Electro-optic modulator

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

${\displaystyle Ae^{i\omega t}.}$

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

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

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

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

to which we apply a simple identity for sine,

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

This expression we interpret to mean that we have the original carrier signal plus two small sidebands, one at ${\displaystyle \omega +\Omega }$ and another at ${\displaystyle \omega -\Omega }$. 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 involving Bessel functions called the Jacobi–Anger expansion which can be used to derive

${\displaystyle Ae^{i\omega t+i\beta \sin(\Omega t)}=Ae^{i\omega t}\left(J_{0}(\beta )+\sum _{k=1}^{\infty }J_{k}(\beta )e^{ik\Omega t}+\sum _{k=1}^{\infty }(-1)^{k}J_{k}(\beta )e^{-ik\Omega t}\right),}$

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,

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

A phase modulating EOM can also be used as an amplitude modulator by using a Mach–Zehnder interferometer. A beam splitter divides the laser light into two paths, one of which has a phase modulator as described above. The beams are then recombined. Changing the electric field on the phase modulating path will then determine whether the two beams interfere constructively or destructively at the output, and thereby control the amplitude or intensity of the exiting light. This device is called a Mach–Zehnder modulator.

Depending on the type and orientation of the nonlinear crystal, and on the direction of the applied electric field, the phase delay can depend on the polarization direction. A Pockels cell can thus be seen as a voltage-controlled waveplate, and it can be used for modulating the polarization state. For a linear input polarization (often oriented at 45° to the crystal axis), the output polarization will in general be elliptical, rather than simply a linear polarization state with a rotated direction.

Polarization modulation in electro-optic crystals can also be used as a technique for time-resolved measurement of unknown electric fields. [1][2] Compared to conventional techniques using conductive field probes and cabling for signal transport to read-out systems, electro-optical measurement is inherently noise resistant as signals are carried by fiber-optics, preventing distortion of the signal by electrical noise sources. The polarization change measured by such techniques is linearly dependent on the electric field applied to the crystal, hence provides absolute measurements of the field, without the need for numerical integration of voltage traces, as is the case for conductive probes sensitive to the time-derivative of the electric field.

• Pockels effect
• Acousto-optic modulator
• Phase modulation
• Dielectric wireless receiver

• Karna, Shashi and Yeates, Alan (ed.) (1996). Nonlinear Optical Materials: Theory and Modeling. Washington, DC: American Chemical Society. pp. 2–3. ISBN 0-8412-3401-9.CS1 maint: multiple names: authors list (link) CS1 maint: extra text: authors list (link)
• Saleh, Teich (first ed.) (1991). Fundamentals of Photonics. New York: Wiley-Interscience Publications. p. 697. ISBN 0-471-83965-5.
•  This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C". (in support of MIL-STD-188)

Notes

• AdvR – Research and custom EO phase and amplitude modulators
• Encyclopedia of Laser Physics and Technology
• Interactive visualization of the transfer characteristic of a Mach–Zehnder modulator for phase and amplitude modulation