Introduction to polarization mode dispersion in optical systems

This introduction covers concepts important to the understanding of polarization mode dispersion (PMD), including optical birefringence, mode coupling in long optical fibers, the Principal States Model, and the time and frequency domain behavior of PMD. Other topics addressed include the concatenation rules, bandwidth of the Principal States, PMD statistics and scaling, PMD system impairments, and PMD outage probability calculations.     

偏振模色散矢量的研究

研究了无损光纤的密勒矩阵,进而得出了偏振模色散矢量的解析表达式、主偏振态对应的斯托克斯矢量的解析表达式,以及高阶偏振模色散矢量的解析表达式.这些解析表达式是由光纤参数决定的.讨论了局部偏振模色散矢量与整体偏振模色散矢量的关系,讨论了利用偏振模色散矢量进行偏振模色散补偿的原理.引入了偏振模色散补偿元件的偏振模色散补偿矢量C,具体计算了正规的非圆光波导类的补偿元件的C.从理论上证明了仅仅利用一个正规的非圆光波导类的补偿元件,例如一根保偏光纤或是一个双折射晶体,是不能实现偏振模色散补偿 关键词： 偏振模色散 密勒矩阵 色散补偿 主偏振态斯托克斯矢量     

Modelling of polarization mode dispersion in optical communications systems

With the rapid increase in the data rates transmitted over optical systems, as well as with the recent extension of terrestrial systems to ultra-long haul reach, polarization mode dispersion (PMD) has become one of the most important and interesting limitations to system performance. This phenomenon originates from mechanical and geometrical distortions that break the cylindrical symmetry of optical fibers and create birefringence. It is the random variations of the local birefringence along the propagation axis of the optical fiber that create the rich and complicated bulk of phenomena that is attributed to PMD. The detailed statistical properties of the local birefringence and its dependence on position are only important as long as the overall system length is comparable with the correlation length of the birefringence in the fiber. In typical systems, however, the latter is smaller by more than three orders of magnitude so that the specific properties of the local birefringence become irrelevant. Instead, the fiber can be viewed as a concatenation of a large number of statistically independent birefringent sections characterized only by the mean square value of their birefringence. This model has been used extensively in the study of PMD and its predictions have been demonstrated to be in excellent agreement with experimental results. This approach opens the door to the world of stochastic calculus, which offers many convenient tools for studying the PMD problem. In this article we review the modelling of PMD and discuss the properties of this phenomenon as a stochastic process. We explain the use of stochastic calculus for the analysis of PMD and describe the derivation of the frequency autocorrelation functions of the PMD vector, its modulus and the principal states. Those quantities are then related to commonly used parameters such as the bandwidth of the first order PMD approximation, the bandwidth of the principal states and to the accuracy of PMD measurements.     

Filter control polarization mode dispersion in dispersion managed soliton systems

In this paper, the dispersion managed soliton (DMS) transmission equation is built on considering the effects of polarization mode dispersion (PMD) and filter control. The DMS transmission of filtering control in constant birefringence fibers is firstly analyzed by varitional method, from which the evolving rules of characteristical DMS parameters are obtained. Secondly, the stability of DMS transmission and its timing jitter are investigated in the random varying birefringence fibers with the conventional model of PMD. The results reveal that filter control DMS system has powerful robustness to PMD effects and DMS's timing jitter can be decreased considerably with the help of filters.     

Effects of residual stress on polarization mode dispersion of fibers made with different types of spinning

We analyze the effects of residual stress on the polarization mode dispersion (PMD) of fibers made with different types of spinning. A theoretical scheme is developed from a previous model by the incorporation of a circular birefringence term contributed by residual torsional stress. It is found that the residual stress can significantly affect the PMD of unidirectionally spun fibers when the fiber birefringence is low, but it has little effect on the PMD of bidirectionally spun fibers.     

Scaling properties of polarization mode dispersion of spun fibers in the presence of random mode coupling

The scaling properties of polarization mode dispersion (PMD) in spun fibers are studied. Simple equations have been obtained to describe the scaling properties of spun fibers as a function of intrinsic fiber birefringence, spin parameters, and mode-coupling length under both optimal and nonoptimal spin conditions. In particular, a counterintuitive result is found for fibers with perfect spin optimization, in which case the fiber PMD increases as the mode-coupling length is shortened. The results are verified with direct numerical modeling.     

