Optical rogue waves
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.
[1][2] These probability distributions are characterized by long tails: large outliers occur rarely, yet much more frequently than expected from Gaussian statistics and intuition.
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.
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.
The initial observations occurred at the University of California, Los Angeles as part of DARPA-funded research[14] 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.
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.
Such dispersive characteristics support modulation instability, which amplifies input noise and forms Stokes and anti-Stokes sidebands around the pump wavelength.
Based on such time-averaged measurements, the spectral profile of the supercontinuum generally appears smooth and relatively featureless, whereas, the spectrum of a single pulse may be highly structured in comparison.
Both the pump power and input noise level are influential in the supercontinuum generation process, determining, e.g., the broadening factor and the onset of soliton-fission.
In the case of large pump power, soliton fission often has been compared to onset of boiling in a superheated liquid in that the transition begins rather suddenly and explosively.
Input noise, or any other stimulus, matching the timing of the sensitive portion of the pump envelope and the frequency shift of modulation instability gain experiences the largest amplification.
A properly positioned detection filter can be used to catch anomalous occurrences, such as a rare soliton that has been liberated due to a small surplus in the key input noise component.
Also, in systems displaying heavy-tailed statistical properties, random input conditions often enter through a seemingly insignificant, nontrivial, or otherwise-hidden variable.
Thus, the appearance of extreme statistics is often striking not only because of their counterintuitive probability assignments, but also because they frequently signify a nontrivial or unexpected sensitivity to initial conditions.
Such a soliton has short duration and high peak intensity, and Raman scattering ensures that it is also redshifted relative to the majority of the input radiation.
In summary, rare solitons may be generated at low pump power or input noise levels, and these events can be identified by their redshifted energy.
[40][41][49] Third-order dispersion and Raman scattering play a central role in the generation of large redshifts, and turbulence treats the statistical properties of weakly-coupled waves with randomized relative phases.
[53][54] Yet, the stochastic nature of rogue waves in optics and hydrodynamics is one of their defining features, but remains an open question for these solutions as well as other postulated analytic forms.
Termed optical rogue wave statistics, this behavior was studied in simulations, which supported an explanation based on pump noise transfer by self-phase modulation.
The occurrence of optical rogue waves are an extreme manifestation of this instability and arise due to a sensitivity to a particular component of input noise.
[1] This sensitivity can be exploited to stabilize and increase the generation efficiency of the spectral broadening process by actively seeding the instability with a controlled signal instead of allowing it to begin from noise.
[18] Added input modulations have also been studied for changing the frequency of rare events[2] and optical feedback can be employed to speed up the spectral broadening process.
[61] Complete single-shot spectral profiles of modulation instability and supercontinuum have been mapped into the time domain with the TS-DFT for capture at megahertz repetition rates.
[20][21][63] These experiments have been used to collect large volumes of spectra data very rapidly, permitting detailed statistical analyses of the underlying dynamics in ways that are exceedingly difficult or impossible to achieve with standard measurement techniques.
In particular, spectral measurements with the TS-DFT have been employed to reveal a number of key aspects of modulation instability in the pulsed (i.e., temporally-confined) scenario.
Such TS-DFT measurements have provided insights into the mechanism that often causes single patterns to dominate a given spatial or temporal region in the various contexts in which modulation instability appears.
