Categories: Fiber optics, Nonlinear dynamics, 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 $$\omega_i(z, t) = -\pdv*{\phi}{t}$$ 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 \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:

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, 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.