From 270adb174e9f536f408296ab0141478666dd1690 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 6 Oct 2024 10:43:21 +0200
Subject: Improve knowledge base

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index 1b6b558..2d81b8f 100644
--- a/source/know/concept/optical-wave-breaking/index.md
+++ b/source/know/concept/optical-wave-breaking/index.md
@@ -1,7 +1,7 @@
 ---
 title: "Optical wave breaking"
 sort_title: "Optical wave breaking"
-date: 2021-02-27
+date: 2024-10-06 # Originally 2021-02-27, major rewrite
 categories:
 - Physics
 - Optics
@@ -10,223 +10,470 @@ categories:
 layout: "concept"
 ---
 
-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).
+In fiber optics, **optical wave breaking (OWB)** is an effect
+that can occur in light pulse envelopes $$A(z, t)$$ governed by
+the [nonlinear Schrödinger equation](/know/concept/nonlinear-schrodinger-equation/):
+
+$$\begin{aligned}
+    0
+    &= i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma_0 |A|^2 A
+\end{aligned}$$
+
+OWB is caused by an interaction between
+the [group velocity dispersion (GVD)](/know/concept/dispersive-broadening/)
+caused by the $$\beta_2$$-term,
+and the [self-phase modulation (SPM)](/know/concept/self-phase-modulation/)
+caused by the $$\gamma_0$$ term.
 It only happens in the normal dispersion regime ($$\beta_2 > 0$$)
-for pulses meeting a certain criterium, as we will see.
+for pulses meeting certain criteria, as we shall 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.
+In short, SPM creates low frequencies at the front of the pulse
+and high ones at the back, and for $$\beta_2 > 0$$,
+GVD makes 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 $$\omega_i(z, t) = -\ipdv{\phi}{t}$$
-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 $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$.
-
-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.
-This leads to steepening at the edges, i.e. the S gradually turns into a Z.
-Shortly before the slope would become infinite,
-small waves start "falling off" the edge of the pulse,
-hence the name *wave breaking*:
-
-{% include image.html file="frequency-full.png" width="100%"
-    alt="Instantaneous frequency profile evolution" %}
-
-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
-(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.
-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.
-Dispersive broadening then continues normally:
+To illustrate the resulting dynamics,
+the simulated power $$|A|^2$$ of a Gaussian pulse with settings
+$$T_0 = 100\:\mathrm{fs}$$, $$P_0 = 5\:\mathrm{kW}$$,
+$$\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$
+is plotted below as a function of $$z$$,
+with the time domain on the left
+and the frequency domain on the right:
+
+$$\begin{aligned}
+    A(0, t)
+    &= \sqrt{P_0} \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg)
+\end{aligned}$$
+
+{% include image.html file="simulation-full.png" width="100%"
+    alt="Plot of optical wave breaking simulation results" %}
+
+OWB occurs at a distance called $$L_\mathrm{WB}$$,
+and until that point things look relatively normal,
+with SPM causing spectral broadening
+and GVD causing subtle internal deformation in the time domain.
+After $$L_\mathrm{WB}$$, the pulse suddenly explodes due to GVD,
+and complicated so-called **sidelobes** appear in the frequency domain,
+which seem to block any further SPM.
+To investigate, we plot a series of spectrograms of the same simulation:
 
 {% include image.html file="spectrograms-full.png" width="100%"
-    alt="Spectrograms of pulse shape evolution" %}
+    alt="Spectrograms of simulated pulse shape evolution" %}
+
+At first, we see the appearance of SPM's typical "S" shape,
+which quickly starts turning into a "Z" due to GVD.
+When the transition to "Z" is complete,
+there are many overlapping frequencies at the edges of the pulse.
+This causes a complicated interaction
+that generates the sidelobes,
+and causes a train of small waves to "fall off"
+the near-vertical pulse edges in the time domain,
+hence the name *wave breaking*.
+Eventually, those small waves melt together,
+leaving behind a curious trapezoid shape
+that gets stretched by GVD as usual.
+
+We would like to theoretically predict
+the distance $$L_\mathrm{WB}$$ at which the wave breaks.
+First we show the general principle,
+and then we apply it to a couple of example pulses.
+
+
+
+## General method
 
