Expand knowledge base with material from BSc thesis

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+---
+title: "Dispersive broadening"
+firstLetter: "D"
+publishDate: 2021-02-27
+categories:
+- Physics
+- Optics
+- Fiber optics
+
+date: 2021-02-27T11:48:34+01:00
+draft: false
+markup: pandoc
+---
+
+# Dispersive broadening
+
+In optical fibers, **dispersive broadening** is a (linear) effect
+where group velocity dispersion (GVD) "smears out" a pulse in the time domain
+due to the different group velocities of its frequencies,
+since pulses always have a non-zero width in the $\omega$-domain.
+No new frequencies are created.
+
+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:
+
+$$\begin{aligned}
+    0
+    = i \pdv{A}{z} - \frac{\beta_2}{2} \pdv[2]{A}{t} + \gamma |A|^2 A
+\end{aligned}$$
+
+We set $\gamma = 0$ to ignore all nonlinear effects,
+and consider a Gaussian initial condition:
+
+$$\begin{aligned}
+    A(0, t)
+    = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
+\end{aligned}$$
+
+By [Fourier transforming](/know/concept/fourier-transform/) in $t$,
+the full analytical solution $A(z, t)$ is found to be as follows,
+where it can be seen that the amplitude
+decreases and the width increases with $z$:
+
+$$\begin{aligned}
+    A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
+    \exp\!\bigg(\! -\!\frac{t^2 / (2 T_0^2)}{1 + \beta_2^2 z^2 / T_0^4} \big( 1 + i \beta_2 z / T_0^2 \big) \bigg)
+\end{aligned}$$
+
+To quantify the strength of dispersive effects,
+we define the dispersion length $L_D$
+as the distance over which the half-width at $1/e$ of maximum power
+(initially $T_0$) increases by a factor of $\sqrt{2}$:
+
+$$\begin{aligned}
+    T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2}
+    \qquad \implies \qquad
+    \boxed{
+        L_D = \frac{T_0^2}{|\beta_2|}
+    }
+\end{aligned}$$
+
+This phenomenon is illustrated below for our example of a Gaussian pulse
+with parameter values $T_0 = 1\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$,
+$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0$:
+
+<img src="pheno-disp.jpg">
+
+The **instantaneous frequency** $\omega_\mathrm{GVD}(z, t)$,
+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 $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$:
+
+$$\begin{aligned}
+    \omega_{\mathrm{GVD}}(z,t)
+    = \pdv{t} \Big( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \Big)
+    = \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
+\end{aligned}$$
+
+This expression is linear in time, and depending on the sign of $\beta_2$,
+frequencies on one side of the pulse arrive first,
+and those on the other side arrive last.
+The effect is stronger for smaller $T_0$:
+this makes sense, since short pulses are spectrally wider.
+
+The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/)
+leads to many interesting effects,
+such as [modulational instability](/know/concept/modulational-instability/)
+and [optical wave breaking](/know/concept/optical-wave-breaking/).
+Of great importance is the sign of $\beta_2$:
+in the **anomalous dispersion regime** ($\beta_2 < 0$),
+lower frequencies travel more slowly than higher ones,
+and vice versa in the **normal dispersion regime** ($\beta_2 > 0$).