# Mapping PMD quantifies system performance

**by Aaron Ayer, Jay Damask, George Gray, and Paul Leo**

Understanding the effects of polarization-mode dispersion on carrier service-level agreements is difficult and has been approached inconsistently. Armed with measurement standards and a process for calculating total outage probability, vendors and carriers can accurately forecast performance in the presence of PMD.

The focus of the telecommunications industry changed substantially in 2001. The economic downturn re-emphasized the importance of the established carrier base, forced these carriers to squeeze maximum efficiency from their networks, and mandated that every possible step be taken to minimize capital outlay.

The performance of a carrier's installed fiber plant thus becomes a critical factor. Using legacy fiber means overcoming a certain amount of optical "baggage." Perhaps the most problematic issue for high-speed transmission on older networks is polarization-mode dispersion (PMD). Polarization-mode dispersion can impose hard limits on the distances that transmission systems can attain, often requiring the addition of uneconomical regeneration equipment.

At 10 Gbit/s, the current transmission rate of choice, PMD is a problem that must be addressed on a significant number of fiber spans. As the best fibers have already been selected for early 10-Gbit deployment, the proportion of PMD-impaired fibers that must be lit grows with time. At 40 Gbit/s, slated for first deployments within the next 18 months, it is widely accepted that PMD will be a limiting impairment on virtually every span.

In response, network planners have begun to consider PMD carefully when considering new capacity deployments. Increasingly, they seek to ensure that the systems deployed in their networks will perform to a certain standard in the presence of PMD. When PMD potentially prevents lighting of capacity, they actively seek qualification of PMD compensation solutions.

Here a problem arises. To what standard should systems be tested? No widely available test exists that translates a given PMD level into quantitative performance metrics for lit optical-communications systems. How can a vendor prove that its solution—with or without PMD compensation—actually works? How can a carrier compare two different PMD compensation solutions? And how can a carrier verify the performance a system will exhibit for specific network conditions?

**TAKING MEANINGFUL MEASUREMENTS**

First, to bring confidence to any PMD stress test of an optical transmission system or subsystem, it is essential to know with certainty the levels of PMD that are introduced during a test. Next, the test should be reproducible—repeatability allows a carrier to compare one system with another, and to validate evaluations performed by the system vendors. Finally, it should be possible to correlate the experimental results with PMD statistics to estimate network outage probability. This total outage probability (TOP) is the definitive measure of performance that a carrier can translate directly into service-level agreements.

The key enabler for testing PMD states is a programmable PMD source (PMDS). Polarization-mode sources that create first-order PMD have been available for some time now.

First-order PMD is the well-known differential group delay (DGD), which is the time slip between two orthogonal polarization components. Differential group delay alone is frequency independent and simply delays one polarization component of a bit stream with respect to the other. This effect alone is sufficient to bring down an optical link, but more complex effects, like second- and higher-order PMD, play an important role.

Second-order PMD (SOPMD) introduces a frequency dependence to the impact PMD has on a bit stream. One of the two second-order components, called polarization-dependent chromatic dispersion (PDCD), alters the DGD magnitude for each frequency component across the pulse bandwidth. The other second-order component, called depolarization, changes the orientation of polarization components from one frequency to another, which creates a "loss of identity" to the bit pattern because no one polarization component retains the original bit pattern. It is generally accepted that depolarization dominates PDCD in terms of bit-pattern impairment.

Sources that generate first-order PMD alone are thus inadequate to suitably reproduce the PMD components generated by installed fiber. In fact, their use can dramatically misrepresent the real-world performance of a receiver. A suitable source is one that provides calibrated, predictable, repeatable, and stable real-world PMD. For example, a PMDS can be built with waveplates and birefringent crystals (see Fig. 1). A significant advantage of such a multistage source is that both first- and second-order PMD can be generated. Operation of the instrument such that two groups of stages are created will generate both the frequency-independent DGD and the frequency-dependent depolarization component of SOPMD. Within a group, the waveplates are aligned with the crystals, generating pure DGD. The PMD source as illustrated spans a DGD range of 0 to 120 ps, and a SOPMD range of 0 to 3600 ps^{2}.

To test with a PMDS, the instrument is inserted between a transmitter and receiver, and several hundred first- and second-order PMD states are generated. At each PMD state, the bit-error rate (BER) is measured. A "PMD map" is thus obtained that indicates how a particular receiver performs over DGD and SOPMD space (see Fig. 2, left). The gradient lines in this map are not vertical, calling out the rather strong SOPMD sensitivity that characterizes this receiver. A second PMD map is shown for the same receiver, but with the addition of a PMD compensator module (see Fig. 2, right). The marked improvement highlights how these maps can be used to compare performance levels. Similarly, mapping can be used to compare receiver designs during development at a system vendor or to compare systems during laboratory evaluation at a carrier.

