Periodic Reporting for period 1 - OMICRON (Optical transmission based on integrability and nonlinear Fourier transform) Periodo di rendicontazione: 2018-01-12 al 2020-01-11 Sintesi del contesto e degli obiettivi generali del progetto "OMICRON is focused on the application of a new advanced mathematical tool – nonlinear Fourier transform (NFT) – in optical communication systems, to bridge that powerful mathematical method with the practical needs arising in optical communication, in particular – for the purpose of nonlinearity mitigation. The motivation to undertake this study is as follows. The exponential surge in global data traffic driven by the proliferation of bandwidth-hungry online services (cloud computing, on-demand HD video streams, etc.) results in the escalating pressure on the speed and quality of information flows interconnecting network participants. The “Internet of Things”, in which consumer and sensor devices will be connected to the Internet, has a projected number of 50 billion devices by 2021. Even without new services, current growth rates are greater than 20% per year. A breakthrough in the functioning of communication networks has been the compensation of the linear dispersion. In the next step, noise and nonlinearity are becoming the key factors that limit the performance of future transmission systems. The alarming observation that the spectral efficiency of fibre channels is limited when using current techniques, and starts to decay at high signal powers due to nonlinearity. It was predicted that within the next decade the existing optical fibre technology will approach the “nonlinear transmission limit"" which caps the achievable rate of error-free data transmission. Therefore, radical innovations and alternative solutions to the functioning mechanism of optical networks are unavoidable. That is exactly why the NFT was proposed as a novel tool for designing effectively nonlinearity-free communication systems to combat the nonlinear impairments. The schematic representation of NFT-based transmission system is given in Fig.1 showing operations of Inverse/Direct NFT and the possibility to use continuous spectrum and solitons as data carriers.The project posed the goal of developing an efficacious solution to overcome limits imposed by the fibre nonlinearity on the capacity of optical transmission systems, by using the NFT signal decomposition and employing the parameters of resulting nonlinear modes as data carriers. On the physical level, the main factor limiting the performance of current NFT-based optical communications is the signal-noise interference occurring in the nonlinear Fourier domain. The study of noise and building the model is, thus, of paramount importance for both the performance optimisation and for the capacity estimation." Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti Several NFT-based optical transmission systems were studied in the project. For these systems, we assumed a zero gain-loss, and so the only source of transmission corruption considered was the in-line optical noise. We proposed the asymptotic analytical theory describing the noise action on nonlinear modes and signal-noise interference phenomena. Then, the leading order results of the approximate theory for the NFT noise were checked against the same data obtained with the use of massive Monte-Carlo simulations.(i) The case of “conventional” nonlinear spectrum modulation. With the outcomes obtained, we were then able to define the limitations of our analytical approach. We showed that the effective nonlinear noise emerging in the NF domain deviates significantly from the progenitor Gaussian noise in the optical domain. We found that the complex NF noise covariance normalised to the propagation distance demonstrates a non-monotonic dependence on the latter and deviates noticeably from its linear limits. In addition, we investigated the impact of solitons or, in other words, eigenvalues on the capacity and performance of an optical transmission system.(ii) For the b-modulated systems, we derived the completely new results regarding the stochastic corruption affecting the data encoded into the b-coefficient. For this advanced NFT system, we performed the similar checks as in (i), and found a very good correspondence between the theory for the nonlinear noise correlators and the results of Monte-Carlo simulations.(iii) For the NFT transmission system employing two polarisations, we developed the fast NFT and inverse NFT signal processing routines in collaboration with the Delft Technical University (Prof. Sander Wahls).(iv) The profound knowledge of the noise properties inside the NFT domain allows us to design special-tailored equalisers. We proposed two types of advanced equalisers based on the machine learning/artificial intelligence. The application of these advanced tools effectively removes nonlinear and noise-induced signal impairments at the receiver resulting in better system performance. These approaches were proposed for the experimental evaluation. However, due to the novelty and complexity of the proposed method, so far, no complete experimental confirmation of the theoretical results has been obtained, and this is the subject of the on-going studies.Overall, The Fellow presented the project outcomes on 5 international scientific conferences, several seminars, 2 workshops and published in 7 peer-reviewed conference proceedings. In addition to the dissemination activities in the scientific community, all results were shared with the general public using social media (Twitter, ResearchGate). Moreover, The Fellow took part in such events as Unifest, BigBang Fair and Chinese Summer school to promote engineering and specifically nonlinear photonic among children. Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora) During the OMICRON implementation we achieved the following state-of-the-art results:(i) We derived the analytical models for the noisy channels hampering the transmutation for the “conventional” NFT-based systems. The analytical results were then compared with the massive Monte-Carlo simulations. It was shown that the theory underestimates the true noise arising in the system. Moreover, it was shown that for some regimes the processing noise arising due to the finite accuracy of NFT operations at the receiver and transmitter. Our results indicate that the NFDM defines a complicated nonlinear channel with the channel law depending in a non-trivial way on the propagation distance and the input state. The observed phenomena shed some new light on the signal noise interaction in the NF domain and our results can be used both for the optimisation of performance in the NFT-based communication systems and for the improvement of analytical models and approaches referring to the description of the NF noise and related information theory measures.(ii) For the first time, we derived the analytical model of noise emerging inside the NF domain for the advanced b-modulated systems. It was shown that this theory complies very well with the results of simulations. Again, the impact of the results obtained is similar to that of the item (i). Moreover, as it was recently shown, the b-modulated systems bring about the best data rate among all NFT-based concepts. Therefore, these results have the highest practical importance and potential for further implementation in real transmission lines. (iii) We developed the fast NFT and INFT routines and the signal processing routine for the case of dual polarisations. (iv) We proposed two different types of machine learning equalisers for the improvement of transmission in various NFT based transmission systems. We applied for the first time such methods as k-nearest neighbors for the NFT communication system. The schematic representation of NFT-based transmission system
