Autoencoder assisted subcarrier optimization for nonlinear frequency division multiplexing.

Nonlinear frequency division multiplexing (NFDM) is a novel optical communication technique that can achieve nonlinear free transmission. However, current design of NFDM is analogous to orthogonal frequency division multiplexing (OFDM), where sinc function is utilized as subcarriers, which may not be optimal for nonlinear spectrums. In this paper, we propose an auto-encoder (AE) assisted subcarrier optimization scheme for dual-polarized (DP) NFDM systems. Numerical verifications show that our scheme can improve the Q-factor by 1.54 dB and 0.62 dB compared to sinc subcarrier and linear minimum mean square error (LMMSE) equalization, respectively, in a 960 km transmission scenario. We also analyze the characteristics of the optimized subcarriers and discuss how they enhance the performance. Furthermore, we demonstrate the robustness of the optimized subcarriers to different modulation formats, transmission distances and bandwidth. Our work provides a new idea in subcarrier design for NFDM.

Physical layer encryption for coherent PDM system based on polarization perturbations using a digital optical polarization scrambler.

In this paper, a security enhanced physical layer encryption scheme is proposed for coherent optical polarization division multiplexing (PDM) systems. The concept of a digital optical polarization scrambler (DOPS) is introduced to apply high speed rotation of state of polarization (RSOP) to the transmitted signal, which enables encryption based on polarization perturbations and offers superior flexibility in polarization management. By utilizing different combinations of digital polarization device matrices and adjusting their key parameters, four encryption modes are designed. The proposed encryption scheme is successfully implemented in a PDM-QPSK system at the data rate of 32 Gbps. Experimental results demonstrate that authorized users can successfully decrypt the received signal, while the eavesdroppers cannot derive useful information with a bit error rate (BER) at approximately 0.5. To enhance system security, a 5-D chaotic system is introduced with superior properties of high sensitivity to initial values and improved uniform distribution, which guarantees the large entropy and further the system's security. This scheme can effectively prevent against brute attacks with the expanded key space of 1060.

ASE noise mitigation with digital frequency offset loading for discrete spectrum nonlinear frequency division multiplexing systems.

We propose an amplified spontaneous emission (ASE) noise mitigation scheme utilizing digital frequency offset loading (DFO-loading) for discrete spectrum nonlinear frequency division multiplexing (DS-NFDM) systems. Firstly, based on the one-to-one mapping relationship between frequency offsets and eigenvalue positions, the transmitter side encodes 4-bit information onto 16 kinds of different digital frequency offsets. Then, a sliding window-assisted eigenvalue position (SWA-EP) decoding technology is further proposed to substitute the classical channel equalization and carrier phase recovery processes, with the purpose of recovering the original information. The numerical and experimental results demonstrate that, compared with b-coefficient 16 quadrature amplitude modulation (QAM) scheme, Q-factor gains are 2.1 dB under 15 dB optical signal-to-noise ratio (OSNR) and 1.8 dB after 800 km fiber transmission, respectively. Moreover, its complexity is only 0.6% of the b-coefficient scheme. The DFO-loading scheme offers an effective and low-complexity way to mitigate ASE noise of DS-NFDM system.

Kalman filter polarization demultiplexing algorithm based on diagonalized matrix treatment.

When we implement the equalizations of polarization effects using a Kalman filter (KF) in a coherent optical fiber communication system, we will require to multiply many matrices. If the state vector describing the system has a dimension of n, the state error covariance matrix P will have the dimension of n × n, and other matrices used in the Kalman filter will also have the dimension of n × l (l is the dimension of the measurement vector). If n is very large, the KF-based algorithm will suffer from significant complexity, which results in an impractical KF-based polarization demultiplexing algorithm. In this paper, we propose a new structured KF-based polarization demultiplexing algorithm in which the state error covariance matrix P is diagonalized, which we call the diagonalized Kalman filter (DKF). We theoretically analyze the rationality of the DKF, and the validity of the DKF was verified in both 64 Gbaud polarization-division multiplexed (PDM) QPSK and 16QAM Nyquist coherent optical simulation systems. Compared with the conventional KF, simulation results proved that under a rotation of state of polarization from 1 to 10 Mrad/s for QPSK and 1 to 5 Mrad/s for 16QAM, a differential group delay from 15 to 75 ps, and a residual chromatic dispersion of 100 ps/nm, the OSNR penalties for the DKF are only within 0.5 dB for QPSK at the threshold BER = 3.8 × 10-3, and within 2 dB for 16-QAM at the threshold BER = 2 × 10-2, respectively, compare to the case of no impairment. In the meantime, for the proposed DKF, a computational complexity reduction of over 30% is achieved, compared with conventional KF, at the expense of about no more than 50 symbols convergence delay.

