From fda947364c33ea7f6273a7f3ad1e8898edbe1754 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 29 Sep 2024 22:15:59 +0200
Subject: Improve knowledge base

---
 .../nonlinear-schrodinger-equation/index.md        | 58 +++++++++++++---------
 1 file changed, 35 insertions(+), 23 deletions(-)

(limited to 'source/know/concept/nonlinear-schrodinger-equation/index.md')

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.
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