Improve knowledge base

5 files changed, 235 insertions, 100 deletions

diff --git a/source/know/concept/convolution-theorem/index.md b/source/know/concept/convolution-theorem/index.md
index 3f9eafb..8462fcc 100644
--- a/source/know/concept/convolution-theorem/index.md
+++ b/source/know/concept/convolution-theorem/index.md
@@ -24,10 +24,10 @@ and $$A$$ and $$B$$ are the constants from its definition:
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
-            A \cdot (f * g)(x)
+            A \: (f * g)(x)
             &= \hat{\mathcal{F}}{}^{-1}\Big\{ \tilde{f}(k) \: \tilde{g}(k) \Big\}
             \\
-            B \cdot (\tilde{f} * \tilde{g})(k)
+            B \: (\tilde{f} * \tilde{g})(k)
             &= \hat{\mathcal{F}}\Big\{ f(x) \: g(x) \Big\}
         \end{aligned}
     }
@@ -45,7 +45,7 @@ $$\begin{aligned}
     \\
     &= A \int_{-\infty}^\infty g(x') \: f(x - x') \dd{x'}
     \\
-    &= A \cdot (f * g)(x)
+    &= A \: (f * g)(x)
 \end{aligned}$$
 
 Then we do the same again,
@@ -59,7 +59,7 @@ $$\begin{aligned}
     \\
     &= B \int_{-\infty}^\infty \tilde{g}(k') \: \tilde{f}(k - k') \dd{k'}
     \\
-    &= B \cdot (\tilde{f} * \tilde{g})(k)
+    &= B \: (\tilde{f} * \tilde{g})(k)
 \end{aligned}$$
 {% include proof/end.html id="proof-fourier" %}
 
diff --git a/source/know/concept/fundamental-solution/index.md b/source/know/concept/fundamental-solution/index.md
index 947aada..4728c6f 100644
--- a/source/know/concept/fundamental-solution/index.md
+++ b/source/know/concept/fundamental-solution/index.md
@@ -11,7 +11,7 @@ layout: "concept"
 Given a linear operator $$\hat{L}$$ acting on $$x \in [a, b]$$,
 its **fundamental solution** $$G(x, x')$$ is defined as the response
 of $$\hat{L}$$ to a [Dirac delta function](/know/concept/dirac-delta-function/)
-$$\delta(x - x')$$ for $$x \in ]a, b[$$:
+$$\delta(x - x')$$ located at $$x' \in \: ]a, b[$$:
 
 $$\begin{aligned}
     \boxed{
@@ -24,7 +24,7 @@ Where $$A$$ is a constant, usually $$1$$.
 Fundamental solutions are often called **Green's functions**,
 but are distinct from the (somewhat related)
 [Green's functions](/know/concept/greens-functions/)
-in many-body quantum theory.
+in quantum mechanics.
 
 Note that the definition of $$G(x, x')$$ generalizes that of
 the [impulse response](/know/concept/impulse-response/).
@@ -44,20 +44,20 @@ $$\begin{aligned}
 
 
 {% include proof/start.html id="proof-solution" -%}
-$$\hat{L}$$ only acts on $$x$$, so $$x' \in ]a, b[$$ is simply a parameter,
+$$\hat{L}$$ only acts on $$x$$, so $$x' \in \: ]a, b[$$ is simply a parameter,
 meaning we are free to multiply the definition of $$G$$
 by the constant $$f(x')$$ on both sides,
 and exploit $$\hat{L}$$'s linearity:
 
 $$\begin{aligned}
     A f(x') \: \delta(x - x')
-    = f(x') \hat{L}\{ G(x, x') \}
+    = f(x') \: \hat{L}\{ G(x, x') \}
     = \hat{L}\{ f(x') \: G(x, x') \}
 \end{aligned}$$
 
 We then integrate both sides over $$x'$$ in the interval $$[a, b]$$,
 allowing us to consume $$\delta(x \!-\! x')$$.
-Note that $$\int \dd{x'}$$ commutes with $$\hat{L}$$ acting on $$x$$:
+Note that integration commutes with $$\hat{L}$$'s action:
 
 $$\begin{aligned}
     A \int_a^b f(x') \: \delta(x - x') \dd{x'}
@@ -72,27 +72,37 @@ satisfies $$\hat{L}\{ u(x) \} = f(x)$$, recognizable here.
 {% include proof/end.html id="proof-solution" %}
 
