Probability densities of second-order polarization mode dispersionincluding polarization dependent chromatic fiber dispersion

We describe experiments and simulation of second-order polarization mode dispersion (PMD) components in optical fibers with emphasis on polarization-dependent chromatic dispersion (PCD). Excellent agreement is found in comparisons of experimental, simulated, and theoretical probability densities. To our knowledge, these are the first such comparisons for the second-order PMD magnitude and the PCD     

On the second-order approximation of PMD

A second-order polarization mode dispersion (PMD) approximation based upon the pulse-width distortion has been studied. It shows that a complete second-order approximation should include the second derivative of the PR-ID vector as well as the first derivative of the PMD vector. Second-order pulse distortions are explicitly expressed including a `first-order' term involving principal states of polarization (PSP) of the pulse and a second-order term involving the beating between fiber chromatic dispersion and effective PMD chromatic dispersion. An analytical result is derived for the probability of second-order PR-ID power penalty. It shows that the mean PMD of the fiber should be restricted to 26 ps and 18 ps, respectively for an optical link with zero and 850 ps/nm chromatic dispersion, in order to maintain a one dB second-order PMD power penalty with a probability below 10^-6 at a data rate of 10 Gb/s. The analysis also indicates that a second-order PMD compensator can be used as a dynamic chromatic dispersion compensator     

Soliton robustness to the polarization-mode dispersion in opticalfibers

The influences of chromatic dispersion, coupling length, and soliton energy on the soliton robustness to polarization-mode dispersion (PMD) are investigated numerically. We find that both chromatic dispersion and soliton energy have significant effects on the soliton robustness to PMD, and by optimizing chromatic dispersion and soliton energy, soliton pulse broadening can be depressed to within 10% even when the differential group delay is about twice the input pulse width. In addition, a recent experimental result on soliton robustness is numerically verified     

First-order PMD compensation in nonlinearly dispersive opticalcommunication systems via transmission over the principal states ofpolarization

Soliton propagation in the presence of polarization mode dispersion (PMD) is studied with an analytical approach. It is shown that there exist two peculiar initial states of polarization that may prevent the soliton to undergo shape distortions. These states coincide with the states that, in the absence of chromatic dispersion and nonlinearity, are usually referred to as principal states of polarization. This fact is used to show that it is possible to improve the performance of an optical transmission system operating in the presence of chromatic dispersion, PMD and Kerr nonlinearity by coupling the input pulses to one of the principal states of polarization     

Statistical correlation between first and second-order PMD

We investigate the statistical correlation between first- and second-order polarization mode dispersion (PMD) effects, which is important for PMD mitigation. The theoretical results are compared to numerical simulations and experimental data from a real high-PMD fiber. A new dependence between first- and second-order PMD is found. We show that the root mean square (RMS) value of the second-order PMD component, perpendicular to the PMD vector, increases with the length of the PMD vector     

Improved Numerical Model for the Average Transfer Matrix of First- and Second-Order PMD

An improved model for the average transfer matrix of first- and second-order polarization-mode dispersion (F&SO-PMD) in optical fibers is derived from numerical simulations of the frequency dependence of the differential phase shifts and cross coupling between signal components that are transmitted in two principal states of polarization (PSPs). The mean differential phase shifts and cross-coupling phases are calculated for various given values of the differential group delay (DGD) and the second-order PMD parameter that characterizes the coupling between PSPs. It is found that the mean differential phase delays are equal to the DGDs only over a narrow optical bandwidth, beyond which they decrease with increasing frequency offset and in some cases even reverse sign. Similarly, the mean cross-coupling phases increase less rapidly with frequency than assumed in other popular models and always approach an asymptotic value, at which half of the optical power is coupled from one PSP to the other. Moreover, it is shown that these mean differential phase shifts and cross-coupling phases define a transfer matrix for F&SO-PMD that nicely predicts the average eye-opening penalties in return-to-zero-formatted digital optical signals that are transmitted in one of the two PSPs. These predictions are particularly accurate when the average polarization-dependent chromatic dispersion (PCD) is included in the PSP phases. Additional simulations of F&SO-PMD compensation reveal that the signal impairments caused by PCD, on average, are substantially smaller than those introduced by the cross coupling between PSPs     

