Advances in modeling optical fiber transmission systems
University of Maryland Baltimore County, Computer Science and Electrical Engineering Department, TRC-201A, 1000 Hilltop Circle, Baltimore, MD 21250, phone: 410-455-3501, fax: 410-455-6500, E-mail: menyuk@umbc.edu
*Also at: Laboratory for Telecommunications Sciences, c/o USARL, 2800 Powder Mill Road, Bldg. 601, Rm. 110, Adelphi, MD 20783-1197
Abstract
Our research group has made important advances in the past year in modeling optical fiber transmission systems. We highlight work on importance sampling and linearization.
Paper
In the past five years, the complexity and cost of optical fiber transmission systems has grown substantially. The greatly increased complexity of modern-day systems has led to a need for modeling methods that will allow engineers to design and optimize their systems in parallel with laboratory testbeds. A number of commercial enterprises now offer modeling tools that partially fulfill this need; however, current modeling methods are not equal to the task of accurately calculating bit error rates. Of course, there is much that current methods can do. They can be used to accurately optimize dispersion maps, modulation formats, channel spacings, and receiver bandwidths. Recently, my own group demonstrated an unprecedented agreement between theory and experiment over a distance of 24,000 km in calculating amplitude margins [1]. In this experiment, we observed voltage margins above which the error rate is greater than 10-6 due to marks being counted as spaces and below which the error rate is greater than 10-6 due to spaces being counted as marks. This achievement required us to use highly accurate models of both the receiver and gain saturation in the amplifiers.
At the same time, the holy grail of finding methods to accurately calculate bit error rates in real-world systems remains elusive. We have made important progress in this quest during the past year, and we will present those results here.
Before describing our recent progress, we note our view that this quest is not only of evident practical importance but is also a highly rewarding area of intellectual endeavor. The task of reconciling theoretical and experimental results, so that they agree to within the accuracy of both, more often than not yields unexpected insights into what is really important in the experiments. The great 19-th century mathematician, Henri Poincaré stated it best when he wrote (slightly abbreviated here), “Il est inutile de demander plus de précision (des calculs théoriques) qu’aux observations; mais on ne doit pas en demander moins.” (There is no point in asking for more accuracy from theory than from the observations, but one should not ask for less.) [2].
The first advance that I will describe is the application of importancesampling to polarization problems in optical fiber transmission. Inparticular, we have addressed the problem of determining the outage probability of an ideal first-order compensator. In practical system design, the usual practice is to allocate some system margin for a particular effect, e.g., 1 dB of margin for polarization mode dispersion (PMD), with an assurance that the power penalty will be less than that margin with some low probability, e.g., 10-6. The challenge has been that it is impossible in practice to directly simulate the billions of fiber realizations that would be needed to show that the outage probability is less than 10-6. Thus, workers in this field have resorted to stopgap procedures like demonstrating that the average differential group delay (DGD) is reduced as much as possible. Using importance sampling [3], we have found ways to bias the distributions of the DGD so that with 100,000 realizations or even less, we can accurately calculate the outage probability. We note that each realization corresponds to the fiber’s state at one moment in time as its state varies due to changing temperature and stress.
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Fig. 1. Outage probability as a function of the power penalty on both linear and logarithmic scales for a 10 Gbs NRZ signal transmitted through an optical fiber with 30 ps of average DGD. Solid lines show the uncompensated results. Dotted lines are results for a compensator with DGDc equal to 30 ps. Dashed lines are results for a compensator with DGDc equal to 75 ps. While choosing DGDc equal to the average DGD of 30 ps leads to a large reduction in the average power, choosing DGDc equal to 75 ps, which is 2.5 times the average DGD, leads to a large reduction of the outage probability at power penalties above 0.8 dB. |
In Fig. 1, we show the complement of the cumulative distribution function as a function of the outage probability for a first-order compensator with a fixed differential group delay element DGDc. The compensator has a variable polarization rotator, whose orientation we choose to maximally reduce the power penalty due to PMD. We show both normal and log scales. We study its ability to compensate for PMD in a fiber, whose accumulated average DGD equals 30 ps. It is evident that the choice DGDc = 30 ps reduces the power penalty more than DGDc = 75 ps. However, if the goal is to ensure that the probability of a power penalty of 1 dB is less than 10-6, the choice DGDc = 50 ps is better. This calculation would not be possible without importance sampling.
The second advance, which we call linearization, is based on the observation that noise beating with itself during the transmission through the optical fiber is normally quite small. One must, of course, take into account the noise beating with itself in the receiver. If noise beating with itself is small during transmission, then the noise components at the end of the transmission line will obey a multivariate Gaussian distribution [4]. We have applied this approach to our experimental dispersion managed soliton system over 24,000 km [1]. This application is a stringent test of our approach because the system is highly nonlinear. The noise components that we use are the Fourier components with phase jitter and timing jitter removed, as is appropriate in a soliton system. The phase jitter has no effect on the received signal, but the timing jitter must be restored at the receiver. We have directly calculated the covariance matrix from Monte Carlo simulations, and we have validated the linearization assumption by computation of the timing jitter and the marginal distributions of the Fourier components. We note that this approach not only allows us to accurately calculate the probability distribution function of the receiver voltage for the marks and the spaces, but it also allows us tocalculate the probability contour plots for the entire eye diagram.
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Fig. 2. (a) Probability density function of the filtered current y of a 10 Gbs dispersion-managed soliton signal afer square-law detection and an 8.6 Ghz filter. The solid lines are the pdf calculated using linearization, and the dashed lines are standard Gaussian fits. (b) Corresponding eye diagram, shown as a contour plot. |
In Fig. 2, we show the theoretically calculated probability distribution of the marks and the spaces in a dispersion-managed-soliton, recirculating-loop experiment. We note that the distribution of the marks and spaces is far from Gaussian. Moreover, because of the timing jitter, it is not a generalized chi-square distribution [4] either. Correctly calculating the probability distribution function is critical for accurately calculating both the optimal decision level and the bit error rates. In the case presented here, the standard Gaussian fit predicts a minimimum error rate of 10-41, while the accurate calculation predicts a minimum error rate of 5x10-13. We also show the corresponding eye diagram. In contrast to the usual numerical practice of generating an eye by superimposing traces of the marks and spaces, we are able to generate complete contour plots of the eye diagram.
In conclusion, good modeling of optical fiber transmission systems has become critical to design them efficiently. Much has been accomplished in recent years, and my own group has contributed in important ways to this development. In the past year, we have applied importance sampling to calculating the outage probability of PMD compensators, and we have shown that it is possible to accurately calculate the probability distribution functions of the receiver voltage for the marks and the spaces. While much remains to be done to achieve the ultimate goal of accurately modeling bit error rates, much has been accomplished.
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