---
title: "Optical wave breaking"
sort_title: "Optical wave breaking"
date: 2021-02-27
categories:
- Physics
- Optics
- Fiber optics
- Nonlinear optics
layout: "concept"
---

In fiber optics, **optical wave breaking** (OWB) is a nonlinear effect
caused by interaction between
[group velocity dispersion](/know/concept/dispersive-broadening/) (GVD) and
[self-phase modulation](/know/concept/self-phase-modulation/) (SPM).
It only happens in the normal dispersion regime ($$\beta_2 > 0$$)
for pulses meeting a certain criterium, as we will see.

SPM creates low frequencies at the front of the pulse, and high ones at the back,
and if $$\beta_2 > 0$$, GVD lets low frequencies travel faster than high ones.
When those effects interact, the pulse gets temporally stretched
in a surprisingly sophisticated way.

To illustrate this, the instantaneous frequency $$\omega_i(z, t) = -\ipdv{\phi}{t}$$
has been plotted below for a theoretical Gaussian input pulse experiencing OWB,
with settings $$T_0 = 100\:\mathrm{fs}$$, $$P_0 = 5\:\mathrm{kW}$$,
$$\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$.

In the left panel, we see the typical S-shape caused by SPM,
and the arrows indicate the direction that GVD is pushing the curve in.
This leads to steepening at the edges, i.e. the S gradually turns into a Z.
Shortly before the slope would become infinite,
small waves start "falling off" the edge of the pulse,
hence the name *wave breaking*:

{% include image.html file="frequency-full.png" width="100%" alt="Instantaneous frequency profile evolution" %}

Several interesting things happen around this moment.
To demonstrate this, spectrograms of the same simulation
have been plotted below, together with pulse profiles
in both the $$t$$-domain and $$\omega$$-domain on an arbitrary linear scale
(click the image to get a better look).

Initially, the spectrum broadens due to SPM in the usual way,
but shortly after OWB, this process is stopped by the appearance
of so-called **sidelobes** in the $$\omega$$-domain on either side of the pulse.
In the meantime, in the time domain,
the pulse steepens at the edges, but flattens at the peak.
After OWB, a train of small waves falls off the edges,
which eventually melt together, leading to a trapezoid shape in the $$t$$-domain.
Dispersive broadening then continues normally:

{% include image.html file="spectrograms-full.png" width="100%" alt="Spectrograms of pulse shape evolution" %}

We call the distance at which the wave breaks $$L_\mathrm{WB}$$,
and would like to analytically predict it.
We do this using the instantaneous frequency $$\omega_i$$,
by estimating when the SPM fluctuations overtake their own base,
as was illustrated earlier.

To get $$\omega_i$$ of a Gaussian pulse experiencing both GVD and SPM,
it is a reasonable approximation, for small $$z$$, to simply add up
the instantaneous frequencies for these separate effects:

$$\begin{aligned}
    \omega_i(z,t)
    &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
    = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
    + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
\end{aligned}$$

Assuming that $$z$$ is small enough such that $$z^2 \approx 0$$, this
expression can be reduced to:

$$\begin{aligned}
    \omega_i(z,t)
    \approx \frac{\beta_2 tz}{T_0^4} \bigg( 1 + 2\frac{\gamma P_0 T_0^2}{\beta_2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
    = \frac{\beta_2 t z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
\end{aligned}$$

Where we have assumed $$\beta_2 > 0$$,
and $$N_\mathrm{sol}$$ is the **soliton number**,
which is defined as:

$$\begin{aligned}
    N_\mathrm{sol}^2 \equiv \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|}
\end{aligned}$$

This quantity is very important in anomalous dispersion,
but even in normal dispesion, it is still a useful measure of the relative strengths of GVD and SPM.
As was illustrated earlier, $$\omega_i$$ overtakes itself at the edges,
so OWB occurs when $$\omega_i$$ oscillates there,
which starts when its $$t$$-derivative,
the **instantaneous chirpyness** $$\xi_i$$,
has *two* real roots for $$t^2$$:

$$\begin{aligned}
    0
    = \xi_i(z,t)
    = \pdv{\omega_i}{t}
    &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
    = \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big)
\end{aligned}$$

Where the function $$f(x)$$ has been defined for convenience. As it turns
out, this equation can be solved analytically using the Lambert $$W$$ function,
leading to the following exact minimum value $$N_\mathrm{min}^2$$ for $$N_\mathrm{sol}^2$$,
such that OWB can only occur when $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$:

$$\begin{aligned}
    \boxed{
        N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12
    }
\end{aligned}$$

If this condition $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$ is not satisfied,
$$\xi_i$$ cannot have two roots for $$t^2$$, meaning $$\omega_i$$ cannot overtake itself.
GVD is unable to keep up with SPM, so OWB will not occur.

