Multicanonical Investigation of Joint Probability Density Function of PMD and PDL

Multicanonical Investigation of Joint Probability Density Function of PMD and PDL David S. Waddy, Liang Chen, Xiaoyi Bao Fiber Optics Group, Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canada K1N 6N5 waddy@ieee.org Abstract: Multicanonical Monte Carlo (MMC) simulations are used to investigate the joint probability of Polarization Mode Dispersion (PMD) and Polarization Dependent Loss (PDL) in a field fiber emulator. A joint PDF of PMD and PDL showing rare events is reported for the first-time. Wang-Landau simulation technique is employed to see how PMD is affected by PDL at low probability events and conversely, how PDL is affected by PMD. The MMC technique is also used to model an optical communications system with PMD and PDL. Introduction Monte Carlo (MC) Simulations have been used in the past to investigate Polarization-mode dispersion (PMD) and polarization dependent loss (PDL) in optical fiber systems [1]. Recently, more advanced techniques: Importance Sampling[2, 3], Brownian Bridge[4], Wang-Landau Monte Carlo (WLMC) [5], and Multicanonical Monte Carlo (MMC) [6, 7] have been introduced to investigate PMD. These advanced simulation techniques are used because they allow one to study the rare PMD events that cause system outages. PMD and PDL are both present in optical fibers, but the PMD usually results in a much greater system impact. PMD and PDL interact and cause different statistical distributions than either acting alone[8]. In this proceeding we investigate the PMD/PDL joint probability density function (PDF). We also use WLMC to study rare events in the PDL and PMD statistical distributions. To our knowledge, this is the first time the WLMC or MMC technique has been used to study PMD and PDL together. The main advantage of MMC and WLMC is it does not need to be specially biased to reach rare statistical events. Importance sampling has been recently used to model PMD and PDL[9], but not the joint PDF. Multicanonical Monte Carlo MMC technique was first applied to PMD simulations by Yevick [10]. MMC is an important numerical simulation techniques because it allows for rare statistical events to be sampled with equal probability. Standard MC simulations sample high frequency events more than low frequency events and converge slowly to a correct answer. MMC allows for rare events to be sampled that would be time prohibitive for a standard MC simulation to sample. MMC is gaining popularity in the PMD community because it can be used to quickly compute the statistics of large communication systems that have many statistical processes interacting. WLMC is also employed. This is another advanced MC simulation technique that can be applied to optical communication systems. This procedure uses two histograms and measures the flatness of one to determine when all events have been properly sampled. Details of the implementation can be seen in [5]. We prefer WLMC to MMC because it converges faster for most cases we have attempted. Joint PDF of PMD and PDL To calculate the joint PDF of PMD and PDL a computer code was written implementing MMC similar to Yevick s[6] which uses the wave-plate model[1]. The PMD/PDL was found by using PMD/PDL recursion relation[1]. The birefringence (DGD) and differential attenuation (PDL) were left constant and distributed evenly along the N=15 sections 488 Photonics North 2004: Photonic Applications in Telecommunications, Sensors, Software, and Lasers, edited by J. Armitage, R. Lessard, G. Lampropoulos, Proc. of SPIE Vol. 5579 (SPIE, Bellingham, WA, 2004) 0277-786X/04/$15 doi: 10.1117/12.567470

15 30 25 10 20 15 5 10 0 0 5 10 15 5 0 0 5 10 15 Figure 1. MMC simulation of joint PMD/PDL PDF for 15 section emulator with PMD=1 ps and PDL=1dB per section. 7 iterations, 4 million in each. Numbers correspond to 1eN probability. Figure 2. MMC simulation of joint PMD/PDL PDF for 15 section emulator with PMD=2 ps and PDL=1dB per section. 7 iterations, 4 million in each. Numbers correspond to 1eN probability. in the model. The θ and φ angles were perturbed by using a Gaussian random variable with standard deviation π 8N centered around the previous accepted θ/φ value. The birefringence and differential attenuation angle are dependent in this simulation. Note that we use PMD to refer to the mean DGD over many sections. Figure 1 shows the joint PMD/PDL PDF for an emulator with 1 ps of PMD and 1 db of PDL on each section. Figure 2 shows the joint PMD/PDL PDF for an emulator with 1 ps of PMD and 2 db of PDL on each section. One can observe that both follow the same general shape. PMD in the Presence of PDL The previous section uses MMC for the simulation, in this section WLMC was chosen because it converges quickly. A mean PMD of 2.5 ps and 32 DGD and PDL sections were used for all simulations. The Wang-Landau parameters were 100 histogram bins, 2000000 angle changes and 5 iterations. Figure 3 shows six simulation runs for each value of PDL (7 db, 8dB, and 9dB). The birefringence and differential attenuation angle are independent in this simulation. The analytical formula (equation 15 of reference [11]) is for PMD of 2.5 ps and null PDL. One can observe that for low amounts of PDL the analytical approximation for null PDL holds. Only for high amounts of PDL (greater than 7 db) and low probability does a difference become apparent. PDL in the Presence of PMD The WLMC technique was used to numerically simulate how PDL is affected by PMD. 100 histogram bins, 2000000 angle changes and 5 iterations were used for the simulation. The number of sections of PMD and PDL was fixed at 500. The mean PDL used was 2.5 db. An analytical formula for PDL in the presence of null PMD was used: equation 6 in reference [12]. Figure 4 shows the statistical distribution of PDL for PMD of 0.0001, 0.01, and 10 ps. The birefringence and differential attenuation angle are independent in this simulation. We verify that PMD is independent of PDL to very low probabilities, which agrees with previous results for lower probabilities [13]. A discrepancy between the analytical plot at 2.5 db and the simulation results at 2.5 db is observed. By fitting a PDL of 2.7 db the analytical Proc. of SPIE Vol. 5579 489

