From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/dispersive-broadening/index.md | 38 +++++++++++-----------
 1 file changed, 19 insertions(+), 19 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 816135a..4e4cf82 100644
--- a/source/know/concept/dispersive-broadening/index.md
+++ b/source/know/concept/dispersive-broadening/index.md
@@ -12,11 +12,11 @@ layout: "concept"
 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 non-zero 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,
-where the parameters $\beta_2$ and $\gamma$ respectively
+A pulse envelope $$A(z, t)$$ inside a fiber must obey the nonlinear Schrödinger equation,
+where the parameters $$\beta_2$$ and $$\gamma$$ respectively
 control dispersion and nonlinearity:
 
 $$\begin{aligned}
@@ -24,7 +24,7 @@ $$\begin{aligned}
     = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A
 \end{aligned}$$
 
-We set $\gamma = 0$ to ignore all nonlinear effects,
+We set $$\gamma = 0$$ to ignore all nonlinear effects,
 and consider a Gaussian initial condition:
 
 $$\begin{aligned}
@@ -32,10 +32,10 @@ $$\begin{aligned}
     = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
 \end{aligned}$$
 
-By [Fourier transforming](/know/concept/fourier-transform/) in $t$,
-the full analytical solution $A(z, t)$ is found to be as follows,
+By [Fourier transforming](/know/concept/fourier-transform/) in $$t$$,
+the full analytical solution $$A(z, t)$$ is found to be as follows,
 where it can be seen that the amplitude
-decreases and the width increases with $z$:
+decreases and the width increases with $$z$$:
 
 $$\begin{aligned}
     A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
@@ -43,9 +43,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 To quantify the strength of dispersive effects,
-we define the dispersion length $L_D$
-as the distance over which the half-width at $1/e$ of maximum power
-(initially $T_0$) increases by a factor of $\sqrt{2}$:
+we define the dispersion length $$L_D$$
+as the distance over which the half-width at $$1/e$$ of maximum power
+(initially $$T_0$$) increases by a factor of $$\sqrt{2}$$:
 
 $$\begin{aligned}
     T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2}
@@ -56,17 +56,17 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This phenomenon is illustrated below for our example of a Gaussian pulse
-with parameter values $T_0 = 1\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$,
-$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0$:
+with parameter values $$T_0 = 1\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$,
+$$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0$$:
 
 <a href="pheno-disp.jpg">
 <img src="pheno-disp-small.jpg" style="width:100%">
 </a>
 
-The **instantaneous frequency** $\omega_\mathrm{GVD}(z, t)$,
+The **instantaneous frequency** $$\omega_\mathrm{GVD}(z, t)$$,
 which describes the dominant angular frequency at a given point in the time domain,
 is found to be as follows for the Gaussian pulse,
-where $\phi(z, t)$ is the phase of $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$:
+where $$\phi(z, t)$$ is the phase of $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$$:
 
 $$\begin{aligned}
     \omega_{\mathrm{GVD}}(z,t)
@@ -74,20 +74,20 @@ $$\begin{aligned}
     = \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
 \end{aligned}$$
 
-This expression is linear in time, and depending on the sign of $\beta_2$,
+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 $T_0$:
+The effect is stronger for smaller $$T_0$$:
 this makes sense, since short pulses are spectrally wider.
 
 The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/)
 leads to many interesting effects,
 such as [modulational instability](/know/concept/modulational-instability/)
 and [optical wave breaking](/know/concept/optical-wave-breaking/).
-Of great importance is the sign of $\beta_2$:
-in the **anomalous dispersion regime** ($\beta_2 < 0$),
+Of great importance is the sign of $$\beta_2$$:
+in the **anomalous dispersion regime** ($$\beta_2 < 0$$),
 lower frequencies travel more slowly than higher ones,
-and vice versa in the **normal dispersion regime** ($\beta_2 > 0$).
+and vice versa in the **normal dispersion regime** ($$\beta_2 > 0$$).
 
 
 
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