-We call the distance at which the wave breaks $$L_\mathrm{WB}$$,
-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.
+We make the following ansatz for the complex envelope $$A(z, t)$$,
+without loss of generality:
 
-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:
+$$\begin{aligned}
+    A(z, t)
+    = \psi(z, t) \exp\!\big(i \phi(z, t)\big)
+\end{aligned}$$
+
+Inserting this into the NLS equation and dividing out $$e^{i \phi}$$ yields:
 
 $$\begin{aligned}
-    \omega_i(z,t)
-    &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
-    = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
-    + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+    0
+    &= i \psi_z - \psi \phi_z - \frac{\beta_2}{2} (\psi_{tt} + 2 i \psi_t \phi_t + i \psi \phi_{tt} - \psi \phi_t^2) + \gamma_0 \psi^3
 \end{aligned}$$
 
-Assuming that $$z$$ is small enough such that $$z^2 \approx 0$$, this
-expression can be reduced to:
+Since $$\psi$$ and $$\phi$$ are real by definition,
+we can split this into its real and imaginary parts:
 
 $$\begin{aligned}
-    \omega_i(z,t)
-    \approx \frac{\beta_2 tz}{T_0^4} \bigg( 1 + 2\frac{\gamma P_0 T_0^2}{\beta_2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
-    = \frac{\beta_2 t z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+    0
+    &= \psi_z - \frac{\beta_2}{2} (2 \psi_t \phi_t + \psi \phi_{tt})
+    \\
+    0
+    &= - \psi \phi_z - \frac{\beta_2}{2} (\psi_{tt} - \psi \phi_t^2) + \gamma_0 \psi^3
 \end{aligned}$$
 
-Where we have assumed $$\beta_2 > 0$$,
-and $$N_\mathrm{sol}$$ is the **soliton number**,
-which is defined as:
+For our purposes, the second equation is enough.
+We divide it by $$\psi$$ to get an expression for $$\phi_z$$:
 
 $$\begin{aligned}
-    N_\mathrm{sol}^2
-    \equiv \frac{L_D}{L_N}
-    = \frac{\gamma P_0 T_0^2}{|\beta_2|}
+    \phi_z
+    &= - \frac{\beta_2}{2} \frac{\psi_{tt}}{\psi} + \frac{\beta_2}{2} \Omega_i^2 + \gamma_0 \psi^2
 \end{aligned}$$
 
-This quantity is very important in anomalous dispersion,
-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,
-the **instantaneous chirpyness** $$\xi_i$$,
-has *two* real roots for $$t^2$$:
+Where $$\Omega_i \equiv -\phi_t$$ is the **instantaneous frequency**,
+also called the **frequency-chirp variation**,
+which describes the dominant frequency component at a given point $$(z, t)$$;
+basically the center of the spectrograms shown earlier.
+For small $$z$$, this gives us a linear approximation of $$\phi$$:
 
 $$\begin{aligned}
-    0
-    = \xi_i(z,t)
-    = \pdv{\omega_i}{t}
-    &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
-    \equiv \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big)
+    \phi(z, t)
+    &\approx \bigg( \!-\! \frac{\beta_2}{2} \frac{\psi_{tt}}{\psi}
+    + \frac{\beta_2}{2} \Omega_i^2 + \gamma_0 \psi^2 \bigg)\bigg|_{z = 0} z
+    + \phi(0, t)
 \end{aligned}$$
 
-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 $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$:
+And therefore $$\Omega_i$$ is as follows,
+assuming no initial chirp variation $$\Omega_i(0, t) = 0$$:
 
 $$\begin{aligned}
     \boxed{
-        N_\mathrm{min}^2
-        = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big)
-        \approx 1.12
+        \Omega_i(z, t)
+        = -\pdv{\phi}{t}
+        \approx \bigg( \frac{\beta_2}{2} \frac{\psi_{ttt}}{\psi}
+        - \frac{\beta_2}{2} \frac{\psi_{tt} \psi_t}{\psi^2}
+        - 2 \gamma_0 \psi \psi_t \bigg) \bigg|_{z = 0} z
     }
 \end{aligned}$$
 
