## 2 Coherent Burst-Mode Receivers for PDM-QPSK and PDM-16 QAM Modulated Payloads
### Coherent Burst-Mode Receiver Based on the Stokes Sphere  
Using the Jones notation, the received optical signal can be expressed as:  
\[\left( \begin{array}{c}\widetilde{X}^{\prime}\\ \widetilde{Y}^{\prime}\end{array} \right)=M\!\left( \begin{array}{c}\widetilde{X }\\ \widetilde{Y}\end{array} \right)+N(t), \tag{1}\]  
where \(\widetilde{X}\) and \(\widetilde{Y}\) are the complex envelopes of the two orthogonal transmitted polarizations, \(\widetilde{X}^{\prime}\) and \(\widetilde{Y}^{\prime}\) are the received complex envelopes and \(N(t)=\left[\widetilde{n}_{x}(t)\;\widetilde{n}_{y}(t)\right]^{T}\) is a complex additive white Gaussian noise (AWGN) added by active electrical and optical components to both polarizations. The matrix \(M\) models the polarization rotation and can be expressed as [29; 30]:  
\[M=\left( \begin{array}{cc}cos(\theta)e^{-i\varphi/2}&-sin(\theta)e^{-i\varphi/ 2}\\ sin(\theta)e^{+i\varphi/2}&cos(\theta)e^{+i\varphi/2}\end{array} \right), \tag{2}\]  
where the angles \(\theta\) in (\(-90^{\circ}\), \(90^{\circ}\)) and \(\varphi\) in (\(-180^{\circ}\), \(180^{\circ}\)) cover the whole Poincare sphere. Note that this model does not include fiber attenuation losses, chromatic dispersion (CD), polarization dependent loss (PDL) and fiber nonlinearities [31]. The matrix \(M\) is unitary and hence its inverse is equal to its transpose conjugate. Consequently, \(M^{-1}\) can be trivially computed if an estimation of the values of the SOP rotation angles \(\theta\) and \(\varphi\) is known. The Stokes parameters of the received optical signal are defined as [32]:  
\[ \begin{split} s_{0}&=|x|^{2}+|y|^{2},\\ s_{1}&=|x|^{2}-|y|^{2},\\ s_{2}&=2\cdot Re(x\overline{y}),\\ s_{3}&=2\cdot Im(x\overline{y}).\end{split}  \tag{3}\]  
When plotted on the Poincare sphere and for a PDM-QAM modulated signal, the Stokes transformation defines a lens-shaped object [33]. For an optical signal digitized at two samples per bit, each of the tributary components defines an eye diagram as illustrated on Figure 2a, where the clock recovery has placed the even samples at the center of the eye (the optimum sampling point) and the odd samples define the transition points. When plotted on the Stokes space, the even samples appear as a cluster of points located close to the sphere surface and the transition samples, i.e., the odd samples appear as smaller clusters with lower absolute value inside the sphere. For a PDM-QPSK modulated signal, the even samples produce four clusters of points on the sphere surface located on the vertexes of a square. High order QAM modulation formats produce clusters of points that approach the lens-shaped object [34]. This is illustrated on Figure 2d for a PDM-16 QAM modulation. Note that the lens-shaped object defines a geodesic line on the Poincare sphere and that this geodesic line defines an intersection plane with the sphere that can be characterized by a normal orthogonal vector \(\overrightarrow{n}\). The SOP rotation angles \(\theta\) and \(\varphi\) can be estimated by finding the best fitting plane for the set of points \(S=\left\{\overrightarrow{s}_{i}\right\}\), where \(\overrightarrow{s}=(s_{1},\;s_{2},s_{3})\) correspond to the points in the Stokes space given by the even samples of the signal. In this work, the singular value decomposition (SVD) [35] was used to estimate the vector \(\overrightarrow{n}=(n_{1},\;n_{2},n_{3})\) normal to this plane. With the knowledge of \(\overrightarrow{n}\), an estimation of the SOP rotation angles is given by:  
\[\hat{\theta}=\frac{\tan^{-1}\sqrt{n_{2}^{2}+n_{3}^{2}}}{n_{1}}, \tag{4}\]  
\[\hat{\varphi}=\tan^{-1}\frac{n_{3}}{n_{2}}. \tag{5}\]The SVD has a high computational cost. However, an accurate estimation of the angles \(\hat{\theta}\) and \(\hat{\varphi}\) can be obtained with only a reduced number of \(S\) points, which may reduce the overall complexity of the SOP estimator compared to alternative DCBMRX designs with iterative algorithms.