Improvement for a full-spectrum modulated nonlinear frequency division multiplexing transmission system.

Nonlinear frequency division multiplexing (NFDM), as a possible technique to overcome the limit imposed by Kerr nonlinearity in conventional coherent optical communication systems, has attracted widespread attention in the communication community in recent years. In order to fully utilize the available degrees of freedom in the nonlinear spectrum, this paper focuses on the full-spectrum (FS) modulated NFDM system. First, we maximize the data rate of discrete spectrum (DS) by optimizing the distribution of eigenvalues in DS part of FS. Then through introducing the probabilistic shaping (PS) into the FS system, and combined with linear minimum mean square (LMMSE) estimators, a 1120 km transmission with BER below the hard decision forward error correction (HD-FEC) threshold at 112 Gbps is achieved, where 128 subcarriers with PS-64QAM are used in the continuous spectrum (CS) and 13 eigenvalues with 64QAM are adopted in the discrete spectrum (DS). The achievable data rate is about 12% higher than that of pure CS modulation. Our work achieves the current FS NFDM system with the largest number of multiplexed eigenvalues, and provides a way to improve the performance of FS systems.

Noise equalization scheme based on complex-valued ANN for multiple-eigenvalue modulated nonlinear frequency division multiplexing systems.

In multiple-eigenvalue modulated nonlinear frequency division multiplexing (NFDM) systems, the noise degrades the accuracy of the nonlinear Fourier transform (NFT) algorithm, resulting in perturbations in the received eigenvalues and the corresponding discrete spectrum. Moreover, with the increase in the number of eigenvalues and the order of the modulation formats, the impact of noise on the performance of the system is even more. A noise equalization scheme based on complex-valued artificial neural network (c-ANN) for multiple-eigenvalue modulated NFDM systems is proposed. This sceheme inputs the eigenvalues perturbation and the impaired discrete spectrum corresponding to the eigenvalues into the c-ANN in complex form. The scheme constructs a complex-valued logic structure with both amplitude and phase information, overlapping reuse input features and, thereby, effectively reducing the effect of noise on the multiple-eigenvalue NFDM system. The effectiveness of the scheme is verified in long-haul seven-eigenvalue modulated single-polarization NFDM simulation systems with 1 GBaud 16APSK/16QAM/64APSK/64QAM modulation formats, and the results show that the scheme outperforms the NFT receiving without equalization by 1 to 2 orders of magnitude in terms of bit error rate (BER). Among them, the transmission distance of the 64APSK signal after equalization exceeds 800 km while the BER meets 7% forward error correction (FEC) threshold, which is 600 km longer than that of the disequilibrium case, and the spectral efficiency (SE) can reach 1.85 bit/s/Hz. Compared with other schemes, the proposed scheme has better equalization performance under the same complexity, and the complexity can be reduced by half or even under the same performance.

Modulation format identification using the Calinski-Harabasz index.

