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

Optical wave breaking

In fiber optics, optical wave breaking (OWB) is an effect that can occur in light pulse envelopes $A(z, t)$ governed by the nonlinear Schrödinger equation:

$$\begin{aligned} 0 &= i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma_0 |A|^2 A \end{aligned}$$

OWB is caused by an interaction between the group velocity dispersion (GVD) caused by the $\beta_2$-term, and the self-phase modulation (SPM) caused by the $\gamma_0$ term. It only happens in the normal dispersion regime ($\beta_2 > 0$) for pulses meeting certain criteria, as we shall see.

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

To illustrate the resulting dynamics, the simulated power $|A|^2$ of a Gaussian pulse with settings $T_0 = 100\:\mathrm{fs}$, $P_0 = 5\:\mathrm{kW}$, $\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0.1/\mathrm{W}/\mathrm{m}$ is plotted below as a function of $z$, with the time domain on the left and the frequency domain on the right:

$$\begin{aligned} A(0, t) &= \sqrt{P_0} \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \end{aligned}$$

OWB occurs at a distance called $L_\mathrm{WB}$, and until that point things look relatively normal, with SPM causing spectral broadening and GVD causing subtle internal deformation in the time domain. After $L_\mathrm{WB}$, the pulse suddenly explodes due to GVD, and complicated so-called sidelobes appear in the frequency domain, which seem to block any further SPM. To investigate, we plot a series of spectrograms of the same simulation:

At first, we see the appearance of SPM’s typical “S” shape, which quickly starts turning into a “Z” due to GVD. When the transition to “Z” is complete, there are many overlapping frequencies at the edges of the pulse. This causes a complicated interaction that generates the sidelobes, and causes a train of small waves to “fall off” the near-vertical pulse edges in the time domain, hence the name wave breaking. Eventually, those small waves melt together, leaving behind a curious trapezoid shape that gets stretched by GVD as usual.

We would like to theoretically predict the distance $L_\mathrm{WB}$ at which the wave breaks. First we show the general principle, and then we apply it to a couple of example pulses.

General method

We make the following ansatz for the complex envelope $A(z, t)$, without loss of generality:

$$\begin{aligned} A(z, t) = \psi(z, t) \exp\!\big(i \phi(z, t)\big) \end{aligned}$$

Inserting this into the NLS equation and dividing out $e^{i \phi}$ yields:

$$\begin{aligned} 0 &= i \psi_z - \psi \phi_z - \frac{\beta_2}{2} (\psi_{tt} + 2 i \psi_t \phi_t + i \psi \phi_{tt} - \psi \phi_t^2) + \gamma_0 \psi^3 \end{aligned}$$

Since $\psi$ and $\phi$ are real by definition, we can split this into its real and imaginary parts:

$$\begin{aligned} 0 &= \psi_z - \frac{\beta_2}{2} (2 \psi_t \phi_t + \psi \phi_{tt}) \\ 0 &= - \psi \phi_z - \frac{\beta_2}{2} (\psi_{tt} - \psi \phi_t^2) + \gamma_0 \psi^3 \end{aligned}$$

For our purposes, the second equation is enough. We divide it by $\psi$ to get an expression for $\phi_z$:

$$\begin{aligned} \phi_z &= - \frac{\beta_2}{2} \frac{\psi_{tt}}{\psi} + \frac{\beta_2}{2} \Omega_i^2 + \gamma_0 \psi^2 \end{aligned}$$

Where $\Omega_i \equiv -\phi_t$ is the instantaneous frequency, also called the frequency-chirp variation, which describes the dominant frequency component at a given point $(z, t)$; basically the center of the spectrograms shown earlier. For small $z$, this gives us a linear approximation of $\phi$:

$$\begin{aligned} \phi(z, t) &\approx \bigg( \!-\! \frac{\beta_2}{2} \frac{\psi_{tt}}{\psi} + \frac{\beta_2}{2} \Omega_i^2 + \gamma_0 \psi^2 \bigg)\bigg|_{z = 0} z + \phi(0, t) \end{aligned}$$

And therefore $\Omega_i$ is as follows, assuming no initial chirp variation $\Omega_i(0, t) = 0$:

$$\begin{aligned} \boxed{ \Omega_i(z, t) = -\pdv{\phi}{t} \approx \bigg( \frac{\beta_2}{2} \frac{\psi_{ttt}}{\psi} - \frac{\beta_2}{2} \frac{\psi_{tt} \psi_t}{\psi^2} - 2 \gamma_0 \psi \psi_t \bigg) \bigg|_{z = 0} z } \end{aligned}$$

Once we have $\Omega_i$ for a known input pulse, we can check whether OWB is even possible under the given circumstances: $\Omega_i$ must be non-monotonic, i.e. $\ipdv{\Omega_i}{t} = 0$ must have a solution. In other words, there must be a sufficiently prominent “bump” in $\Omega_i$ that gets pulled away by GVD faster than its surroundings, until those more-off-center frequencies overtake less-off-center ones and lead to the overlap that generates the sidelobes and other OWB phenomena.

