Nonlinear Optical Channel Research Articles

Enhanced mutual information neural estimators for optical fiber communication.

The accurate estimation of mutual information (MI) plays a vital role in understanding channel capacity and optimizing the performance of optical communications. While MI computations for the additive white Gaussian noise (AWGN) channel are well-established, they fall short when dealing with the challenges posed by nonlinear optical fiber channels due to an unknown channel model. For the first time, to our knowledge, this Letter introduces a mutual information neural estimator (MINE) for MI estimation in optical fiber communications. We propose an enhanced MINE (E-MINE), achieved by enlarging the training batch size to improve estimation accuracy and stability. Our findings reveal that the E-MINE achieves highly accurate estimations in the AWGN channel and maintains strong consistency with symbol-by-symbol MI estimations, comparable to Monte Carlo (MC) methods based on a Gaussian distribution in long-haul optical fiber channels. Furthermore, with multi-symbol estimation, the E-MINE yields a 0.16 bits/4D-symbol improvement in our experiments. We anticipate that our findings will drive further research in the field, opening new possibilities for enhancing communication systems design and performance using deep learning techniques.

Deep Neural Network-Aided Soft-Demapping in Coherent Optical Systems: Regression Versus Classification

We examine here what type of predictive modelling, classification, or regression, using neural networks (NN), fits better the task of soft-demapping based post-processing in coherent optical communications, where the transmission channel is nonlinear and dispersive. For the first time, we present possible drawbacks in using each type of predictive task in a machine learning context, considering the nonlinear coherent optical channel equalization/soft-demapping problem. We study two types of equalizers based on the feed-forward and recurrent NNs, for several transmission scenarios, in linear and nonlinear regimes of the optical channel. We point out that even though from the information theory perspective the cross-entropy loss (classification) is the most suitable option for our problem, the NN models based on the cross-entropy loss function can severely suffer from learning problems. The latter translates into the fact that regression-based learning is typically superior in terms of delivering higher Q-factor and achievable information rates. In short, we show by empirical evidence that loss functions based on cross-entropy may not be necessarily the most suitable option for training communication systems in practical scenarios when overfitting- and vanishing gradients-related problems come into play.

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.

New Lower Bounds on the Capacity of Optical Fiber Channels via Optimized Shaping and Detection

Constellation shaping is a practical and effective technique to improve the performance and the rate adaptivity of optical communication systems. In principle, it could also be used to mitigate the impact of nonlinear effects, possibly increasing the information rate beyond the current limit dictated by fiber nonlinearity. However, this appealing idea is frustrated by the difficulty of designing an effective shaping strategy that takes into account the nonlinearity and long memory of the fiber channel, as well as the possible interplay with other nonlinearity mitigation strategies. As a result, only little progress has been made so far, while the optimal shaping distribution and the ultimate channel capacity remain unknown. In this work, we describe a novel technique to optimize the shaping distribution in a very general setting and high-dimensional space. For a simplified block-memoryless nonlinear optical channel, the capacity lower bound obtained by the proposed technique can be expressed analytically, establishing the conditions for an unbounded growth of capacity with power. In a more realistic scenario, the technique can be implemented by a rejection sampling algorithm driven by a suitable cost function, and the corresponding achievable information rate estimated numerically. The combination of the proposed technique with an improved (non-Gaussian) decoding metric yields a new capacity lower bound for the dual-polarization WDM channel.

Performance comparison of partial-response square shaped signal with direct detection

In this work, we investigate improving spectral efficiency with direct detection based partial-response square shaping, which generalizes partial-response pulse shaping and reduces width of power spectrum of signal under both linear and non-linear model assumption. For linear model, quadrature point bias of Mach–Zender modulator is used. For non-linear model, null point bias of Mach–Zender is used, however, non-linear pre-coding and equalization are required. By comparing results obtained with linear model and non-linear model, we identify the factors influencing performance of short reach transmission. Results of experiment over a 25 km dispersion uncompensated link show following: Under non-linear model, performance issue is revealed with non-linear optical channel. Under linear model, for PAM4 modified-duo-binary with bias signal and PAM4 duo-binary, 2 b/s/Hz is achievable with 7% overhead hard decision decoding. Also, PAM4 modified-duo-binary with bias signal has 1 order lower bit error rate than PAM4 duo-binary, while theoretical analysis confirms the finding for short reach transmission, that non-linear equalization can recover information from coherent item and can suppress interference caused by self-beat item.