-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.
-GVD is unable to keep up with SPM, so OWB will not occur.
+Once we have $$\Omega_i$$ for a known input pulse,
+we can check whether OWB is even possible under the given circumstances:
+$$\Omega_i$$ must be non-monotonic,
+i.e. $$\ipdv{\Omega_i}{t} = 0$$ must have a solution.
+In other words, there must be a sufficiently prominent "bump" in $$\Omega_i$$
+that gets pulled away by GVD faster than its surroundings,
+until those more-off-center frequencies overtake
+less-off-center ones and lead to the overlap
+that generates the sidelobes and other OWB phenomena.
+
+Let us assume that OWB will occur.
+Consider two parts of the pulse, located $$t_1$$ and $$t_2$$ for $$z = 0$$,
+so separated by a small initial interval $$\Delta{t} \equiv t_2 - t_1$$.
+Due to $$\Omega_i$$ there is a frequency difference between these points,
+causing $$\Delta{t}$$ to change by an amount $$\tau$$
+after the pulse has propagated a short distance $$z$$,
+estimated as follows:
+
+$$\begin{alignedat}{2}
+    \tau
+    &\approx z \Delta\beta_1
+    \approx z \pdv{\beta_1}{\Omega} \Delta{\Omega_i}
+    = z \beta_2 \Delta\Omega_i
+    \approx z \beta_2 \pdv{\Omega_i}{t} \Delta{t}
+\end{alignedat}$$
+
+Where $$\Delta\Omega_i \equiv \Omega_i(z,t_2) - \Omega_i(z,t_1)$$,
+and $$\Delta{\beta_1}$$ is the difference in inverse group velocity $$\beta_1(\Omega)$$
+between $$t_2$$ and $$t_1$$, specifically
+$$\Delta\beta_1 \equiv \beta_1(\Omega_i(z,t_2)) - \beta_1(\Omega_i(z,t_1))$$.
+OWB takes place when $$t_1$$ and $$t_2$$ catch up to each other,
+which is when $$\tau = -\Delta{t}$$.
+In that case, we have:
+
+$$\begin{aligned}
+    z
+    = - \frac{1}{\beta_2 \displaystyle\pdv{\Omega_i}{t}}
+\end{aligned}$$
 
-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 $$\Delta t$$,
-estimated by:
+Assuming $$\beta_2 > 0$$,
+this implies that the wave starts breaking first
+at the $$t$$-values where $$\Omega_i$$ has its most negative slope
+(note that for a symmetric input pulse,
+$$\ipdv{\Omega_i}{t}$$ is also symmetric,
+so OWB will occur simultaneous on both sides).
+We can therefore write an equation for $$L_\mathrm{WB}$$ like so,
+valid for any input pulse shape
+for which we know $$\Omega_i(z, t)$$:
 
 $$\begin{aligned}
-    \Delta t
-    &\approx z \Delta\beta_1
-    \qquad
-    &&\Delta\beta_1
-    \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
+    \boxed{
+        L_\mathrm{WB}
+        = - \frac{1}{\beta_2 \: \mathrm{min}_t\bigg\{ \displaystyle\pdv{\Omega_i}{t} \Big|_{z = L_\mathrm{WB}} \bigg\}}
+    }
+\end{aligned}$$
+
+Let us apply this method to a few specific examples:
+a Gaussian input pulse, and a soliton-shaped one
+(keeping in mind that true [bright solitons](/know/concept/optical-soliton/)
+do not exist for $$\beta_2 > 0$$).
+
+
+
+## Gaussian pulse
+
+For a Guassian input, the amplitude $$\psi$$ is as follows
+in our ansatz $$A = \psi e^{i \phi}$$:
+
+$$\begin{aligned}
+    \psi(0, t)
+    &= \sqrt{P_0} \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg)
+\end{aligned}$$
+
+For reference, its relevant $$t$$-derivatives are given by:
+
+$$\begin{aligned}
+    \psi_t(0, t)
+    &= - \frac{\sqrt{P_0}}{T_0^2} t \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg)
     \\
-    &\approx z \beta_2 \Delta\omega_i
-    \qquad
-    &&\Delta\omega_i
-    \equiv \omega_i(z,t_2) - \omega_i(z,t_1)
+    \psi_{tt}(0, t)
+    &= \frac{\sqrt{P_0}}{T_0^2} \bigg( \frac{t^2}{T_0^2} - 1 \bigg) \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg)
     \\
-    &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
-    \qquad \quad
-    &&\Delta\xi_i
-    \equiv \xi_i(z,t_2) - \xi_i(z,t_1)
+    \psi_{ttt}(0, t)
+    &= \frac{\sqrt{P_0}}{T_0^4} \bigg( 3 - \frac{t^2}{T_0^2} \bigg) t \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg)
 \end{aligned}$$
 
