Raman dissipative soliton source of ultrashort pulses in NIR-III spectral window.

We present a novel fiber source of ultrashort pulses at the wavelength of 1660 nm based on the technique of external cavity Raman dissipative soliton generation. The output energy of the generated 30 ps chirped pulses is in the range of 0.5-3.6 nJ with a slope efficiency of 57%. Numerical simulations are in excellent agreement with the experimental results and the shape of the compressed pulses. The compressed pulses consist of a central part with a duration of 300 fs and a weak pedestal. Our results clearly demonstrate the potential to extend the spectral range of the Raman-assisted technique for generating ultra-short pulses to new frequency regions, including biomedical windows. This paves the way for the development of new dissipative soliton sources in these bands.

Soliton control in fiber lasers with a semiconductor optical amplifier by off-set filtering.

The path-averaged model is applied to described soliton characteristics in the anomalous cavity dispersion fiber laser with semiconductor optical amplifier. It is shown that, by off-setting the optical filter relative to the gain spectral maximum, it is possible to control velocity and frequency of both the fundamental optical soliton and chirped dissipative solitons.

Memory-aware end-to-end learning of channel distortions in optical coherent communications.

We implement a new variant of the end-to-end learning approach for the performance improvement of an optical coherent-detection communication system. The proposed solution enables learning the joint probabilistic and geometric shaping of symbol sequences by using auxiliary channel model based on the perturbation theory and the refined symbol probabilities training procedure. Due to its structure, the auxiliary channel model based on the first order perturbation theory expansions allows us performing an efficient parallelizable model application, while, simultaneously, producing a remarkably accurate channel approximation. The learnt multi-symbol joint probabilistic and geometric shaping demonstrates a considerable bit-wise mutual information gain of 0.47 bits/2D-symbol over the conventional Maxwell-Boltzmann shaping for a single-channel 64 GBd transmission through the 170 km single-mode fiber link.

30-GBaud DP 16-QAM transmission in the E-band enabled by bismuth-doped fiber amplifiers.

We report the transmission of five 30-GBaud dual polarization 16-QAM signals over 160 km of standard single-mode fiber in the E-band (1410-1460 nm). The transmission line consists of two 80-km spans and three independent bismuth-doped fiber amplifiers. The developed amplifiers feature a maximum gain of 27.3 dB, 33.8 dB, and 28.3 dB with a minimum noise figure of 4.8 dB, 4.7 dB, and 5.3 dB, respectively. The maximum signal Q2 factor penalty is 4.5 dB, and the overall performance of the system is above the pre-forward-error-correction (FEC) threshold for a 10-15 post-FEC bit error rate. To the best of our knowledge, this is the record experimentally demonstrated transmission length for a coherent detection signal in the E-band.

Experimental implementation of a neural network optical channel equalizer in restricted hardware using pruning and quantization.

The deployment of artificial neural networks-based optical channel equalizers on edge-computing devices is critically important for the next generation of optical communication systems. However, this is still a highly challenging problem, mainly due to the computational complexity of the artificial neural networks (NNs) required for the efficient equalization of nonlinear optical channels with large dispersion-induced memory. To implement the NN-based optical channel equalizer in hardware, a substantial complexity reduction is needed, while we have to keep an acceptable performance level of the simplified NN model. In this work, we address the complexity reduction problem by applying pruning and quantization techniques to an NN-based optical channel equalizer. We use an exemplary NN architecture, the multi-layer perceptron (MLP), to mitigate the impairments for 30 GBd 1000 km transmission over a standard single-mode fiber, and demonstrate that it is feasible to reduce the equalizer's memory by up to 87.12%, and its complexity by up to 78.34%, without noticeable performance degradation. In addition to this, we accurately define the computational complexity of a compressed NN-based equalizer in the digital signal processing (DSP) sense. Further, we examine the impact of using hardware with different CPU and GPU features on the power consumption and latency for the compressed equalizer. We also verify the developed technique experimentally, by implementing the reduced NN equalizer on two standard edge-computing hardware units: Raspberry Pi 4 and Nvidia Jetson Nano, which are used to process the data generated via simulating the signal's propagation down the optical-fiber system.

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.

Intracavity incoherent supercontinuum dynamics and rogue waves in a broadband dissipative soliton laser.

