Expand knowledge base with material from BSc thesis

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+---
+title: "Modulational instability"
+firstLetter: "M"
+publishDate: 2021-02-26
+categories:
+- Physics
+- Fiber optics
+- Optics
+- Perturbation
+- Nonlinear dynamics
+
+date: 2021-02-26T20:36:22+01:00
+draft: false
+markup: pandoc
+---
+
+# Modulational instability
+
+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:
+
+$$\begin{aligned}
+    0
+    &= - P_0 \sqrt{P_0} \gamma - P_0 \gamma \varepsilon + i \pdv{\varepsilon}{z}
+    - \frac{\beta_2}{2} \pdv[2]{\varepsilon}{t}
+    + \gamma \big(\sqrt{P_0} + \varepsilon\big)^2 \big(\sqrt{P_0} + \varepsilon\big)^*
+    \\
+    &= i \pdv{\varepsilon}{z}
+    - \frac{\beta_2}{2} \pdv[2]{\varepsilon}{t}
+    + \gamma \big( P_0 (\varepsilon + \varepsilon^*) + \sqrt{P_0} |\varepsilon|^2
+    + \sqrt{P_0} \varepsilon (\varepsilon + \varepsilon^*) + \varepsilon |\varepsilon|^2 \big)
+    \\
+    &= i \pdv{\varepsilon}{z} - \frac{\beta_2}{2} \pdv[2]{\varepsilon}{t} + \gamma P_0 (\varepsilon + \varepsilon^*)
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    0
+    &= i \pdv{\varepsilon_r}{z} - \pdv{\varepsilon_i}{z}
+    - \frac{\beta_2}{2} \pdv[2]{\varepsilon_r}{t} - i \frac{\beta_2}{2} \pdv[2]{\varepsilon_i}{t}
+    + 2 \gamma P_0 \varepsilon_r
+\end{aligned}$$
+
+Splitting this into its real and imaginary parts gives two PDEs
+relating $\varepsilon_r$ and $\varepsilon_i$:
+
+$$\begin{aligned}
+    \pdv{\varepsilon_r}{z} = \frac{\beta_2}{2} \pdv[2]{\varepsilon_i}{t}
+    \qquad \quad
+    \pdv{\varepsilon_i}{z} = - \frac{\beta_2}{2} \pdv[2]{\varepsilon_r}{t} + 2 \gamma P_0 \varepsilon_r
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    A(0, t)
+    = \sqrt{P_0} \sech\!\Big(\frac{t}{T_0}\Big)
+\end{aligned}$$
+
+<img src="pheno-mi.jpg">
+
+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).
+