## 2 A Modified Michelson Interferometer as a Parity Selector
\begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Incidence**} & \multicolumn{4}{c}{**Polarization**} \\ \cline{3-6} \multicolumn{2}{c}{} & \multicolumn{2}{c}{\(\boldsymbol{r_{\perp}}\)**(TE)**} & & \multicolumn{2}{c}{\(\boldsymbol{r_{\parallel}}\)**(TM)**} \\ \hline \multicolumn{2}{c}{} & \multicolumn{2}{c}{\(\boldsymbol{|r_{\perp}|}\)} & \(\boldsymbol{\phi_{\perp}}\)**(\(\text{{}^{\circ}}\))** & \(\boldsymbol{|r_{\parallel}|}\) & \(\boldsymbol{\phi_{\parallel}}\)**(\(\text{{}^{\circ}}\))** \\ \hline \multirow{3}{*}{Air-Al} & normal & 0.9872 & \(-172.56\) & 0.9872 & 7.44 \\ \cline{2-6}  & at \(45^{\circ}\) & 0.9910 & \(-174.74\) & 0.9820 & 10.52 \\ \hline \multirow{3}{*}{Air-Ag} & normal & 0.9920 & \(-168.65\) & 0.9920 & 11.35 \\ \cline{2-6}  & at \(45^{\circ}\) & 0.9944 & \(-171.98\) & 0.9887 & 16.04 \\ \hline \multirow{3}{*}{BK7-Ag} & normal & 0.9882 & \(-163.04\) & 0.9882 & 16.96 \\ \cline{2-6}  & at \(45^{\circ}\) & 0.9917 & \(-168.03\) & 0.9834 & 23.94 \\ \hline \multirow{3}{*}{BK7-Al} & normal & 0.9810 & \(-168.86\) & 0.9810 & 11.14 \\ \cline{2-6}  & at \(45^{\circ}\) & 0.9865 & \(-172.13\) & 0.9733 & 15.75 \\ \hline \end{tabular}
\end{table} 
Table 2: Amplitude and phase changes acquired after a reflection in some dielectric-metal interfaces at \(\lambda=1550\,\text{nm}\). The following complex refractive indices [18] were assumed: \(n_{\text{Ag}}=0.40960+10.048\text{i}\), \(n_{\text{Al}}=1.5137+15.234\text{i}\), \(n_{\text{BK7}}=1.5007+1.4361\cdot 10^{-7}\text{i}\). but taking into account that \(|r_{\parallel 0}|=|r_{\perp 0}|\) and \(\phi_{\parallel 0}-\phi_{\perp 0}=\pi\), it results that both conditions are the same. They are minimized simultaneously when:  
\[2 kd=\phi_{\parallel 0},\]  
and the residual crosstalk is governed by \(|r_{\parallel 0}|\), being:  
\[\left|\frac{|r_{\parallel 0}|-1}{|r_{\parallel 0}|+1}\right|^{2}, \tag{3}\]  
which leads to much better results: \(-47.9\) dB for a silvered mirror and \(-43.8\) dB for an aluminized one. Importantly, the key to easily synchronizing both polarizations is that the uncompensated reflection occurs at normal incidence, which is not possible in an MZI. The crosstalk in the exit port that is opposite to the source (Equation (3)) is independent of the beam-splitter specifications because the wave from each arm is transmitted and reflected once at it. However, the light returning to the source port is the result of the interference of a wave reflected twice with another transmitted twice by the beam-splitter. So, if \(t^{BS}\
eq r^{BS}\), their amplitudes are unbalanced which prevents the full destructive interference and increases the crosstalk. Fortunately, a difference of 1% between reflectance and transmittance (which can be achieved with commercial beam-splitters) leads to a crosstalk better than \(-39\) dB, a value almost as good as that of the other port. A variant of this interferometer is shown in Figure 2b, where the hollow roof mirror was replaced with a Porro's prism (let us assume that it is made of BK7 glass) which works under total internal reflection. Note that the phases introduced by these total reflections at \(45^{\circ}\) are far away from zero or \(\pi\), but again they are exactly compensated by those appearing in the total internal reflections of the other arm. In this case, the crosstalk is also given by Equation (3), but being \(r_{\parallel 0}\) the Fresnel coefficient for the interface glass-metal. Although the crosstalk is slightly worse than the previous one, it is still theoretically high: \(-44.5\) dB for silver and \(-40.3\) dB for aluminium. Moreover, the Fresnel prism and the compensator can be glued to the beam splitter with optical c  
Figure 2: (**a**) Proposed interferometer which comprises a Fresnel retarder to synchronize both polarizations in the arm containing the hollow roof prism and both a glass plate and a \(45^{\circ}\) metallic mirror in the other arm as compensating elements. The unfolded paths are shown in light colours.