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Two-dimensional statistics and Q-penalty performance under the combination of two major impairments induced by polarization-dependent loss, namely level imbalance and loss of orthogonality between polarization-multiplexed tributaries, for a polarization-division-multiplexing digital coherent transmission at over 100-Gb/s are presented for the first time to estimate the outage probability needed for designing the system.
©2011 Optical Society of America
1. Introduction
Polarization-dependent loss (PDL) is recognized as a major obstacle to the development of polarization-division multiplexing (PDM) digital coherent transmission systems operating at more than 100 Gb/s [1–3]. In a point-to-point transmission system as shown in Fig. 1(a) , PDL is defined as the maximum to minimum transmission ratio similar to that in a single optical device. The former, which is called system PDL, must be treated as a random variable [4] because the state of polarization (SOP) can be randomly converted in the transmission fibers, while the latter, device PDL, is a constant value. In single-polarization transmission systems, the system PDL causes the optical power of the signals to fluctuate with variations in the SOP of the optical signals. The system has therefore been designed based on the statistical nature of system PDL to estimate the outage probability, i.e., the probability of exceeding the tolerable PDL value [5]. Previous studies have revealed that the probability density function of the system PDL is a Maxwellian distribution [4,6]. It should be noted that the system PDL is an optical property of a transmission line, and does not imply the quality of the optical signals. We need to consider signal impairments that are appropriate for a PDM transmission. As illustrated in Fig. 1(b), two tributaries of polarization-multiplexed signals X and Y, which initially have the same power level and are polarized orthogonally, experience PDL in a transmission system, which causes them to suffer a power level imbalance and/or lose orthogonality [7]. These two impairments, namely the level imbalance caused by PDL (LIP) [8] and the loss of orthogonality caused by PDL (LOP), represent the quality of PDM signals and affect the performance when demultiplexing the PDM tributaries with digital signal processing. Similar to the system PDLs, LIP and LOP must be treated as a random variable. It should also be noted that LIP and LOP usually occur simultaneously and must be treated as a two-dimensional random variable for estimating the outage probability when designing PDM transmission systems.
Fig. 1 Two types of transmission systems including PDL devices: (a) a single-polarization transmission and (b) a polarization-division multiplexing (PDM) transmission where Tmax and Tmin are the maximum and minimum transmission ratios, respectively, X' and Y' are the optical fields of the X- and Y-tributaries, respectively, and θ is their polarization angle after transmission. The models are used for performing numerical simulations in this paper.
In this paper, we investigate and present the statistical properties of LIP and LOP and signal quality under a combination of LIP and LOP for the first time. The two-dimensional distribution and Q-penalty map defined over the LIP-LOP plane are introduced via simulation and experiment to estimate the outage probability.
2. Statistical properties of LIP and LOP
2.1 Definitions and simulation model
Let us consider a PDM optical signal that experiences PDL and whose two tributaries have imbalanced amplitudes and are not orthogonally polarized as shown in Fig. 1(b). LIP is defined as the power ratio between the X- and Y-tributaries [8], that is,
where EX' and EY' are the complex amplitudes of the X- and Y-tributaries. The LOP, namely the deviation from a right angle, is expressed aswhere θ is the polarization angle between the X- and Y-tributaries. The optical receiver sees the signal distortion caused by the PDL as the LIP and the LOP. To perform numerical simulations, we used the model presented in Fig. 1(b), which is a transmission line consisting of PDL devices with the same PDL value connected using single-mode fibers [4]. During propagation through each optical fiber, the SOP of an incident light is converted to another SOP to provide uniform distribution over the Poincaré sphere [4,6]. The LIP and LOP statistics were obtained with the Monte Carlo method by performing iterative calculations for various combinations of SOP conversion.2.2 Statistical property of LIP
To describe the statistics of LIP and LOP, let us begin by explaining them separately to comprehend their nature although, in actual situations, LIP and LOP occur simultaneously and must be treated as combined impairments as mentioned in the Introduction. Figure 2 shows histograms of (a) the system PDL and (b) the LIP calculated using the same model where the device PDL is set at a constant value of 0.5 dB, the number of spans is 20, and the trials were performed 100,000 times. The horizontal axes are in decibels. As revealed in previous studies [4,6], the histogram of the system PDL fits well to a Maxwellian function. On the other hand, the LIP histogram was different from that of the system PDL. As a result of a χ2 goodness-of-fit test, which shows that the calculated p-value was 0.58 exceeding the 5% level of significance, the histogram of LIP was found to be consistent with the Gaussian distribution, that is, the distribution of LIP is in good agreement with the Gaussian distribution. This result is very important and implies that we should use the Gaussian function when estimating the outage probability of tolerable LIP. As the result of calculations employing various device-PDL values or numbers of spans, the ratio of the standard deviation of the LIP to the mean PDL [5], which is equal to the expectation value of the system PDL, was approximately 0.6.