-Where $$\beta_1(\omega)$$ is the inverse of the group velocity.
-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 $$L_\mathrm{WB}$$ at which this happens first
-must satisfy the following condition for some value of $$t$$:
+Substituting these into our general linear approximation
+of $$\Omega_i$$ leads us to:
 
 $$\begin{aligned}
-    L_\mathrm{WB} \: \beta_2 \: \xi_i(L_\mathrm{WB}, t)
-    = -1
-    \qquad \implies \qquad
-    L_\mathrm{WB}^2
-    = - \frac{T_0^4}{\beta_2^2 \: f(t^2/T_0^2)}
+    \Omega_i(z, t)
+    &= z \frac{\beta_2 t}{T_0^4} \bigg( 1 + 2 \frac{\gamma_0 P_0 T_0^2}{\beta_2} \exp\!\Big( \!-\!\frac{t^2}{T_0^2} \Big) \bigg)
+\end{aligned}$$
+
+Since we are in the normal dispersion regime, $$\beta_2 > 0$$,
+so we can recognize the **soliton number** $$N_\mathrm{sol}$$ here,
+which is a useful measure of the relative strengths of GVD and SPM:
+
+$$\begin{aligned}
+    N_\mathrm{sol}^2
+    \equiv \frac{\gamma_0 P_0 T_0^2}{|\beta_2|}
+    = \frac{L_D}{L_N}
+\end{aligned}$$
+
+We thus have the following expression for $$\Omega_i$$,
+sketched below for several values of $$N_\mathrm{sol}$$:
+
+$$\begin{aligned}
+    \Omega_i(z, t)
+    &= z \frac{\beta_2 t}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+\end{aligned}$$
+
+{% include image.html file="gauss-omega-full.png" width="75%"
+    alt="Sketch of instantaneous frequency of Gaussian pulse" %}
+
+At a certain value of $$N_\mathrm{sol}$$, which we call $$N_\mathrm{min}$$,
+we see that $$\Omega_i$$ transitions from having no extrema,
+to having a local minimum and maximum with respect to $$t^2$$.
+Those "bumps" get pulled outward by GVD as indicated by the arrows,
+steepening the outer edges until the slope becomes infinite,
+at which point OWB occurs.
+However, for $$N_\mathrm{sol} < N_\mathrm{min}$$,
+the bumps are not prominent enough:
+the peaks cannot catch up to the outer edges,
+so OWB can never happen.
+
+We would like to find $$N_\mathrm{min}$$.
+To do so, we demand that $$\Omega_i$$ has local extrema
+where the derivative $$\ipdv{\Omega_i}{t}$$ vanishes, as illustrated below.
+Abbreviating $$f(x) \equiv (1 - 2x) e^{-x}$$:
+
+$$\begin{aligned}
+    0
+    = \pdv{\Omega_i}{t}
+    &= z \frac{\beta_2}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big)
+    \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+    \\
+    &= z \frac{\beta_2}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \: f\Big(\frac{t^2}{T_0^2}\Big) \bigg)
 \end{aligned}$$
 
-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 $$f(3/2)$$:
+{% include image.html file="gauss-domegadt-full.png" width="75%"
+    alt="Sketch of derivative of instantaneous frequency of Gaussian pulse" %}
+
+Here we see that as $$N_\mathrm{sol}$$ increases,
+it pulls down the minimum of $$f(x)$$ until it hits the horizontal axis
+when $$N_\mathrm{sol} = N_\mathrm{min}$$.
+We should therefore find the location $$x_\mathrm{min}$$ of this minimum:
+
+$$\begin{aligned}
+    0
+    = f'(x)
+    = (2 x - 3) e^{-x}
+    \qquad\implies\qquad
+    x_\mathrm{min}
+    = \frac{3}{2}
+\end{aligned}$$
+
+So the corresponding minimum value of $$f(x)$$ is given by:
 