Let us assume that OWB will occur. Consider two parts of the pulse, located $t_1$ and $t_2$ for $z = 0$, so separated by a small initial interval $\Delta{t} \equiv t_2 - t_1$. Due to $\Omega_i$ there is a frequency difference between these points, causing $\Delta{t}$ to change by an amount $\tau$ after the pulse has propagated a short distance $z$, estimated as follows:

$$\begin{alignedat}{2} \tau &\approx z \Delta\beta_1 \approx z \pdv{\beta_1}{\Omega} \Delta{\Omega_i} = z \beta_2 \Delta\Omega_i \approx z \beta_2 \pdv{\Omega_i}{t} \Delta{t} \end{alignedat}$$

Where $\Delta\Omega_i \equiv \Omega_i(z,t_2) - \Omega_i(z,t_1)$, and $\Delta{\beta_1}$ is the difference in inverse group velocity $\beta_1(\Omega)$ between $t_2$ and $t_1$, specifically $\Delta\beta_1 \equiv \beta_1(\Omega_i(z,t_2)) - \beta_1(\Omega_i(z,t_1))$. OWB takes place when $t_1$ and $t_2$ catch up to each other, which is when $\tau = -\Delta{t}$. In that case, we have:

$$\begin{aligned} z = - \frac{1}{\beta_2 \displaystyle\pdv{\Omega_i}{t}} \end{aligned}$$

Assuming $\beta_2 > 0$, this implies that the wave starts breaking first at the $t$-values where $\Omega_i$ has its most negative slope (note that for a symmetric input pulse, $\ipdv{\Omega_i}{t}$ is also symmetric, so OWB will occur simultaneous on both sides). We can therefore write an equation for $L_\mathrm{WB}$ like so, valid for any input pulse shape for which we know $\Omega_i(z, t)$:

$$\begin{aligned} \boxed{ L_\mathrm{WB} = - \frac{1}{\beta_2 \: \mathrm{min}_t\bigg\{ \displaystyle\pdv{\Omega_i}{t} \Big|_{z = L_\mathrm{WB}} \bigg\}} } \end{aligned}$$

Let us apply this method to a few specific examples: a Gaussian input pulse, and a soliton-shaped one (keeping in mind that true bright solitons do not exist for $\beta_2 > 0$).

Gaussian pulse

For a Guassian input, the amplitude $\psi$ is as follows in our ansatz $A = \psi e^{i \phi}$:

$$\begin{aligned} \psi(0, t) &= \sqrt{P_0} \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \end{aligned}$$

For reference, its relevant $t$-derivatives are given by:

$$\begin{aligned} \psi_t(0, t) &= - \frac{\sqrt{P_0}}{T_0^2} t \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \\ \psi_{tt}(0, t) &= \frac{\sqrt{P_0}}{T_0^2} \bigg( \frac{t^2}{T_0^2} - 1 \bigg) \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \\ \psi_{ttt}(0, t) &= \frac{\sqrt{P_0}}{T_0^4} \bigg( 3 - \frac{t^2}{T_0^2} \bigg) t \exp\!\bigg( \!-\!\frac{t^2}{2 T_0^2} \bigg) \end{aligned}$$

Substituting these into our general linear approximation of $\Omega_i$ leads us to:

$$\begin{aligned} \Omega_i(z, t) &= z \frac{\beta_2 t}{T_0^4} \bigg( 1 + 2 \frac{\gamma_0 P_0 T_0^2}{\beta_2} \exp\!\Big( \!-\!\frac{t^2}{T_0^2} \Big) \bigg) \end{aligned}$$

Since we are in the normal dispersion regime, $\beta_2 > 0$, so we can recognize the soliton number $N_\mathrm{sol}$ here, which is a useful measure of the relative strengths of GVD and SPM:

$$\begin{aligned} N_\mathrm{sol}^2 \equiv \frac{\gamma_0 P_0 T_0^2}{|\beta_2|} = \frac{L_D}{L_N} \end{aligned}$$