Statistical characteristics and effects of polarization mode dispersion in dispersion managed soliton systems

Recently,the dispersion-managed solitons(DMS) areof great interest in soliton communication systems .Be-cause of some prior advantages compared with conven-tional solitons ,such as higher pulse energy and signalnoise ratio,lower averaged dispersionline an…     

光纤偏振模色散和偏振相关损耗对超短光脉冲传输的影响

采用琼斯矩阵与随机耦合的波片模型（Monte—Carlo）方法相结合研究了在偏振模色散（PMD）和偏振相关损耗（PDL）共同作用下,超短光脉冲经光纤传输后脉冲被展宽的统计特性。结果表明,通过与偏振模色散的相互作用,偏振相关损耗可以引起光脉冲的压缩效应,同时脉冲展宽和压缩的概率密度分布与光纤偏振模色散及偏振相关损耗的相对数值存在十分密切的关系。在偏振模色散为0．05ps／km^1/2,光纤长度为75km的情况下,当偏振相关损耗均值从0．07dB／km增加到0．49dB／km时,光脉冲被压缩的概率从4．3％增大到92．9％。无论何种情况下,脉冲压缩和展宽的概率密度均可用瑞利-麦克斯韦分布函数进行描述。     

Statistical description of optical system performances due torandom coupling on the principal states of polarization

Due to polarization-mode dispersion (PMD), the performances of an optical transmission system show a random behavior. We calculate the probability density function of the Q-factor due to PMD. We adopt the principal states of polarization model, and we also consider the clock recovery behavior at the receiver. Theoretical results are reported for a 40-Gb/s system, and are compared with those obtained considering the effect of pulse broadening due to PMD     

RZ码和NRZ码的偏振模色散自适应补偿特性比较

建立了自适应偏振模色散补偿系统,利用偏振度作为反馈信号,对40 Gb/s的RZ和NRZ码分别进行了PMD补偿的数值模拟,结果显示,采用DOP作反馈信号、用二段补偿器和三段补偿器对二种码型的PMD补偿均是有效的.但由于高阶PMD的影响对NRZ码的补偿效果要优于RZ码,特别是存在偏振相关色散的影响时,对RZ码的补偿的影响在明显大于NRZ码,这说明,对于RZ码补偿偏振相关色散是必要的.     

A deterministically controlled four-segment polarization-mode dispersion emulator

In this paper, we propose a novel design of a first- and second-order polarization-mode dispersion (PMD) emulator based on four concatenated first-order PMD segments. Instead of polarization scrambling at each junction, we set these polarization rotators according to a statistical schedule, so as to produce a realistic probability density function of the first- and second-order PMD presented in long-haul transmission fiber.     

System outage probability due to first- and second-order PMD

A theoretical approach is proposed that allows one to quantify the impact of fiber polarization mode dispersion (PMD) on optical binary transmission taking into account not only first-order polarization mode dispersion, but also signal distortion induced by second-order PMD. Using this approach the impact of the spectral signal width on PMD-induced system outage probability could be studied for the first time. An analysis of 10-Gb/s transmission exhibits that, as long as the mean PMD remains below the commonly accepted limit (about 10 ps) for negligible outage, a linear chirp of up to 30 GHz does not lead to an additional increase of the system outage. This result confirms that low bandwidth modulation schemes (external modulator, low chirp laser) do not suffer from additional outage degradation due to second-order PMD     

Optimal choice of compensation PMD vector in feedforward-type second-order PMD compensation

In this paper, the formula of an optimal compensation vector, which is defined as an input polarization-mode dispersion (PMD) vector of the compensator, for second-order PMD compensation has been derived from the minimization of the root-mean-square (rms) pulse-broadening factor. The derived optimal compensation vector is a linear combination of frequency-averaged PMD vectors and output states of polarization, which shows a similar trend to the previously reported optimal first-order PMD compensation. The rms pulse-broadening factor after optimal second-order PMD compensation has been semianalytically calculated and compared with the previously reported simulation result. They are in good agreement. The formula of the optimal second-order compensation vector derived in this paper provides indispensable information for feedforward second-order PMD compensation.     

