From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/self-steepening/index.md | 140 +++++++++++++++++++++ .../concept/self-steepening/pheno-steep-small.jpg | Bin 0 -> 91324 bytes .../know/concept/self-steepening/pheno-steep.jpg | Bin 0 -> 327309 bytes 3 files changed, 140 insertions(+) create mode 100644 source/know/concept/self-steepening/index.md create mode 100644 source/know/concept/self-steepening/pheno-steep-small.jpg create mode 100644 source/know/concept/self-steepening/pheno-steep.jpg (limited to 'source/know/concept/self-steepening') diff --git a/source/know/concept/self-steepening/index.md b/source/know/concept/self-steepening/index.md new file mode 100644 index 0000000..ba0e906 --- /dev/null +++ b/source/know/concept/self-steepening/index.md @@ -0,0 +1,140 @@ +--- +title: "Self-steepening" +date: 2021-02-26 +categories: +- Physics +- Optics +- Fiber optics +- Nonlinear optics +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: + +$$\begin{aligned} + 0 + = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma \Big(1 + \frac{i}{\omega_0} \pdv{}{t} \Big) \big(|A|^2 A\big) +\end{aligned}$$ + +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 $\phi$: + +$$\begin{aligned} + A(z,t) = \sqrt{P(z,t)} \, \exp\!\big(i \phi(z,t)\big) +\end{aligned}$$ + +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 $\varepsilon = \gamma / \omega_0$: + +$$\begin{aligned} + 0 &= i \frac{1}{2} \frac{P_z}{\sqrt{P}} - \sqrt{P} \phi_z + \gamma P \sqrt{P} + i \varepsilon \frac{3}{2} P_t \sqrt{P} - \varepsilon P \sqrt{P} \phi_t +\end{aligned}$$ + +This results in two equations, respectively corresponding to the real +and imaginary parts: + +$$\begin{aligned} + 0 &= - \phi_z - \varepsilon P \phi_t + \gamma P + \\ + 0 &= P_z + \varepsilon 3 P_t P +\end{aligned}$$ + +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, which shows that +more intense parts of the pulse will tend to lag behind compared to the rest: + +$$\begin{aligned} + P(z,t) = f(t - 3 \varepsilon z P) +\end{aligned}$$ + +Where $f$ is the initial power profile: $f(t) = P(0,t)$. +The derivatives $P_t$ and $P_z$ are then given by: + +$$\begin{aligned} + P_t + &= (1 - 3 \varepsilon z P_t) \: f' + \qquad \quad \implies \quad + P_t + = \frac{f'}{1 + 3 \varepsilon z f'} + \\ + P_z + &= (-3 \varepsilon P - 3 \varepsilon z P_z) \: f' + \quad \implies \quad + P_z + = \frac{- 3 \varepsilon P f'}{1 + 3 \varepsilon z f'} +\end{aligned}$$ + +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. +At the propagation distance where this occurs, $L_\mathrm{shock}$, +the pulse will "tip over", creating a discontinuous shock: + +$$\begin{aligned} + \boxed{ + L_\mathrm{shock} = -\frac{1}{3 \varepsilon f_\mathrm{min}'} + } +\end{aligned}$$ + +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. + +A simulation of self-steepening without dispersion is illustrated below +for the following Gaussian initial power distribution, +with $T_0 = 25\:\mathrm{fs}$, $P_0 = 3\:\mathrm{kW}$, +$\beta_2 = 0$ and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$: + +$$\begin{aligned} + f(t) = P(0,t) = P_0 \exp\!\Big(\! -\!\frac{t^2}{T_0^2} \Big) +\end{aligned}$$ + + +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: + +$$\begin{aligned} + f_\mathrm{min}' = - \frac{\sqrt{2} P_0}{T_0} \exp\!\Big(\!-\!\frac{1}{2}\Big) + \quad \implies \quad + L_\mathrm{shock} = \frac{T_0}{3 \sqrt{2} \varepsilon P_0} \exp\!\Big(\frac{1}{2}\Big) +\end{aligned}$$ + +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: + + + + + +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: +the trailing slope becomes extremely steep, and the spectrum +broadens so much that dispersion cannot be neglected anymore. + +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: + +$$\begin{aligned} + \boxed{ + N_\mathrm{ph}(z) = \int_0^\infty \frac{|\tilde{A}(z,\omega)|^2}{\omega} \dd{\omega} + } +\end{aligned}$$ + + +## References +1. B.R. Suydam, [Self-steepening of optical pulses](https://doi.org/10.1007/0-387-25097-2_6), 2006, Springer. diff --git a/source/know/concept/self-steepening/pheno-steep-small.jpg b/source/know/concept/self-steepening/pheno-steep-small.jpg new file mode 100644 index 0000000..bb2a158 Binary files /dev/null and b/source/know/concept/self-steepening/pheno-steep-small.jpg differ diff --git a/source/know/concept/self-steepening/pheno-steep.jpg b/source/know/concept/self-steepening/pheno-steep.jpg new file mode 100644 index 0000000..c369d62 Binary files /dev/null and b/source/know/concept/self-steepening/pheno-steep.jpg differ -- cgit v1.2.3