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diff --git a/source/know/concept/modulational-instability/index.md b/source/know/concept/modulational-instability/index.md new file mode 100644 index 0000000..255a385 --- /dev/null +++ b/source/know/concept/modulational-instability/index.md @@ -0,0 +1,202 @@ +--- +title: "Modulational instability" +date: 2021-02-26 +categories: +- Physics +- Fiber optics +- Optics +- Perturbation +- Nonlinear optics +layout: "concept" +--- + +In fiber optics, **modulational instability** (MI) +is a nonlinear effect that leads to the exponential amplification +of background noise in certain frequency regions. +It only occurs in the [anomalous dispersion regime](/know/concept/dispersive-broadening/) +($\beta_2 < 0$), which we will prove shortly. + +Consider the following simple solution to the nonlinear Schrödinger equation: +a time-invariant constant power $P_0$ at the carrier frequency $\omega_0$, +which is experiencing [self-phase modulation](/know/concept/self-phase-modulation/): + +$$\begin{aligned} + A(z,t) = \sqrt{P_0} \exp( i \gamma P_0 z) +\end{aligned}$$ + +We add a small perturbation $\varepsilon(z,t)$ to this signal, +representing background noise: + +$$\begin{aligned} + A(z,t) = \big(\sqrt{P_0} + \varepsilon(z,t)\big) \exp( i \gamma P_0 z) +\end{aligned}$$ + +We insert this into the nonlinear Schrödinger equation to get a perturbation equation, +which we linearize by assuming that $|\varepsilon|^2$ is negligible compared to $P_0$, +such that all higher-order terms of $\varepsilon$ can be dropped, yielding: + +$$\begin{aligned} + 0 + &= - P_0 \sqrt{P_0} \gamma - P_0 \gamma \varepsilon + i \pdv{\varepsilon}{z} + - \frac{\beta_2}{2} \pdvn{2}{\varepsilon}{t} + + \gamma \big(\sqrt{P_0} + \varepsilon\big)^2 \big(\sqrt{P_0} + \varepsilon\big)^* + \\ + &= i \pdv{\varepsilon}{z} + - \frac{\beta_2}{2} \pdvn{2}{\varepsilon}{t} + + \gamma \big( P_0 (\varepsilon + \varepsilon^*) + \sqrt{P_0} |\varepsilon|^2 + + \sqrt{P_0} \varepsilon (\varepsilon + \varepsilon^*) + \varepsilon |\varepsilon|^2 \big) + \\ + &= i \pdv{\varepsilon}{z} - \frac{\beta_2}{2} \pdvn{2}{\varepsilon}{t} + \gamma P_0 (\varepsilon + \varepsilon^*) +\end{aligned}$$ + +We split the perturbation into real and imaginary parts +$\varepsilon(z,t) = \varepsilon_r(z,t) + i \varepsilon_i(z,t)$, +which we fill in in this equation. +The point is that $\varepsilon_r$ and $\varepsilon_i$ are real functions: + +$$\begin{aligned} + 0 + &= i \pdv{\varepsilon_r}{z} - \pdv{\varepsilon_i}{z} + - \frac{\beta_2}{2} \pdvn{2}{\varepsilon_r}{t} - i \frac{\beta_2}{2} \pdvn{2}{\varepsilon_i}{t} + + 2 \gamma P_0 \varepsilon_r +\end{aligned}$$ + +Splitting this into its real and imaginary parts gives two PDEs +relating $\varepsilon_r$ and $\varepsilon_i$: + +$$\begin{aligned} + \pdv{\varepsilon_r}{z} = \frac{\beta_2}{2} \pdvn{2}{\varepsilon_i}{t} + \qquad \quad + \pdv{\varepsilon_i}{z} = - \frac{\beta_2}{2} \pdvn{2}{\varepsilon_r}{t} + 2 \gamma P_0 \varepsilon_r +\end{aligned}$$ + +We [Fourier transform](/know/concept/fourier-transform/) +these in $t$ to turn them into ODEs relating +$\tilde{\varepsilon}_r(z,\omega)$ and $\tilde{\varepsilon}_i(z,\omega)$: + +$$\begin{aligned} + \pdv{\tilde{\varepsilon}_r}{z} = - \frac{\beta_2}{2} \omega^2 \tilde{\varepsilon}_i + \qquad \quad + \pdv{\tilde{\varepsilon}_i}{z} = \Big(\frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) \tilde{\varepsilon}_r +\end{aligned}$$ + +We are interested in exponential growth, so let us make the following ansatz, +where $k$ may be a function of $\omega$, as long as it is $z$-invariant: + +$$\begin{aligned} + \tilde{\varepsilon}_r(z, \omega) = \tilde{\varepsilon}_r(0, \omega) \exp(k z) + \qquad \quad + \tilde{\varepsilon}_i(z, \omega) = \tilde{\varepsilon}_i(0, \omega) \exp(k z) +\end{aligned}$$ + +With this, we can write the system of ODEs for +$\tilde{\varepsilon}_r(z,\omega)$ and $\tilde{\varepsilon}_i(z,\omega)$ +in matrix form: + +$$\begin{aligned} + \begin{bmatrix} + k & \beta_2 \omega^2 / 2 \\ + \beta_2 \omega^2 / 2 \!+\! 