Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 50 insertions, 50 deletions

diff --git a/source/know/concept/optical-wave-breaking/index.md b/source/know/concept/optical-wave-breaking/index.md
index aa43d25..42064ff 100644
--- a/source/know/concept/optical-wave-breaking/index.md
+++ b/source/know/concept/optical-wave-breaking/index.md
@@ -14,18 +14,18 @@ In fiber optics, **optical wave breaking** (OWB) is a nonlinear effect
 caused by interaction between
 [group velocity dispersion](/know/concept/dispersive-broadening/) (GVD) and
 [self-phase modulation](/know/concept/self-phase-modulation/) (SPM).
-It only happens in the normal dispersion regime ($\beta_2 > 0$)
+It only happens in the normal dispersion regime ($$\beta_2 > 0$$)
 for pulses meeting a certain criterium, as we will see.
 
 SPM creates low frequencies at the front of the pulse, and high ones at the back,
-and if $\beta_2 > 0$, GVD lets low frequencies travel faster than high ones.
+and if $$\beta_2 > 0$$, GVD lets low frequencies travel faster than high ones.
 When those effects interact, the pulse gets temporally stretched
 in a surprisingly sophisticated way.
 
-To illustrate this, the instantaneous frequency $\omega_i(z, t) = -\ipdv{\phi}{t}$
+To illustrate this, the instantaneous frequency $$\omega_i(z, t) = -\ipdv{\phi}{t}$$
 has been plotted below for a theoretical Gaussian input pulse experiencing OWB,
-with settings $T_0 = 100\:\mathrm{fs}$, $P_0 = 5\:\mathrm{kW}$,
-$\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$.
+with settings $$T_0 = 100\:\mathrm{fs}$$, $$P_0 = 5\:\mathrm{kW}$$,
+$$\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$.
 
 In the left panel, we see the typical S-shape caused by SPM,
 and the arrows indicate the direction that GVD is pushing the curve in.
@@ -41,30 +41,30 @@ hence the name *wave breaking*:
 Several interesting things happen around this moment.
 To demonstrate this, spectrograms of the same simulation
 have been plotted below, together with pulse profiles
-in both the $t$-domain and $\omega$-domain on an arbitrary linear scale
+in both the $$t$$-domain and $$\omega$$-domain on an arbitrary linear scale
 (click the image to get a better look).
 
 Initially, the spectrum broadens due to SPM in the usual way,
 but shortly after OWB, this process is stopped by the appearance
-of so-called **sidelobes** in the $\omega$-domain on either side of the pulse.
+of so-called **sidelobes** in the $$\omega$$-domain on either side of the pulse.
 In the meantime, in the time domain,
 the pulse steepens at the edges, but flattens at the peak.
 After OWB, a train of small waves falls off the edges,
-which eventually melt together, leading to a trapezoid shape in the $t$-domain.
+which eventually melt together, leading to a trapezoid shape in the $$t$$-domain.
 Dispersive broadening then continues normally:
 
 <a href="pheno-break-sgram.jpg">
 <img src="pheno-break-sgram-small.jpg" style="width:80%">
 </a>
 
-We call the distance at which the wave breaks $L_\mathrm{WB}$,
+We call the distance at which the wave breaks $$L_\mathrm{WB}$$,
 and would like to analytically predict it.
-We do this using the instantaneous frequency $\omega_i$,
+We do this using the instantaneous frequency $$\omega_i$$,
 by estimating when the SPM fluctuations overtake their own base,
 as was illustrated earlier.
 