 
+In practice, $$G$$ usually only depends on the difference $$x - x'$$,
+in which case the integral shown above becomes a convolution:
+
+$$\begin{aligned}
+    u(x)
+    = \frac{1}{A} \int_a^b f(x') \: G(x - x') \dd{x'}
+    = \frac{1}{A} (f * G)(x)
+\end{aligned}$$
+
 While the impulse response is typically used for initial value problems,
 the fundamental solution $$G$$ is used for boundary value problems.
 Suppose those boundary conditions are homogeneous,
-i.e. $$u(x)$$ or one of its derivatives is zero at the boundaries.
+i.e. $$u$$ or its derivative $$\dot{u}$$ is zero at the boundaries.
 Then:
 
 $$\begin{aligned}
     0
     &= u(a)
     = \frac{1}{A} \int_a^b f(x') \: G(a, x') \dd{x'}
-    \qquad \implies \quad
+    \quad \implies \quad
     G(a, x') = 0
     \\
     0
-    &= u_x(a)
-    = \frac{1}{A} \int_a^b f(x') \: G_x(a, x') \dd{x'}
+    &= \dot{u}(a)
+    = \frac{1}{A} \int_a^b f(x') \: \dot{G}(a, x') \dd{x'}
     \quad \implies \quad
-    G_x(a, x') = 0
+    \dot{G}(a, x') = 0
 \end{aligned}$$
 
-This holds for all $$x'$$, and analogously for the other boundary $$x = b$$.
+Where $$\dot{G}$$ is the derivative of $$G$$ with respect to its first argument.
+This holds for all $$x'$$, and also at the other boundary $$x = b$$.
 In other words, the boundary conditions are built into $$G$$.
 
 What if the boundary conditions are inhomogeneous?
@@ -104,7 +114,7 @@ has homogeneous boundaries again,
 so we can use $$G$$ as usual to find $$u_i(x)$$, and then just add $$u_h(x)$$.
 
 If $$\hat{L}$$ is self-adjoint
-(see e.g. [Sturm-Liouville theory](/know/concept/sturm-liouville-theory/)),
+(see [Sturm-Liouville theory](/know/concept/sturm-liouville-theory/)),
 then the fundamental solution $$G(x, x')$$
 has the following **reciprocity** boundary condition:
 
diff --git a/source/know/concept/no-cloning-theorem/index.md b/source/know/concept/no-cloning-theorem/index.md
index 840a598..9c8b11d 100644
--- a/source/know/concept/no-cloning-theorem/index.md
+++ b/source/know/concept/no-cloning-theorem/index.md
@@ -30,14 +30,14 @@ $$\begin{aligned}
     \ket{0} \ket{?}
     \:\:\longrightarrow\:\:
     \ket{0} \ket{0}
-    \qquad \quad
+    \qquad \qquad
     \ket{1} \ket{?}
     \:\:\longrightarrow\:\:
     \ket{1} \ket{1}
 \end{aligned}$$
 
 If we feed this machine a superposition $$\ket{\psi} = \alpha \ket{0} + \beta \ket{1}$$,
-we *want* the following behaviour:
+we *want* the following behavior:
 
 $$\begin{aligned}
     \Big( \alpha \ket{0} + \beta \ket{1} \Big) \ket{?}
@@ -47,7 +47,7 @@ $$\begin{aligned}
     &= \Big( \alpha^2 \ket{0} \ket{0} + \alpha \beta \ket{0} \ket{1} + \alpha \beta \ket{1} \ket{0} + \beta^2 \ket{1} \ket{1} \Big)
 \end{aligned}$$
 
-Note the appearance of the cross terms with a factor of $$\alpha \beta$$.
+Note the appearance of the cross-terms with a factor of $$\alpha \beta$$.
 The problem is that the fundamental linearity of quantum mechanics
 dictates different behaviour:
 
@@ -59,7 +59,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This is clearly not the same as before: we have a contradiction,
-which implies that such a general cloning machine cannot ever exist.
+which implies that such a general cloning machine cannot exist.
 
