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 Schr\"odinger 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.

Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers

New services and applications are causing an exponential increase in Internet traffic. In a few years, the current fiber optic communication system infrastructure will not be able to meet this demand because fiber nonlinearity dramatically limits the information transmission rate. Eigenvalue communication could potentially overcome these limitations. It relies on a mathematical technique called “nonlinear Fourier transform (NFT)” to exploit the “hidden” linearity of the nonlinear Schrodinger equation as the master model for signal propagation in an optical fiber. We present here the theoretical tools describing the NFT for the Manakov system and report on experimental transmission results for dual polarization in fiber optic eigenvalue communications. A transmission of up to 373.5 km with a bit error rate less than the hard-decision forward error correction threshold has been achieved. Our results demonstrate that dual-polarization NFT can work in practice and enable an increased spectral efficiency in NFT-based communication systems, which are currently based on single polarization channels.

/pdf/dual-polarization-nonlinear-fourier-transform-based-optical-ns5s6z7n6t.pdf

Dual polarization nonlinear Fourier transform-based optical communication system: supplementary material

Nonlinear frequency division multiplexing (NFDM) communication systems that are based on the nonlinear Fourier transform (NFT), have seen a rapid improvement in performance and transmission reach over just a few years. However, such an improvement is now being slowed down by fundamental challenges such as fiber loss and noise. As the NFT theory is defined over a lossless transmission fiber, a strong research focus has been dedicated to either improve the lossless assumption for practical fibers, by adapting the theory to approximately account for the fiber loss, or by devising encoding schemes that increase the robustness of the NFT to the fiber attenuation. However, the proposed solutions provide only minimal benefits to the system performance, especially for long fiber spans as in deployed links. Alternatively, a detection strategy based on replacing a conventional NFT receiver with a time-domain Neural network (NN)-based symbol decisor has been numerically proposed. Here, we extend such an idea by validating it experimentally. In order to apply the method in an experimental environment, the impact of phase noise, and receiver frequency offset needs to be addressed. We, therefore, propose a novel time-domain receiver architecture that combines a two-stage iterative carrier recovery with a NN-based symbol decisor. The carrier recovery, itself based on a NN for phase estimation, is numerically, and experimentally characterized. The proposed receiver has been evaluated for single-polarization two-eigenvalue transmission at 1 GBd. A two-fold increase in the transmission reach is enabled by the NN receiver ( $\approx$ 1600 km) compared to a conventional NFT receiver ( $\approx$ 560 km) for a practical link using 80-km spaced erbium-doped fiber amplifier (EDFAs).

/pdf/experimental-demonstration-of-nonlinear-frequency-division-e1kfk485et.pdf

Experimental Demonstration of Nonlinear Frequency Division Multiplexing Transmission With Neural Network Receiver

We perform geometric constellation shaping with optimized bit labeling using a binary autoencoder including a differential blind phase search (BPS). Our approach enables full end-to-end training of optical coherent transceivers taking into account the digital signal processing. 

/pdf/geometric-constellation-shaping-for-phase-noise-channels-1xfmm9ma.pdf

Geometric Constellation Shaping for Phase-noise Channels Using a Differentiable Blind Phase Search

We review different technologies and architectures for neuromorphic photonic accelerators, spanning from bulk optics to photonic-integrated-circuits (PICs), and assess compute efficiency in OPs/Watt through the lens of a comparative study where key technology aspects are analyzed. With an emphasis on PIC neuromorphic accelerators, we shed light onto the latest advances in photonic and plasmonic modulation technologies for the realization of weighting elements in training and inference applications, and present a recently introduced scalable coherent crossbar layout. Finally, we stress that current technologies face challenges endowing photonic accelerators with compute efficiencies in the PetaOPs/W, and discuss future implementation pathways towards improving performance. 

