Editing Electro-optic modulator (section)

==Phase modulation==

Phase modulation (PM) is a modulation pattern that encodes information as variations in the instantaneous phase of a carrier wave.

The phase of a carrier signal is modulated to follow the changing voltage level (amplitude) of modulation signal. The peak amplitude and frequency of the carrier signal remain constant, but as the amplitude of the information signal changes, the phase of the carrier changes correspondingly. The analysis and the final result (modulated signal) are similar to those of frequency modulation.

A very common application of EOMs is for creating [[sideband]]s in a [[monochromatic]] laser beam. To see how this works, first imagine that the strength of a laser beam with [[frequency]] <math>\omega</math> entering the EOM is given by

:<math>Ae^{i\omega t}.</math>

Now suppose we apply a sinusoidally varying potential voltage to the EOM with frequency <math>\Omega</math> and small amplitude <math>\beta</math>. This adds a time dependent phase to the above expression,

:<math>Ae^{i\omega t + i\beta\sin(\Omega t)}.</math>

Since <math>\beta</math> is small, we can use the [[Taylor expansion]] for the exponential

:<math>Ae^{i\omega t}\left( 1+i\beta\sin(\Omega t)\right) ,</math>

to which we apply a simple identity for [[sine]],

:<math>Ae^{i\omega t}\left( 1 + \frac{\beta}{2}\left(e^{i\Omega t} - e^{-i\Omega t}\right)\right) = A\left( e^{i\omega t}+\frac{\beta}{2}e^{i(\omega+\Omega) t}-\frac{\beta}{2}e^{i(\omega-\Omega) t}\right) .</math>

This expression we interpret to mean that we have the original [[carrier signal]] plus two small sidebands, one at <math>\omega+\Omega</math> and another at <math>\omega-\Omega</math>. 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]] called the [[Jacobi–Anger expansion]] which can be used to derive

:<math>Ae^{i\omega t + i\beta\sin(\Omega t)} = Ae^{i\omega t}\left( J_0(\beta) + \sum_{k=1}^\infty J_k(\beta)e^{ik\Omega t} +  \sum_{k=1}^\infty (-1)^k J_k(\beta)e^{-ik\Omega t}\right) , </math>

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,

:<math>\left( 1 + \beta\sin(\Omega t)\right) Ae^{i\omega t} = Ae^{i\omega t} + \frac{A\beta}{2i}\left( e^{i(\omega+\Omega) t} - e^{i(\omega-\Omega)t} \right) . </math>