in an earlier question, i asked if elliptically-polarised light could be superimposed in a way that allowed the vector(s) $E$ (the jones vectors) to make sense:
superposition of elliptically polarised light
it was very kindly answered by rod as "yes if theta and phi and everything is the same, then the result E1 + E2 is the summation of the major axes".
i note however that yes, if this relation does not hold, all bets are off.
i have an additional case - a scenario where i suspect that the vectors may also similarly be summed, and i believe it's when the two exponent parts of Jones Vectors are offset by pi/4 (90 degrees).
going from the two equations E_1 and E_2 of the former answer (apologies i am not wholly familiar with quora otherwise i would cut/paste them: i will edit if i can work it out... got it!)
$$E_1=\left(\begin{array}{c}e^{i\,\varphi_1}\,\cos\theta_1\\e^{i\,\phi_1}\,\sin\theta_1\end{array}\right)\quad\text{and}\quad E_2=\left(\begin{array}{c}e^{i\,\varphi_2}\,\cos\theta_2\\e^{i\,\phi_2}\,\sin\theta_2\end{array}\right)$$
let us suppose that theta_1 = theta_2 (the cos/sin parts are the same), but that phi_1 = phi_2 + pi/2
my intuition tells me that this results in a simple inversion of the sign of E2 relative to E1, such that the vectors may simply be SUBTRACTED instead of ADDED, because sqrt(-1) squared comes back to -1. but my intuition is also telling me that one of the preconditions may need to be modified, so that theta_1 = theta_2 + pi/2 as well for example. however these are the kinds of things i have difficulty working out.
help greatly appreciated to work out if my suspicions are correct, that there are indeed other special cases (right-angle phase-coherent cases) where Jones Vectors may indeed simply be added (or subtracted).
working on this myself (comments as notes below), so far i have:
$$ E_{\hat{x}} = E_{n\hat{x}} e^{-i \left( \psi_{\hat{x}} \right) } e^{-i \left( kz / 2 \right) } e^{-i \left( - \omega t / 2 \right) } e^{-i \left( \theta/2 \right) } $$
where
$$ E_{n\hat{x}} = E_{0\hat{x}} e^{-i \left( \theta \right) }, \theta = n\tau/12 $$
where of course tau is pi / 2 because i love the tauday manifesto :)
the theta/2 is down to a property of the solution i'm working on (mobius-light), but i think may turn out to be very relevant... still uncertain about all this stuff...