Neuromorphic photonic technologies and architectures: scaling opportunities and performance frontiers [Invited]

Training of autoencoders using the back-propagation algorithm is challenging for non-differential channel models or in an experimental environment where gradients cannot be computed. In this paper, we study a gradient–free training method based on the cubature Kalman filter. To numerically validate the method, the autoencoder is employed to perform geometric constellation shaping on differentiable communication channels, showing the same performance as the back-propagation algorithm. Further investigation is done on a non–differentiable communication channel that includes: laser phase noise, additive white Gaussian noise and blind phase search-based phase noise compensation. Our results indicate that the autoencoder can be successfully optimized using the proposed training method to achieve better robustness to residual phase noise with respect to standard constellation schemes such as Quadrature Amplitude Modulation and Iterative Polar Modulation for the considered conditions.

/pdf/gradient-free-training-of-autoencoders-for-non-2uwk5mlj8t.pdf

Gradient-Free Training of Autoencoders for Non-Differentiable Communication Channels

We propose an autoencoder-based geometric shaping that learns a constellation robust to SNR and laser linewidth estimation errors. This constellation maintains shaping gain in mutual information (up to 0.3 bits/symbol) with respect to QAM over various SNR and laser linewidth values.

/pdf/end-to-end-learning-of-a-constellation-shape-robust-to-3qa9fcla8n.pdf

End-to-end Learning of a Constellation Shape Robust to Variations in SNR and Laser Linewidth.

Training of autoencoders using the back-propagation algorithm is challenging for non-differential channel models or in an experimental environment where gradients cannot be computed. In this paper, we study a gradient-free training method based on the cubature Kalman filter. To numerically validate the method, the autoencoder is employed to perform geometric constellation shaping on differentiable communication channels, showing the same performance as the back-propagation algorithm. Further investigation is done on a non-differentiable communication channel that includes: laser phase noise, additive white Gaussian noise and blind phase search-based phase noise compensation. Our results indicate that the autoencoder can be successfully optimized using the proposed training method to achieve better robustness to residual phase noise with respect to standard constellation schemes such as Quadrature Amplitude Modulation and Iterative Polar Modulation for the considered conditions.

Gradient-free training of autoencoders for non-differentiable communication channels

Vendor interoperability is one of the desired future characteristics of optical networks. This means that the transmission system needs to support a variety of hardware with different components, leading to system uncertainties throughout the network. For example, uncertainties in signal-to-noise ratio and laser linewidth can negatively affect the quality of transmission within an optical network due to e.g. mis-parametrization of the transceiver signal processing parameters. In this paper, we propose to geometrically optimize a constellation shape that is robust to uncertainties in the channel conditions by utilizing end-to-end learning. In the optimization step, the channel model includes additive noise and residual phase noise. In the testing step, the channel model consists of laser phase noise, additive noise and blind phase search as the carrier phase recovery algorithm. Two noise models are considered for the additive noise: white Gaussian noise and nonlinear interference noise model for fiber communication. The latter models the behavior of an optical fiber channel more accurately because it considers the nonlinear effects of the optical fiber. For this model, the uncertainty in the signal-to-noise ratio can be divided between amplifier noise figures and launch power variations. For both noise models, our results indicate that the learned constellations are more robust to uncertainties in channel conditions compared to a standard constellation scheme such as quadrature amplitude modulation and more importantly standard geometric constellation shaping techniques.

End-to-end Learning of a Constellation Shape Robust to Channel Condition Uncertainties

The autoencoder concept for geometric constellation shaping is discussed. Applications in coherent optical fiber communications are presented. Several popular training algorithms are compared. The quantization problem of finite precision DAC and ADC is addressed. © 2022 The Author(s)

Ognjen Jovanovic

Papers

Gradient-Free Training of Autoencoders for Non-Differentiable Communication Channels

End-to-end Learning of a Constellation Shape Robust to Variations in SNR and Laser Linewidth.

Gradient-free training of autoencoders for non-differentiable communication channels

End-to-end Learning of a Constellation Shape Robust to Channel Condition Uncertainties

Recent advances in constellation optimization for fiber-optic channels