diff options
Diffstat (limited to 'source/know/concept/nonlinear-schrodinger-equation')
| -rw-r--r-- | source/know/concept/nonlinear-schrodinger-equation/index.md | 708 |
1 files changed, 708 insertions, 0 deletions
diff --git a/source/know/concept/nonlinear-schrodinger-equation/index.md b/source/know/concept/nonlinear-schrodinger-equation/index.md new file mode 100644 index 0000000..820b361 --- /dev/null +++ b/source/know/concept/nonlinear-schrodinger-equation/index.md @@ -0,0 +1,708 @@ +--- +title: "Nonlinear Schrödinger equation" +sort_title: "Nonlinear Schrodinger equation" # sic +date: 2024-09-15 +categories: +- Physics +- Mathematics +- Fiber optics +- Nonlinear optics +layout: "concept" +--- + +The **nonlinear Schrödinger (NLS) equation** +is a nonlinear 1+1D partial differential equation +that appears in many areas of physics. +It is often given in its dimensionless form, +where it governs the envelope $$u(z, t)$$ +of an underlying carrier wave, +with $$t$$ the transverse coordinate, +and $$r = \pm 1$$ a parameter determining +which of two regimes the equation is intended for: + +$$\begin{aligned} + \boxed{ + i \pdv{u}{z} + \pdvn{2}{u}{t} + r |u|^2 u + = 0 + } +\end{aligned}$$ + +Many variants exist, depending on the conventions used by authors. +The NLS equation is used to describe pulses in fiber optics (as derived below), +waves over deep water, local opening of DNA chains, and much more. +Very roughly speaking, it is a valid description of +"all" weakly nonlinear, slowly modulated waves in physics. + +It exhibits an incredible range of behaviors, +from "simple" effects such as +[dispersive broadening](/know/concept/dispersive-broadening/), +[self-phase modulation](/know/concept/self-phase-modulation/) +and [first-order solitons](/know/concept/optical-soliton/), +to weirder and more complicated phenomena like +[modulational instability](/know/concept/modulational-instability/), +[optical wave breaking](/know/concept/optical-wave-breaking/) +and periodic *higher-order solitons*. +It is also often modified to include additional physics, +further enriching its results with e.g. +[self-steepening](/know/concept/self-steepening/) +and *soliton self-frequency shifting*. + +We only consider fiber optics here; +the NLS equation can be derived in many other ways. +We start from the most general form of the +[electromagnetic wave equation](/know/concept/electromagnetic-wave-equation/), +after assuming the medium cannot be magnetized ($$\mu_r = 1$$): + +$$\begin{aligned} + \nabla \cross \big( \nabla \cross \vb{E} \big) + = - \mu_0 \varepsilon_0 \pdvn{2}{\vb{E}}{t} - \mu_0 \pdvn{2}{\vb{P}}{t} +\end{aligned}$$ + +Using the vector identity +$$\nabla \cross (\nabla \cross \vb{E}) = \nabla (\nabla \cdot \vb{E}) - \nabla^2 \vb{E}$$ +and [Gauss's law](/know/concept/maxwells-equations/) $$\nabla \cdot \vb{E} = 0$$, +and splitting the polarization $$\vb{P}$$ +into linear and nonlinear contributions +$$\vb{P}_\mathrm{L}$$ and $$\vb{P}_\mathrm{NL}$$: + +$$\begin{aligned} + \nabla^2 \vb{E} - \mu_0 \varepsilon_0 \pdvn{2}{\vb{E}}{t} + &= \mu_0 \pdvn{2}{\vb{P}_\mathrm{L}}{t} + \mu_0 \pdvn{2}{\vb{P}_\mathrm{NL}}{t} +\end{aligned}$$ + +In general, $$\vb{P}_\mathrm{L}$$ is given by the convolution +of $$\vb{E}$$ with a second-rank response tensor $$\chi^{(1)}$$: + +$$\begin{aligned} + \vb{P}_\mathrm{L}(\vb{r}, t) + = \varepsilon_0 \int_{-\infty}^\infty \chi^{(1)}(t - t') \cdot \vb{E}(\vb{r}, t') \dd{t'} +\end{aligned}$$ + +In $$\vb{P}_\mathrm{NL}$$ we only include third-order nonlinearities, +since higher orders are usually negligible, +and second-order nonlinear effects only exist in very specific crystals. +So we "only" need to deal with a fourth-rank response tensor $$\chi^{(3)}$$: + +$$\begin{aligned} + \vb{P}_\mathrm{NL}(\vb{r}, t) + = \varepsilon_0 \iiint_{-\infty}^\infty \chi^{(3)}(t \!-\! t_1, t \!-\! t_2, t \!-\! t_3) + \:\vdots\: \vb{E}(\vb{r}, t_1) \vb{E}(\vb{r}, t_2) \vb{E}(\vb{r}, t_3) \dd{t_1} \dd{t_2} \dd{t_3} +\end{aligned}$$ + +In practice, two phenomena contribute to $$\chi^{(3)}$$: +the *Kerr effect* due to electrons' response to $$\vb{E}$$, +and *Raman scattering* due to nuclei's response, +which is slower because of their mass. +But if the light pulses are sufficiently long (>1ps in silica), +both effects can be treated as fast, so: + +$$\begin{aligned} + \chi^{(3)}(t \!-\! t_1, t \!-\! t_2, t \!-\! t_3) + &= \chi^{(3)} \delta(t - t_1) \delta(t - t_2) \delta(t - t_3) +\end{aligned}$$ + +Where $$\delta$$ is the [Dirac delta function](/know/concept/dirac-delta-function/). +To keep things simple, +we consider linearly $$x$$-polarized light $$\vb{E} = \vu{x} |\vb{E}|$$, +such that the tensor can be replaced with its scalar element $$\chi^{(3)}_{xxxx}$$. +Then: + +$$\begin{aligned} + \vb{P}_\mathrm{NL} + = \varepsilon_0 \chi^{(3)}_{xxxx} \big( \vb{E} \cdot \vb{E} \big) \vb{E} +\end{aligned}$$ + +For the same reasons, the linear polarization is reduced to: + +$$\begin{aligned} + \vb{P}_\mathrm{L} + &= \varepsilon_0 \chi^{(1)}_{xx} \vb{E} +\end{aligned}$$ + +Next, we decompose $$\vb{E}$$ as follows, +consisting of a carrier wave $$e^{-i \omega_0 t}$$ +at a constant frequency $$\omega_0$$, +modulated by an envelope $$E$$ +that is assumed to be slowly-varying compared to the carrier, +plus the complex conjugate $$E^* e^{i \omega_0 t}$$: + +$$\begin{aligned} + \vb{E}(\vb{r}, t) + &= \vu{x} \frac{1}{2} \Big( E(\vb{r}, t) e^{- i \omega_0 t} + E^*(\vb{r}, t) e^{i \omega_0 t} \Big) +\end{aligned}$$ + +Note that no generality has been lost in this step. +Inserting it into the polarizations: + +$$\begin{aligned} + \mathrm{P}_\mathrm{L} + &= \vu{x} \frac{1}{2} \varepsilon_0 \chi^{(1)}_{xx} \Big( E(\vb{r}, t) e^{- i \omega_0 t} + E^*(\vb{r}, t) e^{i \omega_0 t} \Big) + \\ + \vb{P}_\mathrm{NL} + &= \vu{x} \frac{1}{8} \varepsilon_0 \chi^{(3)}_{xxxx} \Big( E e^{- i \omega_0 t} + E^* e^{i \omega_0 t} \Big)^{3} + \\ + &= \vu{x} \frac{1}{8} \varepsilon_0 \chi^{(3)}_{xxxx} + \Big( E^3 e^{- i 3 \omega_0 t} + 3 E^2 E^* e^{- i \omega_0 t} + 3 E (E^*)^2 e^{i \omega_0 