**INTRODUCING FIBER STATISTICS**

Polarization-mode dispersion maps for receivers provide a powerful qualitative insight into the performance of a subsystem or system over a swath of PMD space. However, quoting a service-level agreement with confidence requires including the external variable of fiber behavior itself. That's where the statistics of fiber PMD come into play.

Poole and Wagner were the first researchers to define PMD as a quantifiable effect.^{1} Foschini then applied powerful stochastic techniques to arrive at analytic expressions for the probability of differential group delay, second-order PMD, and various components.^{2, 3} For example, it has been established that DGD in a fiber follows a well-known Maxwellian distribution.

While the formulas derived by Foschini are extremely useful, there is a limitation: the joint probability of arriving at a specific combination of DGD and SOPMD is required to analyze the PMD maps. This probability can be calculated directly on a computer, creating a joint-probability density function. Gordon shows how to apply PMD concatenation rules so that the overall PMD can be calculated at one frequency for any number of birefringent sections.^{4} However, billions of calculations are necessary to even begin to approach low-likelihood occurrences.

One such calculation using a billion instances of a 2000-section fiber is shown in Fig. 3. The contours lie in SOPMD/DGD space, and fall off with higher PMD values. Here, the importance of including SOPMD in evaluation is again emphasized: while PMD maps show the impact of SOPMD on a receiver, the joint-probability density function demonstrates statistically that SOPMD is almost always present.

Conveniently, the joint-probability density function of Fig. 3 is universal; both the DGD and SOPMD axes scale with the mean fiber PMD, <τ>. That means once the joint-probability density function has been calculated, it only needs to be scaled to the appropriate fiber.

**PREDICTING NETWORK PERFORMANCE**

The inputs are now complete to calculate total outage probability, a critical metric for a service-level agreement. The TOP calculation combines a PMD map and the universal joint-probability density function, and can be written as

where Pr(τ, τ_{ω}) is the probability of reaching PMD state (τ, τ_{ω}); <τ> is the mean-fiber PMD that is used to scale the joint-probability density function; I() is an indicator function that is unity if the argument is true and zero otherwise; and TOL is a threshold (for example, 1 x 10^{-9}).

In a plot of a TOP curve as a function of mean-fiber PMD, the curve extends down to 15 ps (see Fig. 4). At mean DGD = 15 ps, the predicted outage is 30 sec/yr for the receiver map in Fig. 2 (left). As the mean DGD increases so does the expected number of seconds with severe errors. (Total outage probability values below mean DGD = 15 ps are unavailable because the joint-probability density function plotted in Fig. 3 does not have visibility below Pr = 10^{-4}.)

Figure 4 also plots a simpler quantity called the expected bit-error rate, E[BER]. The expected BER is simply the measured BER at a particular PMD state multiplied by the probability of arriving at that state. The E[BER] measurements have a floor at 10^{-11} only because of the measurement limitations. The dashed line extrapolates beyond the error floor.

**CONCLUSIONS**

Armed with a programmable PMD source, the universal PMD joint-probability density function, and methods of how to combine measurement with statistics, the TOP for a specific optical system can be predicted for any given network PMD level. With this information, systems performance can be objectively and quantitatively compared. Economic analyses can be performed with confidence to determine the advantages of compensated or uncompensated system deployment. Finally, perhaps most importantly for a carrier, engineering rules can be written for their systems and crafts persons can be trained to deploy PMD-mitigated capacity on fiber previously unusable at 10 Gbit/s.

**REFERENCES**

1. C. D. Poole and R. E. Wagner, Electron. Lett. 22(19), 1029 (Sept. 11, 1986).

2. G. J. Foschini and C. D. Poole, J. Lightwave Technol. 9(11), 1439 (November 1991).

3. G. J. Foschini et al., IEEE Photon. Technol. Lett. 12(3), 293 (March 2000).

4. F. Curti et al., J. Lightwave Technol. 8(8), 1162 (August 1990).

**Aaron Ayer** is senior product marketing manager, and Jay Damask, George Gray, and Paul Leo are engineers at Yafo Networks, 1340F Charwood Rd., Hanover, MD 21076. Aaron Ayer can be reached at aayer@yafonet.com.