Categories: Fiber optics, Optics, Physics.

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 nonzero width in the ω\omega-domain. No new frequencies are created.

A pulse envelope A(z,t)A(z, t) inside a fiber must obey the nonlinear Schrödinger equation, where the parameters β2\beta_2 and γ\gamma respectively control dispersion and nonlinearity:

0=iAzβ222At2+γA2A\begin{aligned} 0 = i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A \end{aligned}

We set γ=0\gamma = 0 to ignore all nonlinear effects, and consider a Gaussian initial condition:

A(0,t)=P0exp ⁣( ⁣ ⁣t22T02)\begin{aligned} A(0, t) = \sqrt{P_0} \exp\!\bigg(\!-\!\frac{t^2}{2 T_0^2}\bigg) \end{aligned}

By Fourier transforming in tt, the full analytical solution A(z,t)A(z, t) is found to be as follows, where it can be seen that the amplitude decreases and the width increases with zz:

A(z,t)=P01iβ2z/T02exp ⁣( ⁣ ⁣t2/(2T02)1+β22z2/T04(1+iβ2z/T02))\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 LDL_D as the distance over which the half-width at 1/e1/e of maximum power (initially T0T_0) increases by a factor of 2\sqrt{2}:

T01+β22LD2/T04=T02    LDT02β2\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 \equiv \frac{T_0^2}{|\beta_2|} } \end{aligned}

This phenomenon is illustrated below for our example of a Gaussian pulse with parameter values T0=1psT_0 = 1\:\mathrm{ps}, P0=1kWP_0 = 1\:\mathrm{kW}, β2=10ps2/m\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m} and γ=0\gamma = 0:

Dispersive broadening simulation results

The instantaneous frequency ωGVD(z,t)\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 ϕ(z,t)\phi(z, t) is the phase of A(z,t)=P(z,t)exp(iϕ(z,t))A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t)):

ωGVD(z,t)=t(β2zt2/(2T04)1+β22z2/T04)=β2z/T021+β22z2/T04tT02\begin{aligned} \omega_{\mathrm{GVD}}(z,t) = \pdv{}{t}\bigg( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \bigg) = \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 β2\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 T0T_0: this makes sense, since shorter 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 β2\beta_2: in the anomalous dispersion regime (β2<0\beta_2 < 0), lower frequencies travel more slowly than higher ones, and vice versa in the normal dispersion regime (β2>0\beta_2 > 0).


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 O. Bang, Numerical methods in photonics: lecture notes, 2019, unpublished.