t} + (E^*)^3 e^{i 3 \omega_0 t} \Big) +\end{aligned}$$ + +The terms with $$3 \omega_0$$ represent *third-harmonic generation*, +and only matter if the carrier is phase-matched +to the tripled wave, which is generally not the case, +so they can be ignored. +Now, if we decompose the polarizations in the same was as $$\vb{E}$$: + +$$\begin{aligned} + \vb{P}_\mathrm{L}(\vb{r}, t) + &= \vu{x} \frac{1}{2} \Big( P_\mathrm{L}(\vb{r}, t) e^{- i \omega_0 t} + P_\mathrm{L}^*(\vb{r}, t) e^{i \omega_0 t} \Big) + \\ + \vb{P}_\mathrm{NL}(\vb{r}, t) + &= \vu{x} \frac{1}{2} \Big( P_\mathrm{NL}(\vb{r}, t) e^{- i \omega_0 t} + P_\mathrm{NL}^*(\vb{r}, t) e^{i \omega_0 t} \Big) +\end{aligned}$$ + +Then it is straightforward to see that their envelope functions are given by: + +$$\begin{aligned} + P_\mathrm{L} + &= \varepsilon_0 \chi^{(1)}_{xx} E + \\ + P_\mathrm{NL} + &= \frac{3}{4} \varepsilon_0 \chi^{(3)}_{xxxx} |E|^2 E +\end{aligned}$$ + +The forward carrier $$e^{- i \omega_0 t}$$ +and the backward carrier $$e^{i \omega_0 t}$$ +can be regarded as separate channels, +which only interact via $$P_\mathrm{NL}$$. +From now on, we only consider the forward-propagating wave, +so all terms containing $$e^{i \omega_0 t}$$ are dropped; +by taking the complex conjugate of the resulting equations, +the backward-propagating counterparts can always be recovered, +so no information is really lost. +Therefore, the main wave equation becomes: + +$$\begin{aligned} + 0 + &= \bigg( + \nabla^2 E - \mu_0 \varepsilon_0 \pdvn{2}{E}{t} - \mu_0 \pdvn{2}{P_\mathrm{L}}{t} - \mu_0 \pdvn{2}{P_\mathrm{NL}}{t} + \bigg) e^{-i \omega_0 t} + \\ + &\approx \bigg( + \nabla^2 E - \Big( 1 + \chi^{(1)}_{xx} + \frac{3}{4} \chi^{(3)}_{xxxx} |E|^2 \Big) \mu_0 \varepsilon_0 \pdvn{2}{E}{t} + \bigg) e^{-i \omega_0 t} +\end{aligned}$$ + +Where we have used our assumption that $$E$$ is slowly-varying +to treat $$|E|^2$$ as a constant, +in order to move it outside the $$t$$-derivative. +We thus arrive at: + +$$\begin{aligned} + 0 + &= \bigg( \nabla^2 E - \frac{\varepsilon_r}{c^2} \pdvn{2}{E}{t} \bigg) e^{-i \omega_0 t} +\end{aligned}$$ + +Where $$c = 1 / \sqrt{\mu_0 \varepsilon_0}$$ is the phase velocity of light in a vacuum, +and the relative permittivity $$\varepsilon_r$$ is defined as shown below. +Note that this is a mild abuse of notation, +since the symbol $$\varepsilon_r$$ is usually reserved for linear materials: + +$$\begin{aligned} + \varepsilon_r + \equiv 1 + \chi^{(1)}_{xx} + \frac{3}{4} \chi^{(3)}_{xxxx} |E|^2 +\end{aligned}$$ + +Next, we take the [Fourier transform](/know/concept/fourier-transform/) +$$t \to \omega$$ 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: + +$$\begin{aligned} + 0 + &= \hat{\mathcal{F}}\bigg\{ \bigg( \nabla^2 E - \frac{\varepsilon_r}{c^2} \pdvn{2}{E}{t} \bigg) e^{-i \omega_0 t} \bigg\} + \\ + &= \int_{-\infty}^\infty + \bigg( \nabla^2 E - \frac{\varepsilon_r}{c^2} \pdvn{2}{E}{t} \bigg) + e^{i s (\omega - \omega_0) t} \dd{t} + \\ + &= \nabla^2 E + s^2 (\omega - \omega_0)^2 \frac{\varepsilon_r}{c^2} E +\end{aligned}$$ + +We use $$s^2 = 1$$ and define $$\Omega \equiv \omega - \omega_0$$ +as the frequency shift relative to the carrier wave: + +$$\begin{aligned} + 0 + &= \nabla^2 E + \frac{\Omega^2 \varepsilon_r}{c^2} E +\end{aligned}$$ + +This is a so-called *Helmholtz equation* in 3D, +which we will solve using separation of variables, +by assuming that its solution can be written as: + +$$\begin{aligned} + E(\vb{r}, \Omega) + &= F(x, y) \: A(z, \Omega) \: e^{i \beta_0 z} +\end{aligned}$$ + +Where $$\beta_0$$ is the wavenumber of the carrier, +which will be determined later. +Inserting this ansatz into the Helmholtz equation yields: + +$$\begin{aligned} + 0 + &= \Big( \pdvn{2}{F}{x} + \pdvn{2}{F}{y} \Big) A e^{i \beta_0 z} + + \pdvn{2}{}{z} \Big( A e^{i \beta_0 z} \Big) F + + \frac{\Omega^2 \varepsilon_r}{c^2} F A e^{i \beta_0 z} + \\ + &= \bigg( \Big( \pdvn{2}{F}{x} + \pdvn{2}{F}{y} \Big) A + + \Big( \pdvn{2}{A}{z} + 2 i \beta_0 \pdv{A}{z} - \beta_0^2 A \Big) F + + \frac{\Omega^2 \varepsilon_r}{c^2} F A \bigg) e^{i \beta_0 z} +\end{aligned}$$ + +We divide by $$F A \: e^{i \beta_0 z}$$ +and rearrange the terms in a specific way: + +$$\begin{aligned} + \Big( \pdvn{2}{F}{x} + \pdvn{2}{F}{y} \Big) \frac{1}{F} + \frac{\Omega^2 \varepsilon_r}{c^2} + &= - 2 i \beta_0 \pdv{A}{z} \frac{1}{A} + \beta_0^2 +\end{aligned}$$ + +Now all the $$x$$- and $$y$$-dependence is on the left, +and the $$z$$-dependence is on the right. +We have placed the $$\varepsilon_r$$-term on the left too +because it depends relatively strongly on $$(x, y)$$ +to describe the fiber's internal structure, +and weakly on $$z$$ due to nonlinear effects. +Meanwhile, $$\beta_0$$ is on the right because that will lead to +a nicer equation for $$A$$ later. + +Note that both sides are functions of $$\Omega$$. +Based on the aforementioned dependences, +in order for this equation to have a solution for all $$(x, y, z)$$, +there must exist a quantity $$\beta(\Omega)$$ that is constant in space, +such that we obtain two separated equations for $$F$$ and $$A$$: + +$$\begin{aligned} + \beta(\omega) + &= \bigg( \pdvn{2}{F}{x} + \pdvn{2}{F}{y} \bigg) \frac{1}{F} + \frac{\omega^2 \varepsilon_r}{c^2} + \\ + \beta(\Omega) + &= - 2 i \beta_0 \pdv{A}{z} \frac{1}{A} + \beta_0^2 +\end{aligned}$$ + +Note that we replaced $$\Omega$$ with $$\omega$$ in $$F$$'s equation +(and redefined $$\beta$$ and $$\varepsilon_r$$ accordingly). +This is not an innocent detail: +the idea is that $$\omega \sqrt{\varepsilon_r} / c$$ +would be the light's wavenumber if it had not been trapped in a waveguide, +and that $$\beta$$ is the *confined* wavenumber, +also known as the **propagation constant**. +If we had kept $$\Omega$$, +the meaning of $$\beta$$ would not be so straightforward. + +The difference between $$\beta(\omega)$$ and $$\beta_0$$ +is simply that $$\beta_0 \equiv \beta(\omega_0)$$. +Our ansatz for separating the variables contained $$\beta_0$$, +such that the full carrier wave $$e^{i \beta_0 z - i \omega_0 t}$$ was represented +(with $$e^{- i \omega_0 t}$$ now hidden inside the Fourier transform). +But later, to properly describe how light behaves inside the fiber, +the full dispersion relation $$\beta(\omega)$$ will be needed. + +Multiplying by $$F$$ and $$A$$, +we get the following set of equations, +implicitly coupled via $$\beta$$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + 0 + &= \pdvn{2}{F}{x} + \pdvn{2}{F}{y} + \bigg( \frac{\omega^2 \varepsilon_r}{c^2} - \beta^2 \bigg) F + \\ + 0 + &= 2 i \beta_0 \pdv{A}{z} + \big( \beta^2 - \beta_0^2 \big) A + \end{aligned} + } +\end{aligned}$$ + +The equation for $$F$$ must be solved first. +To do so, we treat the nonlinearity as a perturbation +to be neglected initially. +In other words, we first solve the following eigenvalue problem for $$\beta^2$$, +where $$n(x, y)$$ is the linear refractive index, +with $$n^2 = 1 + \Real\{\chi^{(1)}_{xx}\} \approx \varepsilon_r$$: + +$$\begin{aligned} + \pdvn{2}{F}{x} + \pdvn{2}{F}{y} + \bigg( \frac{\omega^2 n^2}{c^2} - \beta^2 \bigg) F + = 0 +\end{aligned}$$ + +This gives us the allowed values of $$\beta$$; +see [step-index fiber](/know/concept/step-index-fiber/) for an example solution. +Now we add the small index change $$\Delta{n}(x, y)$$ due to nonlinear effects: + +$$\begin{aligned} + \varepsilon_r + = (n + \Delta{n})^2 + \approx n^2 + 2 n \: \Delta{n} +\end{aligned}$$ + +Then it can be shown using first-order +[perturbation theory](/know/concept/time-independent-perturbation-theory/) +that the eigenfunction $$F$$ is not really affected, +and the eigenvalue $$\beta^2$$ is shifted by $$\Delta(\beta^2)$$, given by: + +$$\begin{aligned} + \Delta(\beta^2) + = \frac{2 \omega^2}{c^2} \frac{\displaystyle \iint_{-\infty}^\infty n \: \Delta{n} \: |F|^2 \dd{x} \dd{y}} + {\displaystyle \iint_{-\infty}^\infty |F|^2 \dd{x} \dd{y}} +\end{aligned}$$ + +But we are more interested in the *wavenumber* shift $$\Delta{\beta}$$ +than the *eigenvalue* shift $$\Delta(\beta^2)$$. +They are related to one another as follows: + +$$\begin{aligned} + \beta^2 + \Delta(\beta^2) + = (\beta + \Delta{\beta})^2 + \approx \beta^2 + 2 \beta \Delta{\beta} +\end{aligned}$$ + +Furthermore, we assume that the fiber only consists of materials +with similar refractive indices, or in other words, +that it confines the light using only a small index difference, +in which case we can treat $$n$$ as a constant and move it outside the integral. +Then $$\Delta{\beta}$$ becomes: + +$$\begin{aligned} + \Delta{\beta} + = \frac{\omega^2 n}{\beta c^2} \frac{\displaystyle \iint_{-\infty}^\infty \Delta{n} \: |F|^2 \dd{x} \dd{y}} + {\displaystyle \iint_{-\infty}^\infty |F|^2 \dd{x} \dd{y}} +\end{aligned}$$ + +Recall that $$\beta$$ is the wavenumber of the confined mode: +by solving the unperturbed $$F$$-equation, +it can be shown that $$\beta$$'s value is somewhere +between the bulk wavenumbers of the fiber materials. +Since we just approximated $$n$$ as a constant, +this means that $$\omega n / c \approx \beta$$, leading us to +the general "final" form of $$\Delta{\beta}$$, +with