Identical beam splitters in a two-path interferometer ($P_0=1-{\sin }^2\beta {\cos }^2(\varphi /2)$)

For two identical mixers $U=R_x(\beta )$ with a lower-arm phase shifter $P(\varphi )=\mathrm{diag}(1,e^{i\varphi })$, the upper-input state evolves asThe upper-output interferometric visibility is ${\sin }^2\beta /(1+{\cos }^2\beta )$ and reaches one exactly at $\beta =\pi /2(\mathrm{mod}\pi )$. Replacing the second mixer by its inverse changes the port convention and phase dependence; it is a different specified network.