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Impact of QPSK, star 8-QAM, and 64-QAM modulation formats on the NSD versus input power The system consists of an 80 km NZDSF with 10 Gbaud symbol rate. The modulation format impact is measured by the power gap in dB between the crossing points of the NSD curves for each modulation format with the horizontal black curve representing a constant 0.1% NSD.
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Communication using the optical fibre channel can be challenging due to nonlinear effects that arise in the optical propagation. These effects represent physical processes that originate from light propagation in optical fibres. To obtain fundamental understandings of these processes, mathematical models are typically used. These models are based o...
Citations
... An optical pulse as it propagates in a single mode fiber has been governed via the nonlinear Schrödinger equation [57,58]. Another type of the optical pulse that propagates through a two-level resonant medium without any loss or distortion has been known as the self-induced transparency soliton, modelled via the Maxwell-Bloch system [59][60][61]. ...
In this paper, we concentrate on the higher-order nonlinear Schrödinger-Maxwell-Bloch system with the sextic terms, which could characterize the ultra-short optical pulses in an erbium-doped fiber. Proceeding from the existing Lax pair and one-fold Darboux transformation (DT), we build an N-fold generalized DT with one spectral parameter by means of the limit procedure, and on this basis determine the Nth-order solutions of that system. The second- and third-order degenerate solitons are shown through the second-order and third-order solutions, respectively, and we also present the second-order degenerate breather through the second-order solutions. We obtain the eye-shaped rogue wave involving one hump and two valleys, rogue wave involving four valleys, as well as four-petaled rogue wave involving two humps and two valleys via the first-order solutions. Using the second-order solutions, we obtain the interaction between the two first-order rogue waves and show that the second-order rogue wave divides into three first-order rogue waves which are arranged in the triangle structure. Modifying that generalized DT, we work out the second-order and third-order mixed wave solutions, and then show the interactions between the first-order/second-order rogue wave and first-order breather.
... However, if the wavelength converter is operated in the condition where nonlinear noise generation is not negligible, additional characterization of the nonlinear noise is required to estimate system performance accurately. Although GN model [19] consider the impact of nonlinear noise, it should be noted that the nonlinear noise generated in wavelength converter is not necessarily a result of statistical randomization due to chromatic dispersion [20]. ...
Multi-band WDM transmission beyond the C+L-band is a promising technology for achieving larger capacity transmission by a limited number of installed fibers. In addition to the C- and L-band, we can expect use the S-band as the next band. Although the development of optical components for new bands, particularly transceivers, entails resource dispersion, which is one of the barriers to the realization of multi-band systems, wavelength conversion by transparent all-optical signal processing enables new wavelength band transmission using existing components. Therefore, we proposed a transmission system including a new wavelength band such as the S-band and made it possible to use a transceiver for the existing band by performing the whole-band wavelength conversion without using a transceiver for the new band. As a preliminary verification to demonstrate multi-band WDM transmission including S-band, we investigated the application of a novel wavelength converter between C-band and S-band, which consists of periodically poled lithium niobate waveguide, to the proposed system. We first characterized the conversion efficiency and noise figure of the wavelength converter and estimated the transmission performance of the system through the wavelength converter. Using the evaluated wavelength converters and test signals of 64 channels arranged in the C-band at 75-GHz intervals, we constructed an experimental setup for S-band transmission through an 80-km standard single-mode fiber. We then demonstrated error-free transmission of real-time 400-Gb/s DP-16QAM signals after forward error correction decoding. From the experimental results, it was clarified that thewavelength converter which realizes the uniformlossless conversion covering the whole C-band effectively achieves the S-band WDM transmission, and it was verified that the capacity improvement of the multi-band WDM system including the S-band can be expected by applying it in combination with the C+L-band WDM system.
... In contrast to the numerical approach of DBP, the solution of the Manakov equation can also be analytically approximated using regular perturbation theory [19]- [24]. The perturbation theory-based recursive closed-form approximate solution is in a power series form, and it converges to the exact solution asymptotically [20]. Intending to design a computationally efficient digital NLC technique, the perturbation series truncated to first-order (FO) has been well investigated in the literature for the post-compensation of the nonlinearity effect [24]. ...
... For the triplets-based techniques, we start with a sufficiently large number of triplet features to assess the maximal performance. In particular, the window size N w = 75 is selected, and the number of triplets obtained after applying the truncation criterion in (20)is n t = 1681, and n IFWM = 1378, n ICIXPM = 151, n CIXPM = 151, and n SPM = 1. Similarly, for FL PB-NLC, a bi-LSTM with u = 30 hidden units is used. ...