all the arguments shown for clarity: + +$$\begin{aligned} + \boxed{ + \Delta{\beta}(\omega) + = \frac{\omega}{c \mathcal{A}_\mathrm{mode}} + \iint_{-\infty}^\infty \Delta{n}(x, y, \omega) \: |F(x, y)|^2 \dd{x} \dd{y} + } +\end{aligned}$$ + +Where we have defined the *mode area* $$\mathcal{A}_\mathrm{mode}$$ as shown below. +In order for $$\mathcal{A}_\mathrm{mode}$$ to be in units of area, +$$F$$ must be dimensionless, +and consequently $$A$$ has (SI) units of an electric field. + +$$\begin{aligned} + \mathcal{A}_\mathrm{mode} + \equiv \iint_{-\infty}^\infty |F|^2 \dd{x} \dd{y} +\end{aligned}$$ + +Now we finally turn our attention to the equation for $$A$$. +Before perturbation, it was: + +$$\begin{aligned} + 0 + &= 2 i \beta_0 \pdv{A}{z} + \big( \beta^2 - \beta_0^2 \big) A +\end{aligned}$$ + +Where $$\beta \approx \beta_0$$, so we can replace +$$\beta^2 - \beta_0^2$$ with $$2 \beta_0 (\beta - \beta_0)$$. +Also including $$\Delta{\beta}$$, we get: + +$$\begin{aligned} + 0 + &= i \pdv{A}{z} + \big( \beta + \Delta{\beta} - \beta_0 \big) A +\end{aligned}$$ + +Usually, we do not know a full expression for $$\beta(\omega)$$, +so it makes sense to expand it around the carrier frequency $$\omega_0$$ as follows, +where $$\beta_n = \idvn{n}{\beta}{\omega} |_{\omega = \omega_0}$$: + +$$\begin{aligned} + \beta(\omega) + &= \beta_0 + + (\omega - \omega_0) \beta_1 + + (\omega - \omega_0)^2 \frac{\beta_2}{2} + + (\omega - \omega_0)^3 \frac{\beta_3}{6} + + \: ... +\end{aligned}$$ + +Spectrally, the broader the light pulse, the more terms must be included. +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 in many cases. +Inserting the expansion into $$A$$'s equation: + +$$\begin{aligned} + 0 + &= i \pdv{A}{z} + i \frac{\beta_1}{s} (-i s \Omega) A - \frac{\beta_2}{2 s^2} (- i s \Omega)^2 A + \Delta{\beta}_0 A +\end{aligned}$$ + +Which we have rewritten 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 $$\omega$$, +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 $$A(z, t)$$: + +$$\begin{aligned} + 0 + &= i \pdv{A}{z} + i s \beta_1 \pdv{A}{t} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \Delta{\beta}_0 A +\end{aligned}$$ + +The next step is to insert our expression for $$\Delta{\beta}_0$$, +for which we must first choose a specific form for $$\Delta{n}$$ +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. +The most commonly used form for $$\Delta{n}$$ is therefore: + +$$\begin{aligned} + \Delta{n}(x, y, \omega) + = n_2(\omega) \: I(x, y, \omega) + i \frac{c \alpha(\omega)}{2 \omega} +\end{aligned}$$ + +Where $$I$$ is the intensity (i.e. power per unit area) of the light, +$$n_2$$ is the material's *Kerr coefficient* in units of inverse intensity, +and $$\alpha$$ is the attenuation coefficient +consisting of linear and nonlinear contributions +(see [multi-photon absorption](/know/concept/multi-photon-absorption/)). +Specifically, they are given by: + +$$\begin{aligned} + n_2 + = \frac{3 \Real\{\chi^{(3)}_{xxxx}\}}{4 \varepsilon_0 c n^2} + \qquad + \alpha + = \frac{\omega \Imag\{\chi^{(1)}_{xx}\}}{c n} + + \frac{3 \omega \Imag\{\chi^{(3)}_{xxxx}\}}{2 \varepsilon_0 c^2 n^2} I + \qquad + I + = \frac{\varepsilon_0 c n}{2} |F|^2 |A|^2 +\end{aligned}$$ + +For simplicity we set $$\Imag\{\chi^{(3)}_{xxxx}\} = 0$$, +which is a good approximation for silica fibers. +Inserting this form of $$\Delta{n}$$ into $$\Delta{\beta_0}$$ +and neglecting the $$(x, y)$$-dependence of $$\Delta{n}$$ yields: + +$$\begin{aligned} + \Delta{\beta}_0 + &= i \frac{\alpha}{2} \frac{\mathcal{A}_\mathrm{mode}}{\mathcal{A}_\mathrm{mode}} + + \frac{\omega_0 \varepsilon_0 c n n_2}{2 c \mathcal{A}_\mathrm{mode}} |A|^2 \iint_{-\infty}^\infty |F|^4 \dd{x} \dd{y} + \\ + &= i \frac{\alpha}{2} + + \gamma_0 \frac{\varepsilon_0 c n}{2} \mathcal{A}_\mathrm{mode} |A|^2 +\end{aligned}$$ + +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: + +$$\begin{aligned} + \boxed{ + \gamma(\omega) + \equiv \frac{\omega n_2(\omega)}{c \mathcal{A}_\mathrm{eff}(\omega)} + } + \qquad \qquad + \boxed{ + \mathcal{A}_\mathrm{eff}(\omega) + \equiv \frac{\displaystyle \bigg( \iint_{-\infty}^\infty |F|^2 \dd{x} \dd{y} \bigg)^2} + {\displaystyle \iint_{-\infty}^\infty |F|^4 \dd{x} \dd{y}} + } +\end{aligned}$$ + +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: + +$$\begin{aligned} + 0 + &= i \pdv{A}{z} + i s \beta_1 \pdv{A}{t} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + i \frac{\alpha}{2} A + + \gamma_0 \frac{\varepsilon_0 c n}{2} \mathcal{A}_\mathrm{mode} |A|^2 A +\end{aligned}$$ + +The factor $$\varepsilon_0 c n / 2$$ looks familiar from the intensity $$I$$. +This, combined with $$\mathcal{A}_\mathrm{mode}$$ +and the fact that $$A$$ is an electric field, +suggests that we can redefine $$A \to A'$$ +such that $$|A'|^2$$ is the optical power in watts. +Hence we make the following transformation: + +$$\begin{aligned} + \frac{\varepsilon_0 c n}{2} \mathcal{A}_\mathrm{mode} |A|^2 + \:\:\to\:\: + |A|^2 +\end{aligned}$$ + +We can divide away the transformation factors +from all other terms in the equation, since they are linear, +leading to the full *nonlinear Schrödinger equation*: + +$$\begin{aligned} + \boxed{ + 0 + = i \pdv{A}{z} + i s \beta_1 \pdv{A}{t} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + i \frac{\alpha}{2} A + \gamma_0 |A|^2 A + } +\end{aligned}$$ + +This can be reduced by switching to a coordinate system +where the time axis slides along the propagation axis at a speed $$s v$$, +so we define $$Z \equiv z$$ and $$T \equiv t - s z / v$$ such that: + +$$\begin{aligned} + \pdv{A}{z} + &= \pdv{A}{Z} \pdv{Z}{z} + \pdv{A}{T} \pdv{T}{z} + = \pdv{A}{Z} - \frac{s}{v} \pdv{A}{T} + \\ + \pdv{A}{t} + &= \pdv{A}{Z} \pdv{Z}{t} + \pdv{A}{T} \pdv{T}{t} + = \pdv{A}{T} +\end{aligned}$$ + +We insert this and set $$v = v_g$$, +where $$v_g = 1 / \beta_1$$ is the light's group velocity: + +$$\begin{aligned} + \boxed{ + 0 + = i \pdv{A}{Z} - \frac{\beta_2}{2} \pdvn{2}{A}{T} + i \frac{\alpha}{2} A + \gamma_0 |A|^2 A + } +\end{aligned}$$ + +The NLS equation's name is due to its similarity +to the Schrödinger equation of quantum physics, +if you set $$\alpha = 0$$ and treat $$\gamma_0 |A|^2$$ as a potential. +In fiber optics, the equation is usually rearranged +to highlight that $$Z$$ (or $$z$$) is the propagation direction: + +$$\begin{aligned} + \pdv{A}{Z} + = - i \frac{\beta_2}{2} \pdvn{2}{A}{T} - \frac{\alpha}{2} A + i \gamma_0 |A|^2 A +\end{aligned}$$ + +Next, we want to reduce the equation to its dimensionless form. +To do so, we make the following coordinate transformation, +where $$\tilde{A}$$, $$\tilde{Z}$$ and $$\tilde{T}$$ are unitless, +and $$A_c$$, $$Z_c$$ and $$T_c$$ are dimensioned scale parameters +to be determined later: + +$$\begin{aligned} + \tilde{A}(\tilde{Z}, \tilde{T}) + = \frac{A(Z, T)}{A_c} + \qquad\qquad + \tilde{Z} + = \frac{Z}{Z_c} + \qquad\qquad + \tilde{T} + = \frac{T}{T_c} +\end{aligned}$$ + +We insert this into the NLS equation, +after setting $$\alpha = 0$$ according to convention: + +$$\begin{aligned} + 0 + = i \frac{A_c}{Z_c} \pdv{\tilde{A}}{\tilde{Z}} + - \frac{\beta_2}{2} \frac{A_c}{T_c^2} \pdvn{2}{\tilde{A}}{\tilde{T}} + + \gamma_0 A_c^3 \big|\tilde{A}\big|^2 \tilde{A} +\end{aligned}$$ + +Multiplying by $$Z_c / A_c$$ to make all terms dimensionless leads us to: + +$$\begin{aligned} + 0 + = i \pdv{\tilde{A}}{\tilde{Z}} + - \frac{\beta_2 Z_c}{2 T_c^2} \pdvn{2}{\tilde{A}}{\tilde{T}} + + \gamma_0 A_c^2 Z_c \big|\tilde{A}\big|^2 \tilde{A} +\end{aligned}$$ + +The goal is to remove those constant factors. +In other words, we demand: + +$$\begin{aligned} + \frac{\beta_2 Z_c}{2 T_c^2} + = -1 + \qquad\qquad + \gamma_0 A_c^2 Z_c + = r +\end{aligned}$$ + +Where $$r \equiv \pm 1$$, whose sign choice will be explained shortly. +Note that we have two equations for three unknowns +($$A_c$$, $$Z_c$$ and $$T_c$$), +so one of the parameters needs to fixed manually. +For example, we could choose our "input power" +$$A_c \equiv \sqrt{1\:\mathrm{W}}$$, and then: + +$$\begin{aligned} + Z_c + = - \frac{2 T_c^2}{\beta_2} + \qquad + T_c^2 + = -\frac{r \beta_2}{2 \gamma_0 A_c^2} + \qquad\implies\qquad + Z_c + = \frac{r}{\gamma_0 A_c^2} + \qquad + T_c + = \sqrt{ -\frac{r \beta_2}{2 \gamma_0 A_c^2} } +\end{aligned}$$ + +Because $$T_c$$ must be real, +we should choose $$r \equiv - \sgn(\gamma_0 \beta_2)$$. +We thus arrive at: + +$$\begin{aligned} + \boxed{ + 0 + = i \pdv{\tilde{A}}{\tilde{Z}} + + \pdvn{2}{\tilde{A}}{\tilde{T}} + + r \big|\tilde{A}\big|^2 \tilde{A} + } +\end{aligned}$$ + +In fiber optics, $$\gamma_0 > 0$$ for all materials, +meaning $$r$$ represents the dispersion regime, +so $$r = 1$$ is called *anomalous dispersion* +and $$r = -1$$ *normal dispersion*. +In some other fields, where $$\beta_2 < 0$$ always, +$$r = 1$$ is called a *focusing nonlinearity* +and $$r = -1$$ a *defocusing nonlinearity*. +The famous bright solitons only exist for $$r = 1$$, +so many authors only show that case. + + + +## References + +1. G.P. Agrawal, + *Nonlinear fiber optics*, 6th edition, + Elsevier. +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. |