Expand knowledge base with material from BSc thesis

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+---
+title: "Self-phase modulation"
+firstLetter: "S"
+publishDate: 2021-02-26
+categories:
+- Physics
+- Optics
+- Fiber optics
+- Nonlinear dynamics
+
+date: 2021-02-27T10:09:32+01:00
+draft: false
+markup: pandoc
+---
+
+# Self-phase modulation
+
+In fiber optics, **self-phase modulation** (SPM) is a nonlinear effect
+that gradually broadens pulses' spectra.
+Unlike dispersion, SPM does create new frequencies: in the $\omega$-domain,
+the pulse steadily spreads out with a distinctive "accordion" peak.
+Lower frequencies are created at the front of the
+pulse and higher ones at the back, giving S-shaped spectrograms.
+
+A pulse envelope $A(z, t)$ inside a fiber must obey the nonlinear Schrödinger equation,
+where the parameters $\beta_2$ and $\gamma$ respectively
+control dispersion and nonlinearity:
+
+$$\begin{aligned}
+    0
+    = i \pdv{A}{z} - \frac{\beta_2}{2} \pdv[2]{A}{t} + \gamma |A|^2 A
+\end{aligned}$$
+
+By setting $\beta_2 = 0$ to neglect dispersion,
+solving this equation becomes trivial.
+For any arbitrary input pulse $A_0(t) = A(0, t)$,
+we arrive at the following analytical solution:
+
+$$\begin{aligned}
+    A(z,t) = A_0 \exp\!\big( i \gamma |A_0|^2 z\big)
+\end{aligned}$$
+
+The intensity $|A|^2$ in the time domain is thus unchanged,
+and only its phase is modified.
+It is also clear that the largest phase increase occurs at the peak of the pulse,
+where the intensity is $P_0$.
+To quantify this, it is useful to define the **nonlinear length** $L_N$,
+which gives the distance after which the phase of the
+peak has increased by exactly 1 radian:
+
+$$\begin{aligned}
+    \gamma P_0 L_N = 1
+    \qquad \implies \qquad
+    \boxed{
+        L_N = \frac{1}{\gamma P_0}
+    }
+\end{aligned}$$
+
+SPM is illustrated below for the following Gaussian initial pulse envelope,
+with parameter values $T_0 = 6\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$,
+$\beta_2 = 0$, and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$:
+
+$$\begin{aligned}
+    A(0, t)
+    = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
+\end{aligned}$$
+
+From earlier, we then know the analytical solution for the $z$-evolution:
+
+$$\begin{aligned}
+    A(z, t) = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big) \exp\!\bigg( i \gamma z P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
+\end{aligned}$$
+
+<img src="pheno-spm.jpg">
+
+The **instantaneous frequency** $\omega_\mathrm{SPM}(z, t)$,
+which describes the dominant angular frequency at a given point in the time domain,
+is found to be as follows for the Gaussian pulse,
+where $\phi(z, t)$ is the phase of $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$:
+
+$$\begin{aligned}
+    \omega_{\mathrm{SPM}}(z,t)
+    = - \pdv{\phi}{t}
+    = 2 \gamma z P_0 \frac{t}{T_0^2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
+\end{aligned}$$
+
+This result gives the S-shaped spectrograms seen in the illustration.
+The frequency shift thus not only depends on $L_N$,
+but also on $T_0$: the spectra of narrow pulses broaden much faster.
+
+The interaction between self-phase modulation
+and [dispersion](/know/concept/dispersive-broadening/)
+leads to many interesting effects,
+such as [modulational instability](/know/concept/modulational-instability/)
+and [optical wave breaking](/know/concept/optical-wave-breaking/).