Modulation format identification (MFI) is a key technology in optical performance monitoring for the next-generation optical network, such as the intelligent cognitive optical network. An MFI scheme based on the Calinski-Harabasz index for a polarization-division multiplexing (PDM) optical fiber communication system is proposed. The numerical simulations were carried out on a 28 Gbaud PDM communication system. The results show that the required minimum optical signal-to-noise ratio values of each modulation format to achieve 100% identification accuracy are all equal to or lower than their corresponding 7% forward error correction thresholds, and the proposed scheme is robust to residual chromatic dispersion. Meanwhile, the proposed scheme was further verified by 20 Gbaud PDM-QPSK/16QAM/32QAM long-haul fiber transmission experiments. The results show that the scheme has a good reliability when fiber non-linear impairments exist. In addition, the complexity of the scheme is significantly lower than that of other clustering-based MFI schemes.

Nonlinear frequency domain PMD modeling and equalization for nonlinear frequency division multiplexing transmission.

Polarization mode dispersion (PMD) is one of the fundamental properties of a standard single-mode fiber. It affects the propagating signals and degrades the performance of high-speed optical fiber communication systems. PMD also gives an effect on the nonlinear spectra or scattering data in nonlinear frequency division multiplexing (NFDM) systems. However, PMD is usually described in the linear frequency domain, and there are few investigations about the influence of PMD in the nonlinear frequency domain (NFD). An NFD-PMD model is needed to understand the impact of PMD in the NFD. In this work, using a linear approximation method, we first propose an NFD-PMD model and verify its effectiveness. With the guide of the NFD-PMD model, a blind NFD-PMD equalization scheme is designed. The simulation results indicate that the proposed NFD-PMD equalization scheme has better performance than the training sequence method based on linear frequency domain equalization.

Physical layer encryption scheme based on cellular automata and DNA encoding by hyper-chaos in a CO-OFDM system.

This study proposes an encryption scheme combining cellular automata (CA) and DNA encoding to improve security of a coherent optical orthogonal frequency division multiplexing (CO-OFDM) system, wherein key sequences are generated with good randomness and unpredictability by a 4-dimensional hyper-chaotic system. A base scrambling pseudo random binary sequence (PRBS) generated by the CA is introduced, which results in better scrambling effect and randomness in the conventional complex DNA encoding. The randomness, complexity and security of the system is enhanced due to 6 variable keys (key space of ∼10138). An experiment conducted in a 40 GHz 16QAM CO-OFDM system over an 80 km standard single mode fiber (SSMF) shows that the authorized user can successfully decrypt the received signal, while the eavesdroppers cannot derive useful information with bit error rate (BER) at approximately 0.5. An allowable optical signal to noise ratio (OSNR) penalty of 0.5 dB will be introduced to achieve same BER before and after encryption due to the error propagation of cellular automata.

Probabilistic shaping and neural network-based optimization for a nonlinear frequency division multiplexing system.

A joint scheme introducing probabilistic shaping (PS) at the transmitter and utilizing a neural network (NN) equalizer at the receiver is proposed to improve the performance of the b-modulated nonlinear frequency division multiplexing (NFDM) system. Through a numerical simulation, we demonstrate that PS plays a leading role for low launch power case, which improves the performance of the system effectively, while the NN equalizer's superiority appears in a high launch power region, whose main role is to weaken the correlation among subcarriers for improving system performance. The proposed scheme would enlighten the optimum modulation and detection schemes of the NFDM system.

Cross-coupling effect induced beam shifts for polarized vortex beam at two-dimensional anisotropic monolayer graphene surface.

We investigated beam shifts for an arbitrarily polarized vortex beam reflected and transmitted at two-dimensional (2D) anisotropic monolayer graphene surface. And generalized expressions are theoretically derived for calculating beam shifts of vortex beam. Then, we presented the beam shifts associated with the self-isotropic (SI) effect, self-anisotropic (SA) effect and cross-coupling (XC) effect originated from self-isotropic interaction, self-anisotropic interaction and cross-coupling interaction between isotropic and anisotropic of two-dimensional media, respectively. More importantly, novel optical phenomena resulting from the XC effect are flexibly shown by manipulation OAM. We believe that our results can be extensively extended to 2D anisotropic Dirac semimetals and Weyl semimetals, and expect the results to be significant and contribute to the understanding of the spin and orbit Hall effect of the light.

