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

# Optical wave breaking

In fiber optics, **optical wave breaking** (OWB) is a nonlinear effect
caused by interaction between
group velocity dispersion (GVD) and
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*:

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 want to predict it analytically. 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:

This quantity is very important in anomalous dispersion,
but even in normal dispersion, 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 $t^2$:

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 $N_\mathrm{sol}^2 > N_\mathrm{min}^2$:

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. For a certain choice of $t_1$ and $t_2$, OWB occurs when they catch up to each other, which is when $-\Delta t = (t_2 - t_1)$. The distance $L_\mathrm{WB}$ at which this happens first must satisfy the following condition for some 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:

Because all spectral broadening up to $L_\mathrm{WB}$ is caused by SPM, whose $\omega$-domain 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

- D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro, Wave breaking in nonlinear-optical fibers, 1992, Optical Society of America.
- A.M. Heidt, A. Hartung, H. Bartelt, Generation of ultrashort and coherent supercontinuum light pulses in all-normal dispersion fibers, 2016, Springer Media.