---
title: "Dispersive broadening"
date: 2021-02-27
categories:
- Physics
- Optics
- Fiber optics
layout: "concept"
---

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:

$$\begin{aligned}
    0
    = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A
\end{aligned}$$

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 $\gamma = 0$:

<a href="pheno-disp.jpg">
<img src="pheno-disp-small.jpg" style="width:100%">
</a>

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** ($\beta_2 > 0$).



## References
1.  O. Bang,
    *Numerical methods in photonics: lecture notes*, 2019,
    unpublished.