Independently tunable first- and second-order polarization-mode dispersion emulator

By solving the equation set for the first- and second-order polarization-mode dispersion (PMD) in retro-verse manner, we show that the independent control of first- and second-order PMD is possible, with only two control parameters. The analysis on the emulation algorithm as well as the operation range/resolution shows excellent agreement with the experimental demonstration operated with a sample emulator constructed of a fixed differential group delay (DGD), a polarization rotator, and a variable-DGD element.     

偏振模色散统计特性的研究

利用动态方程研究了偏振模色散矢量的统计特性 ,分别用统计系综和频率系综两种方法进行数值模拟。数值结果表明 ,统计系综和频率系综两种方法的结果一致。在大归一化距离条件下 ,均方根时延的平方随归一化距离线性变化 ;偏振模色散矢量的各个分量服从高斯分布 ,差分群时延服从麦克斯韦分布 ;偏振模色散矢量弥散于整个邦加球     

Optical Equalization: System Modeling and Performance Evaluation

Optical equalization in fiber-optic systems is theoretically investigated. First, a wide variety of optical equalizers is presented by giving their equivalent models, by describing the basic principles of operation, and by showing analogies and differences among different structures and configurations. Then, evaluation of the bit error rate and outage probability, in the presence of optical noise, chromatic dispersion (CD), and polarization mode dispersion (PMD), is discussed by comparing different methods based either on the PMD vector and its related statistics or on Markov chain Monte Carlo (MCMC) simulations applied to a random waveplate model. As the first relevant result, the accuracy and complexity of different PMD models in compensated systems are tested and compared, and a novel reduced Bruyegravere-Kogelnik (BK) model, which is more accurate and simple than the complete BK model or other second-order models, is proposed. As an additional result, the ultimate theoretical performance for different configurations and numbers of stages of the equalizers is given, and it is shown that when a single equalizer is used for a pair of polarization-multiplexed channels, the system tolerance to PMD is highly increased     

偏振模色散补偿对脉冲展宽的影响

在高速光纤通信系统中,偏振模色散(PMD)已被认为是限制光纤通信系统性能的一个主要因素,而脉冲的均方根展宽是表征信号传输性能的一个重要的物理量.研究了偏振模色散对脉冲展宽的影响,推导出含有一阶偏振模色散补偿的光纤链路中偏振模色散矢量的自相关函数的分析表达式.在此表达式的基础上,推出了以高斯脉冲为例的一阶偏振模色散补偿系统中的均方根脉宽解析表达式.     

PMD second-order effects on pulse propagation in single-modeoptical fibers

The time-space varying birefringence in single-mode optical fibers causes the polarization mode dispersion (PMD) to be a serious problem in high bit-rate transmissions. The PMD first- and second-order statistics are well known in the literature, but second-order PMD-induced pulse distortions have still to be clarified for sequences of pulses of arbitrary shape. We give, for the first time, the exact PMD time impulse response, up to second order. We show, both numerically and experimentally, that the effect of the rotation of the principal states of polarization (PSP) is to generate power overshoots on the “1” and “0” bit sequences     

Emulation and inversion of polarization-mode dispersion

When a fiber is characterized by measured polarization mode dispersion (PMD) vector data, inversion of these data is required to determine the frequency dependence of the fiber's Jones matrix and, thereby, its pulse response. This tutorial reviews the principal concepts and theory employed in approaches to PMD inversion and in the closely related emulation of PMD. We discuss three second-order emulator models and the distinction between the PMD vectors and the eigenvectors of the fiber's Jones matrix. We extend emulation and inversion to fourth-order and sixth-order PMD using higher order concatenation rules, rotations of higher power designating higher rates of acceleration with frequency, and representation of these rotations by Stokes' vectors.