Understanding Electric Field Changes Passing Through a Waveplate

[raisin_raisin]

Hello,
I am trying to figure out how the electric field changes before and after passing through a waveplate but my notes don't make any sense.
I think
E^ext_in=Ecos$$\theta$$ and E^ord_in=Esin$$\theta$$
where theta is the angle the electric field makes with the optic axis.

I can't tell how the amplitude changes on Eout though.

I know they pick up a relative phase but I can't tell if this is in addition to a phase anyway.

i.e is it just
E^ext_in=Ecos$$\theta$$
and E^ord_in=Esin$$\theta$$ e^{i$$\Delta\varphi$$}

or if it should be the same but with a different amplitude i.e E' not E afterwards (in which case how do I calculate this?) or whether it should be

E^ext_out=Ecos$$\theta$$e^{i$$\varphi$$_e}
and E^ord_out=Esin$$\theta$$ e^{i$$\varphi$$_o}e^{i$$\Delta\varphi$$}

Or should it be this with E' instead of E?
This is what my notes suggest but I am hoping it is wrong since delta phi is $$\pi$$ so it would make it easier if it wasn't.

Thanks for any help.

 

[Redbelly98]

Assuming no absorption or reflective losses, E^ext and E^ord keep the same amplitudes. They do get a relative phase shift, so just tack on a factor exp(iφ) to one of the amplitudes as you had originally.

I know they pick up a relative phase but I can't tell if this is in addition to a phase anyway.

Yes, there is an overall phase as well, but that is not important as it does not affect the amplitude or polarization of the output wave.