The modulation of the polarization of a laser to me is the most interesting and applicable use of an EoM. I recall from before this equation

(1)  

This is the first key modulating the polarization this means once you give it the correct voltage to your EoM it acts like a half wave plate. A half wave plate is an optical element that that allows you to rotate your electromagnetic wave. The corresponding Jones matrix to this is . I will go a little more in depth on different angles effects later on

Okay so before the laser enters the EoM it must be either horizontally or vertically polarized using some kind of polarizer. For example the use of the beam splitter in the figure below

Let’s use the horizontally polarized beam to be the one to go into the EoM so the Jones vector is and lets orient the EoM to making the half wave plate Jones matrix so the effect looks like this

Notice the is now in the y direction making the laser beam vertically polarized

Lets look at how a half wave plate acts for different . Below is a graph of the amount of , the blue line and the amount of , the red line that will exit the half wave plate as a function of

As you can see that when you have only a predicted above. From this graph you can also see that when you have equal amounts and

Here is an animation of How the electric field direction will change with different

Mathmatica was giving me a lot of trouble to do this simple task. This still shows how there exists a that will make the electric field switch directions