Frequency Logarithmic Perturbation on the Group-Velocity Dispersion Parameter With Applications to Passive Optical Networks

Signal propagation in an optical fiber can be described by the nonlinear Schr\"odinger equation (NLSE). The NLSE has no known closed-form solution, mostly due to the interaction of dispersion and nonlinearities. In this paper, we present a novel closed-form approximate model for the nonlinear optical channel, with applications to passive optical networks. The proposed model is derived using logarithmic perturbation in the frequency domain on the group-velocity dispersion (GVD) parameter of the NLSE. The model can be seen as an improvement of the recently proposed regular perturbation (RP) on the GVD parameter. RP and logarithmic perturbation (LP) on the nonlinear coefficient have already been studied in the literature, and are hereby compared with RP on the GVD parameter and the proposed LP model. As an application of the model, we focus on passive optical networks. For a 20 km PON at 10 Gbaud, the proposed model improves upon LP on the nonlinear coefficient by 1.5 dB. For the same system, a detector based on the proposed LP model reduces the uncoded bit-error-rate by up to 5.4 times at the same input power or reduces the input power by 0.4 dB at the same information rate.

Experimental realization of dual task processing with a photonic reservoir computer

We experimentally demonstrate the possibility to process two tasks in parallel with a photonic reservoir computer based on a vertical-cavity surface-emitting laser (VCSEL) as a physical node with time-delay optical feedback. The two tasks are injected optically by exploiting the polarization dynamics of the VCSEL. We test our reservoir with the very demanding task of nonlinear optical channel equalization as an illustration of the performance of the system and show the recover of two signals simultaneously with an error rate of 0.3% (3%) for a 25 km-fiber distortion (50 km-fiber distortion) at a processing speed of 51.3 Mb/s.

Non-Orthogonal Signal Transmission Over Nonlinear Optical Channels

The performance of spectrally efficient frequency division multiplexing (SEFDM) in optical communication systems is investigated considering the impact of fiber nonlinearities. Relative to orthogonal frequency division multiplexing (OFDM), sub-carriers within SEFDM signals are packed closer at a frequency spacing less than the symbol rate. In order to recover the data, a specially designed sphere decoding detector is used at the receiver end to compensate for the self-created inter carrier interference encountered in SEFDM signals. Our research demonstrated the benefits of the use of sphere decoding in SEFDM and also demonstrates the performance improvement of long-haul optical communication systems using SEFDM compared to the use of conventional OFDM, when fiber nonlinearities are considered. Different modulation formats ranging from 4QAM to 32QAM are studied and it is shown that, for the same spectral efficiency and information rate, SEFDM signals allow a significant increase in the transmission distance compared to conventional OFDM signals.

Introducing Enumerative Sphere Shaping for Optical Communication Systems With Short Blocklengths

Probabilistic shaping based on constant composition distribution matching (CCDM) has received considerable attention as a way to increase the capacity of fiber optical communication systems. CCDM suffers from significant rate loss at short blocklengths and requires long blocklengths to achieve high shaping gain, which makes its implementation very challenging. In this paper, we propose to use enumerative sphere shaping (ESS) and investigate its performance for the nonlinear fiber optical channel. ESS has lower rate loss than CCDM at the same shaping rate, which makes it a suitable candidate to be implemented in real-time high-speed optical systems. In this paper, we first show that finite blocklength ESS and CCDM exhibit higher effective signal-to-noise ratio than their infinite blocklength counterparts. These results show that for the nonlinear fiber optical channel, large blocklengths should be avoided. We then show that for a 400 Gb/s dual-polarization 64-QAM WDM transmission system, ESS with short blocklengths outperforms both uniform signaling and CCDM. Gains in terms of both bit-metric decoding rate and bit-error rate are presented. ESS with a blocklength of 200 is shown to provide an extension reach of about 200 km in comparison with CCDM with the same blocklength. The obtained reach increase of ESS with a blocklength of 200 over uniform signaling is approximately 450 km (approximately 19%)

Origin of optical losses in gallium arsenide disk whispering gallery resonators.

Whispering gallery modes in GaAs disk resonators reach half a million of optical quality factor. These high Qs remain still well below the ultimate design limit set by bending losses. Here we investigate the origin of residual optical dissipation in these devices. A Transmission Electron Microscope analysis is combined with an improved Volume Current Method to precisely quantify optical scattering losses by roughness and waviness of the structures, and gauge their importance relative to intrinsic material and radiation losses. The analysis also provides a qualitative description of the surface reconstruction layer, whose optical absorption is then revealed by comparing spectroscopy experiments in air and in different liquids. Other linear and nonlinear optical loss channels in the disks are evaluated likewise. Routes are given to further improve the performances of these miniature GaAs cavities.

