Analysis of laser radiation using the Nonlinear Fourier transform

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.

Single-Photon Level Dispersive Fourier Transform: Ultrasensitive Characterization of Noise-Driven Nonlinear Dynamics

Dispersive Fourier transform is a characterization technique that allows directly extracting an optical spectrum from a time domain signal, thus providing access to real-time characterization of the signal spectrum. However, these techniques suffer from sensitivity and dynamic range limitations, hampering their use for special applications in, e.g., high-contrast characterizations and sensing. Here, we report on a novel approach to dispersive Fourier transform-based characterization using single-photon detectors. In particular, we experimentally develop this approach by leveraging mutual information analysis for signal processing and hold a performance comparison with standard dispersive Fourier transform detection and statistical tools. We apply the comparison to the analysis of noise-driven nonlinear dynamics arising from well-known modulation instability processes. We demonstrate that with this dispersive Fourier transform approach, mutual information metrics allow for successfully gaining insight into the fluctuations associated with modulation instability-induced spectral broadening, providing qualitatively similar signatures compared to ultrafast photodetector-based dispersive Fourier transform but with improved signal quality and spectral resolution (down to 53 pm). The technique presents an intrinsically unlimited dynamic range and is extremely sensitive, with a sensitivity reaching below the femtowatt (typically 4 orders of magnitude better than ultrafast dispersive Fourier transform detection). We show that this method can not only be implemented to gain insight into noise-driven (spontaneous) frequency conversion processes but also be leveraged to characterize incoherent dynamics seeded by weak coherent optical fields.

Polarization dynamics of vector solitons in a fiber laser

We investigate the polarization dynamics of vector solitons in a fiber laser mode-locked by a saturable absorber (SA). Three types of vector solitons were obtained in the laser, including group velocity locked vector solitons (GVLVS), polarization locked vector solitons (PLVS), and polarization rotation locked vector solitons (PRLVS). Their polarization evolution during intracavity propagation is discussed. Pure vector solitons are obtained from the continuous wave (CW) background by soliton distillation, and the characteristics of the vector solitons without and with distillation are analyzed, respectively. Numerical simulations suggest that the features of vector solitons in a fiber laser could be assemble to those generated in fibers.

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.

Linear optical sampling enabled soliton nonlinear frequency spectrum classification

Nonlinear Fourier transform (NFT) is a powerful tool for characterizing optical soliton dynamics, which, however, suffers from fundamental limitations that ultra-wide bandwidth photodetectors and ultra-high sampling rate analog-to-digital converters should be used when accessing the full-field information of an ultrafast optical pulse. Herein, we report on the experimental demonstration of the linear optical sampling (LOS) enabled nonlinear frequency spectrum classification of ultrashort optical pulses, which could break this limitation. Instead of traditional coherent detection, the LOS overcomes the ultra-wide bandwidth constraint of commercially available optoelectrical devices. By finely adjusting the repetition rate difference between the soliton to be characterized and the sampling pulsed source, a 55.56-TSa/s equivalent sampling rate arising in the LOS can be secured, where only 400-MHz balanced photodetectors and 5-GSa/s analog-to-digital converter are used. Meanwhile, according to the nonlinear frequency spectrum calculated from the accurate full-field information, the promising concept of soliton distillation has been experimentally verified for the first time. The LOS-enabled NFT technique provides an alternative and efficient characterization tool for ultrafast fiber lasers, which facilities comprehensive insight into soliton dynamics.

Neural networks for computing and denoising the continuous nonlinear Fourier spectrum in focusing nonlinear Schrödinger equation

We combine the nonlinear Fourier transform (NFT) signal processing with machine learning methods for solving the direct spectral problem associated with the nonlinear Schrödinger equation. The latter is one of the core nonlinear science models emerging in a range of applications. Our focus is on the unexplored problem of computing the continuous nonlinear Fourier spectrum associated with decaying profiles, using a specially-structured deep neural network which we coined NFT-Net. The Bayesian optimisation is utilised to find the optimal neural network architecture. The benefits of using the NFT-Net as compared to the conventional numerical NFT methods becomes evident when we deal with noise-corrupted signals, where the neural networks-based processing results in effective noise suppression. This advantage becomes more pronounced when the noise level is sufficiently high, and we train the neural network on the noise-corrupted field profiles. The maximum restoration quality corresponds to the case where the signal-to-noise ratio of the training data coincides with that of the validation signals. Finally, we also demonstrate that the NFT b-coefficient important for optical communication applications can be recovered with high accuracy and denoised by the neural network with the same architecture.