Robust neural network receiver for multiple-eigenvalue modulated nonlinear frequency division multiplexing system.

Nonlinear frequency division multiplexing (NFDM) has been shown to be promising in overcoming the fiber Kerr nonlinearity limit. In multiple-eigenvalue modulated NFDM systems, the transmission capacity increases with the number of modulated eigenvalues. However, as the number of modulated eigenvalues increases, the complexities of the signal waveform and the nonlinear Fourier transform (NFT) algorithm for demodulation increase dramatically as well, while the accuracy drops significantly. Meanwhile, impairments such as amplifier spontaneous emission noise and phase noise in practical channels would perturb the eigenvalues and the corresponding nonlinear spectra during transmission. Coupled with an increase in the modulation format order, it is difficult for NFT algorithm-based receivers to recover information. To enable the use of multiple-eigenvalue modulated NFDM systems, we propose an innovative receiver based on regression neural networks (NNs), which can demodulate information correctly for both single- and dual-polarization NFDM systems. The results show that it has strong robustness and has a certain tolerance to the impairments of communication systems. In the contrast that the poor demodulation performance of the NFT and the Euclidean minimum distance (MD) receivers for multi-eigenvalue modulated NFDM systems, our proposed NN receiver can achieve low bit error rate with 2 GBaud 16QAM modulation over 1,000 km transmission in four-eigenvalue modulated single-polarization NFDM systems. The performance of three receivers (NFT, MD and NN) in a two-eigenvalue modulated NFDM system are also compared, the NN receiver shows the best performance and appears more suitable for higher-order modulation formats.

Symmetric spin splitting of elliptically polarized vortex beams reflected at air-gold interface via pseudo-Brewster angle.

A simple expression of the transverse spatial spin splitting of light-carrying intrinsic orbital angular momentum (IOAM) is theoretically derived for reflections at strong absorbing media surfaces. By introducing an asymmetric spin splitting (ASS) factor, the transverse spatial symmetric spin splitting (SSS) and ASS of an arbitrary polarized vortex beam can be distinguished. Here, the transverse spatial SSS of an elliptically polarized vortex beam with a phase difference of 90° is predicted when the incident angle is close to the pseudo-Brewster angle. Remarkably, the larger transverse spatial SSS reaches 1100 nm for the incident circularly polarized LG beam with l=3. It is noteworthy that the transverse spatial SSS can be flexibly manipulated by changing the polarized angle, meaning it is theoretically possible to realize fully polarization-controllable transverse spatial SSS for elliptically polarized incident vortex beams. These results could potentially be applied to precision polarization metrology and edge-enhanced imaging.

Nonlinear-frequency-packing nonlinear frequency division multiplexing transmission.

A nonlinear frequency division multiplexing (NFDM) transmission system, designed specifically for nonlinear fiber channel, has the potential to overcome the nonlinear Shannon capacity limit. However, the spectral efficiency (SE) of the current proven NFDM transmission systems is still lower than that of the analogous orthogonal frequency division multiplexing system. It is extremely necessary to explore effective modulation scheme for the aim of increasing the SE of NFDM system. In this study, we first propose the nonlinear-frequency-packing nonlinear frequency division multiplexing (NFP-NFDM) transmission system. In NFP-NFDM, the spacing of nonlinear subcarriers is squeezed and more nonlinear subcarriers can be packed, but the inter carrier interference (ICI) is introduced. The method of NFP in nonlinear Fourier domain is carefully designed to reduce the complexity of ICI cancellation. Through numerical simulation, we illustrate the feasibility of NFP-NFDM transmission, and higher SE in NFP-NFDM than that of NFDM system is also demonstrated. The upper bound of the normalized SE for NFP-NFDM is estimated, which is higher than that of current NFDM system. Besides, we find out that the NFP scheme may have the advantage of reducing the signal-noise interaction in fiber transmission scenario, which indicates there may be a better way to load the data into the nonlinear Fourier domain.

Frequency offset estimation for nonlinear frequency division multiplexing with discrete spectrum modulation.

