Observation of Optical Undular Bores in Multiple Four-Wave Mixing

We demonstrate that wave-breaking dramatically affects the dynamics of nonlinear frequency conversion processes that operate in the regime of high efficiency (strong multiple four-wave mixing). In particular, by exploiting an all-optical-fiber platform, we show that input modulations propagating in standard telecom fibers in the regime of weak normal dispersion lead to the formation of undular bores (dispersive shock waves) that mimic the typical behavior of dispersive hydrodynamics exhibited, e.g., by gravity waves and tidal bores. Thanks to the nonpulsed nature of the beat signal employed in our experiment, we are able to clearly observe how the periodic nature of the input modulation forces adjacent undular bores to collide elastically.

Popular Summary

“Morning glory clouds,” a rare meteorological phenomenon, or “the Mascaret,” large tidal waves traveling upstream in river estuaries, are spectacular examples in nature of nonlinear wave phenomena called undular bores or dispersive shock waves: a vertical wave front followed by spontaneous emission of a fast-expanding wave train. The basic ingredient underlying such phenomena is the nonlinearity of the wave-forming medium that induces an initial wave steepening, combined with a weak dispersion. Such conditions can also be met in the optical context, for example, for an intense pulse propagating along an optical fiber with normal color dispersion or for a laser beam propagating in a medium with a nonlinear response of a defocusing type. In this paper, we report the first experimental demonstration of multiple optical undular bores generated in a frequency-conversion process occurring in standard telecom optical fibers.

In our experiment, the light-carrying medium is a standard optically nonlinear telecom fiber. We exploit multiple four-wave mixing (FWM)—a cascade (comb) of new equispaced frequency lines generated by mixing two or three injected frequency lines produced by deep amplitude modulation of a continuous-wave (nonpulsed) laser source. We carefully design the experiment in such a way that dispersive effects are much weaker than nonlinear ones. In this regime, the overall cascaded optical field behaves like a fluid undergoing the formation of an array of twin breaking (shock) points from which multiple temporal undular bores emerge. A complex temporal pattern is observed to emerge because of unavoidable collisions of adjacent undular bores.

Multiple four-wave mixing is currently being exploited in photonic applications such as signal regeneration, noiseless amplification, or frequency-comb generation. Our all-fiber experiment not only sheds new light on this nonlinear phenomenon but also demonstrates its potential as a platform for creating and investigating, in laboratory settings, fascinating wave phenomena whose natural counterparts are much more challenging to characterize and repeat.

Figure 1

(a) Sketch of the input in frequency domain showing the triple-frequency (solid blue arrows) and dual-frequency (dashed brown arrows) cases, respectively. (b) Generic mechanism of wave breaking via a gradient catastrophe (length of the arrows is proportional to the local velocity at the point of application over the input; the dashed vertical line is the classical shock wave). (c) Initial power profile $|u_0(t){|}^2$ and instantaneous incremental velocity $\delta v(t)$ resulting from the nonlinear self-phase modulation for triple-frequency input [$\eta =0.8$, $\alpha =0$, $m=1$ in Eq. (1)].

Figure 2

Breaking mechanism for an input modulated carrier (three-wave input, $\eta =0.8$ and $\alpha =0$). (a) Normalized power $\rho =|u{|}^2$ and velocity $v$ at wave-breaking distance $z=z_b\simeq 0.37$ from the reduced hydrodynamical model [Eqs. (3)]; the dashed curves stand for the input. (b) Surface plot of power $|u(z,t){|}^2$ evolving according to the NLS equation [Eq. (2)] for $ϵ=0.04$. (c) Normalized breaking distance $z_b$ vs pump power fraction $\eta$.

Figure 3

Experimental setup. ECL: external cavity laser; IM: intensity modulators; PM: phase modulator; PPG: pulse pattern generator; OSO: optical sampling oscilloscope; OSA: optical spectrum analyzer; NZDSF: nonzero dispersion shifted fiber. The insets show traces of the input and output, as seen on the OSO.

Figure 4

Measured UB dynamics for symmetric dual-pumped FWM (cosine type of input, $\eta =\alpha =0$). (a) 3D surface of output power vs time and input power. (b),(c) Color-level plot from the output power profiles at different input powers from measured data (b), compared with numerical simulations of the dimensional NLS equation [(c), color level plot of $|u{|}^2$]. (d),(e) Temporal OSO traces at power levels: (d) $P=25.6\mathrm{dBm}$, close to breaking; (e) $P=34.6\mathrm{dBm}$, beginning of UB collision (the input is reported as a solid blue line). (f) Output spectra in the low-power ($P=13\mathrm{dBm}$, preshock) and high-power ($P=34.6\mathrm{dBm}$, postshock) regimes.

Figure 5

Measured UB dynamics for the pump-signal input configuration (asymmetric dual-pumped FWM, $\eta =0$, $\alpha =0.3$). (a) 3D surface of measured output power vs time and input power. (b),(d) Color-level plot of the output power profile at different input powers from measured data (b) and numerical simulations of the dimensional NLS equation (d). (c) Temporal OSO trace at $P=30.6\mathrm{dBm}$, showing the onset of asymmetric oscillations (the solid blue line stands for the input).

Figure 6

As in Fig. 5, with FWM produced by the three-wave input with $\eta =0.7$ and $\alpha =0$. The temporal OSO trace in (c) corresponds to a power $P=32\mathrm{dBm}$.