Different Propagation Regimes

In the previous subsection (Section 2), we found several equations governing pulse propagation in fibers called the Nonlinear Schrodinger Equation, with different equations valid for different physical conditions. For pulse widths $\geq$ 1 ps, the following form is apt for our purposes:

\begin{equation} i\frac{\partial{A}}{\partial{z}} = -\frac{i}{2}\alpha{A} + \frac{1}{2}\beta_2\frac{\partial^2A}{\partial{T^2}} – \gamma|A|^2A \end{equation}

As discussed in the previous subsection, “$\alpha$” is the constant that governs absorption, “$\beta_2$” governs second order dispersion, and $\gamma$ is the constant that governs the nonlinearity. The strength of dispersion vs nonlinearity for pulses propagating in the fiber depends on the initial width $T_0$ and peak power $P_0$ of the incident pulse; for different values of both parameters one effect may dominate over another. Typically, then, length scales are used to compare these effects. These are the dispersion length $L_D$ and the nonlinear length $L_{NL}$. Let’s investigate what these are.

First, we can create a time scale normalized to the initial pulse width $T_0$ through:

\begin{equation} \tau = \frac{T}{T_0} = \frac{t-z/v_g}{T_0} \end{equation}

and also create a normalize amplitude U using its relationship to $A(z,\tau)$ as:

\begin{equation} A(z,\tau) = \sqrt{P_0}e^{-\alpha{z}/2}U(z,\tau) \end{equation}

We can then use all of these equations to create a normalized propagation equation, given as:

\begin{equation} i\frac{\partial{U}}{\partial{z}} = \frac{\pm\beta_2}{2L_D}\frac{\partial^2U}{\partial{\tau^2}}-\frac{e^{-\alpha{z}}}{L_{NL}}|U|^2U \end{equation}

where $L_D$ and $L_{NL}$ are given as:

\begin{equation} L_D = \frac{T_0^2}{|\beta_2|} \end{equation}

\begin{equation} L_{NL} = \frac{1}{\gamma{P_0}} \end{equation}

Using these two lengths, we can compare the relative importance of the nonlinearity vs dispersion for a length of fiber L. With L << $L_D$, $L_{NL}$, neither dispersive nor nonlinear effects play a significant role in pulse propagation. In this regime, the pulse maintains its shape, and the fiber effectively is a mere transporter of pulses through space. The only effect of the fiber would then be to reduce the energy of the pulses due to fiber losses. This regime is useful for communications systems.

When the fiber length is longer or comparable to both the dispersion and nonlinear length, both effects play a significant role in the pulse propagation and must be considered in conjunction with one another. In the normal dispersion regime, GVD and SPM can be used to compress a pulse. In the anomalous dispersion regime, GVD and SPM can act to “balance” each other through the creation of solitons.

%d bloggers like this: