For a laser pulse travelling through an optical fiber, its intensity is highest at its peak, so the Kerr effect will be strongest there. This means that the peak travels slightly slower than the rest of the pulse, leading to self-steepening of its trailing edge. Mathematically, this is described by adding a new term to the nonlinear Schrödinger equation:
Where is the angular frequency of the pump. We will use the following ansatz, consisting of an arbitrary power profile with a phase :
For a long pulse travelling over a short distance, it is reasonable to neglect dispersion (). Inserting the ansatz then gives the following, where :
This results in two equations, respectively corresponding to the real and imaginary parts:
The phase is not so interesting, so we focus on the latter equation for . As it turns out, it has a general solution of the form below (you can verify this yourself), which shows that more intense parts of the pulse will lag behind compared to the rest:
Where is the initial power profile: . The derivatives and are given by:
These derivatives both go to infinity when their denominator is zero, which, since is positive, will happen earliest where has its most negative value, called , which is located on the trailing edge of the pulse. At the propagation distance where this occurs, , the pulse will “tip over”, creating a discontinuous shock:
In practice, however, this will never actually happen, because by the time is reached, the pulse spectrum will have become so broad that dispersion can no longer be neglected.
A simulation of self-steepening without dispersion is illustrated below for the following Gaussian initial power distribution, with , , and :
Its steepest points are found to be at , so and are given by:
This example Gaussian pulse therefore has a theoretical , which turns out to be accurate, although the simulation breaks down due to insufficient resolution:
Unfortunately, self-steepening cannot be simulated perfectly: as the pulse approaches , its spectrum broadens to infinite frequencies to represent the singularity in its slope. The simulation thus collapses into chaos when the edge of the frequency window is reached. Nevertheless, the general trends are nicely visible: the trailing slope becomes extremely steep, and the spectrum broadens so much that dispersion cannot be neglected anymore.
- B.R. Suydam, Self-steepening of optical pulses, 2006, Springer.