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Diffstat (limited to 'source')
| -rw-r--r-- | source/know/concept/convolution-theorem/index.md | 8 | ||||
| -rw-r--r-- | source/know/concept/fundamental-solution/index.md | 34 | ||||
| -rw-r--r-- | source/know/concept/no-cloning-theorem/index.md | 8 | ||||
| -rw-r--r-- | source/know/concept/nonlinear-schrodinger-equation/index.md | 58 | ||||
| -rw-r--r-- | source/know/concept/self-steepening/index.md | 227 |
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. |