Fiber nonlinearity effects cap achievable rates and ranges in long-haul optical fiber communication links. Conventional nonlinearity compensation methods, such as perturbation theory-based nonlinearity compensation (PB-NLC), attempt to compensate for the nonlinearity by approximating analytical solutions to the signal propagation over optical fibers. However, their practical usability is limited by model mismatch and the immense computational complexity associated with the analytical computation of perturbation triplets and the nonlinearity distortion field. Recently, machine learning techniques have been used to optimise parameters of PB-based approaches, which traditionally have been determined analytically from physical models. It has been claimed in the literature that the learned PB-NLC approaches have improved performance and/or reduced computational complexity over their non-learned counterparts. In this paper, we first revisit the acclaimed benefits of the learned PB-NLC approaches by carefully carrying out a comprehensive performance-complexity analysis utilizing state-of-the-art complexity reduction methods. Interestingly, our results show that least squares-based PB-NLC with clustering quantization has the best performance-complexity trade-off among the learned PB-NLC approaches. Second, we advance the state-of-the-art of learned PB-NLC by proposing and designing a fully learned structure. We apply a bi-directional recurrent neural network for learning perturbation triplets that are alike those obtained from the analytical computation and are used as input features for the neural network to estimate the nonlinearity distortion field. Finally, we demonstrate through numerical simulations that our proposed fully learned approach achieves an improved performance-complexity trade-off compared to the existing learned and non-learned PB-NLC techniques.
... Finally, we comment on modeling fiber transmission when approaching the zero-dispersion wavelength λ zd . When the chromatic dispersion approaches zero, the nature of the problem is modified, the perturbative approach must be modified as proposed in [56], and relevant parametric phenomena may impact propagation [20]. Moreover, in such a spectral region, considering the statistical variations of λ zd in cables [57] becomes crucial because of possible chromatic dispersion magnitude variations being comparable with its nominal value. ...
In the evolution of optical networks, enhancement of spectral efficiency (SE) enhancement has been the most cost-efficient and thus the main driver for capacity enhancementincrease for decades. As a result, the development of optical transport systems has been focused on the
$C$
-and
$L$
-bands, where silica optical fiber exhibits the lowest attenuation, and erbium-doped fiber amplifiers provide an efficient solution forto compensatinge for the optical loss. With a gradual maturity in the SE growth, however, the extension of the optical bandwidth beyond the
$C$
$+$
$L$
-band is expected to play a significant role in the future capacity upgrades of optical networks and, thus, attracting increasing research interests. In this article, we discuss the merits and challenges of ultrawideband optical transport systems and networks beyond conventional bands.
... Alternatively, the solution of the signal propagation equation can be analytically approximated using the first-order (FO) perturbation theory [20]. The NLC technique using this approximate analytical solution is referred to as the FO perturbation theory-based NLC (FO-PB-NLC) [21]- [32]. It is interesting to mention that there is an intrinsic relation between the time-domain perturbation theory and the frequency-domain Volterra series-based approach. ...
The first-order (FO) perturbation theory-based nonlinearity compensation (PB-NLC) technique has been widely investigated to combat the detrimental effects of the intra-channel Kerr nonlinearity in polarization-multiplexed (Pol-Mux) optical fiber communication systems. However, the NLC performance of the FO-PB-NLC technique is significantly limited in highly nonlinear regimes of the Pol-Mux long-haul optical transmission systems. In this paper, we extend the FO theory to second-order (SO) to improve the NLC performance. This technique is referred to as the SO-PB-NLC. A detailed theoretical analysis is performed to derive the SO perturbative field for a Pol-Mux optical transmission system. Following that, we investigate a few simplifying assumptions to reduce the implementation complexity of the SO-PB-NLC technique. The numerical simulations for a single-channel system show that the SO-PB-NLC technique provides an improved bit-error-rate performance and increases the transmission reach, in comparison with the FO-PB-NLC technique. The complexity analysis demonstrates that the proposed SO-PB-NLC technique has a reduced computational complexity when compared to the digital back-propagation with one step per span.
... One of the most popular approximations is based on a regular perturbation (RP) expansion on the nonlinear parameter γ and truncated to the first-order term. Considerable amount of validation work has been carried out on a waveform level [4], [2], [5], [6] for this first-order RP. However, the question of how this accuracy translates to the discrete-time domain has been, to the best of our knowledge, not well addressed in the literature. ...
The accuracy of a discrete-time channel model based on regular perturbation is numerically studied for unamplified links. We analyse the distance between discrete nonlinear interference points and show that such distance can be used to estimate the effective channel memory.
... RP and LP on γ cover the area under the yellow and blue curves in Fig. 1, respectively. Recently, we proposed RP on β 2 in [7] and compared with RP on γ. RP on β 2 provided more accuracy in the weakly dispersive and highly nonlinear regimes, represented by the area under the red curve in Fig. 1. ...
... We recently proposed the RP on β 2 in [7]. The same procedure as in (2) can be applied by considering A as a power series of β 2 , i.e., ...
... Analogously to RP on γ, the function A (β2) 0 in (10) for RP on β 2 is an accurate model when dispersion is negligible, and is called the NLPN model [3], [7]. The first-order RP on β 2 in (9) is accurate for low accumulated dispersion and is illustrated as the red curve in Fig. 1. ...
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.
