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/self-phase-modulation/index.md | 30 +++++++++++-----------
 1 file changed, 15 insertions(+), 15 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 3b46c4e..f13ad2f 100644
--- a/source/know/concept/self-phase-modulation/index.md
+++ b/source/know/concept/self-phase-modulation/index.md
@@ -12,13 +12,13 @@ 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,
+Unlike dispersion, SPM does create new frequencies: in the $$\omega$$-domain,
 the pulse steadily spreads out with a distinctive "accordion" peak.
 Lower frequencies are created at the front of the
 pulse and higher ones at the back, giving S-shaped spectrograms.
 
-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:
 
 $$\begin{aligned}
@@ -26,20 +26,20 @@ $$\begin{aligned}
     = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A
 \end{aligned}$$
 
-By setting $\beta_2 = 0$ to neglect dispersion,
+By setting $$\beta_2 = 0$$ to neglect dispersion,
 solving this equation becomes trivial.
-For any arbitrary input pulse $A_0(t) = A(0, t)$,
+For any arbitrary input pulse $$A_0(t) = A(0, t)$$,
 we arrive at the following analytical solution:
 
 $$\begin{aligned}
     A(z,t) = A_0 \exp\!\big( i \gamma |A_0|^2 z\big)
 \end{aligned}$$
 
-The intensity $|A|^2$ in the time domain is thus unchanged,
+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,
-where the intensity is $P_0$.
-To quantify this, it is useful to define the **nonlinear length** $L_N$,
+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:
 
@@ -52,15 +52,15 @@ $$\begin{aligned}
 \end{aligned}$$
 
 SPM is illustrated below for the following Gaussian initial pulse envelope,
-with parameter values $T_0 = 6\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$,
-$\beta_2 = 0$, and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$:
+with parameter values $$T_0 = 6\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$,
+$$\beta_2 = 0$$, and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$:
 
 $$\begin{aligned}
     A(0, t)
     = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
 \end{aligned}$$
 
-From earlier, we then know the analytical solution for the $z$-evolution:
+From earlier, we then know the analytical solution for the $$z$$-evolution:
 
 $$\begin{aligned}
     A(z, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \exp\!\bigg( i \gamma z P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
@@ -70,10 +70,10 @@ $$\begin{aligned}
 <img src="pheno-spm-small.jpg" style="width:100%">
 </a>
 
-The **instantaneous frequency** $\omega_\mathrm{SPM}(z, t)$,
+The **instantaneous frequency** $$\omega_\mathrm{SPM}(z, t)$$,
 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 $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$$:
 
 $$\begin{aligned}
     \omega_{\mathrm{SPM}}(z,t)
@@ -82,8 +82,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This result gives the S-shaped spectrograms seen in the illustration.
-The frequency shift thus not only depends on $L_N$,
-but also on $T_0$: the spectra of narrow pulses broaden much faster.
+The frequency shift thus not only depends on $$L_N$$,
+but also on $$T_0$$: the spectra of narrow pulses broaden much faster.
 
 The interaction between self-phase modulation
 and [dispersion](/know/concept/dispersive-broadening/)
-- 
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