 $$\begin{aligned}
     f_\mathrm{min}
-    = f(3/2)
-    = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2)
-    = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2
+    = f(x_\mathrm{min})
+    = -2 e^{-3/2}
+\end{aligned}$$
+
+Inserting this into our demand that $$\ipdv{\Omega_i}{t} = 0$$
+yields a simple expression for $$N_\mathrm{min}$$:
+
+$$\begin{aligned}
+    0
+    = 1 + 2 N_\mathrm{min}^2 \: f_\mathrm{min}
+    \qquad\implies\qquad
+    \boxed{
+        N_\mathrm{min}^2
+        = \frac{e^{3/2}}{4}
+        \approx 1.12
+    }
+\end{aligned}$$
+
+If $$N_\mathrm{sol}^2 < N_\mathrm{min}^2$$,
+then our demand cannot be satisfied:
+$$\Omega_i$$ cannot overtake itself,
+GVD is unable to keep up with SPM, and OWB cannot occur.
+From now on, we assume $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$.
+
+We now have everything we need to calculate the OWB distance $$L_\mathrm{WB}$$
+using its general recipe.
+Inserting $$\ipdv{\Omega_i}{t}$$,
+whose minimum we already know, we get:
+
+$$\begin{aligned}
+    L_\mathrm{WB}^2
+    = - \frac{T_0^4}{\beta_2^2 (1 + 2 N_\mathrm{sol}^2 f_\mathrm{min})}
+    = \frac{T_0^4}{\beta_2^2 (N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1)}
 \end{aligned}$$
 
-Clearly, $$f_\mathrm{min} \ge 0$$ when $$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$$,
-which, when inserted above, leads to an imaginary $$L_\mathrm{WB}$$,
-confirming that OWB cannot occur in that case.
-Otherwise, if $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$, then:
+Leading to the following prediction for $$L_\mathrm{WB}$$,
+which appears to agree well with the OWB
+observed in the simulation shown earlier.
+Note that if $$N_\mathrm{sol} < N_\mathrm{min}$$
+then $$L_\mathrm{WB}$$ is imaginary,
+confirming that OWB is not possible in that situation:
 
 $$\begin{aligned}
     \boxed{
         L_\mathrm{WB}
-        = \frac{T_0^2}{\beta_2 \, \sqrt{- f_\mathrm{min}}}
-        = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
+        = \frac{T_0^2}{\beta_2 \sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
     }
 \end{aligned}$$
 
-This prediction for $$L_\mathrm{WB}$$ appears to agree well
-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 $$\omega$$-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:
+## Soliton-shaped pulse
+
+Although solitons do not exist in the normal dispersion regime,
+we can still create pulses with the same shape, given by:
 
 $$\begin{aligned}
-    \omega_{\mathrm{SPM}}(L_\mathrm{WB},t)
-    = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
+    \psi(0, t)
+    &= \sqrt{P_0} \sech\!\Big( \frac{t}{T_0} \Big)
 \end{aligned}$$
 
-Assuming that $$N_\mathrm{sol}^2$$ is large in the denominator, this can
-be approximately reduced to:
+For reference, we also calculate its relevant $$t$$-derivatives:
 
 $$\begin{aligned}
-    \omega_\mathrm{SPM}(L_\mathrm{WB}, t)
-    \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
-    = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
+    \psi_t(0, t)
+    &= - \frac{\sqrt{P_0}}{T_0} \tanh\!\Big( \frac{t}{T_0} \Big) \sech\!\Big( \frac{t}{T_0} \Big)
+    \\
+    \psi_{tt}(0, t)
+    &= \frac{\sqrt{P_0}}{T_0^2} \bigg( \tanh^2\!\Big( \frac{t}{T_0} \Big) - \sech^2\!\Big( \frac{t}{T_0} \Big) \bigg)
+    \sech\!\Big( \frac{t}{T_0} \Big)
+    \\
+    \psi_{ttt}(0, t)
+    &= \frac{\sqrt{P_0}}{T_0^3} \bigg( 5 \sech^2\!\Big( \frac{t}{T_0} \Big) - \tanh^2\!\Big( \frac{t}{T_0} \Big) \bigg)
+    \tanh\!\Big( \frac{t}{T_0} \Big) \sech\!\Big( \frac{t}{T_0} \Big)
 \end{aligned}$$
 
