The simplest kind of EOM consists of a crystal, such as Lithium niobate, whose refractive index is a function of the strength of the local electric field. That means that if lithium niobate is exposed to an electric field, light will travel more slowly through it. But the phase of the light leaving the crystal is directly proportional to the length of time it took that light to pass through it. Therefore, the phase of the laser light exiting an EOM can be controlled by changing the electric field in the crystal.
Note that the electric field can be created placing a parallel plate capacitor across the crystal. Since the field inside a parallel plate capacitor depends linearly on the potential, the index of refraction depends linearly on the field (for crystals where Pockel's effect dominates), and the phase depends linearly on the index of refraction, the phase modulation must depend linearly on the potential applied to the EOM.
Liquid crystal devices are electro-optical phase modulators if no polarizers are used.
Once one can make a phase modulating EOM, it's a fairly simple matter to turn that into an amplitude modulating EOM by using a Mach-Zehnder interferometer. Simply use a beam splitter to divide the laser light into two paths, one of which has a phase modulator as described above, and then recombine the two beams. By changing the electric field on the phase modulating path, one can control whether the two beams constructively or destructively interfere and thereby control the amplitude or intensity of the exiting light.
A very common application of EOMs is for creating sidebands in a monochromatic laser beam. To see how this works, first imagine that the strength of a laser beam with frequency leaving the EOM is given by
Now suppose we apply a sinusoidally varying potential to the EOM with frequency and small amplitude . This adds a time dependent phase to the above expression,
Since is small, we can use the Taylor expansion for the exponential
to which we apply a simple identity for sine,
This expression we interpret to mean that we have the original carrier frequency plus two small sidebands, one at and another at . Notice however that we only used the first term in the Taylor expansion - in truth there are an infinite number of sidebands. There is a useful identity involving Bessel functions
which gives the amplitudes of all the sidebands. Notice that if one modulates the amplitude instead of the phase, one gets only the first set of sidebands,