---
title: "Optical wave breaking"
firstLetter: "O"
publishDate: 2021-02-27
categories:
- Physics
- Optics
- Fiber optics
- Nonlinear dynamics
date: 2021-02-27T10:09:46+01:00
draft: false
markup: pandoc
---
# Optical wave breaking
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 $\omega_i(z, t) = -\pdv*{\phi}{t}$
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*:
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:
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:
$$\begin{aligned}
\omega_i(z,t)
&\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
\\
% &= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
% + \frac{2\gamma P_0 z}{T_0^2} t \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
% \\
&= \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
+ 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
\end{aligned}$$
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:
$$\begin{aligned}
N_\mathrm{sol}^2 = \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|}
\end{aligned}$$
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 only occurs when $\omega_i$ is not monotonic,
which is 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 $N_\mathrm{sol}^2 > N_\mathrm{min}^2$:
$$\begin{aligned}
N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12
\end{aligned}$$
Now, consider two times $t_1$ 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:
$$\begin{aligned}
\Delta t
&\approx z \Delta\beta_1
\qquad
&&\Delta\beta_1 = \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
\\
&\approx z \beta_2 \Delta\omega_i
\qquad
&&\Delta\omega_i = \omega_i(z,t_2) - \omega_i(z,t_1)
\\
&\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
\qquad \quad
&&\Delta\xi_i = \xi_i(z,t_2) - \xi_i(z,t_1)
\end{aligned}$$
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, $z = L_\mathrm{WB}$,
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 in the
condition above, confirms that OWB cannot occur in that case. Otherwise,
if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then:
$$\begin{aligned}
L_\mathrm{WB}
= - \frac{T_0^2}{\beta_2 \, \sqrt{f_\mathrm{min}}}
= \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
= \frac{L_D}{\sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}}
\end{aligned}$$
This prediction for $L_\mathrm{WB}$ appears to agree well
with the OWB observed in the simulation:
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:
$$\begin{aligned}
\omega_\mathrm{SPM}(L_\mathrm{WB}, t)
% = \frac{2 \gamma P_0 t \exp(-t^2 / T_0^2)}{\beta_2 \sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
\approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
= 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
\end{aligned}$$
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.
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:
$$\begin{aligned}
\omega_i(z,t)
&\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
\\
% &= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
% + \frac{2\gamma P_0 z}{T_0^2} t \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
% \\
&= \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
+ 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
\end{aligned}$$
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:
$$\begin{aligned}
N_\mathrm{sol}^2 = \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|}
\end{aligned}$$
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 only occurs when $\omega_i$ is not monotonic,
which is 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 $N_\mathrm{sol}^2 > N_\mathrm{min}^2$:
$$\begin{aligned}
N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12
\end{aligned}$$
Now, consider two times $t_1$ 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:
$$\begin{aligned}
\Delta t
&\approx z \Delta\beta_1
\qquad
&&\Delta\beta_1 = \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
\\
&\approx z \beta_2 \Delta\omega_i
\qquad
&&\Delta\omega_i = \omega_i(z,t_2) - \omega_i(z,t_1)
\\
&\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
\qquad \quad
&&\Delta\xi_i = \xi_i(z,t_2) - \xi_i(z,t_1)
\end{aligned}$$
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, $z = L_\mathrm{WB}$,
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 in the
condition above, confirms that OWB cannot occur in that case. Otherwise,
if $N_\mathrm{sol}^2 > N_\mathrm{min}^2$, then:
$$\begin{aligned}
L_\mathrm{WB}
= - \frac{T_0^2}{\beta_2 \, \sqrt{f_\mathrm{min}}}
= \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
= \frac{L_D}{\sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}}
\end{aligned}$$
This prediction for $L_\mathrm{WB}$ appears to agree well
with the OWB observed in the simulation:
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:
$$\begin{aligned}
\omega_\mathrm{SPM}(L_\mathrm{WB}, t)
% = \frac{2 \gamma P_0 t \exp(-t^2 / T_0^2)}{\beta_2 \sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
\approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
= 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\frac{t^2}{T_0^2}\Big)
\end{aligned}$$
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.