## 2 Coherent Burst-Mode Receivers for PDM-QPSK and PDM-16 QAM Modulated Payloads
### Methodology and Experimental Results  
Note that the experimental setup of Figure 4a does not include a polarization scrambler to rotate the SOP of the optical packets before the receiver. Instead, a set of 128 packets having a known SOP were captured, and then the SOP was estimated using the whole of the optical packet in order to  
Figure 4: (**a**) Experimental setup for the characterization of the proposed coherent burst-mode receiver; and (**b**) diagram of the generated optical packets.  
Figure 3: Digital signal processing (DSP) stages of the digital coherent burst-mode receiver.  
minimize the estimation errors, and the SOP was digitally corrected. Then, each packet was chopped to the desired header length and the SOP was digitally rotated using Equation (2) with \(\varphi\) in the range [-180\({}^{\circ}\), 180\({}^{\circ}\)] and \(\theta\) in the range of [-90\({}^{\circ}\), 90\({}^{\circ}\)] to simulate any arbitrary SOP rotation state over the whole Poincare sphere. Before the digital SOP rotation, synthetic AWGN, independent for every optical packet and every SOP rotation, was added to the signal in order to produce a payload BER of roughly \(10^{-3}\).  
Figure 5a shows the averaged payload-BER of 128 optical packets as a function of the rotation angles \(\theta\) and \(\varphi\) covering the whole Poincare sphere surface. In this case, no synthetic noise was added and the equalizer was a standard CMA equalizer initialized with taps:  
\[\begin{split} H_{xx}&=H_{yy}=[0\;0\;0\;1\;0\;0],\\ H_{xy}&=H_{yx}=[0\;0\;0\;0\;0\;0].\end{split} \tag{7}\]  
Inspection of Figure 5a clearly reveals that the CMA equalizer exhibits a singularity when the SOP rotation angle is \(\theta=[+45^{\circ},-45^{\circ}]\) for any value of \(\varphi\). We repeated this measurement and added synthetic noise to the experimental data with a power such as the payload-BER equaled the forward error correction (FEC) limit of \(10^{-3}\). Figure 5b shows that the effect of noise, compared to Figure 5a, is to unsmooth the curves, but the CMA equalizer singularity is still present. Next, the proposed SOP estimation and equalizer initialization algorithm were tested for header sizes of [4; 8; 16; 32; 64] samples. The results of this measurement are plotted in Figure 5c. Inspection of the figure reveals that the average BER stays in a range of approximately [\(10^{-3}\), \(5\times 10^{-3}\)], independently of the header size, which demonstrates that the proposed algorithm solves the CMA singularity problem.  
Finally, we assessed the equalizer convergence time by doing a time-resolved BER measurement averaging the errors of each of the four PDM-QPSK PRBS tributaries over the 128 captured packets. Figure 6a shows the time-resolved BER in the packet payload, for the standard CMA receiver and the proposed new receiver algorithm, averaged over all SOP rotations covering the Poincare sphere. The red line shows the average BER for all 128 packets without any SOP rotation and can be considered as the calibrated BER value. For the standard CMA receiver, all the cases where the singularity was detected were not computed in order to not artificially increase the number of errors. Figure 6b shows the curves of Figure 6a filtered with a median filter having a window size of 31 to smooth the curves. Results demonstrate that, on average, the CMA receiver takes up to 3000 symbols to converge. However, the CMA equalizer initialized with the algorithm proposed in this work converges instantly for an estimation size of 16 or greater and completely avoids the singularity problem. Under a noise loading that yields a payload BER of \(10^{-3}\), initialization of the CMA equalizer is successful with only 16 estimation points of the packet header.  
Figure 5: (**a**) BER probability as a function of the SOP angles \(\theta\) and \(\varphi\) for the standard CMA algorithm and no additional noise, (**b**) same as (**a**) but with synthetic AWGN, and (**c**) same parameters as (**a**) but for the proposed algorithm and varying-size header lengths.