From d6f086e33d143ec6e84b0058e7d8832c166f4427 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 17 Jul 2022 09:06:31 +0200
Subject: Minor fixes

---
 content/know/concept/coupled-mode-theory/index.pdc | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

(limited to 'content/know/concept/coupled-mode-theory/index.pdc')

diff --git a/content/know/concept/coupled-mode-theory/index.pdc b/content/know/concept/coupled-mode-theory/index.pdc
index c9e9ad4..581dce4 100644
--- a/content/know/concept/coupled-mode-theory/index.pdc
+++ b/content/know/concept/coupled-mode-theory/index.pdc
@@ -109,6 +109,7 @@ 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 $S_\ell^\mathrm{in}$:
+
 $$\begin{aligned}
     A(t)
     = A e^{-i \omega_0 t + t / \tau_\ell}
@@ -129,7 +130,7 @@ $$\begin{aligned}
     = \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|
+    = \frac{4}{\tau_\ell^2} |A|^2
 \end{aligned}$$
 
 But thanks to energy conservation,
@@ -203,7 +204,8 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \dv{A}{t}
-            &= - i \omega_0 A - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A
+            &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) 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}
-- 
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