We thus have the following expression for $\Omega_i$, sketched below for several values of $N_\mathrm{sol}$:

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

At a certain value of $N_\mathrm{sol}$, which we call $N_\mathrm{min}$, we see that $\Omega_i$ transitions from having no extrema, to having a local minimum and maximum with respect to $t^2$. Those “bumps” get pulled outward by GVD as indicated by the arrows, steepening the outer edges until the slope becomes infinite, at which point OWB occurs. However, for $N_\mathrm{sol} < N_\mathrm{min}$, the bumps are not prominent enough: the peaks cannot catch up to the outer edges, so OWB can never happen.

We would like to find $N_\mathrm{min}$. To do so, we demand that $\Omega_i$ has local extrema where the derivative $\ipdv{\Omega_i}{t}$ vanishes, as illustrated below. Abbreviating $f(x) \equiv (1 - 2x) e^{-x}$:

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

Here we see that as $N_\mathrm{sol}$ increases, it pulls down the minimum of $f(x)$ until it hits the horizontal axis when $N_\mathrm{sol} = N_\mathrm{min}$. We should therefore find the location $x_\mathrm{min}$ of this minimum:

$$\begin{aligned} 0 = f'(x) = (2 x - 3) e^{-x} \qquad\implies\qquad x_\mathrm{min} = \frac{3}{2} \end{aligned}$$

So the corresponding minimum value of $f(x)$ is given by:

$$\begin{aligned} f_\mathrm{min} = f(x_\mathrm{min}) = -2 e^{-3/2} \end{aligned}$$

Inserting this into our demand that $\ipdv{\Omega_i}{t} = 0$ yields a simple expression for $N_\mathrm{min}$:

$$\begin{aligned} 0 = 1 + 2 N_\mathrm{min}^2 \: f_\mathrm{min} \qquad\implies\qquad \boxed{ N_\mathrm{min}^2 = \frac{e^{3/2}}{4} \approx 1.12 } \end{aligned}$$

If $N_\mathrm{sol}^2 < N_\mathrm{min}^2$, then our demand cannot be satisfied: $\Omega_i$ cannot overtake itself, GVD is unable to keep up with SPM, and OWB cannot occur. From now on, we assume $N_\mathrm{sol}^2 > N_\mathrm{min}^2$.

We now have everything we need to calculate the OWB distance $L_\mathrm{WB}$ using its general recipe. Inserting $\ipdv{\Omega_i}{t}$, whose minimum we already know, we get:

$$\begin{aligned} L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 (1 + 2 N_\mathrm{sol}^2 f_\mathrm{min})} = \frac{T_0^4}{\beta_2^2 (N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1)} \end{aligned}$$

Leading to the following prediction for $L_\mathrm{WB}$, which appears to agree well with the OWB observed in the simulation shown earlier. Note that if $N_\mathrm{sol} < N_\mathrm{min}$ then $L_\mathrm{WB}$ is imaginary, confirming that OWB is not possible in that situation:

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

Soliton-shaped pulse

Although solitons do not exist in the normal dispersion regime, we can still create pulses with the same shape, given by:

$$\begin{aligned} \psi(0, t) &= \sqrt{P_0} \sech\!\Big( \frac{t}{T_0} \Big) \end{aligned}$$

For reference, we also calculate its relevant $t$-derivatives:

$$\begin{aligned} \psi_t(0, t) &= - \frac{\sqrt{P_0}}{T_0} \tanh\!\Big( \frac{t}{T_0} \Big) \sech\!\Big( \frac{t}{T_0} \Big) \\ \psi_{tt}(0, t) &= \frac{\sqrt{P_0}}{T_0^2} \bigg( \tanh^2\!\Big( \frac{t}{T_0} \Big) - \sech^2\!\Big( \frac{t}{T_0} \Big) \bigg) \sech\!\Big( \frac{t}{T_0} \Big) \\ \psi_{ttt}(0, t) &= \frac{\sqrt{P_0}}{T_0^3} \bigg( 5 \sech^2\!\Big( \frac{t}{T_0} \Big) - \tanh^2\!\Big( \frac{t}{T_0} \Big) \bigg) \tanh\!\Big( \frac{t}{T_0} \Big) \sech\!\Big( \frac{t}{T_0} \Big) \end{aligned}$$

Substituting these into our general linear approximation of $\Omega_i$, and once again recognizing the soliton number $N_\mathrm{sol}$, leads us to the following function, sketched below:

$$\begin{aligned} \Omega_i(z, t) &= z \frac{2 \beta_2}{T_0^3} \big( 1 + N_\mathrm{sol}^2 \big) \sech^2\!\Big( \frac{t}{T_0} \Big) \tanh\!\Big( \frac{t}{T_0} \Big) \end{aligned}$$

Curiously, this $\Omega_i$ is non-monotonic for all $N_\mathrm{sol}$, so OWB occurs even in the linear limit $N_\mathrm{sol} \to 0$. This suggests that OWB is not an inherently nonlinear effect, instead happening as long as there are bumps in $\Omega_i$, regardless of their origin (SPM or simply the pulse shape).