10 0 analytical simulation 10 5 Normalized Frequency of Occurence 10 10 15 10 0 7dB 8dB 9dB 0 2 4 6 8 10 12 14 16 18 Figure 3. Wang-Landau simulation showing PMD of 2.5 ps for PDL of 7 db, 8 db, and 9 db. The analytical plot is for PMD of 2.5 ps and PDL of 0 db. formula was observed to fit. An approximation is made in reference [12] and we believe this to be the cause of the discrepancy. The analytical formula is observed to diverge at lower probabilities. It is plausible that the difference is not noticeable at the lower probabilities previously tested. Further investigation is needed. 10 10 analytical 2.5 db analytical 2.7 db pmd=0.0001 pmd=0.01 pmd=10 Normalized Frequency of Occurence 10 10 10 10 10 14 10 16 10 18 2 4 6 8 10 12 14 16 Figure 4. Wang-Landau simulation showing PDL of 2.5 db for PMD of 0.0001 ps, 0.01 ps, and 10 ps. Analytical results are for 2.5 db and 2.7 db of PDL and null PMD. Optical Communications System An optical communication system is modelled using the MMC simulation technique. The system consists of a pseudorandom bit sequence (PRBS) generator driving a LiNO3 modulator through which a laser is launched. The PRBS is 64 bits with 128 samples per bit. A bit frequency of 10 GHz and a laser frequency of 193.1 THz (~ 1550 nm) is used. As figure 5 shows an erbium doped fiber amplifier (EDFA) is modelled to noise load the system with amplified spontaneous emission (ASE) noise. This is done to simulate a real system that is noise limited. An attenuator is used to lower the optical signal to noise ratio (OSNR). A PMD/PDL emulator is modelled as above with a mean PMD of 490 Proc. of SPIE Vol. 5579

13 ps and mean PDL of 2 db. The emulator uses 12 sections. Optical and electrical filters are used to remove noise and open the electrical eye. Sampling is performed at a fixed point, which is deemed to be optimal at the beginning of the simulation. Noise caused by the the optical detector and EDFA are modelled as Gaussian white noise. The MMC portion of the simulation only consists of the PMD/PDL emulator, not all statistical effects interacting. In other words, the other effects are just used in a MC context, not MMC. We hope to perform full MMC simulations with all effects interacting in the future. It would be too lengthy to go into details on the full optical system, but formulas used are standard and almost exclusively taken from Agrawal [14]. Figure 6 shows the Q factor result for PMD and PDL. The MMC simulation used 4 iterations and 5000 angle changes. One can observe that the plot is still quite jagged and needs more simulation iterations and angle changes to converge. Unfortunately, the simulation is slow and needs to be optimized before it is practical to get a better plot. Work is in progress to improve the results. Conclusions Figure 5. Schematic of the optical communication system modelled by the simulation. The joint PDF of PMD/PDL was modeled for the first time using MMC which allows the rare events to be discerned in greater detail than previous MC simulations. This aids system designers in knowing the rare combined effects of PMD and PDL. PMD was investigated in the presence of varying PDL. PDL of greater than 7 db was found to effect PMD for rare event cases. PDL was investigated in the presence of differing PMD. No discernible effect was observed for rare events. However, a small discrepancy with the analytical result was observed. The MMC technique was used to simulate an optical communications system for the first time, to our knowledge. More work is in progress to improve the quality of this result. Acknowledgement It is a pleasure to acknowledge the financial support of NSERC (Natural Sciences and Engineering Research Council of Canada), CIPI (Canadian Institute for Photonic Innovations) and AAPN (Agile All Photonic Network). Proc. of SPIE Vol. 5579 491

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