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/optical-wave-breaking/index.md | 51 ++++++++++++++--------
 1 file changed, 32 insertions(+), 19 deletions(-)

(limited to 'source/know/concept/optical-wave-breaking')

diff --git a/source/know/concept/optical-wave-breaking/index.md b/source/know/concept/optical-wave-breaking/index.md
index 882749f..3509bc2 100644
--- a/source/know/concept/optical-wave-breaking/index.md
+++ b/source/know/concept/optical-wave-breaking/index.md
@@ -54,7 +54,7 @@ Dispersive broadening then continues normally:
 {% include image.html file="spectrograms-full.png" width="100%" alt="Spectrograms of pulse shape evolution" %}
 
 We call the distance at which the wave breaks $$L_\mathrm{WB}$$,
-and would like to analytically predict it.
+and want to predict it analytically.
 We do this using the instantaneous frequency $$\omega_i$$,
 by estimating when the SPM fluctuations overtake their own base,
 as was illustrated earlier.
@@ -84,11 +84,13 @@ and $$N_\mathrm{sol}$$ is the **soliton number**,
 which is defined as:
 
 $$\begin{aligned}
-    N_\mathrm{sol}^2 \equiv \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|}
+    N_\mathrm{sol}^2
+    \equiv \frac{L_D}{L_N}
+    = \frac{\gamma P_0 T_0^2}{|\beta_2|}
 \end{aligned}$$
 
 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.
+but even in normal dispersion, 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,
@@ -100,17 +102,19 @@ $$\begin{aligned}
     = \xi_i(z,t)
     = \pdv{\omega_i}{t}
     &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
-    = \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big)
+    \equiv \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big)
 \end{aligned}$$
 
 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,
+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 $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$:
 
 $$\begin{aligned}
     \boxed{
-        N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12
+        N_\mathrm{min}^2
+        = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big)
+        \approx 1.12
     }
 \end{aligned}$$
 
@@ -129,28 +133,33 @@ $$\begin{aligned}
     \Delta t
     &\approx z \Delta\beta_1
     \qquad
-    &&\Delta\beta_1 \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
+    &&\Delta\beta_1
+    \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
     \\
     &\approx z \beta_2 \Delta\omega_i
     \qquad
-    &&\Delta\omega_i \equiv \omega_i(z,t_2) - \omega_i(z,t_1)
+    &&\Delta\omega_i
+    \equiv \omega_i(z,t_2) - \omega_i(z,t_1)
     \\
     &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
     \qquad \quad
-    &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1)
+    &&\Delta\xi_i
+    \equiv \xi_i(z,t_2) - \xi_i(z,t_1)
 \end{aligned}$$
 
 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,
+For a certain choice of $$t_1$$ and $$t_2$$,
+OWB occurs when they 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 $$t$$:
+The distance $$L_\mathrm{WB}$$ at which this happens first
+must satisfy the following condition for some value of $$t$$:
 
 $$\begin{aligned}
-    L_\mathrm{WB} \, \beta_2 \, \xi_i(L_\mathrm{WB}, t) = -1
+    L_\mathrm{WB} \: \beta_2 \: \xi_i(L_\mathrm{WB}, t)
+    = -1
     \qquad \implies \qquad
-    L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 \, f(t^2/T_0^2)}
+    L_\mathrm{WB}^2
+    = - \frac{T_0^4}{\beta_2^2 \: f(t^2/T_0^2)}
 \end{aligned}$$
 
 The time $$t$$ of OWB must be where $$\omega_i(t)$$ has its steepest slope,
@@ -158,7 +167,8 @@ which is at the minimum value of $$\xi_i(t)$$, and by extension $$f(x)$$.
 This turns out to be $$f(3/2)$$:
 
 $$\begin{aligned}
-    f_\mathrm{min} = f(3/2)
+    f_\mathrm{min}
+    = f(3/2)
     = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2)
     = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2
 \end{aligned}$$
@@ -182,8 +192,9 @@ with the OWB observed in the simulation:
 {% include image.html file="simulation-full.png" width="100%" alt="Optical wave breaking simulation results" %}
 
 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.
+whose $$\omega$$-domain 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:
 
 $$\begin{aligned}
@@ -205,7 +216,8 @@ $$\pm 1 / \sqrt{2 e}$$ at $$x^2 = 1/2$$. The maximum SPM frequency shift
 achieved at $$L_\mathrm{WB}$$ is therefore given by:
 
 $$\begin{aligned}
-    \omega_\mathrm{max} = \sqrt{\frac{2 \gamma P_0}{e \beta_2}}
+    \omega_\mathrm{max}
+    = \sqrt{\frac{2 \gamma P_0}{e \beta_2}}
 \end{aligned}$$
 
 Interestingly, this expression does not contain $$T_0$$ at all,
@@ -214,6 +226,7 @@ is independent of the pulse width,
 for sufficiently large $$N_\mathrm{sol}$$.
 
 
+
 ## References
 1.  D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro,
     [Wave breaking in nonlinear-optical fibers](https://doi.org/10.1364/JOSAB.9.001358),
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