... For the Manakov equation (Eq. (1)), perturbative methods describing the field evolution in the highly nonlinear regime [16]- [19] have been developed under some limiting assumptions. Conversely, we suggest adding trainable general nonlinearity functions to the memory-less solution provided by Eq. (2): additions Θ 1 , Θ 2 to the exponential power, responsible for phase distortion, and additions Ξ 1 , Ξ 2 , responsible for amplitude distortion. ...
Nonlinearity compensation is considered as a key enabler to increase channel transmission rates in the installed optical communication systems. Recently, data-driven approaches — motivated by modern machine learning techniques — have been proposed for optical communications in place of traditional model-based counterparts. In particular, the application of neural networks (NN) allows improving the performance of complex modern fiber-optic systems without relying on any a priori knowledge of their specific parameters. In this work, we introduce a novel design of complex-valued NN for optical systems and examine its performance in standard single mode fiber (SSMF) and large effective-area fiber (LEAF) links operating in relatively high nonlinear regime. First, we present a methodology to design a new type of NN based on the assumption that the channel model is more accurate in the nonlinear regime. Second, we implement a Bayesian optimizer to jointly adapt the size of the NN and its number of input taps depending on the different fiber properties and total length. Finally, the proposed NN is numerically and experimentally validated showing an improvement of 1.7 dB in the linear regime, 2.04 dB at the optimal optical power and 2.61 at the max available power on Q-factor when transmitting a WDM 30×200G DP-16QAM signal over a 620 km SSMF legacy link. The results highlight that the NN is able to mitigate not only part of the nonlinear impairments caused by optical fiber propagation but also imperfections resulting from using low-cost legacy transceiver components, such as digital-to-analog converter (DAC) and Mach-Zehnder modulator.
... In [16,17] the NLI was modelled by manipulating the non-linear Schrödinger equation (NLSE) in the frequency domain as a Four-Wave Mixing (FWM)-like noise, approaching the problem from a spectrally aggregated perspective. Theoretical approaches to the calculation of NLI are still being developed -we highlight in particular [18], which presents an analytical solution to the NLSE using a regular perturbation. In [19][20][21][22] the self-channel (SC) and cross-channel (XC) components of the NLI are treated separately, considering the NLI generation in the time domain and observing collisions among pulses. ...
Lightpaths within optical line systems (OLS)s that deploy coherent optical technologies are mainly impaired by two additive Gaussian disturbances: the amplified spontaneous emission (ASE) noise from the optical amplifiers and the non-linear interference (NLI) from fiber propagation, together with some amount of phase noise, typically compensated for by the carrier phase estimator module within the digital signal processing (DSP) unit. The main obstacle in accurately modelling the physical layer of a disaggregated optical network arises from the spatially-coherent and spectrally-aggregated general behavior of the NLI generation. Within this paper, we perform an accurate split-step Fourier method (SSFM) physical layer simulation campaign over a wide range of fiber chromatic dispersion values that range from 2 to 16.7 ps / (nm·km) and channel symbol rates from 32 GBd to 85 GBd. For all the explored scenarios, we first show that the NLI generation in an OLS can be spectrally disaggregated in a practical manner by considering a superposition of self-channel (SC) and cross-channel (XC) NLI components only. Secondly, by considering the span-by-span generalized signal-to-noise ratio (GSNR) deterioration, we show that the XC-NLI accumulation components can also be considered as spatially disaggregated, leaving the SC-NLI as the only spatial coherency contribution. Consequently, by appropriately managing these coherent NLI contributions, we find that it is possible to produce a conservative physical layer model that is both spectrally and spatially disaggregated.
... Whereas for simple transmission channels, such as additive white Gaussian noise (AWGN) channels, it is possible to analytically derive optimal modulation and detection strategies, such an optimization is particularly challenging for the nonlinear optical channel. As a closed-form expression to describe the signal propagation through an optical fiber is not available, current research relies on approximate analytical or numerical models achieving only a limited degree of accuracy [5]. Even following this direction though, no definitive answer to optimal modulation and detection strategies is know. ...
Optimizing modulation and detection strategies for a given channel is critical to maximize the throughput of a communication system. Such an optimization can be easily carried out analytically for channels that admit closed-form analytical models. However, this task becomes extremely challenging for nonlinear dispersive channels such as the optical fiber. End-to-end optimization through autoencoders (AEs) can be applied to define symbol-to-waveform (modulation) and waveform-to-symbol (detection) mappings, but so far it has been mainly shown for systems relying on approximate channel models. Here, for the first time, we propose an AE scheme applied to the full optical channel described by the nonlinear Schr\{"o}dinger equation (NLSE). Transmitter and receiver are jointly optimized through the split-step Fourier method (SSFM) which accurately models an optical fiber. In this first numerical analysis, the detection is performed by a neural network (NN), whereas the symbol-to-waveform mapping is aided by the nonlinear Fourier transform (NFT) theory in order to simplify and guide the optimization on the modulation side. This proof-of-concept AE scheme is thus benchmarked against a standard NFT-based system and a threefold increase in achievable distance (from 2000 to 6640 km) is demonstrated.
