Now consider a beam that is linearly polarized in the horizontal orientation shining through a single hole in the first screen. In this case, the diffraction intensity and phase on the second (x-y) screen are identical to that of the "unpolarized" beam from the previous page:
You'll have to take my word that the other two cross-sections produce the same 1-hole pattern from the side and above as in the scalar case. However, something new that does not exist in the scalar case is the resulting 2d distribution of polarization in the down-beam field. In this case, the polarization is simply linear horizontal everywhere, in all three cross-sections. Here is just the x-y plane, but the other two cross-sections are identical:
A beam that is linearly polarized in the vertical orientation also produces the same diffraction intensity and phase patterns on the second screen:
In this case, the polarization is everywhere vertical:
You'll have to take my word that any linear polarization at the source will produce the same 1-hole intensity and phase patterns, and a uniform 2d distribution of down-beam polarization.