Polarization mode dispersion of spun fibers: an analytical solution

A simple analytical solution is derived from coupled-mode theory to describe the evolution of polarization mode dispersion (PMD) in spun fibers. For practical fibers with a beat length greater than a few meters, the solution is valid for a whole category of periodic spin profiles. We find that the PMD change factor is independent of the intrinsic birefringence of the fiber and the fiber PMD scales linearly with the fiber length in the short length regime. This solution allows us to determine phase-matching conditions for spun fibers, in which the PMD evolves periodically along the fiber. An example of determining the phase-matching conditions of sinusoidal-type spin profiles is given.     

Statistics of polarization mode dispersion-induced gain fluctuations in Raman amplified optical transmissions

A systematic experimental evaluation of polarization mode dispersion (PMD) -induced polarization-dependent gain (PDG) in forward pumped Raman amplification in dispersion-shifted and in dispersion-compensating fiber was performed. Good agreement was obtained between the measured statistical parameters and the current analytical model for PDG fluctuation statistics. The probability distribution of the PDG was approximately Maxwellian within the range PMD >0.05 ps/km(1/2). The interplay between PMD and gain fluctuations is discussed; random birefringence strongly reduces PDG fluctuations. However, the trade-off between reduction of the power penalties for PDG and increase of the penalties for PMD distortion precludes the use of PMD instead of source depolarization techniques for reduction of PDG.     

The PMD of sinusoidally spun fibers in the presence of random birefringence perturbations

Although fiber spinning is known to reduce polarization mode dispersion (PMD) effects in optical fibers, relatively few studies have been performed of the dependence of the reduction factor on the strength of random birefringence fluctuations. In this paper, we apply a general mathematical model of random fiber birefringence to sinusoidally spun fibers. We find that while even in the presence of random birefringence perturbations the maximum reduction of PMD is still obtained when the phase matching condition is satisfied, the degree of PMD reduction and the probability distribution function of the DGD both vary with the random birefringence profiles.     

Polarization mode dispersion and vectorial modulational instability in air-silica microstructure fiber

The birefringence of an air-silica microstructure fiber has been studied by measurement of the fiber polarization mode dispersion (PMD) over the wavelength range 545-640 nm. The experimental results are shown to be in good agreement with vectorial numerical calculations, assuming an elliptical core with an eccentricity of 7%. We also report controlled experiments studying nonlinear vectorial modulation instability in the fiber, yielding 3.9-THz modulational instability sideband shifts that are in good agreement with theoretical predictions based on the calculated fiber dispersion and PMD characteristics.     

Polarization mode dispersion in long single-mode-fiber links: A review

Polarization mode dispersion (PMD) is the interplay of birefringence (both systematic and random) and coupling between orthogonal polarizations, in a single-mode fiber. PMD will set the ultimate limit to the fiber length × capacity product, when all the deterministic causes of dispersion are under control. We review the state of the art in this field, with emphasis on experiments, which demonstrate that the behavior of long links can be predicted with full confidence, in a statistical sense, from factory-based tests on individual cables. Successful application of the same experimental technique to detecting fiber stresses is also demonstrated.     

Birefringence and polarization mode dispersion in graded-core stress-applied polarization-maintaining fibers

We present a simple analytical method to study birefringence and polarization mode dispersion (PMD) of graded-core stress-applied polarization-maintaining fibers. It is based on the equivalent step-index-fiber method. It is shown that the agreement between present results and the results obtained using the finite-element method is very good. Polarization characteristics of stress-applied polarization-maintaining fibers having a dip in the refractive index at the center of the core are also investigated.     

Origin and System Effects of Polarization Mode Dispersion in Chirped Bragg Gratings

Polarization mode dispersion of chirped Bragg gratings is analyzed in terms of key birefringence phenomena and impact on telecommunication systems performance. The influence on polarization mode dispersion (PMD) of fiber birefringence, grating chromatic dispersion, and ripples of the group delay curve is pointed out. Polarization mode dispersion influence on systems performance is investigated by numerical simulations and transmission experiments at 10 Gbit/s. The deterministic nature of Bragg gratings PMD determines a moderate, upper-limited system penalty for a transmission line employing a single compensating device. However, in the case of broadband components, a non-negligible PMD penalty may be observed due to the difficulty of controlling accurately the group delay linearity.     

