Analysis of the Polarization-Mode-Dispersion Vector Distribution for Linearly Birefringent Optical Fibers

We analyze the polarization-mode-dispersion (PMD) vector distribution for linearly birefringent optical fibers. We assume the linear birefringence vector components as white Gaussian processes. From the corresponding Fokker-Planck equation, we find an asymptotic solution for the probability density function of the PMD vector. Our analysis shows that the PMD vector distribution is dependent on the polar angle during its transient and the distribution tail for the magnitude of the PMD vector is higher than the Maxwellian.     

Multiple importance sampling for first- and second-order polarization-mode dispersion

A simulation method that targets all possible combinations of first- and second-order polarization-mode dispersion (PMD) is described. Use of this method in importance-sampled Monte Carlo simulations yields a more comprehensive determination of PMD-induced system penalties than first-order biasing alone and significantly speeds up the calculation of outage probabilities, particularly when PMD compensation is employed. The technique is demonstrated by using it to calculate the probability distribution function (pdf) of second-order PMD and the joint pdf of the magnitude of first- and second-order PMD.     

A MAP equalizer for the optical communications channel

A maximum a posteriori (MAP) equalizer is presented for optical fiber communication systems. Assuming that the span of the intersymbol interference (ISI) does not extend beyond neighboring bits-typically the case for the distortion introduced by polarization-mode dispersion (PMD)-we derive the conditional probability density function (pdf) in the electrical domain in the presence of PMD and amplified spontaneous emission (ASE)-dominated noise. Simulation results with an accurate receiver model and all-order PMD show the success of the MAP equalizer in reducing the bit error rate (BER) degradation due to PMD.     

Statistical PMD emulator using variable DGD elements

We report on a novel polarization-mode dispersion (PMD) emulator with exact Maxwellian statistics. The proposed PMD emulator was implemented by using variable differential group-delay elements and polarization controllers. Unlike the previous PMD emulators, the proposed PMD emulator could generate not only the first-order PMD with exact Maxwellian distribution, but also the second-order PMD.     

On the Utility of Laguerre Series for the Envelope PDF in Multipath Fading Channels

It is well known that multipath fading significantly affects the performance of communication systems. In order to incorporate the impact of this random phenomenon on system analysis and design, in many cases, we need to calculate the probability density function (pdf) of the received signal envelope in multipath fading channels. In this paper, we consider a general multipath fading channel with arbitrary number of paths, where the amplitudes of multipath components are arbitrary correlated positive random variables, independent of phases, whereas the phases are independent and identically distributed random variables with uniform distributions. Since the integral form of the envelope pdf for such a general channel model is too complicated to be used for analytic calculations, we propose two infinite expansions for the pdf: a Laguerre series and a power series. Based on the tight uniform upper bounds on the truncation error of these two infinite series, we show that the Laguerre series is superior to the power series due to the fact that for a fixed number of terms, it yields a smaller truncation error. This Laguerre series with a finite number of terms, which expresses the envelope pdf just in terms of simple polynomial–exponential kernels, is particularly useful for mathematical performance prediction of communication systems in those indoor and outdoor multipath propagation environments, where the number of strong multipath components is small.      

Polarization-mode-dispersion emulator using variable differential-Group-delay (DGD) elements and its use for experimental importance sampling

We demonstrate a practical polarization-mode-dispersion (PMD) emulator using programmable differential-group-delay (DGD) elements. The output PMD statistics of the emulator can be chosen by varying the average of the Maxwellian DGD distribution applied to each element. The emulator exhibits good stability and repeatability in a laboratory environment. In addition, we demonstrate how this emulator may be used to experimentally employ the powerful technique of importance sampling to quickly generate extremely low probability events. This technique is used to measure the Q-factor degradation due to both average and rare PMD values in a 10-Gb/s transmission system.     

Polarization mode dispersion in a single mode fiber

The Jones matrix method is used to measure the polarization mode dispersion (PMD) of a large variety of single mode fibers in the 1500 nm range. The dependence of PMD on wavelength, time, and temperature are studied in two different regimes: adiabatic and isothermal. In the adiabatic regime, time dependent stresses are introduced in the fiber by subjecting it to large and rapid changes in temperature. In this regime it is shown that the probability density function of PMD, obtained as a function of temperature and wavelength, fits very closely the theoretically predicted Maxwellian function. In the isothermal regime, the temperature of the fiber is held constant and the stresses are allowed to relax to their long term steady state conditions. In this regime the PMD exhibits a strong dependence on wavelength but otherwise is a bounded function which is nearly stationary with time. Test and analysis of the deterministic PMD in a specially constructed polarization maintaining fiber are used to study the dependence of PMD on temperature and wavelength. Finally, the system implications of this PMD study are described briefly     