We do not care where those local extrema are, only that they exist, so we move on immediately to finding where $\Omega_i$ has its most negative slope, which is at some (but not all) solutions of:

$$\begin{aligned} 0 &= \pdvn{2}{\Omega_i}{t} \\ &= z \frac{8 \beta_2}{T_0^5} \big( 1 + N_\mathrm{sol}^2 \big) \bigg( \tanh^2\!\Big( \frac{t}{T_0} \Big) - 2 \sech^2\!\Big( \frac{t}{T_0} \Big) \bigg) \sech^2\!\Big( \frac{t}{T_0} \Big) \tanh\!\Big( \frac{t}{T_0} \Big) \end{aligned}$$

One solution is clearly $t = 0$ because $\tanh(0) = 0$, but from the plot we can see that $\Omega_i$’s slope is positive there, so we must continue our search. The next candidate is:

$$\begin{aligned} 0 &= \tanh^2(x) - 2 \sech^2(x) \\ &= 3 \tanh^2(x) - 2 \end{aligned}$$

Where we have used the standard identity $\sech^2(x) + \tanh^2(x) = 1$. Isolating for $x$ and writing out $\tanh^{-1}(x)$ as a logarithm yields:

$$\begin{aligned} x &= \tanh^{-1}\!\bigg( \!\pm\!\sqrt{\frac{2}{3}}\bigg) \\ &= \frac{1}{2} \ln\!\bigg( \frac{1 \pm \sqrt{2/3}}{1 \mp \sqrt{2/3}} \bigg) \\ &= \frac{1}{2} \ln\!\bigg( \frac{\sqrt{3} \pm \sqrt{2}}{\sqrt{3} \mp \sqrt{2}} \bigg) \\ &= \frac{1}{2} \ln\!\bigg( \frac{(\sqrt{3} \pm \sqrt{2})^2}{(\sqrt{3} \mp \sqrt{2}) (\sqrt{3} \pm \sqrt{2})} \bigg) \\ &= \frac{1}{2} \ln(5 \pm 2 \sqrt{6}) \end{aligned}$$

Note that $\ln(5 \!+\! 2 \sqrt{6}) = - \ln(5 \!-\! 2 \sqrt{6}) \equiv 2 x_0$. The values of $\sech$ and $\tanh$ are given by:

$$\begin{aligned} \sech(\pm x_0) = \frac{1}{\sqrt{3}} \qquad\qquad \tanh(\pm x_0) = \pm \sqrt{\frac{2}{3}} \end{aligned}$$

The minimum value of the slope $\ipdv{\Omega_i}{t}$ is therefore as follows:

$$\begin{aligned} \mathrm{min}_t\bigg\{ \displaystyle\pdv{\Omega_i}{t} \bigg\} &= z \frac{2 \beta_2}{T_0^4} (1 + N_\mathrm{sol}^2) \bigg( \sech^2\!\Big( \frac{t}{T_0} \Big) - 2 \tanh^2\!\Big( \frac{t}{T_0} \Big) \bigg) \sech^2\!\Big( \frac{t}{T_0} \Big) \bigg|_{t = x_0 T_0} \\ &= - z \frac{2 \beta_2}{3 T_0^4} \big( 1 + N_\mathrm{sol}^2 \big) \end{aligned}$$

Inserting this into $L_\mathrm{WB}$’s general equation, we find that OWB occurs at a distance with a similar $T_0^2 / \beta_2$-dependence as for the Gaussian pulse, confirming that OWB is mostly linear:

$$\begin{aligned} \boxed{ L_\mathrm{WB} = \frac{\sqrt{3} T_0^2}{\beta_2 \sqrt{2 + 2 N_\mathrm{sol}^2}} } \end{aligned}$$

References

1. D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro, Wave breaking in nonlinear-optical fibers, 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, 2016, Springer Media.

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