 
 
diff --git a/source/know/concept/nonlinear-schrodinger-equation/index.md b/source/know/concept/nonlinear-schrodinger-equation/index.md
index 2ea1b23..820b361 100644
--- a/source/know/concept/nonlinear-schrodinger-equation/index.md
+++ b/source/know/concept/nonlinear-schrodinger-equation/index.md
@@ -212,20 +212,20 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Next, we take the [Fourier transform](/know/concept/fourier-transform/)
-$$t \to (\omega\!-\!\omega_0)$$ of the wave equation,
-once again treating $$|E|^2$$ (inside $$\varepsilon_r$$) as a constant.
+$$t \to \omega$$ of the wave equation,
+again treating $$|E|^2$$ (inside $$\varepsilon_r$$) as a constant.
 The constant $$s = \pm 1$$ is included here
 to deal with the fact that different authors use different sign conventions:
 
 $$\begin{aligned}
     0
-    &= \hat{\mathcal{F}}\bigg\{ \nabla^2 E - \frac{\varepsilon_r}{c^2} \pdvn{2}{E}{t} \bigg\}
+    &= \hat{\mathcal{F}}\bigg\{ \bigg( \nabla^2 E - \frac{\varepsilon_r}{c^2} \pdvn{2}{E}{t} \bigg) e^{-i \omega_0 t} \bigg\}
     \\
     &= \int_{-\infty}^\infty
     \bigg( \nabla^2 E - \frac{\varepsilon_r}{c^2} \pdvn{2}{E}{t} \bigg)
     e^{i s (\omega - \omega_0) t} \dd{t}
     \\
-    &= \nabla^2 E + s^2 \frac{\varepsilon_r}{c^2} (\omega - \omega_0)^2 E
+    &= \nabla^2 E + s^2 (\omega - \omega_0)^2 \frac{\varepsilon_r}{c^2} E
 \end{aligned}$$
 
 We use $$s^2 = 1$$ and define $$\Omega \equiv \omega - \omega_0$$
@@ -392,8 +392,8 @@ with all the arguments shown for clarity:
 $$\begin{aligned}
     \boxed{
         \Delta{\beta}(\omega)
-        = \frac{\omega}{c \mathcal{A}_\mathrm{mode}(\omega)}
-        \iint_{-\infty}^\infty \Delta{n}(x, y, \omega) \: |F(x, y, \omega)|^2 \dd{x} \dd{y}
+        = \frac{\omega}{c \mathcal{A}_\mathrm{mode}}
+        \iint_{-\infty}^\infty \Delta{n}(x, y, \omega) \: |F(x, y)|^2 \dd{x} \dd{y}
     }
 \end{aligned}$$
 
@@ -403,8 +403,8 @@ $$F$$ must be dimensionless,
 and consequently $$A$$ has (SI) units of an electric field.
 
 $$\begin{aligned}
-    \mathcal{A}_\mathrm{mode}(\omega)
-    \equiv \iint_{-\infty}^\infty |F(x, y, \omega)|^2 \dd{x} \dd{y}
+    \mathcal{A}_\mathrm{mode}
+    \equiv \iint_{-\infty}^\infty |F|^2 \dd{x} \dd{y}
 \end{aligned}$$
 
 Now we finally turn our attention to the equation for $$A$$.
@@ -442,7 +442,7 @@ Recall that earlier, in order to treat $$\chi^{(3)}$$ as instantaneous,
 we already assumed a temporally broad
 (spectrally narrow) pulse.
 Hence, for simplicity, we can cut off this Taylor series at $$\beta_2$$,
-which is good enough for many cases.
+which is good enough in many cases.
 Inserting the expansion into $$A$$'s equation:
 
 $$\begin{aligned}
@@ -450,10 +450,11 @@ $$\begin{aligned}
     &= i \pdv{A}{z} + i \frac{\beta_1}{s} (-i s \Omega) A - \frac{\beta_2}{2 s^2} (- i s \Omega)^2 A + \Delta{\beta}_0 A
 \end{aligned}$$
 
