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

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 \(\omega_i(z, t) = -\pdv*{\phi}{t}\) 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 \(N_\mathrm{sol}\).

- 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.

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