Electro-optic modulator

The electro-optic effect describes two phenomena, the change of absorption and the change in the refractive index of a material, resulting from the application of a DC or an electric field with much lower frequency than the optical carrier. This is caused by forces that distort the position, orientation, or shape of the molecules constituting the material. Generally, a nonlinear optical material, such as ferroelectrics like lithium niobate (LiNbO₃) or barium titanate (BaTiO₃), polymers, or organic electro-optic materials, with an incident static or low frequency optical field will see a modulation of its refractive index.

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 takes 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 by 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 Pockels 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.

The voltage required for inducing a phase change of ${\displaystyle \pi }$ is called the half-wave voltage (${\displaystyle V_{\pi }}$). For a Pockels cell, it is usually hundreds or even thousands of volts, so that a high-voltage amplifier is required. Suitable electronic circuits can switch such large voltages within a few nanoseconds, allowing the use of EOMs as fast optical switches.

Liquid crystal devices are electro-optical phase modulators if no polarizers are used.

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 sidebands in a monochromatic laser beam. To see how this works, first imagine that the strength of a laser beam with frequency ${\displaystyle \omega }$ entering the EOM is given by

${\displaystyle A{e}^{i\omega t}.}$

Now suppose we apply a sinusoidally varying potential voltage to the EOM with frequency ${\displaystyle \mathrm{\Omega }}$ and small amplitude ${\displaystyle \beta }$. This adds a time dependent phase to the above expression,

${\displaystyle A{e}^{i\omega t+i\beta \sin (\mathrm{\Omega }t)}.}$

Since ${\displaystyle \beta }$ is small, we can use the Taylor expansion for the exponential

${\displaystyle A{e}^{i\omega t}\left(1+i\beta \sin (\mathrm{\Omega }t)\right),}$

to which we apply a simple identity for sine,

${\displaystyle A{e}^{i\omega t}\left(1+\frac{\beta }{2}\left(e^{i\mathrm{\Omega }t}-e^{-i\mathrm{\Omega }t}\right)\right)=A\left(e^{i\omega t}+\frac{\beta }{2}{e}^{i(\omega +\mathrm{\Omega })t}-\frac{\beta }{2}{e}^{i(\omega -\mathrm{\Omega })t}\right).}$

This expression we interpret to mean that we have the original carrier signal plus two small sidebands, one at ${\displaystyle \omega +\mathrm{\Omega }}$ and another at ${\displaystyle \omega -\mathrm{\Omega }}$. 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

${\displaystyle A{e}^{i\omega t+i\beta \sin (\mathrm{\Omega }t)}=A{e}^{i\omega t}\left(J_0(\beta )+\sum _{k=1}^{\mathrm{\infty }}{J}_k(\beta )e^{ik\mathrm{\Omega }t}+\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k{J}_k(\beta )e^{-ik\mathrm{\Omega }t}\right),}$

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,

${\displaystyle \left(1+\beta \sin (\mathrm{\Omega }t)\right)A{e}^{i\omega t}=A{e}^{i\omega t}+\frac{A\beta }{2i}\left(e^{i(\omega +\mathrm{\Omega })t}-e^{i(\omega -\mathrm{\Omega })t}\right).}$