Maximum Likelihood Sequence Detection Receivers for Nonlinear Optical Channels

The space-time whitened matched filter (ST-WMF) maximum likelihood sequence detection (MLSD) architecture has been recently proposed (Maggio et al., 2014). Its objective is reducing implementation complexity in transmissions over nonlinear dispersive channels. The ST-WMF-MLSD receiver (i) drastically reduces the number of states of the Viterbi decoder (VD) and (ii) offers a smooth trade-off between performance and complexity. In this work the ST-WMF-MLSD receiver is investigated in detail. We show that thespace compressionof the nonlinear channel is an instrumental property of the ST-WMF-MLSD which results in a major reduction of the implementation complexity in intensity modulation and direct detection (IM/DD) fiber optic systems. Moreover, we assess the performance of ST-WMF-MLSD in IM/DD optical systems with chromatic dispersion (CD) and polarization mode dispersion (PMD). Numerical results for a 10 Gb/s, 700 km, and IM/DD fiber-optic link with 50 ps differential group delay (DGD) show that the number of states of the VD in ST-WMF-MLSD can be reduced ~4 times compared to an oversampled MLSD. Finally, we analyze the impact of the imperfect channel estimation on the performance of the ST-WMF-MLSD. Our results show that the performance degradation caused by channel estimation inaccuracies is low and similar to that achieved by existing MLSD schemes (~0.2 dB).

Strong nonlinear saturation absorption-induced optical pinhole channel and super-resolution effects: a multi-layer system model.

This work focuses on the strong nonlinear saturation absorption (NSA)-induced optical super-resolution effect. A multi-layer system model is proposed to understand the strong NSA-induced formation of an optical pinhole channel and the generation of a super-resolution spot. Taking a Sb2Te3 thin film as an example, numerical simulations were conducted. The results illustrate that an optical pinhole channel is clearly formed by the NSA characteristics. This pinhole channel is similar to a near-field light probe. Light travels through the pinhole channel, and a super-resolution spot is generated at its apex. The near-field spot scanning experimental results show that the reduction ratio of the spot is approximately 44.8%, which is basically consistent with the numerical simulation result of 43%. This work is helpful for understanding optical nonlinear super-resolution effects and developing nanolithography, nanodata storage, high-resolution optical imaging technologies with nonlinear thin films.

Capacity of a Nonlinear Optical Channel With Finite Memory

The channel capacity of a nonlinear, dispersive fiber-optic link is revisited. To this end, the popular Gaussian noise (GN) model is extended with a parameter to account for the finite memory of realistic fiber channels. This finite-memory model is harder to analyze mathematically but, in contrast to previous models, it is valid also for nonstationary or heavy-tailed input signals. For uncoded transmission and standard modulation formats, the new model gives the same results as the regular GN model when the memory of the channel is about ten symbols or more. These results confirm previous results that the GN model is accurate for uncoded transmission. However, when coding is considered, the results obtained using the finite-memory model are very different from those obtained by previous models, even when the channel memory is large. In particular, the peaky behavior of the channel capacity, which has been reported for numerous nonlinear channel models, appears to be an artifact of applying models derived for independent input in a coded (i.e., dependent) scenario.

Information Transmission Using the Nonlinear Fourier Transform, Part III: Spectrum Modulation

Motivated by the looming "capacity crunch" in fiber-optic networks, information transmission over such systems is revisited. Among numerous distortions, inter-channel interference in multiuser wavelength-division multiplexing (WDM) is identified as the seemingly intractable factor limiting the achievable rate at high launch power. However, this distortion and similar ones arising from nonlinearity are primarily due to the use of methods suited for linear systems, namely WDM and linear pulse-train transmission, for the nonlinear optical channel. Exploiting the integrability of the nonlinear Schr\"odinger (NLS) equation, a nonlinear frequency-division multiplexing (NFDM) scheme is presented, which directly modulates non-interacting signal degrees-of-freedom under NLS propagation. The main distinction between this and previous methods is that NFDM is able to cope with the nonlinearity, and thus, as the the signal power or transmission distance is increased, the new method does not suffer from the deterministic cross-talk between signal components which has degraded the performance of previous approaches. In this paper, emphasis is placed on modulation of the discrete component of the nonlinear Fourier transform of the signal and some simple examples of achievable spectral efficiencies are provided.