-Which we have rewritten as preparation for taking the inverse Fourier transform,
+Which we have rewritten in preparation for taking the inverse Fourier transform,
 by introducing $$s$$ and by replacing $$\Delta{\beta}(\omega)$$
 with $$\Delta{\beta_0} \equiv \Delta{\beta}(\omega_0)$$
-in order to remove all explicit dependence on $$\omega$$.
+in order to remove all explicit dependence on $$\omega$$,
+i.e. we only keep the first term of $$\Delta{\beta}$$'s Taylor expansion.
 After transforming and using $$s^2 = 1$$,
 we get the following equation for $$A(z, t)$$:
 
@@ -468,11 +469,11 @@ according to which effects we want to include.
 Earlier, we approximated $$\varepsilon_r \approx n^2$$,
 so if we instead say that $$\varepsilon_r = (n \!+\! \Delta{n})^2$$,
 then $$\Delta{n}$$ should include absorption and nonlinearity.
-A simple and commonly used form for $$\Delta{n}$$ is therefore:
+The most commonly used form for $$\Delta{n}$$ is therefore:
 
 $$\begin{aligned}
-    \Delta{n}
-    = n_2 I + i \frac{\alpha c}{2 \omega}
+    \Delta{n}(x, y, \omega)
+    = n_2(\omega) \: I(x, y, \omega) + i \frac{c \alpha(\omega)}{2 \omega}
 \end{aligned}$$
 
 Where $$I$$ is the intensity (i.e. power per unit area) of the light,
@@ -491,12 +492,13 @@ $$\begin{aligned}
     + \frac{3 \omega \Imag\{\chi^{(3)}_{xxxx}\}}{2 \varepsilon_0 c^2 n^2} I
     \qquad
     I
-    = \frac{\varepsilon_0 c n}{2} |E|^2
+    = \frac{\varepsilon_0 c n}{2} |F|^2 |A|^2
 \end{aligned}$$
 
-For simplicity, we set $$\Imag\{\chi^{(3)}_{xxxx}\} = 0$$,
-which is a good approximation for fibers made of silica.
-Inserting this form of $$\Delta{n}$$ into $$\Delta{\beta_0}$$ then yields:
+For simplicity we set $$\Imag\{\chi^{(3)}_{xxxx}\} = 0$$,
+which is a good approximation for silica fibers.
+Inserting this form of $$\Delta{n}$$ into $$\Delta{\beta_0}$$
+and neglecting the $$(x, y)$$-dependence of $$\Delta{n}$$ yields:
 
 $$\begin{aligned}
     \Delta{\beta}_0
@@ -507,24 +509,31 @@ $$\begin{aligned}
     + \gamma_0 \frac{\varepsilon_0 c n}{2} \mathcal{A}_\mathrm{mode} |A|^2
 \end{aligned}$$
 
-Where we have defined the nonlinear parameter $$\gamma_0$$ like so,
+Where we have defined the parameter $$\gamma_0 \equiv \gamma(\omega_0)$$ like so,
 involving the **effective mode area** $$\mathcal{A}_\mathrm{eff}$$,
 which contains all information about $$F$$ needed for solving $$A$$'s equation:
 
 $$\begin{aligned}
     \boxed{
-        \gamma_0
-        = \gamma(\omega_0)
-        \equiv \frac{\omega_0 n_2}{c \mathcal{A}_\mathrm{eff}}
+        \gamma(\omega)
+        \equiv \frac{\omega n_2(\omega)}{c \mathcal{A}_\mathrm{eff}(\omega)}
     }
     \qquad \qquad
     \boxed{
-        \mathcal{A}_\mathrm{eff}(\omega_0)
+        \mathcal{A}_\mathrm{eff}(\omega)
         \equiv \frac{\displaystyle \bigg( \iint_{-\infty}^\infty |F|^2 \dd{x} \dd{y} \bigg)^2}
         {\displaystyle \iint_{-\infty}^\infty |F|^4 \dd{x} \dd{y}}
     }
 \end{aligned}$$
 
