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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. |