# Self-Steepening

Thus far in our discussion of SPM, we have only been considering the effects of the third order nonlinear effects. However, for pulses shorter than 100 fs, it is necessary to include the effects of higher-order dipsersion and nonlinear effects in our generalized propagation equation in a fiber. One of these important higher order nonlinear effects is known as self-steepening, and results from an intensity dependent group velocity.

1. Relevant Equation
2. Optical Shock Front
3. Effects of GVD on Optical Shocks

RELEVANT EQUATION

If we can neglect the contribution of molecular vibrations (the Raman response) to the susceptibility, we can define our new normalized propagation equation as:

\begin{equation} i\frac{\partial{U}}{\partial{z}} = \frac{\pm\beta_2}{2L_D}\frac{\partial^2U}{\partial\tau^2} + i\frac{\pm\beta_3}{6L_D}\frac{\partial^3U}{\partial\tau^3} – \frac{e^{-\alpha{z}}}{L_{NL}}[|U|^2U + is\frac{\partial}{\partial{\tau}}(|U|^2U)] \end{equation}

The self-steepening effects are considered in the last term with s defined as:

\begin{equation} s = \frac{1}{\omega_0T_0} = \frac{T_{opt}}{2\pi{T_0}} \end{equation}

OPTICAL SHOCK FRONT