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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. |