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+---
+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*:
+
+<a href="pheno-break-inst.jpg">
+<img src="pheno-break-inst-small.jpg" style="width:100%">
+</a>
+
+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:
+
+<a href="pheno-break-sgram.jpg">
+<img src="pheno-break-sgram-small.jpg" style="width:80%">
+</a>
+
+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:
+
+<a href="pheno-break.jpg">
+<img src="pheno-break-small.jpg" style="width:100%">
+</a>
+
+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.
+