Loop-synchronous polarization scrambling technique for simulating polarization effects using recirculating fiber loops

A loop-synchronous polarization-scrambling technique has been proposed for the purpose of simulating polarization effects in straight-line systems using recirculating loops. This technique uses a fast polarization controller within a fiber loop. The polarization controller changes its transmission matrix after each round trip of the optical signal circulating through the loop; thus, the periodic polarization transform of the loop is avoided. Moreover, the polarization controller generates a series of random uncorrelated transmission matrixes. Therefore, the mean-square value of differential group delay (DGD) or polarization-dependent loss (PDL) increases linearly with the number of circulations. The matrix expression for a random polarization transform that scatters the state of polarization (SOP) uniformly on the Poincare sphere for any input SOP was also found. Experiments were performed for a 94-km fiber loop that contains a fixed DGD or PDL element. The histograms of polarisation mode dispersion (PMD) induced power penalties at 10/sup -9/ bit error rate (BER) were measured. There is a good agreement between experimental and theoretical results. Using loop-synchronous polarization scrambling, accurate reproduction of the Maxwellian distribution of DGD can be realized when the background PMD of transmission fiber is much smaller than the PMD intentionally introduced into the loop.     

Polarization-dependent gain due to signal-signal Raman interaction in low-PMD fiber

We study polarization-dependent gain (PDG) due to signal-signal Raman interaction (SSRI). We find that SSRI-induced PDG can be significant when the fiber polarization-mode dispersion (PMD) is low. If the PMD from the discrete optical components is also small, the accumulated PDG grows almost linearly with the number of the span.     

高速光纤通信系统中高阶PMD补偿技术研究

随着光放大器和色度色散补偿技术的不断提高，光纤的偏振模色散（PMD）已经成为超高速、超长距离光纤通信系统发展的主要障碍，在40Gbit/s或更高速率的光纤通信系统中，PMD的影响已不可忽略，必须考虑PMD的补偿问题，从高阶PMD对40Gbit/s NRZ系统影响的数值模拟发现，当光纤中PMD高阶效应比较明显时，将严重劣化一阶PMD补偿的效果，另外，通过对两种高阶PMD补偿器的比较介绍，认为两段级联的高队PMD补偿系统是一种比较现实的补偿方法。     

First- and second-order PMD statistical properties of constantly spun randomly birefringent fibers

Polarization-mode dispersion (PMD) causes significant impairment for high bit-rate optical telecommunications systems. It is known that PMD can be strongly reduced by spinning the fiber as it is drawn. In this paper, we focus on the case of randomly birefringent fibers spun at a constant rate, providing analytical expressions for the asymptotic statistical properties of PMD. In particular, we investigate the behavior of the first- and second-order PMD, demonstrating that a constantly spun fiber behaves asymptotically as an unspun fiber. Conversely, we show that the distance at which the PMD reaches its asymptotic trend increases with the spin rate up to lengths of several kilometers.     

Time evolution of PMD due to tides and sun radiation on submarine fibers

Using a fast polarization mode dispersion (PMD) set of 15-s interval and long measurement duration ($>24\text{h}$), we found the correlation between polarization effects, tides and sun radiation on submarine fibers. We also found Maxwellian distribution for the differential group delay (DGD) even with strong biased output states of polarization (SOP) over 1 h measurement. PMD bandwidth and the fastest decorrelation time were calculated to be 165 GHz and less than 15 s.     

Degree-of-polarization-based PMD monitoring for subcarrier-multiplexed signals via equalized carrier/sideband filtering

We demonstrate a method for differential-group-delay (DGD) and polarization-mode-dispersion (PMD) monitoring using the degree-of-polarization (DOP) in subcarrier-multiplexed (SCM) systems. Traditional SCM signal show very little DOP sensitivity to DGD/PMD due to the low modulation depth used for generating SCM signals. We use a narrow-band optical filter to equalize the power of the carrier and one of the sidebands by offsetting the filter from the carrier, enabling PMD and DGD monitoring and more than tripling the DOP sensitivity to DGD/PMD. Our technique is simple, uses only a single optical filter, and can be applied to both single- and double-sideband (SSB and DSB) SCM signals as well as single and multisubcarrier systems. Additionally, we show that our monitoring technique is robust to the chromatic dispersion-induced radio-frequency (RF) power fading effect seen in DSB SCM signals. Using this technique to enhance the DOP sensitivity to DGD/PMD and generate a feedback signal to a PMD compensator (PMDC), we obtain an 11-dB improvement in the 5% RF power tail.     

