## 2 A Modified Michelson Interferometer as a Parity Selector  
Let us consider that one of the mirrors of a Michelson interferometer is replaced with two perpendicular plane mirrors which form a hollow roof prism mirror whose edge is along the \(Y\)-axis as shown Figure 1. We also consider a zero path difference, which is achieved if the image of the prism edge through the beam splitter matches the plane mirror of the opposite arm. When an incoming plane wave illuminates this mirror normally, a uniform interference field is obtained at the output. In particular, by assuming an ideal phase change at each mirror reflection of \(\pi\) for TE polarization (electric field parallel to \(Y\)-axis), this configuration results in a destructive interference of the plane wave at the output port opposite to the light source because one reflection more occurs in the arm containing the hollow roof prism. Now, let us consider an incident wave whose amplitude changes slowly along its wavefront. The hollow roof mirror spatially inverts the wavefront, exchanging its right and left sides symmetrically around its edge. That is of no consequence if the wave has also a symmetrical amplitude, for example a gaussian beam centred on the edge. However, if the wave amplitude is antisymmetric (such as in an odd Hermite-Gaussian beam), the spatial inversion is equivalent to a sign change of the amplitude, that is, to an extra \(\pi\) phase. Therefore the antisymmetric waves will interfere destructively in the source port and constructively in the opposite one, unlike the symmetrical waves do. In order to avoid that the symmetric wavefronts return exactly to the light source, the incoming wave must propagate slightly out of the \(X\!Z\)-plane. In other words, the rays shown in Figure 1 actually are the projections of real rays on the \(X\!Z\)-plane. In short, this Michelson interferometer separates the symmetric and antisymmetric part of the incoming TE-polarized wave, sending each part to a different exit port. Conversely, the same interferometer acts as multiplexer when symmetric and antisymmetric wavefronts enter from different ports. For TM polarization, the reflection phases at metal mirrors are near zero; consequently symmetrical and antisymmetric wavefronts emerge from exchanged ports with respect to the TE case. In order to make a polarization-insensitive (de)multiplexer, a quarter-wave plate have to be included in one arm of the interferometer. As the wave crosses twice this plate, it introduces another \(\pi\)  
Figure 1: (**a**) Michelson interferometer with a hollow roof mirror. (**b**) The zero path difference is indicated by the dotted line. The behavior for TE polarization assuming reflection phases of \(\pi\) is shown for different transverse fields. phase between both polarizations. Thus, they will become synchronized, that is, both polarizations will interfere constructively in the same output port for the same wavefront. Non-Ideal MirrorsWhen we take into account the non-ideal reflectance of the metal mirrors, the amplitudes of the interfering waves become unbalanced and the phase difference is not exactly 0 or \(\pi\) for the zero geometrical path difference configuration. Consequently, the destructive interferences are not complete, which results in a crosstalk between symmetrical and antisymmetric wavefronts. The phase difference can usually be compensated with geometrical path for a monochromatic source. However, in our case, it can not be done simultaneously for both polarizations since the difference between their reflection phases at an incidence of 45\({}^{\circ}\) is not exactly \(\pi\), unlike in normal incidence (Table 2). Instead it is 185.27\({}^{\circ}\) for aluminium and 188.02\({}^{\circ}\) for silver; these mismatches have a greater influence in crosstalk than the amplitude difference of waves from both arms.