Fig. 2 Histograms (blue dots) of the calculated (a) system PDL and (b) the level imbalance caused by PDL (LIP) where the device PDLs are set to 0.5 dB, and there are 20 spans, and 100,000 trials. The red curves are the fitting curves: (a) Maxwellian function and (b) Gaussian function. The horizontal axes are both in decibels.
2.3 Statistical property of LOP
Figure 3(a) shows a LOP histogram, which was calculated using the same model as in Fig. 1(b). The distribution function of the histogram is similar to the Rayleigh function but unidentified at present. To help comprehend the LOP distribution, let us present another point of view. Figure 3(b) shows the histogram of a newly introduced index, namely, the polarization crosstalk induced by PDL (PXP), which is defined with the following expression:
Fig. 3 Histograms (blue dots) of (a) LOP (loss of orthogonality induced by PDL) and (b) PXP (polarization crosstalk induced by PDL) under the same conditions as in Fig. 1. The red curve in (b) is a fitting curve, which is a Rayleigh function.
As a result of a χ2 goodness-of-fit test, which shows that the calculated p-value was 0.42 thus exceeding the 5% level of significance, the histogram of PXP was found to be consistent with the Rayleigh distribution, that is, the distribution of PXP is in good agreement with the Rayleigh distribution.
2.4 Statistical property of the combination of LIP and LOP
Now let us describe the statistical property of the combination of LIP and LOP. Figure 4 shows two-dimensional histograms of LIP and LOP under various conditions where (a), (b) and (c) are for the same device PDL of 0.5 dB and a different number of spans, that is, 5, 10, 20 spans, respectively. (d), (e) and (f) are for the same number of span (20) and different device PDLs of 0.2, 0.3 and 0.4 dB, respectively. To calculate the outage probability, information on the signal quality, for example, the Q-penalty under the combination of LIP and LOP is required as well as the histogram.
Fig. 4 Two-dimensional histograms of LIP and LOP under various conditions where (a), (b) and (c) are for the same device PDL of 0.5 dB and spans of 5, 10, 20, respectively. (d), (e) and (f) are for 20 spans and different device PDLs of 0.2, 0.3 and 0.4 dB, respectively. There were 100,000 trials.
3. Signal quality vs. combination of LIP and LOP
The transmission performance under the combination of LIP and LOP was experimentally evaluated as a Q-penalty by employing the setup shown in Fig. 5 . The two optical tributaries generated from a QPSK modulator (QPSK mod), which was driven using two 32-Gb/s electrical signals consisting of pseudo-random bit sequences with lengths of 211-1, were polarization-multiplexed to produce a 128-Gb/s PDM-QPSK signal after setting the LIP with a variable optical attenuator (VOA) and the LOP with a polarization controller (PC1). The optical frequency was 193.4 THz (1550.12 nm). The polarization-multiplexed signal was input into a receiver (PMD-QPSK RX) after loading optical noise from an erbium-doped fiber amplifier (ASE). The received signal were stored as 4-channel intradyne-detected data with a digital oscilloscope and post-processed with the off-line processing [9] to evaluate as the Q-factor.