Although fruitful studies have been conducted on carrier frequency offset (CFO) estimations in linear coherent optical fiber communication systems, there are few studies on CFO estimations and recoveries in the systems based on the nonlinear Fourier transform (NFT). Although the CFO is originated from the linear frequency domain, it definitely has effects on nonlinear spectra, including the shift of the nonlinear frequency and the phase rotations of the scattering data, which are similar to its effects on linear spectra. This work indicates that it is feasible to estimate frequency offset (FO) by capturing symbol variations in the nonlinear frequency domain (NFD) rather than in the linear frequency domain; the latter was usually exploited in the literature. Based on a thorough investigation of the FO induced behavior that appears in a nonlinear frequency division multiplexing (NFDM) system, we proposed a nonlinear frequency domain estimation method aided by training symbols (TS) using an angle search algorithm after NFT operations at the receiver. The discussions in this paper prove that the proposed method is generally applicable to the NFDM systems regardless of whether using single or multiple eigenvalues. A performance comparison between the NFD method and the conventional method in the linear frequency domain is performed with different modulation formats for both single and multiple eigenvalue NFDM transmission systems. The analysis results show that the proposed method holds the better stability and estimation accuracy in contrast with the linear domain estimation method. The TS overhead can also be deduced dramatically, which implies better transmission efficiency. Therefore, the NFD method is more powerful for eigenvalue NFDM transmission systems, especially for the scenarios where high order modulation formats and multiple eigenvalues are utilized.

Micro-fiber Mach-Zehnder interferometer based on ring-core fiber.

A micro-fiber Mach-Zehnder interferometer (MZI), with a thousands-µm-long ring-core fiber (RCF), is demonstrated, and its performance investigation is also implemented. In this paper, the proposed MZI is manufactured by ends-splicing the short RCF segment with single-mode fiber (SMF-28), respectively. The scheme of the MZI is a typically core-mismatch structure, which has the advantages of miniaturization and simplification. Due to the core mismatch between RCF and SMF, the light from the SMF can be well separated into ring core (RC) and silica center (SC) of the RCF at the first splicing point. After transmitting through the RC and SC, the two separated light beams encounter each other and interfere at the second splicing point. Different from conventional micro-fiber MZIs using SMFs or few-mode fibers, the RCF has a higher numerical aperture, which can generate a larger optical path-length difference with a short length fiber, accumulates a higher extinction ratio and suppresses the crosstalk between the core and cladding modes. Therefore, our proposed MZI is more stable and the best extinction ratios can reach up to 18.2 dB. Meanwhile, owing to the core structure of RCF (where SC is surrounded by high-index ring core), the power propagating through low-index area of RCF is mostly confined into SC (termed the silica-center modes). These characteristics would lead to the lower sensitivity to external disturbances.

Joint blind equalization of CD and RSOP using a time-frequency domain Kalman filter structure in Stokes vector direct detection system.

We propose a joint blind equalization method for chromatic dispersion (CD) and ultra-fast rotation of state-of-polarization (RSOP) in a Stokes vector direct detection (SV-DD) system based on a new time-frequency domain Kalman filter structure. In an SV-DD system, the impairments induced by CD and RSOP possess a nonlinear form. Therefore, CD and RSOP cannot be treated sequentially, which causes difficulty in jointly equalizing these two impairments using an ordinary algorithm. The Kalman filter was proven to be effective in equalizing polarization effects in a coherent receiver. However, this approach has inherent limitations given that the Kalman filter was originally presented as a method implemented in the time domain whereas CD is eventually induced in the frequency domain. In this report, the proposed time-frequency domain Kalman filter can facilitate CD compensation in the frequency domain and RSOP equalization in the time domain by exploiting a sliding window structure. Both the CD compensation and the RSOP equalization are conducted in Stokes space when the proposed method is utilized, which is specially designed for an SV-DD system. The presented approach was checked using a 28 Gbaud 16-QAM SV-DD system simulation platform. The simulation results confirm that the method is very effective and has strong tolerance to CD (more than 2550 ps/nm, equivalent to a 150 km G. 652 fiber) combined with ultra-fast RSOP (up to 2 Mrad/s) for application in extreme polarization environments, like the transient lightning in a rainy day.

