Categories: Fiber optics, Nonlinear optics, Optics, Physics.


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=iAzβ222At2+γ(1+iω0t)A2A\begin{aligned} 0 = i\pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma \Big(1 + \frac{i}{\omega_0} \pdv{}{t} \Big) |A|^2 A \end{aligned}

Where ω0\omega_0 is the angular frequency of the pump. We will use the following ansatz, consisting of an arbitrary power profile PP with a phase ϕ\phi:

A(z,t)=P(z,t)exp ⁣(iϕ(z,t))\begin{aligned} A(z,t) = \sqrt{P(z,t)} \, \exp\!\big(i \phi(z,t)\big) \end{aligned}

For a long pulse travelling over a short distance, it is reasonable to neglect dispersion (β2=0\beta_2 = 0). Inserting the ansatz then gives the following, where ε=γ/ω0\varepsilon = \gamma / \omega_0:

0=i12PzPPϕz+γPP+iε32PtPεPPϕt\begin{aligned} 0 &= i \frac{1}{2} \frac{P_z}{\sqrt{P}} - \sqrt{P} \phi_z + \gamma P \sqrt{P} + i \varepsilon \frac{3}{2} P_t \sqrt{P} - \varepsilon P \sqrt{P} \phi_t \end{aligned}

This results in two equations, respectively corresponding to the real and imaginary parts:

0=ϕzεPϕt+γP0=Pz+ε3PtP\begin{aligned} 0 &= - \phi_z - \varepsilon P \phi_t + \gamma P \\ 0 &= P_z + \varepsilon 3 P_t P \end{aligned}

The phase ϕ\phi is not so interesting, so we focus on the latter equation for PP. 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(t3εzP)\begin{aligned} P(z,t) = f(t - 3 \varepsilon z P) \end{aligned}

Where ff is the initial power profile: f(t)=P(0,t)f(t) = P(0,t). The derivatives PtP_t and PzP_z are given by:

Pt=(13εzPt)f    Pt=f1+3εzfPz=(3εP3εzPz)f    Pz=3εPf1+3εzf\begin{aligned} P_t &= (1 - 3 \varepsilon z P_t) \: f' \qquad \quad \implies \quad P_t = \frac{f'}{1 + 3 \varepsilon z f'} \\ P_z &= (-3 \varepsilon P - 3 \varepsilon z P_z) \: f' \quad \implies \quad P_z = \frac{- 3 \varepsilon P f'}{1 + 3 \varepsilon z f'} \end{aligned}

These derivatives both go to infinity when their denominator is zero, which, since ε\varepsilon is positive, will happen earliest where ff' has its most negative value, called fminf_\mathrm{min}', which is located on the trailing edge of the pulse. At the propagation distance zz where this occurs, LshockL_\mathrm{shock}, the pulse will “tip over”, creating a discontinuous shock:

0=1+3εzfmin    Lshock13εfmin\begin{aligned} 0 = 1 + 3 \varepsilon z f_\mathrm{min}' \qquad \implies \qquad \boxed{ L_\mathrm{shock} \equiv -\frac{1}{3 \varepsilon f_\mathrm{min}'} } \end{aligned}

In practice, however, this will never actually happen, because by the time LshockL_\mathrm{shock} 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=25fsT_0 = 25\:\mathrm{fs}, P0=3kWP_0 = 3\:\mathrm{kW}, β2=0\beta_2 = 0 and γ=0.1/W/m\gamma = 0.1/\mathrm{W}/\mathrm{m}:

f(t)=P(0,t)=P0exp ⁣( ⁣ ⁣t2T02)\begin{aligned} f(t) = P(0,t) = P_0 \exp\!\Big(\! -\!\frac{t^2}{T_0^2} \Big) \end{aligned}

Its steepest points are found to be at 2t2=T022 t^2 = T_0^2, so fminf_\mathrm{min}' and LshockL_\mathrm{shock} are given by:

fmin=2P0T0exp ⁣( ⁣ ⁣12)    Lshock=T032εP0exp ⁣(12)\begin{aligned} f_\mathrm{min}' = - \frac{\sqrt{2} P_0}{T_0} \exp\!\Big(\!-\!\frac{1}{2}\Big) \quad \implies \quad L_\mathrm{shock} = \frac{T_0}{3 \sqrt{2} \varepsilon P_0} \exp\!\Big(\frac{1}{2}\Big) \end{aligned}

This example Gaussian pulse therefore has a theoretical Lshock=0.847mL_\mathrm{shock} = 0.847\,\mathrm{m}, which turns out to be accurate, although the simulation breaks down due to insufficient resolution:

Self-steepening simulation results

Unfortunately, self-steepening cannot be simulated perfectly: as the pulse approaches LshockL_\mathrm{shock}, 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.


  1. B.R. Suydam, Self-steepening of optical pulses, 2006, Springer.