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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

All-fiber highly chirped dissipative soliton generation in the telecom range.

A high-energy (0.93 nJ) all-fiber erbium femtosecond oscillator operating in the telecom spectral range is proposed and realized. The laser cavity, built of commercially available fibers and components, combines polarization maintaining (PM) and non-PM parts providing stable generation of highly chirped (chirp parameter 40) pulses compressed in an output piece of standard PM fiber to 165 fs. The results of the numerical simulation agree well with the experiment. The analyzed intracavity pulse dynamics enables the classification of the generated pulses as dissipative solitons.

Mitigation of nonlinear transmission effects for OFDM 16-QAM optical signal using adaptive modulation.

The impact of the fiber Kerr effect on error statistics in the nonlinear (high power) transmission of the OFDM 16-QAM signal over a 2000 km EDFA-based link is examined. We observed and quantified the difference in the error statistics for constellation points located at three power-defined rings. Theoretical analysis of a trade-off between redundancy and error rate reduction using probabilistic coding of three constellation power rings decreasing the symbol-error rate of OFDM 16-QAM signal is presented. Based on this analysis, we propose to mitigate the nonlinear impairments using the adaptive modulation technique applied to the OFDM 16-QAM signal. We demonstrate through numerical modelling the system performance improvement by the adaptive modulation for the large number of OFDM subcarriers (more than 100). We also show that a similar technique can be applied to single carrier transmission.

Periodic nonlinear Fourier transform for fiber-optic communications, Part I: theory and numerical methods.

In this work, we introduce the periodic nonlinear Fourier transform (PNFT) method as an alternative and efficacious tool for compensation of the nonlinear transmission effects in optical fiber links. In the Part I, we introduce the algorithmic platform of the technique, describing in details the direct and inverse PNFT operations, also known as the inverse scattering transform for periodic (in time variable) nonlinear Schrödinger equation (NLSE). We pay a special attention to explaining the potential advantages of the PNFT-based processing over the previously studied nonlinear Fourier transform (NFT) based methods. Further, we elucidate the issue of the numerical PNFT computation: we compare the performance of four known numerical methods applicable for the calculation of nonlinear spectral data (the direct PNFT), in particular, taking the main spectrum (utilized further in Part II for the modulation and transmission) associated with some simple example waveforms as the quality indicator for each method. We show that the Ablowitz-Ladik discretization approach for the direct PNFT provides the best performance in terms of the accuracy and computational time consumption.

Periodic nonlinear Fourier transform for fiber-optic communications, Part II: eigenvalue communication.

In this paper we propose the design of communication systems based on using periodic nonlinear Fourier transform (PNFT), following the introduction of the method in the Part I. We show that the famous "eigenvalue communication" idea [A. Hasegawa and T. Nyu, J. Lightwave Technol. 11, 395 (1993)] can also be generalized for the PNFT application: In this case, the main spectrum attributed to the PNFT signal decomposition remains constant with the propagation down the optical fiber link. Therefore, the main PNFT spectrum can be encoded with data in the same way as soliton eigenvalues in the original proposal. The results are presented in terms of the bit-error rate (BER) values for different modulation techniques and different constellation sizes vs. the propagation distance, showing a good potential of the technique.

Nonlinear inverse synthesis technique for optical links with lumped amplification.

The nonlinear inverse synthesis (NIS) method, in which information is encoded directly onto the continuous part of the nonlinear signal spectrum, has been proposed recently as a promising digital signal processing technique for combating fiber nonlinearity impairments. However, because the NIS method is based on the integrability property of the lossless nonlinear Schrödinger equation, the original approach can only be applied directly to optical links with ideal distributed Raman amplification. In this paper, we propose and assess a modified scheme of the NIS method, which can be used effectively in standard optical links with lumped amplifiers, such as, erbium-doped fiber amplifiers (EDFAs). The proposed scheme takes into account the average effect of the fiber loss to obtain an integrable model (lossless path-averaged model) to which the NIS technique is applicable. We found that the error between lossless path-averaged and lossy models increases linearly with transmission distance and input power (measured in dB). We numerically demonstrate the feasibility of the proposed NIS scheme in a burst mode with orthogonal frequency division multiplexing (OFDM) transmission scheme with advanced modulation formats (e.g., QPSK, 16QAM, and 64QAM), showing a performance improvement up to 3.5 dB; these results are comparable to those achievable with multi-step per span digital back-propagation.

Adiabatic soliton laser.

The key to generating stable optical pulses is mastery of nonlinear light dynamics in laser resonators. Modern techniques to control the buildup of laser pulses are based on nonlinear science and include classical solitons, dissipative solitons, parabolic pulses (similaritons) and various modifications and blending of these methods. Fiber lasers offer remarkable opportunities to apply one-dimensional nonlinear science models for the design and optimization of very practical laser systems. Here, we propose a new concept of a laser based on the adiabatic amplification of a soliton pulse in the cavity-the adiabatic soliton laser. The adiabatic change of the soliton parameters during evolution in the resonator relaxes the restriction on the pulse energy inherent in traditional soliton lasers. Theoretical analysis is confirmed by extensive numerical modeling.

Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers.

In linear communication channels, spectral components (modes) defined by the Fourier transform of the signal propagate without interactions with each other. In certain nonlinear channels, such as the one modelled by the classical nonlinear Schrödinger equation, there are nonlinear modes (nonlinear signal spectrum) that also propagate without interacting with each other and without corresponding nonlinear cross talk, effectively, in a linear manner. Here, we describe in a constructive way how to introduce such nonlinear modes for a given input signal. We investigate the performance of the nonlinear inverse synthesis (NIS) method, in which the information is encoded directly onto the continuous part of the nonlinear signal spectrum. This transmission technique, combined with the appropriate distributed Raman amplification, can provide an effective eigenvalue division multiplexing with high spectral efficiency, thanks to highly suppressed channel cross talk. The proposed NIS approach can be integrated with any modulation formats. Here, we demonstrate numerically the feasibility of merging the NIS technique in a burst mode with high spectral efficiency methods, such as orthogonal frequency division multiplexing and Nyquist pulse shaping with advanced modulation formats (e.g., QPSK, 16QAM, and 64QAM), showing a performance improvement up to 4.5 dB, which is comparable to results achievable with multi-step per span digital back propagation.

All-fiber passively mode-locked Tm-doped NOLM-based oscillator operating at 2-µm in both soliton and noisy-pulse regimes.

A self-starting all-fiber passively mode-locked Tm(3+)-doped fiber laser based on nonlinear loop mirror (NOLM) is demonstrated. Stable soliton pulses centered at 2017.33 nm with 1.56 nm FWHM were produced at a repetition rate of 1.514 MHz with pulse duration of 2.8 ps and pulse energy of 83.8 pJ. As increased pump power, the oscillator can also operate at noise-like (NL) regime. Stable NL pulses with coherence spike width of 341 fs and pulse energy of up to 249.32 nJ was achieved at a center wavelength of 2017.24 nm with 21.33 nm FWHM. To the best of our knowledge, this is the first 2 µm region NOLM-based mode-locked fiber laser operating at two regimes with the highest single pulse energy for NL pulses.