Joint equalization scheme of ultra-fast RSOP and large PMD compensation in presence of residual chromatic dispersion.

Polarization demultiplexing is generally carried out by a multiple-input multiple-output (MIMO) based algorithm in polarization division multiplexing (PDM) coherent systems. However, in some extreme environments, the MIMO algorithm becomes inapplicable due to the ultra-fast rotation of the state of polarization (RSOP) and large polarization mode dispersion (PMD). In addition, the residual chromatic dispersion (RCD) is always present because of the mismatch of the compensated chromatic dispersion and real value induced in the optical fiber channel. According to the literature, the Kalman filter-based polarization demultiplexing algorithms possess very weak RCD tolerance. Faced with this dilemma, in this paper, a new Kalman filter structure is proposed, which can jointly compensate ultra-fast RSOP, large PMD and RCD. This Kalman filter structure enables the equalization of the RSOP in the time domain and compensation for RCD and PMD in the frequency domain. We verified the performance of the proposed Kalman scheme in the 28 Gbaud PDM-QPSK/16 QAM coherent system, with a comparison to constant modulus algorithm/multiple modulus algorithm (CMA/MMA). The simulation results confirm that, compared with CMA/MMA, the proposed Kalman scheme can provide a significant performance enhancement to cope with ultra-fast RSOP (up to 3 Mrad/s) and large PMD (more than 200 ps) with a large tolerance to RCD (over the range of ± 820 ps/nm in PDM-QPSK and ± 500 ps/nm in PDM-16 QAM).

High sensitivity micro-fiber Mach-Zehnder interferometric temperature sensors with a high index ring layer.

The influence of the high index ring layer (HIRL) in a tapered fiber Mach-Zehnder interferometer (MZI) on the interference observed, and thus on its potential applications in temperature sensing, has been investigated. The MZI was comprised of a tapered Ring Core Fiber (RCF), spliced between two single mode fibers (SMF). Since part of core mode from the SMF was converted into cladding modes in the RCF, due to the mismatch in the cores between the RCF and SMF, the residual power enters and then propagates along the center of the RCF (silica). The difference in phase between the radiation travelling along these different paths is separated by the HIRL to generate an interference effect. Compared with fiber interferometers based on core and cladding mode interference, the thin fiber HIRL is capable of separating the high order cladding modes and the silica core mode, under grazing incident conditions. Therefore, the optical path difference (OPD) and the sensitivity are both substantially improved over what is seen in conventional devices, showing their potential for interferometric temperature sensor applications. The optimum temperature sensitivity obtained was 186.6 pm/°C, which is ∼ 11.7 times higher than has been reported previously.

All-fiber broadband multiplexer based on an elliptical ring core fiber structure mode selective coupler.

A novel all-fiber broadband mode multiplexer based on an elliptical ring core fiber structure mode selective coupler (MSC) is proposed and analyzed numerically. The MSC is composed of a core-doped single-mode fiber and a polarization-maintaining elliptical ring-shaped core few-mode fiber (FMF). The mode multiplexing can be implemented by a single LP01 mode excitation that does not introduce insertion loss, which can simplify and promote the multiplexer. The high refractive index difference between the elliptical annular region and the cladding of the FMF, as well as the ellipticity of the ring-shaped core of the FMF could not only decompose the degenerate modes, but also reduce the mode coupling of the adjacent spatial modes to eliminate the multiple-input-multiple-output digital-signal processing in the receiver of the mode division multiplexing system. Finally, a seven-mode multiplexer is modeled and optimized for operation covering the C+L band. Furthermore, the multiplexer provides a high coupling efficiency (>64% for the worst case) for the seven spatial modes, LP01, LP11a, LP11b, LP21a, LP21b, LP31a, and LP31b, and relatively high mode extinction ratios of more than 15 dB in the C-band.