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Polarized diffraction of four point sources: horizontal and vertical orientations

Here are the x-y intensity and phase patterns for another 4 polarized sources. The two sources at upper left and lower right of center are polarized horizontally, while the two sources at lower left and upper right are polarized vertically:

In this case, we again see intensity interference between the two horizontal sources, and between the two vertical sources, at opposite corners, but not between the horizontal and vertical sources. Here is the z-y intensity and phase:

The x-z intensity and phase:

Polarization in the x-y plane:

In this case, we see polarization interference between the pairs of horizontal and vertical sources at top, bottom, left, and right. Here is the polarization in the z-y plane:

Polarization in the x-z plane:

These few examples should give you the sense that an enormous variety of 2d polarization patterns can be created from just a few sources distributed in simple ways. Furthermore, it might be possible to generate "almost" any desired 2d polarization pattern by careful manipulation of sources in 2d and 3d (these examples are only 2d). The "trick", then, will be to determine, in a systematic way, which 2d or 3d distribution of sources generates a particular pattern. And although these particular patterns are static, it will be changes in 2d patterns --changes over time-- that perform the actual computations. Physically, these operations will be performed by optical elements that can transform one 2d polarization pattern into another.

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