Nonlinear Fourier transform for analysis of optical spectral combs

Nonlinear Fourier classification of 663 rogue waves measured in the Philippine Sea

Rogue waves are sudden and extreme occurrences, with heights that exceed twice the significant wave height of their neighboring waves. The formation of rogue waves has been attributed to several possible mechanisms such as linear superposition of random waves, dispersive focusing, and modulational instability. Recently, nonlinear Fourier transforms (NFTs), which generalize the usual Fourier transform, have been leveraged to analyze oceanic rogue waves. Next to the usual linear Fourier modes, NFTs can additionally uncover nonlinear Fourier modes in time series that are usually hidden. However, so far only individual oceanic rogue waves have been analyzed using NFTs in the literature. Moreover, the completely different types of nonlinear Fourier modes have been observed in these studies. Exploiting twelve years of field measurement data from an ocean buoy, we apply the nonlinear Fourier transform (NFT) for the nonlinear Schrödinger equation (NLSE) (referred to NLSE-NFT) to a large dataset of measured rogue waves. While the NLSE-NFT has been used to analyze rogue waves before, this is the first time that it is systematically applied to a large real-world dataset of deep-water rogue waves. We categorize the measured rogue waves into four types based on the characteristics of the largest nonlinear mode: stable, small breather, large breather and (envelope) soliton. We find that all types can occur at a single site, and investigate which conditions are dominated by a single type at the measurement site. The one and two-dimensional Benjamin-Feir indices (BFIs) are employed to examine the four types of nonlinear spectra. Furthermore, we verify on a part of the data set that for the localized types, the largest nonlinear Fourier mode can be attributed directly to the rogue wave, and investigate the relation between the height of the rogue waves and that of the dominant nonlinear Fourier mode. While the dominant nonlinear Fourier mode in general only contributes a small fraction of the rogue wave, we find that soliton modes can contribute up to half of the rogue wave. Since the NLSE does not account for directional spreading, the classification is repeated for the first quartile with the lowest directional spreading for each type. Similar results are obtained.

Optical Darboux Transformer

The optical Darboux transformer for solitons is introduced as a photonic device that performs the Darboux transformation directly in the optical domain. This enables two major advances for optical signal processing based on the nonlinear Fourier transform: (i) the multiplexing of solitonic waveforms corresponding to different discrete eigenvalues of the Zakharov-Shabat system, and (ii) the selective filtering of an arbitrary number of individual solitons too. The optical Darboux transformer can be built using existing commercially available photonic technology components and constitutes a universal tool for signal processing, optical communications, optical rogue waves generation, and waveform shaping and control in the nonlinear Fourier domain.

Characterization of sidebands in fiber lasers based on nonlinear Fourier transformation

Phase evolution of soliton and that of first-order sidebands in a fiber laser are investigated by using nonlinear Fourier transform (NFT). Development from dip-type sidebands to peak-type (Kelly) sidebands is presented. The phase relationship between the soliton and the sidebands calculated by the NFT are in good agreement with the average soliton theory. Our results suggest that NFT can be an effective tool for the analysis of laser pulses.