Optical rogue waves

Optical rogue waves are rare pulses of light analogous to rogue or freak ocean waves. The term optical rogue waves was coined to describe rare pulses of broadband light arising during the process of supercontinuum generation—a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input waveform—in nonlinear optical fiber. In this context, optical rogue waves are characterized by an anomalous surplus in energy at particular wavelengths (e.g., those shifted to the red of the input waveform) or an unexpected peak power. These anomalous events have been shown to follow heavy-tailed statistics, also known as L-shaped statistics, fat-tailed statistics, or extreme-value statistics. These probability distributions are characterized by long tails: large outliers occur rarely, yet much more frequently than expected from Gaussian statistics and intuition. Such distributions also describe the probabilities of freak ocean waves and various phenomena in both the man-made and natural worlds. Despite their infrequency, rare events wield significant influence in many systems. Aside from the statistical similarities, light waves traveling in optical fibers are known to obey the similar mathematics as water waves traveling in the open ocean (the nonlinear Schrödinger equation), supporting the analogy between oceanic rogue waves and their optical counterparts. More generally, research has exposed a number of different analogies between extreme events in optics and hydrodynamic systems. A key practical difference is that most optical experiments can be done with a table-top apparatus, offer a high degree of experimental control, and allow data to be acquired extremely rapidly. Consequently, optical rogue waves are attractive for experimental and theoretical research and have become a highly studied phenomenon. The particulars of the analogy between extreme waves in optics and hydrodynamics may vary depending on the context, but the existence of rare events and extreme statistics in wave-related phenomena are common ground.

History Optical rogue waves were initially reported in 2007 based on experiments investigating the stochastic properties of supercontinuum generation from a train of nearly-identical picosecond input pulses. In the experiments, radiation from a mode-locked laser (megahertz pulse train) was injected into a nonlinear optical fiber and characteristics of the output radiation were measured at the single-shot level for thousands of pulses (events). These measurements revealed that the attributes of individual pulses can be markedly different from those of the ensemble average. Consequently, these attributes are normally averaged out or hidden in time-averaged observations. The initial observations occurred at the University of California, Los Angeles as part of DARPA-funded research aiming to harness supercontinuum for time-stretch A/D conversion and other applications in which stable white light sources are required (e.g., real-time spectroscopy). The study of optical rogue waves ultimately showed that stimulated supercontinuum generation (as described further below) provides a means of becalming such broadband sources. Pulse-resolved spectral information was obtained by extracting wavelengths far from that of the input pulse using a longpass filter and detecting the filtered light with a photodiode and a real-time digital oscilloscope. The radiation can also be spectrally resolved with the time-stretch dispersive Fourier transform (TS-DFT), which produces a wavelength-to-time mapping such that the temporal traces collected for each event correspond to the actual spectral profile over the filtered bandwidth. The TS-DFT has subsequently been used to stretch the complete (unfiltered) output spectra of such broadband pulses, thereby allowing measurement of full pulse-resolved spectra at the megahertz repetition rate of the source (see below).

Pulse-resolved measurements showed that a fraction of the pulses had much more redshifted energy content than the majority of events. In other words, the energy passed by the filter was much larger for a small fraction of the events, and the fraction of events with anomalous energy content in this spectral band could be increased by raising the power of the input pulses. Histograms of this energy content showed heavy-tailed properties. In some scenarios, the vast majority of events had a negligible amount of energy within the filter bandwidth (i.e., below the measurement noise floor), while a small number of events had energies at least 30–40 times the average value, making them very clearly visible. The analogy between these extreme optical events and hydrodynamic rogue waves was initially developed by noting a number of parallels, including the role of solitons, heavy-tailed statistics, dispersion, modulation instability, and frequency downshifting effects. Additionally, forms of the nonlinear Schrödinger equation are used to model both optical pulse propagation in nonlinear fiber and deep water waves, including hydrodynamic rogue waves. Simulations were then conducted with the nonlinear Schrödinger equation in an effort to model the optical findings. For each trial or event, the initial conditions consisted of an input pulse and a minute amount of broadband input noise. The initial conditions (i.e., pulse power and noise level) were chosen so that the spectral broadening was relatively limited in the typical events. Collecting the results from the trials, very similar filtered energy statistics were observed compared with those seen experimentally. The simulations showed that rare events had experienced significantly more spectral broadening than the others because a soliton had been ejected in the former class of events, but not in the vast majority of events. By applying a correlation analysis between the redshifted output energy and the input noise, it was observed that a particular component of the input noise was elevated each time a surplus in the redshifted noise was generated. The critical noise component has specific frequency and timing relative to the pulse envelope—a noise component that efficiently seeds modulation instability and can, therefore, accelerate the onset of soliton fission.

Principles

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