-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:
+Substituting these into our general linear approximation of $$\Omega_i$$,
+and once again recognizing the soliton number $$N_\mathrm{sol}$$,
+leads us to the following function, sketched below:
 
 $$\begin{aligned}
-    \omega_\mathrm{max}
-    = \sqrt{\frac{2 \gamma P_0}{e \beta_2}}
+    \Omega_i(z, t)
+    &= z \frac{2 \beta_2}{T_0^3} \big( 1 + N_\mathrm{sol}^2 \big)
+    \sech^2\!\Big( \frac{t}{T_0} \Big) \tanh\!\Big( \frac{t}{T_0} \Big)
 \end{aligned}$$
 
-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 $$N_\mathrm{sol}$$.
+{% include image.html file="sech-omega-full.png" width="75%"
+    alt="Sketch of instantaneous frequency of soliton-shaped pulse" %}
+
+Curiously, this $$\Omega_i$$ is non-monotonic for all $$N_\mathrm{sol}$$,
+so OWB occurs even in the linear limit $$N_\mathrm{sol} \to 0$$.
+This suggests that OWB is not an inherently nonlinear effect,
+instead happening as long as there are bumps in $$\Omega_i$$,
+regardless of their origin (SPM or simply the pulse shape).
+
+We do not care where those local extrema are, only that they exist,
+so we move on immediately to finding where $$\Omega_i$$
+has its most negative slope,
+which is at some (but not all) solutions of:
+
+$$\begin{aligned}
+    0
+    &= \pdvn{2}{\Omega_i}{t}
+    \\
+    &= z \frac{8 \beta_2}{T_0^5} \big( 1 + N_\mathrm{sol}^2 \big)
+    \bigg( \tanh^2\!\Big( \frac{t}{T_0} \Big) - 2 \sech^2\!\Big( \frac{t}{T_0} \Big) \bigg)
+    \sech^2\!\Big( \frac{t}{T_0} \Big) \tanh\!\Big( \frac{t}{T_0} \Big)
+\end{aligned}$$
+
+One solution is clearly $$t = 0$$ because $$\tanh(0) = 0$$,
+but from the plot we can see that $$\Omega_i$$'s slope is positive there,
+so we must continue our search.
+The next candidate is:
+
+$$\begin{aligned}
+    0
+    &= \tanh^2(x) - 2 \sech^2(x)
+    \\
+    &= 3 \tanh^2(x) - 2
+\end{aligned}$$
+
+Where we have used the standard identity $$\sech^2(x) + \tanh^2(x) = 1$$.
+Isolating for $$x$$ and writing out $$\tanh^{-1}(x)$$ as a logarithm yields:
+
+$$\begin{aligned}
+    x
+    &= \tanh^{-1}\!\bigg( \!\pm\!\sqrt{\frac{2}{3}}\bigg)
+    \\
+    &= \frac{1}{2} \ln\!\bigg( \frac{1 \pm \sqrt{2/3}}{1 \mp \sqrt{2/3}} \bigg)
+    \\
+    &= \frac{1}{2} \ln\!\bigg( \frac{\sqrt{3} \pm \sqrt{2}}{\sqrt{3} \mp \sqrt{2}} \bigg)
+    \\
+    &= \frac{1}{2} \ln\!\bigg( \frac{(\sqrt{3} \pm \sqrt{2})^2}{(\sqrt{3} \mp \sqrt{2}) (\sqrt{3} \pm \sqrt{2})} \bigg)
+    \\
+    &= \frac{1}{2} \ln(5 \pm 2 \sqrt{6})
+\end{aligned}$$
+
+Note that $$\ln(5 \!+\! 2 \sqrt{6}) = - \ln(5 \!-\! 2 \sqrt{6}) \equiv 2 x_0$$.
+The values of $$\sech$$ and $$\tanh$$ are given by:
+
+$$\begin{aligned}
+    \sech(\pm x_0)
+    = \frac{1}{\sqrt{3}}
+    \qquad\qquad
+    \tanh(\pm x_0)
+    = \pm \sqrt{\frac{2}{3}}
+\end{aligned}$$
+
+The minimum value of the slope $$\ipdv{\Omega_i}{t}$$ is therefore as follows:
+
+$$\begin{aligned}
+    \mathrm{min}_t\bigg\{ \displaystyle\pdv{\Omega_i}{t} \bigg\}
+    &= z \frac{2 \beta_2}{T_0^4} (1 + N_\mathrm{sol}^2)
+    \bigg( \sech^2\!\Big( \frac{t}{T_0} \Big) - 2 \tanh^2\!\Big( \frac{t}{T_0} \Big) \bigg) \sech^2\!\Big( \frac{t}{T_0} \Big)
+    \bigg|_{t = x_0 T_0}
+    \\
+    &= - z \frac{2 \beta_2}{3 T_0^4} \big( 1 + N_\mathrm{sol}^2 \big)
+\end{aligned}$$
+
+Inserting this into $$L_\mathrm{WB}$$'s general equation,
+we find that OWB occurs at a distance with a similar
+$$T_0^2 / \beta_2$$-dependence as for the Gaussian pulse,
+confirming that OWB is mostly linear:
+
+$$\begin{aligned}
+    \boxed{
+        L_\mathrm{WB}
+        = \frac{\sqrt{3} T_0^2}{\beta_2 \sqrt{2 + 2 N_\mathrm{sol}^2}}
+    }
+\end{aligned}$$
 