Evidence of Chaotic Dynamics in Three-Soliton Collisions

We observe chaotic optical wave dynamics characterized by erratic energy transfer and soliton annihilation and creation in the aftermath of a three-soliton collision in a photorefractive crystal. Irregular dynamics are found to be mediated by the nonlinear Raman effect, a coherent interaction that leads to nonreciprocal soliton energy exchange. Results extend the analogy between solitons and particles to the emergence of chaos in three-body physics and provide new insight into the origin of the irregular dynamics that accompany extreme and rogue waves.

Nearly integrable turbulence and rogue waves in disordered nonlinear Schrödinger systems

Integrable nonlinear Schrödinger (NLS) systems provide a platform for exploring the propagation and interaction of nonlinear waves. Extreme events such as rogue waves (RWs) are currently of particular interest. However, the presence of disorder in these systems is sometimes unavoidable, for example, in the forms of turbulent current in the ocean and random fluctuation in optical media, and its influence remains less understood. Here, we report numerical experiments of two nearly-integrable NLS equations with the effect of disorder showing that the probability of RW occurrence can be significantly increased by adding weak system noise. Linear and nonlinear spectral analyses are proposed to qualitatively explain those findings. Our results are expected to shed light on the understanding of the interplay between disorder and nonlinearity, and may motivate new experimental works in hydrodynamics, nonlinear optics, and Bose-Einstein condensates.

Vector soliton dynamics in a high-repetition-rate fiber laser

The existence of vector solitons that arise from the birefringence nature of optical fibers has been increasingly of interest for the stability of mode-locked fiber lasers, particularly for those operating in the high-fundamental-repetition-rate regime, where a large amount of fiber birefringence is required to restore the phase relation between the orthogonally polarized vector solitons, resulting in stable mode-locking free of polarization rotation. These vector solitons can exhibit diverse time-varying polarization dynamics, which prevent industrial and scientific applications requiring stable and uniform pulse trains at high fundamental repetition rates. This pressing issue, however, has so far been rarely studied. To this end, here we theoretically and experimentally dissect the formation of vector solitons in a GHz-repetition-rate fiber laser and investigate effective methods for suppressing roundtrip-to-roundtrip polarization dynamics. Our numerical model can predict both dynamic and stable regimes of high-repetition-rate mode-locking by varying the amount of fiber birefringence, resulting in the polarization rotation vector soliton (PRVS) and linearly polarized soliton (LPS), respectively. These dynamic behaviors are further studied by using an analytical approach. Interestingly, our theoretical results indicate a cavity-induced locking effect, which can be a complementary soliton trapping mechanism for the co-propagating solitons. Finally, these theoretical predications are experimentally verified, and we obtain both PRVS and LPS by adjusting the intracavity fiber birefringence.

Nonlinear Fourier transform for analysis of optical spectral combs

The nonlinear Fourier transform (NFT) is used to characterize the optical combs in the Lugiato-Lefever equation with both anomalous and normal dispersion. We demonstrate that the NFT signal processing technique can simplify analysis of the formation of dissipative dark solitons and regimes exploiting modulation instability for a generation of coherent structures, by approximating the comb with several discrete eigenvalues, providing a platform for the analytical description of dissipative coherent structures.

Nonlinear Fourier transform for characterization of the coherent structures in optical microresonators

We propose and demonstrate, in the framework of the generic mean-field model, the application of the nonlinear Fourier transform (NFT) signal processing based on the Zakharov-Shabat spectral problem to the characterization of the round trip scale dynamics of radiation in optical fiber- and microresonators.