System Penalty From Second-Order PMD Approximation: Reference Frame Implication

In this paper, we show that second-order PMD approximations, when derived using the input frame and output frame of an optical link, result in two different PMD systems represented by two distinct Jones matrices. In contrast to an all-order representation of a system, which is independent of reference frame, it is found that a finite second-order approximation cannot be obtained simultaneously in both input and output reference frames, except in some limited special cases. The consequences of this are illustrated by analyzing the difference between these two approximations in pulsewidth distortion. It is shown that the second-order PMD approximation in the input frame corresponds to a truncated second-order expansion of the pulse distortion, whereas the second-order PMD approximation in the output frame leads to an infinite-order pulse distortion     

Pulse distortion induced by polarization-mode dispersion and polarization-dependent loss in lightwave transmission systems

The combined effects of polarization-mode dispersion (PMD) and polarization-dependent loss (PDL) on pulse broadening in optical transmission links for both PMD uncompensated and compensated systems are investigated by numerical simulation. It is shown that, when PDL is large, PDL considerably reduces the efficiency of PMD compensation, which, we find, is mainly due to the increased higher order PMD caused by the combined effects of PMD and PDL, as well as nonorthogonal principal states of polarization.     

160 Gb/s-Based Field Transmission Experiments Using Polarizer-Based PMD Compensator With Optical Power Monitor

Higher bit-rate transmission is attractive for improving network resource efficiency and reducing the complexity of network management in future transmission systems. However, chromatic dispersion and polarization mode dispersion (PMD) are one of the most serious impairments. In particular, PMD should be compensated for dynamically because it changes rapidly according to environmental variations such as temperature change and mechanical vibration. Therefore, an adaptive PMD compensator is indispensable for higher bit-rate transmission systems. In this paper, we employed a simple and bit-rate independent PMD compensator based on a polarizer with an optical power monitoring scheme in 160 Gb/s-based field experiments. By using the PMD compensator, the single channel transmission of a 160 Gb/s return-to-zero differential phase-shift-keying modulation signal over an installed fiber link with buried and aerial cable routes was successfully achieved. Approximately 1 dB of Q-factor was improved by using the PMD compensator when PMD impairment was maximized. Through these experiments, the effectiveness of the PMD compensator in the higher bit-rate transmission systems was confirmed in the field environment. Furthermore, single-polarization 8times160 Gb/s wavelength division multiplexing transmission over the installed 200 km standard single mode fiber without polarization demultiplexing was successfully achieved by using the simple PMD compensator.     

Convergence of differential entropies

Calculation of the differential entropy of the limiting density of a sequence of probability density functions (pdf) is an interesting mathematical problem and is important in asymptotic analysis of communication systems. In such cases, it would be of interest to know if the limit of the differential entropies H/sub n/, corresponding to the sequence of pdf f/sub n/, is equal to the differential entropy H, of the limiting pdf f. In this correspondence, we establish sufficient conditions under which H/sub n/ /spl rarr/ H.     

DOP ellipsoids for systems with frequency-dependent principalstates

The effects of polarization mode dispersion (PMD) in optical communication systems limit the practical length of fiber optic links. Characterization of the PMD of an optical system can be performed by observing the surface obtained from the vector whose length is the degree of polarization and direction is defined by the output state of polarization for a randomly varying input polarization. An analytical solution to the ellipsoid is derived for a simple higher order PMD system that is composed of two polarization-maintaining fibers. An approximate inverse relationship between the ellipsoid radii and the first- and second-order components of the PMD vector is also derived     

2.5 Gb/s偏振模色散自动补偿对偏振模色散容限值的影响

在我国现有的光纤通信骨干网中,绝大部分速率为2.5 Gb/s。今后如在我国现有的2.5 Gb/s网络系统上进行密集波分复用(DWDM)升级,亦需要考虑偏振模色散(PMD)对系统容量升级的影响。采用十段高双折射光纤级联而成的偏振模色散模拟器模拟实际光纤,从信号中提取基带频率分量作为反馈信号,对2.5 Gb/s系统进行了偏振模色散自动补偿实验,并对反馈前后的系统进行了系统代价的测量和比较。实验结果表明,偏振模色散自动补偿能较大幅度地提高系统的偏振模色散容限值。