-To get $\omega_i$ of a Gaussian pulse experiencing both GVD and SPM,
-it is a reasonable approximation, for small $z$, to simply add up
+To get $$\omega_i$$ of a Gaussian pulse experiencing both GVD and SPM,
+it is a reasonable approximation, for small $$z$$, to simply add up
 the instantaneous frequencies for these separate effects:
 
 $$\begin{aligned}
@@ -74,7 +74,7 @@ $$\begin{aligned}
     + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
 \end{aligned}$$
 
-Assuming that $z$ is small enough such that $z^2 \approx 0$, this
+Assuming that $$z$$ is small enough such that $$z^2 \approx 0$$, this
 expression can be reduced to:
 
 $$\begin{aligned}
@@ -83,8 +83,8 @@ $$\begin{aligned}
     = \frac{\beta_2 t z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
 \end{aligned}$$
 
-Where we have assumed $\beta_2 > 0$,
-and $N_\mathrm{sol}$ is the **soliton number**,
+Where we have assumed $$\beta_2 > 0$$,
+and $$N_\mathrm{sol}$$ is the **soliton number**,
 which is defined as:
 
 $$\begin{aligned}
@@ -93,11 +93,11 @@ $$\begin{aligned}
 
 This quantity is very important in anomalous dispersion,
 but even in normal dispesion, it is still a useful measure of the relative strengths of GVD and SPM.
-As was illustrated earlier, $\omega_i$ overtakes itself at the edges,
-so OWB occurs when $\omega_i$ oscillates there,
-which starts when its $t$-derivative,
-the **instantaneous chirpyness** $\xi_i$,
-has *two* real roots for $t^2$:
+As was illustrated earlier, $$\omega_i$$ overtakes itself at the edges,
+so OWB occurs when $$\omega_i$$ oscillates there,
+which starts when its $$t$$-derivative,
+the **instantaneous chirpyness** $$\xi_i$$,
+has *two* real roots for $$t^2$$:
 
 $$\begin{aligned}
     0
@@ -107,10 +107,10 @@ $$\begin{aligned}
     = \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big)
 \end{aligned}$$
 
-Where the function $f(x)$ has been defined for convenience. As it turns
-out, this equation can be solved analytically using the Lambert $W$ function,
-leading to the following exact minimum value $N_\mathrm{min}^2$ for $N_\mathrm{sol}^2$,
-such that OWB can only occur when $N_\mathrm{sol}^2 > N_\mathrm{min}^2$:
+Where the function $$f(x)$$ has been defined for convenience. As it turns
+out, this equation can be solved analytically using the Lambert $$W$$ function,
+leading to the following exact minimum value $$N_\mathrm{min}^2$$ for $$N_\mathrm{sol}^2$$,
+such that OWB can only occur when $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$:
 
 $$\begin{aligned}
     \boxed{
@@ -118,15 +118,15 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If this condition $N_\mathrm{sol}^2 > N_\mathrm{min}^2$ is not satisfied,
-$\xi_i$ cannot have two roots for $t^2$, meaning $\omega_i$ cannot overtake itself.
+If this condition $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$ is not satisfied,
+$$\xi_i$$ cannot have two roots for $$t^2$$, meaning $$\omega_i$$ cannot overtake itself.
 GVD is unable to keep up with SPM, so OWB will not occur.
 
-Next, consider two points at $t_1$ and $t_2$ in the pulse,
-separated by a small initial interval $(t_2 - t_1)$.
-The frequency difference between these points due to $\omega_i$
+Next, consider two points at $$t_1$$ and $$t_2$$ in the pulse,
+separated by a small initial interval $$(t_2 - t_1)$$.
+The frequency difference between these points due to $$\omega_i$$
 will cause them to displace relative to each other
-after a short distance $z$ by some amount $\Delta t$,
+after a short distance $$z$$ by some amount $$\Delta t$$,
 estimated by:
 
 $$\begin{aligned}
@@ -144,12 +144,12 @@ $$\begin{aligned}
     &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1)
 \end{aligned}$$
 
-Where $\beta_1(\omega)$ is the inverse of the group velocity.
-OWB takes place when $t_2$ and $t_1$ catch up to each other,
-which is when $-\Delta t = (t_2 - t_1)$.
-The distance where this happens first, $z = L_\mathrm{WB}$,
+Where $$\beta_1(\omega)$$ is the inverse of the group velocity.
+OWB takes place when $$t_2$$ and $$t_1$$ catch up to each other,
+which is when $$-\Delta t = (t_2 - t_1)$$.
+The distance where this happens first, $$z = L_\mathrm{WB}$$,
 must therefore satisfy the following condition
-for a particular value of $t$:
+for a particular value of $$t$$:
 
 $$\begin{aligned}
     L_\mathrm{WB} \, \beta_2 \, \xi_i(L_\mathrm{WB}, t) = -1
@@ -157,9 +157,9 @@ $$\begin{aligned}
     L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 \, f(t^2/T_0^2)}
 \end{aligned}$$
 
-The time $t$ of OWB must be where $\omega_i(t)$ has its steepest slope,
-which is at the minimum value of $\xi_i(t)$, and by extension $f(x)$.
-This turns out to be $f(3/2)$:
+The time $$t$$ of OWB must be where $$\omega_i(t)$$ has its steepest slope,
+which is at the minimum value of $$\xi_i(t)$$, and by extension $$f(x)$$.
+This turns out to be $$f(3/2)$$:
 
 $$\begin{aligned}
     f_\mathrm{min} = f(3/2)
@@ -167,10 +167,10 @@ $$\begin{aligned}
     = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2
 \end{aligned}$$
 
-Clearly, $f_\mathrm{min} \ge 0$ when $N_\mathrm{sol}^2 \le N_\mathrm{min}^2$,
-which, when inserted above, leads to an imaginary $L_\mathrm{WB}$,
+Clearly, $$f_\mathrm{min} \ge 0$$ when $$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$$,
+which, when inserted above, leads to an imaginary $$L_\mathrm{WB}$$,
 confirming that OWB cannot occur in that case.
-Otherwise, if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then:
+Otherwise, if $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$, then:
 
 $$\begin{aligned}
     \boxed{
@@ -180,24 +180,24 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-This prediction for $L_\mathrm{WB}$ appears to agree well
+This prediction for $$L_\mathrm{WB}$$ appears to agree well
 with the OWB observed in the simulation:
 
 <a href="pheno-break.jpg">
 <img src="pheno-break-small.jpg" style="width:100%">
 </a>
 
-Because all spectral broadening up to $L_\mathrm{WB}$ is caused by SPM,
+Because all spectral broadening up to $$L_\mathrm{WB}$$ is caused by SPM,
 whose frequency behaviour is known, it is in fact possible to draw
 some analytical conclusions about the achieved bandwidth when OWB sets in.
-Filling $L_\mathrm{WB}$ in into $\omega_\mathrm{SPM}$ gives:
+Filling $$L_\mathrm{WB}$$ in into $$\omega_\mathrm{SPM}$$ gives:
 
 $$\begin{aligned}
     \omega_{\mathrm{SPM}}(L_\mathrm{WB},t)
     = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
 \end{aligned}$$
 
-Assuming that $N_\mathrm{sol}^2$ is large in the denominator, this can
+Assuming that $$N_\mathrm{sol}^2$$ is large in the denominator, this can
 be approximately reduced to:
 
 $$\begin{aligned}
@@ -206,18 +206,18 @@ $$\begin{aligned}
     = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
 \end{aligned}$$
 
-The expression $x \exp(-x^2)$ has its global extrema
-$\pm 1 / \sqrt{2 e}$ at $x^2 = 1/2$. The maximum SPM frequency shift
-achieved at $L_\mathrm{WB}$ is therefore given by:
+The expression $$x \exp(-x^2)$$ has its global extrema
+$$\pm 1 / \sqrt{2 e}$$ at $$x^2 = 1/2$$. The maximum SPM frequency shift
+achieved at $$L_\mathrm{WB}$$ is therefore given by:
 
 $$\begin{aligned}
     \omega_\mathrm{max} = \sqrt{\frac{2 \gamma P_0}{e \beta_2}}
 \end{aligned}$$
 
-Interestingly, this expression does not contain $T_0$ at all,
+Interestingly, this expression does not contain $$T_0$$ at all,
 so the achieved spectrum when SPM is halted by OWB
 is independent of the pulse width,
-for sufficiently large $N_\mathrm{sol}$.
+for sufficiently large $$N_\mathrm{sol}$$.
 
 
 ## References