a/source/know/concept/fabry-perot-cavity/index.md b/source/know/concept/fabry-perot-cavity/index.md
index 3f47c3e..980fa54 100644
--- a/source/know/concept/fabry-perot-cavity/index.md
+++ b/source/know/concept/fabry-perot-cavity/index.md
@@ -15,9 +15,9 @@ which may transmit some of the incoming light.
 Such a setup can be used as e.g. an interferometer or a laser cavity.
 
 Below, we calculate its quasinormal modes in 1D.
-We divide the $x$-axis into three domains: left $L$, center $C$, and right $R$.
-The cavity $C$ has length $\ell$ and is centered on $x = 0$.
-Let $n_L$, $n_C$ and $n_R$ be the respective domains' refractive indices:
+We divide the $$x$$-axis into three domains: left $$L$$, center $$C$$, and right $$R$$.
+The cavity $$C$$ has length $$\ell$$ and is centered on $$x = 0$$.
+Let $$n_L$$, $$n_C$$ and $$n_R$$ be the respective domains' refractive indices:
 
 <a href="cavity.png">
 <img src="cavity.png" style="width:70%">
@@ -28,8 +28,8 @@ Let $n_L$, $n_C$ and $n_R$ be the respective domains' refractive indices:
 
 In its simplest "microscopic" form, the reflection at the boundaries
 is simply caused by the index differences there.
-Consider this ansatz for the [electric field](/know/concept/electric-field/) $E_m(x)$,
-where $m$ is the mode:
+Consider this ansatz for the [electric field](/know/concept/electric-field/) $$E_m(x)$$,
+where $$m$$ is the mode:
 
 $$\begin{aligned}
     E_m(x)
@@ -40,9 +40,9 @@ $$\begin{aligned}
     \end{cases}
 \end{aligned}$$
 
-The goal is to find the modes' wavenumbers $k_m$.
-First, we demand that $E_m$ and its derivative $\idv{E_m}{x}$
-are continuous at the boundaries $x = \pm \ell/2$:
+The goal is to find the modes' wavenumbers $$k_m$$.
+First, we demand that $$E_m$$ and its derivative $$\idv{E_m}{x}$$
+are continuous at the boundaries $$x = \pm \ell/2$$:
 
 $$\begin{aligned}
     A_1 e^{i k_m n_L \ell/2}
@@ -95,8 +95,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We want non-trivial solutions, where we
-cannot simply satisfy the system by setting $A_1$, $A_2$, $A_3$ and
-$A_4$; this constraint will give us an equation for $k_m$. Therefore, we
+cannot simply satisfy the system by setting $$A_1$$, $$A_2$$, $$A_3$$ and
+$$A_4$$; this constraint will give us an equation for $$k_m$$. Therefore, we
 demand that the system matrix is singular, i.e. its determinant is zero:
 
 $$\begin{aligned}
@@ -106,7 +106,7 @@ $$\begin{aligned}
     &+ (n_C^2 + n_L n_R) \big( e^{i k_m (2 n_C - n_L - n_R) \ell/2} - e^{- i k_m (2 n_C + n_L + n_R) \ell/2} \big)
 \end{aligned}$$
 
-We multiply by $e^{i k_m (n_L + n_R) \ell / 2}$ and
+We multiply by $$e^{i k_m (n_L + n_R) \ell / 2}$$ and
 decompose the exponentials into sines and cosines:
 
 $$\begin{aligned}
@@ -125,8 +125,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Thanks to linearity, we can choose one of the amplitudes
-$A_1$, $A_2$, $A_3$ or $A_4$ freely,
-and then the others are determined by $k_m$ and the field's continuity.
+$$A_1$$, $$A_2$$, $$A_3$$ or $$A_4$$ freely,
+and then the others are determined by $$k_m$$ and the field's continuity.
 
 
 ## Macroscopic cavity
@@ -134,7 +134,7 @@ and then the others are determined by $k_m$ and the field's continuity.
 Next, consider a "macroscopic" Fabry-Pérot cavity
 with complex mirror structures at boundaries, e.g. Bragg reflectors.
 If the cavity is large enough, we can neglect the mirrors' thicknesses,
-and just use their reflection coefficients $r_L$ and $r_R$.
+and just use their reflection coefficients $$r_L$$ and $$r_R$$.
 We use the same ansatz:
 
 $$\begin{aligned}
@@ -147,10 +147,10 @@ $$\begin{aligned}
     \end{cases}
 \end{aligned}$$
 
-On the left, $A_3$ is the reflection of $A_2$,
-and on the right, $A_2$ is the reflection of $A_3$,
+On the left, $$A_3$$ is the reflection of $$A_2$$,
+and on the right, $$A_2$$ is the reflection of $$A_3$$,
 where the reflected amplitudes are determined
-by the coefficients $r_L$ and $r_R$, respectively:
+by the coefficients $$r_L$$ and $$r_R$$, respectively:
 
 $$\begin{aligned}
     A_3 e^{- i k_m n_C \ell/2}
@@ -185,8 +185,8 @@ $$\begin{aligned}
     &= 1 - r_L r_R e^{i 2 k_m n_C \ell}
 \end{aligned}$$
 
-Isolating this for $k_m$ yields the following modes,
-where $m$ is an arbitrary integer:
+Isolating this for $$k_m$$ yields the following modes,
+where $$m$$ is an arbitrary integer:
 
 $$\begin{aligned}
     \boxed{
@@ -195,12 +195,12 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-These $k_m$ satisfy the matrix equation above.
-Thanks to linearity, we can choose one of $A_2$ or $A_3$,
+These $$k_m$$ satisfy the matrix equation above.
+Thanks to linearity, we can choose one of $$A_2$$ or $$A_3$$,
 and then the other is determined by the corresponding reflection equation.
 
 Finally, we look at the light transmitted through the mirrors,
-according to $1 \!-\! r_L$ and $1 \!-\! r_R$:
+according to $$1 \!-\! r_L$$ and $$1 \!-\! r_R$$:
 
 $$\begin{aligned}
     A_1 e^{i k_m n_L \ell/2}
@@ -210,7 +210,7 @@ $$\begin{aligned}
     &= (1 - r_R) A_3 e^{i k_m n_C \ell/2}
 \end{aligned}$$
 
-We simply isolate for $A_1$ and $A_4$ respectively,
+We simply isolate for $$A_1$$ and $$A_4$$ respectively,
 yielding the following amplitudes:
 
 $$\begin{aligned}