- Electro-optic modulator
**Electro-optic modulator**(EOM) is an optical device in which a signal-controlled element displayingelectro-optic effect is used to modulate abeam oflight . Themodulation may be imposed on the phase,frequency ,amplitude , or direction of the modulated beam. Modulation bandwidths extending into thegigahertz range are possible with the use oflaser -controlled modulators.:Generally a nonlinear optical material (

organic polymer s have the fastest response rates, and thus are best for this application) with an incident static or low frequencyoptical field will see a modulation of itsrefractive index ."Source:

Federal Standard 1037C andMIL-STD-188 "**Types of EOMs****Phase modulation**The simplest kind of EOM consists of a crystal, such as

Lithium niobate , whose refractive index is a function of the strength of the localelectric 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 dependslinearly on the potential, the index of refraction depends linearly on the field (for crystals wherePockel'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.

**Amplitude modulation**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.**Applications**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 withfrequency $omega$ leaving the EOM is given by:$Ae^\{iomega\; t\}.$

Now suppose we apply a sinusoidally varying potential to the EOM with frequency $Omega$ and small amplitude $eta$. This adds a time dependent phase to the above expression,

:$Ae^\{iomega\; t\; +\; ietasin(Omega\; t)\}.$

Since $eta$ is small, we can use the

Taylor expansion for the exponential:$Ae^\{iomega\; t\}left(\; 1+ietasin(Omega\; t)\; ight)\; ,$

to which we apply a simple identity for

sine ,:$Ae^\{iomega\; t\}left(\; 1+frac\{eta\}\{2\}(e^\{iOmega\; t\}\; -\; e^\{-iOmega\; t\})\; ight)\; =\; Aleft(\; e^\{iomega\; t\}+frac\{eta\}\{2\}e^\{i(omega+Omega)\; t\}-frac\{eta\}\{2\}e^\{i(omega-Omega)\; t\}\; ight)\; .$

This expression we interpret to mean that we have the original

carrier frequency plus two small sidebands, one at $omega+Omega$ and another at $omega-Omega$. 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 involvingBessel functions :$Ae^\{iomega\; t\; +\; ietasin(Omega\; t)\}\; =\; Ae^\{iomega\; t\}left(\; J\_0(eta)\; +\; sum\_\{k=1\}^\{infty\}J\_k(eta)e^\{ikOmega\; t\}\; +\; sum\_\{k=1\}^\{infty\}(-1)^k\; J\_k(eta)e^\{-ikOmega\; t\}\; ight)\; ,$

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,

:$left(\; 1\; +\; etasin(Omega\; t)\; ight)\; Ae^\{iomega\; t\}\; =\; Ae^\{iomega\; t\}\; +\; frac\{Aeta\}\{2i\}left(\; e^\{i(omega+Omega)\; t\}\; -\; e^\{i(omega-Omega)t\}\; ight)\; .$

**ee also***

Pockels effect

*Acousto-optic modulator

*Phase modulation **References****External links*** [

*http://www.advr-inc.com/modulators.html AdvR - Electro-optic phase and amplitude modulators*]

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