From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 20 Dec 2022 20:11:25 +0100
Subject: More improvements to knowledge base

---
 source/know/concept/dispersive-broadening/index.md | 19 +++++++++++--------
 1 file changed, 11 insertions(+), 8 deletions(-)

(limited to 'source/know/concept/dispersive-broadening/index.md')

diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md
index 746eb6d..9642737 100644
--- a/source/know/concept/dispersive-broadening/index.md
+++ b/source/know/concept/dispersive-broadening/index.md
@@ -9,10 +9,10 @@ categories:
 layout: "concept"
 ---
 
-In optical fibers, **dispersive broadening** is a (linear) effect
+In optical fibers, **dispersive broadening** is a linear effect
 where group velocity dispersion (GVD) "smears out" a pulse in the time domain
 due to the different group velocities of its frequencies,
-since pulses always have a non-zero width in the $$\omega$$-domain.
+since pulses always have a nonzero width in the $$\omega$$-domain.
 No new frequencies are created.
 
 A pulse envelope $$A(z, t)$$ inside a fiber must obey the nonlinear Schrödinger equation,
@@ -29,7 +29,7 @@ and consider a Gaussian initial condition:
 
 $$\begin{aligned}
     A(0, t)
-    = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
+    = \sqrt{P_0} \exp\!\bigg(\!-\!\frac{t^2}{2 T_0^2}\bigg)
 \end{aligned}$$
 
 By [Fourier transforming](/know/concept/fourier-transform/) in $$t$$,
@@ -38,7 +38,8 @@ where it can be seen that the amplitude
 decreases and the width increases with $$z$$:
 
 $$\begin{aligned}
-    A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
+    A(z,t)
+    = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
     \exp\!\bigg(\! -\!\frac{t^2 / (2 T_0^2)}{1 + \beta_2^2 z^2 / T_0^4} \big( 1 + i \beta_2 z / T_0^2 \big) \bigg)
 \end{aligned}$$
 
@@ -48,10 +49,12 @@ as the distance over which the half-width at $$1/e$$ of maximum power
 (initially $$T_0$$) increases by a factor of $$\sqrt{2}$$:
 
 $$\begin{aligned}
-    T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2}
+    T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4}
+    = T_0 \sqrt{2}
     \qquad \implies \qquad
     \boxed{
-        L_D = \frac{T_0^2}{|\beta_2|}
+        L_D
+        \equiv \frac{T_0^2}{|\beta_2|}
     }
 \end{aligned}$$
 
@@ -68,7 +71,7 @@ where $$\phi(z, t)$$ is the phase of $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t
 
 $$\begin{aligned}
     \omega_{\mathrm{GVD}}(z,t)
-    = \pdv{}{t}\Big( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \Big)
+    = \pdv{}{t}\bigg( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \bigg)
     = \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
 \end{aligned}$$
 
@@ -76,7 +79,7 @@ This expression is linear in time, and depending on the sign of $$\beta_2$$,
 frequencies on one side of the pulse arrive first,
 and those on the other side arrive last.
 The effect is stronger for smaller $$T_0$$:
-this makes sense, since short pulses are spectrally wider.
+this makes sense, since shorter pulses are spectrally wider.
 
 The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/)
 leads to many interesting effects,
-- 
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