 
 
@@ -237,4 +484,3 @@ for sufficiently large $$N_\mathrm{sol}$$.
 2.  A.M.  Heidt,  A.  Hartung,  H.  Bartelt,
     [Generation of ultrashort and coherent supercontinuum light pulses in all-normal dispersion fibers](https://doi.org/10.1007/978-1-4939-3326-6_6),
     2016, Springer Media.
-
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diff --git a/source/know/concept/self-steepening/index.md b/source/know/concept/self-steepening/index.md
index 80d9fcb..409f6c9 100644
--- a/source/know/concept/self-steepening/index.md
+++ b/source/know/concept/self-steepening/index.md
@@ -74,10 +74,10 @@ but in fact most authors make a small additional approximation.
 Let us write out the derivative of $$\gamma(\omega)$$:
 
 $$\begin{aligned}
-    \pdv{\gamma}{\omega}
+    \dv{\gamma}{\omega}
     = \frac{n_2}{c A_\mathrm{eff}}
-    + \frac{\omega}{c A_\mathrm{eff}} \pdv{n_2}{\omega}
-    - \frac{\omega n_2}{c A_\mathrm{eff}^2} \pdv{A_\mathrm{eff}}{\omega}
+    + \frac{\omega}{c A_\mathrm{eff}} \dv{n_2}{\omega}
+    - \frac{\omega n_2}{c A_\mathrm{eff}^2} \dv{A_\mathrm{eff}}{\omega}
 \end{aligned}$$
 
 In practice, the $$\omega$$-dependence of $$n_2$$ and $$A_\mathrm{eff}$$
@@ -102,11 +102,10 @@ is still conserved, defined as:
 $$\begin{aligned}
     \boxed{
         N(z)
-        \equiv \int_{-\infty}^\infty \frac{|\tilde{A}(z, \Omega)|^2}{\Omega} \dd{\Omega}
+        \equiv \int_0^\infty \frac{|A(z, \omega)|^2}{\omega} \dd{\omega}
     }
 \end{aligned}$$
 
-
 A pulse's intensity is highest at its peak,
 so the nonlinear index shift is strongest there,
 meaning that the peak travels slightly slower than the rest of the pulse,
@@ -244,19 +243,6 @@ Nevertheless, the trend is nicely visible:
 the trailing slope becomes extremely steep, and the spectrum
 broadens so much that dispersion can no longer be neglected.
 
-{% comment %}
-When self-steepening is added to the nonlinear Schrödinger equation,
-it no longer conserves the total pulse energy $$\int |A|^2 \dd{t}$$.
-Fortunately, the photon number $$N_\mathrm{ph}$$ is still
-conserved, which for the physical envelope $$A(z,t)$$ is defined as:
-
-$$\begin{aligned}
-    \boxed{
-        N_\mathrm{ph}(z) = \int_0^\infty \frac{|\tilde{A}(z,\omega)|^2}{\omega} \dd{\omega}
-    }
-\end{aligned}$$
-{% endcomment %}
-
 
 
 ## References
-- 
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