## 2 A Modified Michelson Interferometer as a Parity Selector
Indeed, the interfering TE waves emerging from the opposite port to the source are proportional to:  
\[t_{\perp}^{BS}\,r_{\perp 0}\,r_{\perp}^{BS}\qquad\text{and}\qquad r_{\perp}^{ BS}\,r_{\perp 45}^{2}\,t_{\perp}^{BS}\,\mathbf{e}^{\mathrm{i}2 kd},\]  
where \(t_{\perp}^{BS}\) and \(r_{\perp}^{BS}\) are the [PERSON]'s transmittance and reflectance coefficients of the beam-splitter; \(r_{\perp 0}\) (\(r_{\perp 45}\)) is the Fresnel coefficient of the mirrors at normal (45\({}^{\circ}\)) incidence; \(k\) is the wavenumber; and \(d\) is the arm length difference. The amplitude of the interference is proportional to the sum of those values for symmetric wavefronts; but it is their difference if the wavefront is antisymmetric. Therefore the ratio of intensities between symmetric and antisymmetric wavefronts (crosstalk) is:  
\[\left|\frac{r_{\perp 0}+r_{\perp 45}^{2}\,\mathbf{e}^{\mathrm{i}2 kd}}{r_{\perp 0 }-r_{\perp 45}^{2}\,\mathbf{e}^{\mathrm{i}2 kd}}\right|^{2}\simeq\left|\frac{1+ \,\mathbf{e}^{\mathrm{i}(2\phi_{\perp 45}-\phi_{\perp 0}+2 kd)}}{1-\,\mathbf{e}^{ \mathrm{i}(2\phi_{\perp 45}-\phi_{\perp 0}+2 kd)}}\right|^{2},\]  
where we have neglected the mirror absorptions:  
\[r_{\perp 0} =|r_{\perp 0}|\,\mathbf{e}^{\mathrm{i}\phi_{\perp 0}}\simeq e^{ \mathrm{i}\phi_{\perp 0}},\] \[r_{\perp 45} =|r_{\perp 45}|\,\mathbf{e}^{\mathrm{i}\phi_{\perp 45}}\simeq e^{ \mathrm{i}\phi_{\perp 45}}.\]  
This ratio cancels if argument of the complex exponentials is a odd multiple of \(\pi\):  
\[2\phi_{\perp 45}-\phi_{\perp 0}+2 kd=(2m+1)\pi\quad m\in\mathbb{Z}.\]  
Since \(\phi_{\perp 45}\simeq\phi_{\perp 0}\simeq-\pi\), the solution for the smallest \(d\) happens for \(m\ =\ -1\):  
\[2\phi_{\perp 45}-\phi_{\perp 0}+2 kd=-\pi. \tag{1}\]  
Analogously, the crosstalk for TM polarization is given approximately by:  
\[\left|\frac{1+\,\mathbf{e}^{\mathrm{i}\left(2\phi_{\parallel 45}-\phi_{\parallel 0 }+2 kd+\pi\right)}}{1-\,\mathbf{e}^{\mathrm{i}\left(2\phi_{\parallel 45}-\phi_{ \parallel 0}+2 kd+\pi\right)}}\right|^{2}\]  
where a \(\pi\) phase was included to account for the twice crossed \(\lambda/2\) wave plate. In this case, the smaller \(d\) that cancels the last ratio is:  
\[2\phi_{\parallel 45}-\phi_{\perp 0}+2 kd=\pi, \tag{2}\]  
where the relationship: \(\phi_{\parallel 0}-\phi_{\perp 0}=\pi\) was used. As Equations (1) and (2) are not compatible, we can choose the mean value for \(d\) as a trade-off:  
Figure 1:0Therefore, we obtain the same crosstalk for TE and TM polarizations:  
Figure 1:1  
By replacing \(\phi_{\parallel 45}\) and \(\phi_{\perp 45}\) with the values of Table 2, the crosstalk between symmetric and antisymmetric wavefronts achieves \(-26.7\) dB and \(-23.0\) dB for aluminium and silver mirrors respectively. These values are rather modest as theoretical limits. To improve them, a custom retarder wave plate should be made to finely synchronize TE and TM polarizations; but this solution is cumbersome. Instead, we propose the inclusion of a mirror oriented at \(45^{\circ}\) in the arm that only had one mirror as shown in Figure 2a. This compensates for the phases and the amplitude losses of the hollow roof prism at any wavelength. The mirror illuminated under normal incidence remains uncompensated, but its reflection phases for TE and TM polarization differs exactly \(180^{\circ}\), therefore a standard quarter wave retarder is enough to adjust them. In particular, we propose a Fresnel retarder because it is achromatic, and it is advisable that this device works in a band as wide as possible. Moreover, its dispersion can also be compensated in the other arm with a piece of the same glass type with the proper length. To be more specific, the exact crosstalk for TE polarization is now:  
Figure 1:2  
Analogously, the crosstalk for TM polarization is:  
Figure 1:3