Understanding dynamical complexity is one of the most important challenges in science. Significant progress has recently been made in optics through the study of dissipative soliton laser systems, where dynamics are governed by a complex balance between nonlinearity, dispersion, and energy exchange. A particularly complex regime of such systems is associated with noise-like pulse multiscale instabilities, where sub-picosecond pulses with random characteristics evolve chaotically underneath a much longer envelope. However, although observed for decades in experiments, the physics of this regime remains poorly understood, especially for highly-nonlinear cavities generating broadband spectra. Here, we address this question directly with a combined numerical and experimental study that reveals the physical origin of instability as nonlinear soliton dynamics and supercontinuum turbulence. Real-time characterisation reveals intracavity extreme events satisfying statistical rogue wave criteria, and both real-time and time-averaged measurements are in quantitative agreement with modelling.

Convolutional long short-term memory neural network equalizer for nonlinear Fourier transform-based optical transmission systems.

We evaluate improvement in the performance of the optical transmission systems operating with the continuous nonlinear Fourier spectrum by the artificial neural network equalisers installed at the receiver end. We propose here a novel equaliser designs based on bidirectional long short-term memory (BLSTM) gated recurrent neural network and compare their performance with the equaliser based on several fully connected layers. The proposed approach accounts for the correlations between different nonlinear spectral components. The application of BLSTM equaliser leads to a 16x improvement in terms of bit-error rate (BER) compared to the non-equalised case. The proposed equaliser makes it possible to reach the data rate of 170 Gbit/s for one polarisation conventional nonlinear Fourier transform (NFT) based system at 1000 km distance. We show that our new BLSTM equalisers significantly outperform the previously proposed scheme based on a feed-forward fully connected neural network. Moreover, we demonstrate that by adding a 1D convolutional layer for the data pre-processing before BLSTM recurrent layers, we can further enhance the performance of the BLSTM equaliser, reaching 23x BER improvement for the 170 Gbit/s system over 1000 km, staying below the 7% forward error correction hard decision threshold (HD-FEC).

Fast mode decomposition in few-mode fibers.

Retrieval of the optical phase information from measurement of intensity is of a high interest because this would facilitate simple and cost-efficient techniques and devices. In scientific and industrial applications that exploit multi-mode fibers, a prior knowledge of spatial mode structure of the fiber, in principle, makes it possible to recover phases using measured intensity distribution. However, current mode decomposition algorithms based on the analysis of the intensity distribution at the output of a few-mode fiber, such as optimization methods or neural networks, still have high computational costs and high latency that is a serious impediment for applications, such as telecommunications. Speed of signal processing is one of the key challenges in this approach. We present a high-performance mode decomposition algorithm with a processing time of tens of microseconds. The proposed mathematical algorithm that does not use any machine learning techniques, is several orders of magnitude faster than the state-of-the-art deep-learning-based methods. We anticipate that our results can stimulate further research on algorithms beyond popular machine learning methods and they can lead to the development of low-cost phase retrieval receivers for various applications of few-mode fibers ranging from imaging to telecommunications.

Soliton-sinc optical pulses.

We introduce a new, to the best of our knowledge, type of band-limited optical pulse-soliton-sinc tailored to the nonlinear Schrödinger (NLS) equation. The idea behind the soliton-sinc pulse is to combine, even if approximately, a property of a fundamental soliton to propagate without distortions in nonlinear systems governed by the NLS equation with a compact band-limited spectrum of a Nyquist pulse. Though the shape preserving propagation feature is not exact, such soliton-sinc pulses are more robust against nonlinear signal distortions compared to a Nyquist pulse.

Author Correction: Analysis of laser radiation using the Nonlinear Fourier transform.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Tailoring of spatial coherence in a multimode fiber by selectively exciting groups of eigenmodes.

Control of the properties of speckle patterns produced by mutual interference of light waves is important for various applications of multimode optical fibers. It has been shown previously that a high signal-to-noise ratio in a multimode fiber can be achieved by preferential excitation of lower order spatial eigenmodes in optical fiber communication. Here we demonstrate that signal spatial coherence can be tailored by changing relative contributions of the lower and higher order multimode fiber eigenmodes for the research of speckle formation and spatial coherence. It is found that higher order spatial eigenmodes are more conducive to the final speckle formation. The minimum speckle contrast occurs in the lower order spatial eigenmodes dominated regime. This work paves the way for control and manipulation of the spatial coherence of light in a multimode fiber varying from partially coherent or totally incoherent light.

Combining nonlinear Fourier transform and neural network-based processing in optical communications.

We propose a method to improve the performance of the nonlinear Fourier transform (NFT)-based optical transmission system by applying the neural network post-processing of the nonlinear spectrum at the receiver. We demonstrate through numerical modeling about one order of magnitude bit error rate improvement and compare this method with machine learning processing based on the classification of the received symbols. The proposed approach also offers a way to improve numerical accuracy of the inverse NFT; therefore, it can find a range of applications beyond optical communications.

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.

Analysis of laser radiation using the Nonlinear Fourier transform.