Fig. 5 Experimental setup for investigating Q-penalty vs. LIP and LOP where VOA: variable optical attenuator for setting LIP, PC1: polarization controller for setting LOP, CPL: optical coupler, PC2: polarization controller for scrambling polarization, ASE: optical noise source consisting of an erbium-doped fiber amplifier. PDM-QPSK-TX and -RX are a polarization-division-multiplexing QPSK transmitter and receiver, respectively.
Figure 6(a) shows a contour map of the experimentally obtained Q-penalties, which were the differences between the Q-factors at LIP = 0 dB and LOP = 0 degrees over the LIP-LOP plane. The optical signal-to-noise ratio (OSNR) was set at 18 dB at the receiver. The Q-penalty data were interpolated as a quadratic function of LIP and LOP to calculate the contours. Figure 6(b) shows another Q-penalty map obtained from the numerical data of simulated signals impaired with LIP and LOP by employing the same off-line processing as that used in the experiment. These results were in good agreement in spite of some fitting errors.
Fig. 6 Contour maps of Q-penalty vs. LIP and LOP obtained from (a) experimental data and (b) numerical simulation data. The Q-penalty data were interpolated as a quadratic function of LIP and LOP to calculate the contours. The OSNR of the received signal was set at 18 dB.
4. Outage probability
We can estimate the probability of the outage exceeding a given tolerable Q-penalty related to the LIP and LOP with the following steps: (1) Divide the two-dimensional histogram into two regions according to whether the LIP-LOP impairment is tolerated for the given Q-penalty, that is, divide the histogram by the equi-Q-penalty curve for the given Q-penalty as shown in Fig. 6. (2) Integrate the histogram over the region shown by the hatching in Fig. 7 where the LIP-LOP impairment is not tolerated for the given Q-penalty. (3) The outage probability is the ratio of the integration calculated now to the integration over the entire region.
Fig. 7 Calculation of outage probability for a given Q-penalty. The two-dimensional histogram over the LIP-LOP plane is bounded by an equi-Q-penalty curve (dashed curve) for a given Q-penalty obtained from the Q-penalty map in Fig. 6. The hatched area is an integral region over which the outage probability is calculated.
Figure 8 shows the resulting outage probability vs. Q-penalty under various conditions: (a) for the same device PDL of 0.5 dB and spans of 5, 10 and 20. (b) for the same number of spans and different device PDLs of 0.2, 0.3, 0.4 and 0.5 dB. The two-dimensional histograms for 100,000 trials were calculated under the above conditions that correspond to Fig. 4. The variation in the outage probability with the Q-penalty was obtained by calculating the outage with respect to various equi-Q-penalty curves. The simulation results agreed well with the experimental results. With the outage graphs, for a given outage probability and a tolerable Q-penalty induced by LIP and LOP, we can estimate the tolerable device PDL value or the transmissible number of spans.
Fig. 8 Resulting outage probability vs. Q-penalty under various conditions: (a) for the same device PDL of 0.5 dB and spans of 5, 10 and 20. (b) for 20 spans and different device PDLs of 0.2, 0.3, 0.4 and 0.5 dB. The solid and dotted curves indicate the outage probabilities obtained with the Q-penalty maps of the experimental result (Fig. 6(a)) and the simulation result (Fig. 6(b)), respectively.
5. Conclusion
We investigated for the first time the statistical properties and the signal quality under the combination of two major impairments induced by PDL, which are the level imbalance caused by PDL (LIP) and the loss of orthogonality caused by PDL (LOP) for a polarization-division-multiplexing digital coherent transmission of over 100-Gb/s. The LIP distribution provided a good fit with a Gaussian function while that of the system PDL was a Maxwellian function. The two-dimensional histogram and Q-penalty map defined over the LIP-LOP plane were introduced via simulation and experiment to estimate the outage probability, which is needed for designing the system. The simulation results were in good agreement with the experimental results. The system requirements for PDL can be estimated from the outage probability based on the introduced two-dimensional statistics and Q-penalty map.
Acknowledgments
We are grateful to Yoichi Fukada, Mitsunori Fukutoku, Tetsuro Inui, and Yohei Sakamaki for fruitful discussions.
References and links
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