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

1 files changed, 46 insertions, 46 deletions

diff --git a/source/know/concept/coupled-mode-theory/index.md b/source/know/concept/coupled-mode-theory/index.md
index ecae1bb..6a5ec1b 100644
--- a/source/know/concept/coupled-mode-theory/index.md
+++ b/source/know/concept/coupled-mode-theory/index.md
@@ -10,21 +10,21 @@ layout: "concept"
 
 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:
+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 $t$.
+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 $A$.
+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:
 
 $$\begin{aligned}
@@ -36,35 +36,35 @@ $$\begin{aligned}
     &= \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 $S_\ell^\mathrm{out}$.
+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 $\omega_0$;
+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,
+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 $\omega_0$),
+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 $\tau_0$:
+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:
+To determine $$\gamma_\ell$$, we use energy conservation.
+If all $$S_{\ell'}^\mathrm{in} = 0$$,
+then the energy in $$A$$ decays as:
 
 $$\begin{aligned}
     \dv{|A|^2}{t}
@@ -77,7 +77,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Since all other mechanisms are neglected,
-all this energy must go into $S_\ell^\mathrm{out}$, meaning:
+all this energy must go into $$S_\ell^\mathrm{out}$$, meaning:
 
 $$\begin{aligned}
     |S_\ell^\mathrm{out}|^2
@@ -85,32 +85,32 @@ $$\begin{aligned}
     = \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 $\gamma_\ell = \sqrt{2 / \tau_\ell}$.
+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
+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 $S_\ell^\mathrm{out}$:
+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,
+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 $S_\ell^\mathrm{in}$:
+$$A$$, $$S_\ell^\mathrm{out}$$, 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:
+We insert this expression for $$A(t)$$ into its original differential equation, yielding:
 
 $$\begin{aligned}
     \dv{A}{t}
@@ -118,7 +118,7 @@ $$\begin{aligned}
     = \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:
+Isolating this for $$A$$ leads us to the following power balance equation:
 
 $$\begin{aligned}
     A
@@ -129,7 +129,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 But thanks to energy conservation,
-all power delivered by $S_\ell^\mathrm{in}$ ends up in $A$, so we know:
+all power delivered by $$S_\ell^\mathrm{in}$$ ends up in $$A$$, so we know:
 
 $$\begin{aligned}
     |S_\ell^\mathrm{in}|^2
@@ -138,20 +138,20 @@ $$\begin{aligned}
 \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 $\alpha_\ell = \sqrt{2 / \tau_\ell} = \gamma_\ell$.
+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.
+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:
+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 $A$,
+Meanwhile, using the differential equation for $$A$$,
 we find the following relation:
 
 $$\begin{aligned}
@@ -161,7 +161,7 @@ $$\begin{aligned}
     &= - \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 $\idv{|A|^2}{t}$
+By isolating both of the above relations for $$\idv{|A|^2}{t}$$
 and equating them, we arrive at:
 
 $$\begin{aligned}
@@ -169,9 +169,9 @@ $$\begin{aligned}
     &= - \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 $\alpha_\ell = \gamma_\ell$:
+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 $$\alpha_\ell = \gamma_\ell$$:
 
 $$\begin{aligned}
     |S_\ell^\mathrm{in}|^2 - |S_\ell^\mathrm{out}|^2
@@ -191,8 +191,8 @@ $$\begin{aligned}
     &\quad\; + (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 $\alpha_\ell = \gamma_\ell = \sqrt{2 / \tau_\ell}$,
+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}