Modern high-power lasers exhibit a rich diversity of nonlinear dynamics, often featuring nontrivial co-existence of linear dispersive waves and coherent structures. While the classical Fourier method adequately describes extended dispersive waves, the analysis of time-localised and/or non-stationary signals call for more nuanced approaches. Yet, mathematical methods that can be used for simultaneous characterisation of localized and extended fields are not yet well developed. Here, we demonstrate how the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem can be applied as a signal processing tool for representation and analysis of coherent structures embedded into dispersive radiation. We use full-field, real-time experimental measurements of mode-locked pulses to compute the nonlinear pulse spectra. For the classification of lasing regimes, we present the concept of eigenvalue probability distributions. We present two field normalisation approaches, and show the NFT can yield an effective model of the laser radiation under appropriate signal normalisation conditions.

Gain-through-filtering enables tuneable frequency comb generation in passive optical resonators.

Optical frequency combs (OFCs), consisting of a set of phase-locked, equally spaced laser frequency lines, have enabled a great leap in precision spectroscopy and metrology since seminal works of Hänsch et al. Nowadays, OFCs are cornerstones of a wealth of further applications ranging from chemistry and biology to astrophysics and including molecular fingerprinting and light detection and ranging (LIDAR) systems, among others. Driven passive optical resonators constitute the ideal platform for OFC generation in terms of compactness and low energy footprint. We propose here a technique for the generation of OFCs with a tuneable repetition rate in externally driven optical resonators based on the gain-through-filtering process, a simple and elegant method, due to asymmetric spectral filtering on one side of the pump wave. We demonstrate a proof-of-concept experimental result in a fibre resonator, pioneering a new technique that does not require specific engineering of the resonator dispersion to generate frequency-agile OFCs.

Rogue bioelectrical waves in the brain: the Hurst exponent as a potential measure for presurgical mapping in epilepsy.

OBJECTIVE: Brain electromagnetic activity in patients with epilepsy is characterized by abnormal high-amplitude transient events (spikes) and abnormal patterns of synchronization of brain rhythms that accompany epileptic seizures. With the aim of improving methods for identifying epileptogenic sources in magnetoencephalographic (MEG) recordings of brain data, we applied methods previously used in the study of oceanic 'rogue waves' and other freak events in complex systems. APPROACH: For data from three patients who were awaiting surgical treatment for epilepsy, we used a beamformer source model to produce volumetric maps showing areas with a high proportion of spikes that could be classified as 'rogue waves', and areas with high Hurst exponent (HE). The HE describes the extent to which a system is exhibiting persistent behavior, may predict the likelihood of freak events. These measures were compared with the more standard measure of kurtosis, which has been shown to be a reliable method for localizing interictal spikes. MAIN RESULTS: There was partial concordance between the three different volumetric maps indicating that each measure provides different information about the underlying brain data. The HE, when combined with a simple connectivity analysis based on phase slope index (PSI), was able to identify the probable epileptogenic zone in all three patients, despite very different patterns of abnormal activity. The differences between distributions of high HE and high kurtosis values indicates that while spikes are propagated through cortex from the epileptogenic zone, the persistent dynamical conditions under which the spikes are generated may not be propagated in a similar way. Finally, the patterns of persistent activity, indicating a departure from 'healthy criticality' in brain networks may explain the wide range of social and cognitive impairments that are seen in epilepsy patients. SIGNIFICANCE: The HE is a potentially useful addition to the clinician's battery of measures which may be used convergently to guide surgical intervention.

On the theory of adiabatic field dynamics in the Kerr medium with distributed gain and dispersion.

A general theory is presented for the adiabatic field evolution in a nonlinear Kerr medium with distributed amplification and varying dispersion. Analytical expression is derived linking parameters of the adiabaticity, gain distribution, and dispersion profile. As a particular example, an optical pulse compressor based on the adiabatic dynamics is examined.

Polarization-multiplexed nonlinear inverse synthesis with standard and reduced-complexity NFT processing: erratum.

We correct a formula for the numerical nonlinear Fourier transform in [1]. The conclusions of our work are unchanged.

Contour integrals for numerical computation of discrete eigenvalues in the Zakharov-Shabat problem.

We propose a novel algorithm for the numerical computation of discrete eigenvalues in the Zakharov-Shabat problem. Our approach is based on contour integrals of the nonlinear Fourier spectrum function in the complex plane of the spectral parameter. The reliability and performance of the new approach are examined in application to a single eigenvalue, multiple eigenvalues, and the degenerate breather's multiple eigenvalue. We also study the impact of additive white Gaussian noise on the stability of numerical eigenvalues computation.