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:
0=i∂z∂A−2β2∂t2∂2A+γ(1+ω0i∂t∂)∣A∣2A
Where ω0 is the angular frequency of the pump.
We will use the following ansatz,
consisting of an arbitrary power profile P with a phase ϕ:
A(z,t)=P(z,t)exp(iϕ(z,t))
For a long pulse travelling over a short distance, it is reasonable to
neglect dispersion (β2=0).
Inserting the ansatz then gives the following, where ε=γ/ω0:
0=i21PPz−Pϕz+γPP+iε23PtP−εPPϕt
This results in two equations, respectively corresponding to the real
and imaginary parts:
00=−ϕz−εPϕt+γP=Pz+ε3PtP
The phase ϕ is not so interesting, so we focus on the latter equation for P.
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:
P(z,t)=f(t−3εzP)
Where f is the initial power profile: f(t)=P(0,t).
The derivatives Pt and Pz are given by:
These derivatives both go to infinity when their denominator is zero,
which, since ε is positive, will happen earliest where f′
has its most negative value, called fmin′,
which is located on the trailing edge of the pulse.
At the propagation distance z where this occurs, Lshock,
the pulse will “tip over”, creating a discontinuous shock:
0=1+3εzfmin′⟹Lshock≡−3εfmin′1
In practice, however, this will never actually happen, because by the time
Lshock 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 T0=25fs, P0=3kW,
β2=0 and γ=0.1/W/m:
f(t)=P(0,t)=P0exp(−T02t2)
Its steepest points are found to be at 2t2=T02, so
fmin′ and Lshock are given by:
This example Gaussian pulse therefore has a theoretical
Lshock=0.847m,
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 Lshock, 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.