Next, consider two points at $$t_1$$ and $$t_2$$ in the pulse,
separated by a small initial interval $$(t_2 - t_1)$$.
The frequency difference between these points due to $$\omega_i$$
will cause them to displace relative to each other
after a short distance $$z$$ by some amount $$\Delta t$$,
estimated by:

$$\begin{aligned}
    \Delta t
    &\approx z \Delta\beta_1
    \qquad
    &&\Delta\beta_1 \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
    \\
    &\approx z \beta_2 \Delta\omega_i
    \qquad
    &&\Delta\omega_i \equiv \omega_i(z,t_2) - \omega_i(z,t_1)
    \\
    &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
    \qquad \quad
    &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1)
\end{aligned}$$

Where $$\beta_1(\omega)$$ is the inverse of the group velocity.
OWB takes place when $$t_2$$ and $$t_1$$ catch up to each other,
which is when $$-\Delta t = (t_2 - t_1)$$.
The distance where this happens first, $$z = L_\mathrm{WB}$$,
must therefore satisfy the following condition
for a particular value of $$t$$:

$$\begin{aligned}
    L_\mathrm{WB} \, \beta_2 \, \xi_i(L_\mathrm{WB}, t) = -1
    \qquad \implies \qquad
    L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 \, f(t^2/T_0^2)}
\end{aligned}$$

The time $$t$$ of OWB must be where $$\omega_i(t)$$ has its steepest slope,
which is at the minimum value of $$\xi_i(t)$$, and by extension $$f(x)$$.
This turns out to be $$f(3/2)$$:

$$\begin{aligned}
    f_\mathrm{min} = f(3/2)
    = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2)
    = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2
\end{aligned}$$

Clearly, $$f_\mathrm{min} \ge 0$$ when $$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$$,
which, when inserted above, leads to an imaginary $$L_\mathrm{WB}$$,
confirming that OWB cannot occur in that case.
Otherwise, if $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$, then:

$$\begin{aligned}
    \boxed{
        L_\mathrm{WB}
        = \frac{T_0^2}{\beta_2 \, \sqrt{- f_\mathrm{min}}}
        = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
    }
\end{aligned}$$

This prediction for $$L_\mathrm{WB}$$ appears to agree well
with the OWB observed in the simulation:

{% include image.html file="simulation-full.png" width="100%" alt="Optical wave breaking simulation results" %}

Because all spectral broadening up to $$L_\mathrm{WB}$$ is caused by SPM,
whose frequency behaviour is known, it is in fact possible to draw
some analytical conclusions about the achieved bandwidth when OWB sets in.
Filling $$L_\mathrm{WB}$$ in into $$\omega_\mathrm{SPM}$$ gives:

$$\begin{aligned}
    \omega_{\mathrm{SPM}}(L_\mathrm{WB},t)
    = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
\end{aligned}$$

Assuming that $$N_\mathrm{sol}^2$$ is large in the denominator, this can
be approximately reduced to:

$$\begin{aligned}
    \omega_\mathrm{SPM}(L_\mathrm{WB}, t)
    \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
    = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
\end{aligned}$$

The expression $$x \exp(-x^2)$$ has its global extrema
$$\pm 1 / \sqrt{2 e}$$ at $$x^2 = 1/2$$. The maximum SPM frequency shift
achieved at $$L_\mathrm{WB}$$ is therefore given by:

$$\begin{aligned}
    \omega_\mathrm{max} = \sqrt{\frac{2 \gamma P_0}{e \beta_2}}
\end{aligned}$$

Interestingly, this expression does not contain $$T_0$$ at all,
so the achieved spectrum when SPM is halted by OWB
is independent of the pulse width,
for sufficiently large $$N_\mathrm{sol}$$.


## References
1.  D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro,
    [Wave breaking in nonlinear-optical fibers](https://doi.org/10.1364/JOSAB.9.001358),
    1992, Optical Society of America.
2.  A.M.  Heidt,  A.  Hartung,  H.  Bartelt,
    [Generation of ultrashort and coherent supercontinuum light pulses in all-normal dispersion fibers](https://doi.org/10.1007/978-1-4939-3326-6_6),
    2016, Springer Media.