Sending elliptically polarised beams into a reversed beam splitter

If I send two elliptically polarised beams into the outputs of a (say 50:50) beam splitter which is at some angle $$\theta$$ to the $$x$$-axis, what sort of beam emerges? As far as I'm aware they do not simply add.

Furthermore:

• What if the two beams are not orthogonal in their elliptical polarisations?
• What if the two beams have passed through different optical elements so may not be coherent?
• What if the beam splitter is a polarising beam splitter?

Currently, I am using the fact that antipodal points on the Poincare sphere represent orthogonal polarisations of light. Since a PBS run 'forward' produces orthogonal polarisations of light, I'd imagine any two orthogonal beams of light, regardless of their polarisation, would have the same effect. For this reason, I think the wave resulting from passing two beams of light $$\textbf{E_{1}}$$ and $$\textbf{E_{1}}$$ would be $$(\textbf{E_{1}} + \textbf{E_{1}})$$cos$$(2\Psi)$$cos$$(2\chi)$$ with $$\Psi, \chi$$ defined in the image below (if one input wave is along $$S_{1}$$ and the other at the marked point.) (Image taken from Wikipedia).

• I know the name Poincare should have an accent, I'm just not sure how to add it since using the LaTeX version doesn't seem to work. – Kay Tukendorf Feb 20 at 8:32