If one puts in front of a double slit two quarter-wave circular polarizers (QWP) in front of each slit, say one with 45° and the other with -45° orientation (fast and slow axis - i.e. clock- and anti-clockwise circular polarization) and shines linear polarized light on the slits, one does not get the typical interference pattern but the Gaussian-shaped diffraction pattern. This is the result of the overlap of the fringe- and anti-fringe patterns which, summed up, furnish the typical bell-shaped figure (these result from the two QWP induced pi-phase shift which causes the interference pattern with the central peak to be shifted left and right, respectively, with maxima overlapping minima and vice-versa). One can see this also from the perspective of the which-way question in quantum mechanics which states that by placing the two QWP in front of the slits we have 'marked' the photons and we could recover the which-way information which leads to the loss of the interference pattern. However, what happens if one places only one QWP in front of only one slit? As I understand it, the interference pattern would reappear as only one fringe or anti-fringe (right? Well, maybe not... this is part of my question). I'm in doubt, because if one would analyze this in the which-way perspective photons are still 'labeled', that is, photons going through the slit without the QWP would be linearly polarized, those going through the slit with the QWP are circularly polarized and one could in principle recover the which way information, i.e. one expects still only the diffraction without interference pattern. So, a contradiction arises that hints possibly at a mistake or misunderstanding from my side, but can't find out where it is. Can anyone help?
One way to think of this is to use the "the photon wave function" concept. A polarizer effectively destroys the original photon and creates a new function so now the photons passing thru the slits are not coherent anymore, thus the loss of the diffraction pattern. It is like having 2 separate sources one in front of each slit. When one polarizer is removed a portion of the photons passing thru both slits are coherent again and the partial pattern is seen.
The polarization state of the light illuminating the two slits should be taken into account. Interference between the portions of light passing through the two slits, when the polarization of light passing through one of the slits is rotated, results in a spatial modulation of the polarization state of photons detected at the screen. The pattern of polarization modulation may be calculated simply by summing the amplitudes of light waves (including polarization) from the two slits.