Xiaojun Liang

Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links

A recursive perturbation theory to model the fiber-optic system is developed. Using this perturbation theory, a multi-stage compensation technique to mitigate the intra-channel nonlinear impairments is investigated. The technique is validated by numerical simulations of a single-polarization single-channel fiber-optic system operating at 28 Gbaud, 32-quadrature amplitude modulation (32-QAM), and 40 × 80 km transmission distance. It is found that, with 2 samples per symbol, the multi-stage scheme with eight compensation stages increases the Q-factor as compared with linear compensation by 4.5 dB; as compared with single-stage compensation, the computational complexity is reduced by a factor of 1.3 and the required memory for storing perturbation coefficients is decreased by a factor of 13.

Ideal optical backpropagation of scalar NLSE using dispersion-decreasing fibers for WDM transmission

An ideal optical backpropagation (OBP) scheme to compensate for dispersion and nonlinear effects of the transmission fibers is proposed. The scheme consists of an optical phase conjugator (OPC), N spans of dispersion-decreasing fibers (DDFs) and amplifiers, placed at the end of the fiber optic link. In order to compensate for the nonlinear effects of the transmission fibers exactly, the nonlinear coefficient of the backpropagation fiber has to increase exponentially with distance or equivalently the power in the backpropagation fiber should increase exponentially with distance if the nonlinear coefficient is constant. In this paper, it is shown that a combination of DDFs and amplifiers can compensate for the nonlinear effects exactly. An analytical expression for the dispersion profile of the DDF is derived. Numerical simulation of a long haul wavelength division multiplexing (WDM) fiber optic system with the proposed OBP scheme shows that the system reach can be enhanced by 54% as compared to digital backpropagation (DBP).

Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems

A digital compensation scheme based on a perturbation theory for mitigation of cross-phase modulation (XPM) distortions is developed for dispersion-managed fiber-optic communication systems. It is a receiver-side scheme that uses a hard-decision unit to estimate data for the calculation of XPM fields using the perturbation technique. The intra-channel nonlinear distortions are removed by intra-channel digital backward propagation (DBP) based on split-step Fourier scheme before the hard-decision unit. The perturbation technique is shown to be effective in mitigating XPM distortions. However, wrong estimations in the hard-decision unit result in performance degradation. A hard-decision correction method is proposed to correct the wrong estimations. Numerical simulations show that the hybrid compensation scheme with DBP for dispersion and intra-channel nonlinear impairments compensation and the perturbation technique for XPM compensation brings up to 3.7 dBQ and 1.7 dBQ improvements as compared with the schemes of linear compensation only and intra-channel DBP, respectively. The perturbation technique for XPM compensation requires only one-stage (or two-stage when hard-decision correction is applied) compensation and symbol-rate signal processing.

Analytical modeling of cross-phase modulation in coherent fiber-optic system.

An analytical model to calculate the variance of cross-phase modulation (XPM) distortion in a wavelength-division multiplexed (WDM) fiber-optic system is developed. The method is based on the first order perturbation technique and it is applicable for both dispersion managed and dispersion uncompensated systems. For dispersion managed systems, it is shown that the variance of XPM distortion scales as Nx where N is the number of spans and x ∈ [1, 2] depending on the amount of inline-dispersion compensation. The analytical model is found to be in good agreement with simulations in most of the cases.

Analytical modeling of XPM in dispersion-managed coherent fiber-optic systems

An analytical model for the cross-phase modulation (XPM) variance in dispersion-managed coherent fiber-optic systems is developed based on the first order perturbation theory. The XPM variance is analytically calculated for arbitrary pulse shapes. For a non-Gaussian pulse, the summation of time-shifted Gaussian pulses is used to fit the target pulse shape, which not only provides a good approximation of the non-Gaussian pulse but also allows explicit derivation of the XPM variance. The analytically estimated XPM variance is found to be in good agreement with numerical simulations.