Origin and System Effects of Polarization Mode Dispersion in Chirped Bragg Gratings

Polarization mode dispersion of chirped Bragg gratings is analyzed in terms of key birefringence phenomena and impact on telecommunication systems performance. The influence on polarization mode dispersion (PMD) of fiber birefringence, grating chromatic dispersion, and ripples of the group delay curve is pointed out. Polarization mode dispersion influence on systems performance is investigated by numerical simulations and transmission experiments at 10 Gbit/s. The deterministic nature of Bragg gratings PMD determines a moderate, upper-limited system penalty for a transmission line employing a single compensating device. However, in the case of broadband components, a non-negligible PMD penalty may be observed due to the difficulty of controlling accurately the group delay linearity.     

Jones matrix for second-order polarization mode dispersion

A Jones matrix is constructed for a fiber that exhibits first- and second-order polarization mode dispersion (PMD). It permits the modeling of pulse transmission for fibers whose PMD vectors have been measured or whose statistics have been determined by established PMD theory. The central portion of our model is a correction to the Bruyère model.     

Polarization mode dispersion in single mode optical fibers due to core-ellipticity

The variation of polarization mode dispersion (PMD) with V-parameter in single mode optical fibers due to core-ellipticity is studied by performing numerical simulations taking into account both geometrical and thermal-stress-induced birefringences as well as the variation of fiber refractive indices with wavelength. Simple empirical relations are given for calculating the mean PMD for any value of core-ellipticity and V-parameter of a standard single mode fiber. It is observed that the mean PMD saturates for V ? 1.8 leading to very small second order PMD.     

A deeper analysis of the second order polarization mode dispersion in optical communications systems

This article presents a new approach to the combined analysis of the first and second-order polarization mode dispersion (SOPMD) and shows their importance in optical communication systems performance. How it affects the relation between second-order polarization mode dispersion (PMD) and the differential group delay (DGD) in a single mode fiber is discussed. The analysis is based on time or wavelength and temperature variations changing/impacting PMD and DGD measured values. We present long term statistical characteristics of second-order PMD over a PMD emulator, and investigate the correlation between SOPMD, depolarization (DEP) and polarization dependent chromatic dispersion (PCD). Some authors calculate these modes from measurements obtained from the first order polarization mode dispersion, assuming a positive correlation between these effects, but this not the real relationship between them. A new interpretation of SOPMD was used to analyze the problem of the spectral stability in terms of the temperature. Actual techniques for determination SOPMD, consequently DEP and PCD, use the variation in time and/or wavelength. In some studies the results obtained for SOPMD are correlated to first-order PMD (FOPMD). This paper shows, based on measurements, that the correlation between first and second order Polarization Mode Dispersion (PMD) in the case where the last one is a function of the temperature can assume positive or negative values.     

光纤光栅双折射效应的实验研究

本文对均匀光纤光栅和线性啁啾光纤光栅的双折射效应分别进行了实验研究. 利用压电陶瓷的压电效应实施对均匀光纤光栅和线性啁啾光纤光栅的侧向挤压, 使之产生双折射, 通过改变施加在压电陶瓷上的电压值, 可以实现对光纤光栅双折射大小的控制. 侧向挤压线性啁啾光纤光栅可以补偿光纤通信系统中的偏振模色散.     

Measurement of polarization mode dispersion in single mode fibre

Polarization mode dispersion (PMD) in fibre seriously limits the high-capacity optical fibre communication system. The measurement of PMD was realized by three methods — wavelength scanning method, modified OTDR method and modified York S-38 dispersion meter method. Different types of fibres have been measured using the above three methods and the results have been found to be consistent. Finally, we also found that the measured results of longer non-polarization-maintaining fibre satisfied the Maxwellian probability density function and the results of polarization maintaining fibre were derived. Supported by The National Natural Science Foundations of P. R. China.     

Statistical properties of polarization mode dispersion

Polarization mode dispersion (PMD) is one of the effects which limit bit rates in optical communications systems. The statistics of PMD is complicated because it is multi-dimensional, however, many useful results have been obtained. Much work has been done, but the subject is not yet closed, as witnessed by the existence of this conference. In this talk I will review the way some statistical equations are treated. I will briefly discuss the methods of Ito and Lax, and their application to the PMD problem.