+Note the $$\omega$$-dependence of $$A_\mathrm{eff}$$:
+so far we have conveniently ignored that $$F$$ also depends on $$\omega$$,
+because it is a parameter in its eigenvalue equation.
+This is valid for spectrally narrow pulses, so we will stick with it.
+Just beware that some people make the ad-hoc generalization
+$$\gamma_0 \to \gamma(\omega)$$, which is not correct in general
+(this is an advanced topic, see Lægsgaard).
+
 Substituting $$\Delta{\beta_0}$$ into the main problem
 yields a prototype of the NLS equation:
 
@@ -694,3 +703,6 @@ so many authors only show that case.
 2.  O. Bang,
     *Nonlinear mathematical physics: lecture notes*,
     2020, unpublished.
+3.  J. Lægsgaard,
+    [Mode profile dispersion in the generalized nonlinear Schrödinger equation](https://doi.org/10.1364/OE.15.016110),
+    2007, Optica.
diff --git a/source/know/concept/self-steepening/index.md b/source/know/concept/self-steepening/index.md
index f96c020..80d9fcb 100644
--- a/source/know/concept/self-steepening/index.md
+++ b/source/know/concept/self-steepening/index.md
@@ -1,7 +1,7 @@
 ---
 title: "Self-steepening"
 sort_title: "Self-steepening"
-date: 2021-02-26
+date: 2024-09-29 # Originally 2021-02-26, major rewrite
 categories:
 - Physics
 - Optics
@@ -10,121 +10,228 @@ categories:
 layout: "concept"
 ---
 
-For a laser pulse travelling through an optical fiber,
-its intensity is highest at its peak, so the Kerr effect will be strongest there.
-This means that the peak travels slightly slower
-than the rest of the pulse, leading to **self-steepening** of its trailing edge.
-Mathematically, this is described by adding a new term to the
-nonlinear Schrödinger equation:
+A laser pulse travelling in an optical fiber
+causes a nonlinear change of the material's refractive index,
+and the resulting dynamics are described by
+the [nonlinear Schrödinger (NLS) equation](/know/concept/nonlinear-schrodinger-equation/),
+given in its most basic form by:
 
 $$\begin{aligned}
     0
-    = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma \Big(1 + \frac{i}{\omega_0} \pdv{}{t} \Big) |A|^2 A
+    = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma_0 |A|^2 A
 \end{aligned}$$
 
