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 ($$\beta_2 > 0$$).

## References

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 O. Bang, Numerical methods in photonics: lecture notes, 2019, unpublished.