Digital Back Propagation With Optimal Step Size for Polarization Multiplexed Transmission

A digital back propagation (DBP) scheme with optimal step size for polarization division multiplexed transmission system is proposed. For a fixed number of steps in DBP, the optimum step size is calculated by minimizing the mismatch between the area under the exponentially increasing nonlinearity profile and its stepwise approximation. In simulations, the vector nonlinear Schödinger equation or Manakov equations are used for forward propagation and Manakov equations are used for backward propagation. The simulation results show that the proposed scheme using the optimum step size outperforms that using the uniform step size at the same computational cost.

Comparison of Split-Step Fourier Schemes for Simulating Fiber Optic Communication Systems

This paper mainly focuses on efficient schemes for simulating propagation in optical fibers. Various schemes based on split-step Fourier techniques to solve the nonlinear Schrödinger equation (NLSE), which describes the propagation in optical fibers, are compared. In general, the schemes in which the loss operator is combined with nonlinearity operator are found to be more computationally efficient than the schemes in which the loss is combined with dispersion. When the global error is large, the schemes with variable step size outperform the ones with uniform step size. The schemes based on local error and/or minimum area mismatch (MAM) further improve the computational efficiency. In this scheme, by minimizing the area mismatch between the exponential profile and its stepwise approximation, an optimal step size distribution is found. The optimization problem is solved by the steepest descent algorithm. The number of steps to get the desired accuracy is determined by the local error method. The proposed scheme is found to have higher computational efficiency than the other schemes studied in this paper. For QPSK systems, when the global error is 10-8, the number of fast Fourier transforms (FFTs) needed for the conventional scheme (loss combined with dispersion and uniform step size) is 5.8 times that of the proposed scheme. When the global error is 10-6, the number of FFTs needed for the conventional scheme is 3.7 times that of the proposed scheme.

Correlated digital back propagation based on perturbation theory

We studied a simplified digital back propagation (DBP) scheme by including the correlation between neighboring signal samples. An analytical expression for calculating the correlation coefficients is derived based on a perturbation theory. In each propagation step, nonlinear distortion due to phase-dependent terms in the perturbative expansion are ignored which enhances the computational efficiency. The performance of the correlated DBP is evaluated by simulating a single-channel single-polarization fiber-optic system operating at 28 Gbaud, 32-quadrature amplitude modulation (32-QAM), and 40 × 80 km transmission distance. As compared to standard DBP, correlated DBP reduces the total number of propagation steps by a factor of 10 without performance penalty. Correlated DBP with only 2 steps per link provides about one dB improvement in Q-factor over linear compensation.

Impulse response of nonlinear Schrödinger equation and its implications for pre-dispersed fiber-optic communication systems

In the presence of pre-dispersion, an exact solution of nonlinear Schrödinger equation (NLSE) is derived for impulse input. The phase factor of the exact solution is obtained in a closed form using the exponential integral. The nonlinear interaction among periodically placed impulses launched at the input is investigated, and the condition under which these pulses do not exchange energy is examined. It is found that if the complex weights of the impulses at the input have a secant-hyperbolic envelope and a proper chirp factor, they will propagate over long distances without exchanging energy. To describe their interaction, a discrete version of NLSE is derived. The derived equation is a form of discrete self-trapping (DST) equation, which is found to admit fundamental and higher order soliton solutions in the presence of high pre-dispersion. Nonlinear eigenmodes derived here may be useful for description of signal propagation and nonlinear interaction in highly pre-dispersion fiber-optic systems.

Optical back propagation for compensating nonlinear impairments in fiber optic links with ROADMs

An optical back propagation (OBP) technique is investigated to compensate for nonlinear impairments in fiber optic communication systems with reconfigurable optical add-drop multiplexers (ROADMs). An OBP module consisting of an optical phase conjugator (OPC), amplifiers and dispersion-decreasing fibers (DDFs) fully compensates for the nonlinear impairments of a transmission fiber. The OBP module can be placed after each transmission fiber (inline OBP case) or at each network node (node OBP case). For a wavelength division multiplexing (WDM) system with 2400 km transmission distance and 32-quadrature amplitude modulation (QAM) format, inline OBP and node OBP bring Q-factor improvements of 4.9 dB and 5.6 dB as compared with linear compensation, respectively. In contrast, receiver-side digital back propagation (DBP) only provides 1.3 dB Q-factor gain, due to its incapability of mitigating inter-channel nonlinear effects in fiber optic networks.