# Polarimetry for Photonic Integrated Circuits
## 2 Measurement Methodology  
### Setup Calibration  
The basic approach in this work was to, first, fully characterize a measurement setup in Stokes space in the absence of any device under test (DUT). Then, this characterization was used as a calibration to characterize actual DUTs. The measurement setup is shown in Figure 1. It consisted of a tunable laser source (TLS), a polarization controller (PC), tapered fibers coupling in and out of the DUT, and a polarimeter (PM). All devices were in series in that order, interconnected by single-mode fiber (SMF). Note that no polarization maintaining SMF was used. We define Stokes vectors as follows: \(\tensor{s}{t}_{\textit{Is}}\) is incident to the PC described by \(\underline{M}_{\text{PC}}\), \(\tensor{s}{s}_{\textit{in}}\) is incident to the DUT, \(\tensor{s}{s}_{\textit{out}}\) exits the DUT, and \(\tensor{s}{s}_{\textit{meas}}\) is the PM measurement. The polarization rotations of the SMFs are described by the Muller matrices \(\underline{M}_{\textit{fiber,in}}\) and \(\underline{M}_{\textit{fiber,out}}\) for the fibers before and after the DUT, respectively. SMF Muller matrices are modelled via  
\[\underline{M}_{\textit{SMF}}=\underline{M}_{\frac{3}{4}}(\theta_{3})\underline {M}_{\frac{3}{2}}(\theta_{2})\underline{M}_{\frac{3}{4}}(\theta_{1}) \tag{1}\]  
where, \(\underline{M}_{\frac{3}{4}}\) and \(\underline{M}_{\frac{3}{2}}\) are the Muller matrices of quarter- and half-wave plates, respectively. They are given in Appendix A. This model is inspired by the popular \"mickey mouse\" polarization controllers, which consist of three fiber coils of different winding number that can be rotated relative to one another. Thus, the two fibers in the setup are parametrized by three angles each. The model accounts for any possible polarization rotation that can occur [8]. The particular PC used here (HP 8169A) uses a linear polarizer LP, and a quarter- and a half-wave plate (in that order) to control the SOP. These three elements can be rotated by the respective angles \(\theta_{LP}\), \(\theta_{\lambda/4}\) and \(\theta_{\lambda/2}\). We found that the modelling accuracy was improved significantly by taking into account their residual ellipticity. In addition, the two waveplates were found to be not exactly quarter- and half-wave retarders. For example, the fast axis of the HWP should be \(\left( \begin{array}{cccc}1&1&0&0\end{array} \right)^{T}\) and it should retard by half a wavelength. Therefore, its fast axis \(\tensor{u}{u}\) would have spherical angles \(\psi_{\tensor{u}{u}}=0\) and \(\chi_{\tensor{u}{u}}=0\) and the retardation would be \(\delta=\pi\). The imperfections of the waveplate are modelled via \(\psi_{\tensor{u}{u}}^{\rightarrow}=\Delta\psi_{\frac{3}{2}}\) and \(\chi_{\tensor{u}{u}}^{\rightarrow}=\Delta\chi_{\frac{3}{2}}\) for its fast axis and \(\delta=\pi+\Delta\delta_{\frac{3}{2}}\) for its retardation. The QWP is treated analogously. Similarly, the fast axis of the LP has residual angles \(\Delta\psi_{LP}\) and \(\Delta\chi_{LP}\). We call the Muller matrices of the PC LP, QWP, and HWP \(\underline{M}_{LP}\), \(\underline{M}_{\frac{3}{4}}\), and \(\underline{M}_{\frac{3}{4}}\), respectively. The tilde denotes the inclusion of residual ellipticity and retardation errors. A final refinement of the model is to include the polarization dependent loss (PDL) of the PC. It is modelled by an elliptical diattenuator (see Appendix A and [9]) with \(D_{PDL}\), \(\psi_{PDL}\), and \(\chi_{PDL}\).