-Where $$\omega_0$$ is the angular frequency of the pump.
-We will use the following ansatz,
-consisting of an arbitrary power profile $$P$$ with a phase $$\phi$$:
+Where $$A(z, t)$$ is the modulation profile of the carrier wave,
+$$\beta_2$$ is the group velocity dispersion
+at the carrier frequency $$\omega_0$$,
+and $$\gamma_0 \equiv \gamma(\omega_0)$$ is a nonlinear parameter
+involving the material's Kerr coefficient $$n_2$$
+and the transverse mode's effective area $$A_\mathrm{eff}$$:
+
+$$\begin{aligned}
+    \gamma(\omega)
+    \equiv \frac{\omega n_2(\omega)}{c A_\mathrm{eff}(\omega)}
+\end{aligned}$$
+
+As a consequence of treating $$\gamma_0$$ as frequency-independent,
+only the nonlinear *phase* velocity change is represented,
+but not the *group* velocity change.
+Unfortunately, this form of the NLS equation
+does not allow us to include the full $$\gamma(\omega)$$
+(this is an advanced topic, see Lægsgaard),
+but a decent approximation is to simply Taylor-expand $$\gamma(\omega)$$ around $$\omega_0$$:
+
+$$\begin{aligned}
+    \gamma(\omega)
+    = \gamma_0 + \gamma_1 \Omega + \frac{\gamma_2}{2} \Omega^2 + \frac{\gamma_3}{6} \Omega^2 + ...
+\end{aligned}$$
+
+Where $$\Omega \equiv \omega - \omega_0$$
+and $$\gamma_n \equiv \ipdvn{n}{\gamma}{\omega}|_{\omega=\omega_0}$$.
+For pulses with a sufficiently narrow spectrum,
+we only need the first two terms.
+We insert this into the [Fourier transform (FT)](/know/concept/fourier-transform/)
+$$\hat{\mathcal{F}}$$ of the equation,
+where $$s = \pm 1$$ is the sign of the FT exponent,
+which might vary from author to author
+($$s = +1$$ corresponds to a forward-propagating carrier wave and vice versa):
+
+$$\begin{aligned}
+    0
+    = i\pdv{A}{z} - \frac{\beta_2}{2} (-i s \Omega)^2 A + (\gamma_0 + \gamma_1 \Omega) \hat{\mathcal{F}}\big\{ |A|^2 A \big\}
+\end{aligned}$$
+
+If we now take the inverse FT,
+the factor $$\Omega$$ becomes an operator $$i s \ipdv{}{t}$$:
+
+$$\begin{aligned}
+    0
+    = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \Big( \gamma_0 + i s \gamma_1 \pdv{}{t} \Big) |A|^2 A
+\end{aligned}$$
+
+In theory, this is the desired new NLS equation,
+but in fact most authors make a small additional approximation.
+Let us write out the derivative of $$\gamma(\omega)$$:
+
+$$\begin{aligned}
+    \pdv{\gamma}{\omega}
+    = \frac{n_2}{c A_\mathrm{eff}}
+    + \frac{\omega}{c A_\mathrm{eff}} \pdv{n_2}{\omega}
+    - \frac{\omega n_2}{c A_\mathrm{eff}^2} \pdv{A_\mathrm{eff}}{\omega}
+\end{aligned}$$
+
+In practice, the $$\omega$$-dependence of $$n_2$$ and $$A_\mathrm{eff}$$
+is relatively weak, so the first term is dominant
+and hence sufficient for our purposes.
+We therefore have $$\gamma_1 \approx \gamma_0 / \omega_0$$, leading to:
+
+$$\begin{aligned}
+    \boxed{
+        0
+        = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma_0 \Big( 1 + i \frac{s}{\omega_0} \pdv{}{t} \Big) |A|^2 A
+    }
+\end{aligned}$$
+
+Beware that this NLS equation does not conserve the total energy
+$$E \equiv \int_{-\infty}^\infty |A|^2 \dd{t}$$ anymore,
+which is often used to quantify simulation errors.
+Fortunately, another value can then be used instead:
+it can be shown that the "photon number" $$N$$
+is still conserved, defined as:
+
+$$\begin{aligned}
+    \boxed{
+        N(z)
+        \equiv \int_{-\infty}^\infty \frac{|\tilde{A}(z, \Omega)|^2}{\Omega} \dd{\Omega}
+    }
+\end{aligned}$$
+
+
+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,
+leading to **self-steepening** of its trailing edge;
+an effect exhibited by our modified NLS equation.
+Note that $$s$$ controls which edge is regarded as the trailing one.
+
+Let us make the ansatz below,
+consisting of an arbitrary power profile $$P$$ with phase $$\phi$$:
 
 $$\begin{aligned}
     A(z,t)
     = \sqrt{P(z,t)} \, \exp\!\big(i \phi(z,t)\big)
 \end{aligned}$$
 
-For a long pulse travelling over a short distance, it is reasonable to
-neglect dispersion ($$\beta_2 = 0$$).
-Inserting the ansatz then gives the following, where $$\varepsilon = \gamma / \omega_0$$:
+We assume that $$A$$ has a sufficiently narrow spectrum
+that we can neglect dispersion $$\beta_2 = 0$$ over a short distance.
+Inserting the ansatz into the NLS equation
+with $$\varepsilon \equiv \gamma_0 / \omega_0$$ gives:
 
 $$\begin{aligned}
     0
-    &= i \frac{1}{2} \frac{P_z}{\sqrt{P}} - \sqrt{P} \phi_z + \gamma P \sqrt{P} + i \varepsilon \frac{3}{2} P_t \sqrt{P} - \varepsilon P \sqrt{P} \phi_t
+    &= i \frac{1}{2} \frac{P_z}{\sqrt{P}} - \sqrt{P} \phi_z + \gamma_0 P \sqrt{P}
+    + i s \varepsilon \frac{3}{2} P_t \sqrt{P} - s \varepsilon P \sqrt{P} \phi_t
 \end{aligned}$$
 
