Categories: Fiber optics, Optics, Physics.

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 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 \(\gamma = 0\):

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 leads to many interesting effects, such as modulational instability and 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** (\(\beta_2 > 0\)).

- O. Bang,
*Numerical methods in photonics: lecture notes*, 2019, unpublished.

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