2 \gamma P_0 & - k + \end{bmatrix} + \cdot + \begin{bmatrix} \tilde{\varepsilon}_r(0, \omega) \\ \tilde{\varepsilon}_i(0, \omega) \end{bmatrix} + = + \begin{bmatrix} 0 \\ 0 \end{bmatrix} +\end{aligned}$$ + +This has non-zero solutions if the system matrix' determinant is zero, +which is true when: + +$$\begin{aligned} + k = \pm \sqrt{ - \frac{\beta_2}{2} \omega^2 \Big( \frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) } +\end{aligned}$$ + +To get exponential growth, it is essential that $\mathrm{Re}\{k\} > 0$, +so we discard the negative sign, +and get the following condition for MI: + +$$\begin{aligned} + - \frac{\beta_2}{2} \omega^2 \Big( \frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) > 0 + \quad \implies \quad + \boxed{ + \omega^2 < -\frac{4 \gamma P_0}{\beta_2} + } +\end{aligned}$$ + +Since $\omega^2$ is positive, $\beta_2$ must be negative, +so MI can only occur in the ADR. +It is worth noting that $\beta_2 = \beta_2(\omega_0)$, +meaning there can only be exponential +noise growth when the "parent pulse" is in the anomalous dispersion regime, +but that growth may appear in areas of normal dispersion, +as long as the above condition is satisfied by the parent. + +This result has been derived using perturbation, +so only holds as long as $|\varepsilon|^2 \ll P_0$. +Over time, the noise gets amplified so greatly +that this approximation breaks down. + +Next, we define the **gain** $g(\omega)$, +which expresses how quickly the +perturbation grows as a function of the frequency offset $\omega$: + +$$\begin{aligned} + \boxed{ + g(\omega) + = \mathrm{Re}\{k\} + = \mathrm{Re} \bigg\{ \sqrt{ - \frac{\beta_2}{2} \omega^2 \Big( \frac{\beta_2}{2} \omega^2 + 2 \gamma P_0 \Big) } \bigg\} + } +\end{aligned}$$ + +The frequencies with maximum gain are then found as extrema of $g(\omega)$, +which satisfy: + +$$\begin{aligned} + g'(\omega_\mathrm{max}) = 0 + \qquad \implies \qquad + \boxed{ + \omega_\mathrm{max} = \pm \sqrt{\frac{2 \gamma P_0}{-\beta_2}} + } +\end{aligned}$$ + +A simulation of MI is illustrated below. +The pulse considered was a solition of the following form +with settings $T_0 = 10\:\mathrm{ps}$, $P_0 = 10\:\mathrm{kW}$, +$\beta = -10\:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$, +whose peak is approximately flat, so our derivation is valid there, +hence it "wrinkles" in the $t$-domain: + +$$\begin{aligned} + A(0, t) + = \sqrt{P_0} \sech\!\Big(\frac{t}{T_0}\Big) +\end{aligned}$$ + +<a href="pheno-mi.jpg"> +<img src="pheno-mi-small.jpg" style="width:100%"> +</a> + +Where $L_\mathrm{NL} = 1/(\gamma P_0)$ is the characteristic length of nonlinear effects. +Note that no noise was added to the simulation; +what you are seeing are pure numerical errors getting amplified. + +If one of the gain peaks accumulates a lot of energy quickly ($L_\mathrm{NL}$ is small), +and that peak is in the anomalous dispersion regime, +then it can in turn also cause MI in its own surroundings, +leading to a cascade of secondary and tertiary gain areas. +This is seen above for $z > 30 L_\mathrm{NL}$. + +What we described is "pure" MI, but there also exists +a different type caused by Raman scattering. +In that case, amplification occurs at the strongest peak of the Raman gain $\tilde{g}_R(\omega)$, +even when the parent pulse is in the NDR. +This is an example of stimulated Raman scattering (SRS). + + + +## References +1. O. Bang, + *Numerical methods in photonics: lecture notes*, 2019, + unpublished. +2. O. Bang, + *Nonlinear mathematical physics: lecture notes*, 2020, + unpublished. diff --git a/source/know/concept/modulational-instability/pheno-mi-small.jpg b/source/know/concept/modulational-instability/pheno-mi-small.jpg Binary files differnew file mode 100644 index 0000000..995ec81 --- /dev/null +++ b/source/know/concept/modulational-instability/pheno-mi-small.jpg diff --git a/source/know/concept/modulational-instability/pheno-mi.jpg b/source/know/concept/modulational-instability/pheno-mi.jpg Binary files differnew file mode 100644 index 0000000..e45f074 --- /dev/null +++ b/source/know/concept/modulational-instability/pheno-mi.jpg |