-This results in two equations, respectively corresponding to the real
-and imaginary parts:
+Since $$P$$ is real, this results in two equations,
+for the real and imaginary parts:
 
 $$\begin{aligned}
     0
-    &= - \phi_z - \varepsilon P \phi_t + \gamma P
+    &= - \phi_z  + \gamma_0 P - s \varepsilon P \phi_t
     \\
     0
-    &= P_z + \varepsilon 3 P_t P
+    &= P_z + 3 s \varepsilon P_t P
 \end{aligned}$$
 
 The phase $$\phi$$ is not so interesting, so we focus on the latter equation for $$P$$.
-As it turns out, it has a general solution of the form below (you can verify this yourself),
-which shows that more intense parts of the pulse
-will lag behind compared to the rest:
+You can easily show (by insertion) that it has a general solution of the form below,
+which says that more intense parts of the pulse
+lag behind the rest, as expected:
 
 $$\begin{aligned}
     P(z,t)
-    = f(t - 3 \varepsilon z P)
+    = f(t - 3 s \varepsilon z P)
 \end{aligned}$$
 
-Where $$f$$ is the initial power profile: $$f(t) = P(0,t)$$.
+Where $$f(t) \equiv P(0,t)$$ is the initial power profile.
 The derivatives $$P_t$$ and $$P_z$$ are given by:
 
 $$\begin{aligned}
     P_t
-    &= (1 - 3 \varepsilon z P_t) \: f'
-    \qquad \quad \implies \quad
-    P_t
-    = \frac{f'}{1 + 3 \varepsilon z f'}
+    &= (1 - 3 s \varepsilon z P_t) \: f'
+    \qquad\quad\!\! = \frac{f'}{1 + 3 s \varepsilon z f'}
     \\
     P_z
-    &= (-3 \varepsilon P - 3 \varepsilon z P_z) \: f'
-    \quad \implies \quad
-    P_z
-    = \frac{- 3 \varepsilon P f'}{1 + 3 \varepsilon z f'}
+    &= (-3 s \varepsilon P - 3 s \varepsilon z P_z) \: f'
+    = \frac{- 3 s \varepsilon P f'}{1 + 3 s \varepsilon z f'}
 \end{aligned}$$
 
-These derivatives both go to infinity when their denominator is zero,
-which, since $$\varepsilon$$ is positive, will happen earliest where $$f'$$
-has its most negative value, called $$f_\mathrm{min}'$$,
-which is located on the trailing edge of the pulse.
+Both expressions blow up when their denominator goes to zero,
+which, since $$\varepsilon > 0$$, happens earliest at an extremum of $$f'$$;
+either its minimum ($$s = +1$$) or maximum ($$s = -1$$).
+Let us call this value $$f_\mathrm{extr}'$$,
+located on the trailing edge of the pulse.
 At the propagation distance $$z$$ where this occurs, $$L_\mathrm{shock}$$,
-the pulse will "tip over", creating a discontinuous shock:
+the pulse "tips over", creating a discontinuous shock:
 
 $$\begin{aligned}
     0
-    = 1 + 3 \varepsilon z f_\mathrm{min}'
+    = 1 + 3 s \varepsilon z f_\mathrm{extr}'
     \qquad \implies \qquad
-    \boxed{
+    z
+    = \boxed{
         L_\mathrm{shock}
-        \equiv -\frac{1}{3 \varepsilon f_\mathrm{min}'}
+        \equiv -\frac{\omega_0}{3 s \gamma_0 f_\mathrm{extr}'}
     }
 \end{aligned}$$
 
-In practice, however, this will never actually happen, because by the time
-$$L_\mathrm{shock}$$ is reached, the pulse spectrum will have become so
-broad that dispersion can no longer be neglected.
+In practice, however, this never actually happens,
+because as the pulse approaches $$L_\mathrm{shock}$$,
+its spectrum becomes so broad that dispersion cannot be neglected:
+[dispersive broadening](/know/concept/dispersive-broadening/)
+pulls the pulse apart before a shock can occur.
+The early steepening is observable though.
 