Reduced Complexity MLSD Receivers for Nonlinear Optical Channels

We present a novel maximum likelihood sequence detection (MLSD) receiver structure for nonlinear channels. This scheme is derived by treating the NLC as a multiple input/multiple output system. Then, orthogonal signal components are computed using a special form of space-time whitened matched filter (ST-WMF) obtained by a modified Gram-Schmidt orthogonalization of the Volterra kernels of the NLC. The MLSD receiver consists of the ST-WMF followed by a Viterbi detector (VD) with multidimensional branch metrics. The space orthogonalization and noise whitening achieved by the ST-WMF provide an efficient way to reduce the receiver complexity in the presence of highly dispersive NLC. Complexity reduction is crucial in practical applications such as intensity modulation/direct detection (IM/DD) optical channels. As an example, the number of states of the VD in ST-WMF-MLSD required on a 10 Gb/s, 700 km, IM/DD fiber-optic link is reduced eight times compared with an oversampled MLSD.

All-optical intensity modulation of near infrared light in a liquid crystal channel waveguide

We demonstrate a nonlinear optical channel waveguide made of E7 nematic liquid crystal infiltrated in a silica on silicon groove. Near infrared light at the wavelength of 1560 nm fiber coupled to the core of the liquid crystal waveguide was optically modulated by an optical beam with power below 25 mW by exploiting the optical Freedericks transition. By modeling the optical molecular reorientation in the nematic liquid crystal confined in a waveguiding geometry we are able to reproduce the experimental results.

Estimation and equalization of fiber-wireless uplink for multiuser CDMA 4G networks

Fiber-wireless (Fi-Wi) access fronts can support 100s of Mb/s envisioned by 4G networks. However, a major issue associated with Fi-Wi links is the nonlinear distortion of the radio-over-fiber (ROF) link coupled with the multipath dispersion of the wireless channel. Estimation and subsequent equalization of the concatenated fiber-wireless channel needs to be done, especially at high bit rates. The uplink is severely affected due to large fluctuations in the radio signal. This paper proposes an estimation and subsequent equalization algorithm for the Fi-Wi CDMA uplink. The estimation employs the properties of pseudo noise (PN) sequences and the equalization uses a novel Hammerstein type decision feedback equalizer (HDFE). The estimation and equalization are performed in the presence of multiple access interference (MAI) and wireless and optical channel noise. The cumulative effects of multiuser interference, multipath dispersion, nonlinear distortion, and noise are all considered in our analysis. Correlation properties of white-noise like PN sequences enable decoupling of the linear (wireless) and nonlinear (optical) channel portions. Furthermore, we propose a unique algorithm to mitigate MAI. Numerical evaluations show a good estimation and equalization of both the linear and nonlinear channels. Bit error rate (BER) simulations show that this algorithm leaves only small residual MAI.

Principles for Electronic Equalization of Polarization-Mode Dispersion

System performance of optical fiber transmission in the presence of polarization-mode dispersion (PMD) can be improved by various types of electronic equalizers, which will be discussed in this paper. After a brief review of equalization and detection theory, equalizer concepts are adapted to the nonlinear optical channel and the architecture is optimized for high bit rate applications. Simulation results accounting for implementation losses in state-of-the-art technologies and realistic transmission conditions indicate that these concepts provide adequate solutions for equalization in lightwave transmission systems.

Combining nonlinear equalization and simple detection for high-density optical recording channels

This paper discusses new systems for nonlinear optical recording channels. The systems combine neural network structure into simple detectors such as the multilevel decision-feedback equalizer (MDFE) and the discrete matched filter (DMF). The latter (denoted as DFNE/DMF) provides powerful nonlinear tolerance, while the former (denoted as NMDFE) shows poor tolerance because of conditional training property in the MDFE. When compared with a partial response neural equalizer with maximum-likelihood (PRNE/ML), the proposed DFNE/DMF proves to be very hardware-efficient and able to support high data rates. Simulation results show that the DFNE/DMF increases bloom tolerance up to roughly 30% at a density of S = 6.

Fabrication technique of a nonlinear optical structure using optical polymeric films by direct electron-beam irradiation

A simple fabrication technique of nonlinear optical structures for use with dye-doped polymer is described. Polymethylmethacrylate, U-100 polymer, and polystyrene were used as the host matrices to fabricate the nonlinear optical waveguide. The periodically poled nonlinear optical polymer structures and ridge-type channel structures were fabricated by direct electron-beam irradiation. The electron beam with 25 kV of energy was exposed directly onto the polymer films containing the nonlinear optical chromophores. We can also demonstrate the fabrication technique of the domain-inverted grating of dye-doped polystyrene film.