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/self-phase-modulation/index.md | 15 +++++++++------
 1 file changed, 9 insertions(+), 6 deletions(-)

(limited to 'source/know/concept/self-phase-modulation')

diff --git a/source/know/concept/self-phase-modulation/index.md b/source/know/concept/self-phase-modulation/index.md
index 48ea20b..931e10b 100644
--- a/source/know/concept/self-phase-modulation/index.md
+++ b/source/know/concept/self-phase-modulation/index.md
@@ -12,8 +12,8 @@ layout: "concept"
 
 In fiber optics, **self-phase modulation** (SPM) is a nonlinear effect
 that gradually broadens pulses' spectra.
-Unlike dispersion, SPM does create new frequencies: in the $$\omega$$-domain,
-the pulse steadily spreads out with a distinctive "accordion" peak.
+Unlike dispersion, SPM creates frequencies: in the $$\omega$$-domain,
+the pulse steadily spreads out in a distinctive "accordion" shape.
 Lower frequencies are created at the front of the
 pulse and higher ones at the back, giving S-shaped spectrograms.
 
@@ -32,22 +32,25 @@ For any arbitrary input pulse $$A_0(t) = A(0, t)$$,
 we arrive at the following analytical solution:
 
 $$\begin{aligned}
-    A(z,t) = A_0 \exp\!\big( i \gamma |A_0|^2 z\big)
+    A(z,t)
+    = A_0 \exp\!\big( i \gamma |A_0|^2 z\big)
 \end{aligned}$$
 
 The intensity $$|A|^2$$ in the time domain is thus unchanged,
 and only its phase is modified.
-It is also clear that the largest phase increase occurs at the peak of the pulse,
+Clearly, the largest phase shift increase occurs at the peak,
 where the intensity is $$P_0$$.
 To quantify this, it is useful to define the **nonlinear length** $$L_N$$,
 which gives the distance after which the phase of the
 peak has increased by exactly 1 radian:
 
 $$\begin{aligned}
-    \gamma P_0 L_N = 1
+    \gamma P_0 L_N
+    = 1
     \qquad \implies \qquad
     \boxed{
-        L_N = \frac{1}{\gamma P_0}
+        L_N
+        \equiv \frac{1}{\gamma P_0}
     }
 \end{aligned}$$
 
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