 A simulation of self-steepening without dispersion is illustrated below
-for the following Gaussian initial power distribution,
+for the following initial power distribution,
 with $$T_0 = 25\:\mathrm{fs}$$, $$P_0 = 3\:\mathrm{kW}$$,
-$$\beta_2 = 0$$ and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$:
+$$\beta_2 = 0$$, $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$,
+and a vacuum carrier wavelength $$\lambda_0 \approx 73\:\mathrm{nm}$$
+(the latter determined by the simulation's resolution settings):
 
 $$\begin{aligned}
     f(t)
-    = P(0,t) = P_0 \exp\!\Big(\! -\!\frac{t^2}{T_0^2} \Big)
+    = P(0,t) = P_0 \exp\!\bigg(\!-\!\frac{t^2}{T_0^2} \bigg)
 \end{aligned}$$
 
+The first and second derivatives of this Gaussian $$f$$ are as follows:
 
-Its steepest points are found to be at $$2 t^2 = T_0^2$$, so
-$$f_\mathrm{min}'$$ and $$L_\mathrm{shock}$$ are given by:
+$$\begin{aligned}
+    f'(t)
+    &= - \frac{2 P_0}{T_0^2} t \exp\!\bigg(\!-\!\frac{t^2}{T_0^2} \bigg)
+    \\
+    f''(t)
+    &= \frac{2 P_0}{T_0^2} \bigg( \frac{2 t^2}{T_0^2} - 1 \bigg) \exp\!\bigg(\!-\!\frac{t^2}{T_0^2} \bigg)
+\end{aligned}$$
+
+The steepest points of $$f'$$ are the roots of $$f''$$,
+clearly located at $$2 t^2 = T_0^2$$,
+meaning that $$f_\mathrm{extr}'$$ and $$L_\mathrm{shock}$$
+are in this case given by:
 
 $$\begin{aligned}
-    f_\mathrm{min}'
-    = - \frac{\sqrt{2} P_0}{T_0} \exp\!\Big(\!-\!\frac{1}{2}\Big)
-    \quad \implies \quad
+    f_\mathrm{extr}'
+    = \mp \sqrt{2} e^{-1/2} \frac{P_0}{T_0}
+    \qquad \implies \qquad
     L_\mathrm{shock}
-    = \frac{T_0}{3 \sqrt{2} \varepsilon P_0} \exp\!\Big(\frac{1}{2}\Big)
+    = \frac{e^{1/2}}{3 \sqrt{2}} \frac{\omega_0 T_0}{\gamma_0 P_0}
 \end{aligned}$$
 
 This example Gaussian pulse therefore has a theoretical
 $$L_\mathrm{shock} = 0.847\,\mathrm{m}$$,
-which turns out to be accurate,
-although the simulation breaks down due to insufficient resolution:
+which seems to be accurate based on these plots,
+although the simulation breaks down just before that point due to insufficient resolution:
 
 {% include image.html file="simulation-full.png" width="100%"
     alt="Self-steepening simulation results" %}
@@ -133,9 +240,9 @@ Unfortunately, self-steepening cannot be simulated perfectly: as the
 pulse approaches $$L_\mathrm{shock}$$, its spectrum broadens to infinite
 frequencies to represent the singularity in its slope.
 The simulation thus collapses into chaos when the edge of the frequency window is reached.
-Nevertheless, the general trends are nicely visible:
+Nevertheless, the trend is nicely visible:
 the trailing slope becomes extremely steep, and the spectrum
-broadens so much that dispersion cannot be neglected anymore.
+broadens so much that dispersion can no longer be neglected.
 
 {% comment %}
 When self-steepening is added to the nonlinear Schrödinger equation,
@@ -153,4 +260,10 @@ $$\begin{aligned}
 
 
 ## References
-1. B.R. Suydam, [Self-steepening of optical pulses](https://doi.org/10.1007/0-387-25097-2_6), 2006, Springer.
+
+1.  B.R. Suydam,
+    [Self-steepening of optical pulses](https://doi.org/10.1007/0-387-25097-2_6),
+    2006, Springer.
+2.  J. Lægsgaard,
+    [Mode profile dispersion in the generalized nonlinear Schrödinger equation](https://doi.org/10.1364/OE.15.016110),
+    2007, Optica.