Electro-optic modulator (EOM)

An electro-optic modulator (EOM) changes the phase, amplitude or polarization of a light beam in response to an applied electric field. The classic device uses the linear electro-optic (Pockels) effect, an index change proportional to the field that follows it beyond 100 GHz; the term also covers semiconductor modulators that work through carriers or field-induced absorption. The central figure is the half-wave voltage $V_\pi$, the voltage that shifts the optical phase by π. A 2 cm titanium-diffused lithium niobate waveguide phase modulator with a 10 µm electrode gap needs about 5 V at 1550 nm, and integrated thin-film devices need 1–3 V.

Types

Phase modulator. A single waveguide or crystal between electrodes, with phase $\phi = \pi V/V_\pi$. A sinusoidal drive at $f_m$ with peak phase $\beta$ (the modulation index) creates sidebands at multiples of $f_m$ with field amplitudes $J_k(\beta)$. For $\beta = 1$ rad, $J_0 = 0.765$ and $J_1 = 0.440$: the carrier keeps 58.6% of the power, each first-order sideband carries 19.4%, 4.8 dB below the carrier. The carrier vanishes at $\beta = 2.405$.

Amplitude modulator. Two phase-modulated arms in an interferometer form a Mach-Zehnder modulator, with transmission

$$T(V) = \cos^2\!\left(\frac{\pi V}{2V_\pi}\right)$$

for a balanced device with no bias phase. Analog links hold it at quadrature ($T = 0.5$, the steepest point); digital links swing between maximum and null.

Polarization modulators and Pockels cells. A field that changes a crystal's birefringence changes the polarization state; between polarizers this is an amplitude switch. Free-space Pockels cells of this kind run at kilovolt levels.

Semiconductor modulators. The electro-absorption modulator changes absorption near a band edge through the Franz-Keldysh or quantum-confined Stark effect. A microring modulator shifts a ring resonance by a fraction of its linewidth, giving a small, low-capacitance device with a narrow optical bandwidth. Silicon modulators use plasma-dispersion phase shifters, where depletion or injection of free carriers changes the index along with the loss. Barium titanate grown on silicon supplies a Pockels coefficient far larger than lithium niobate's.

Half-wave voltage

For a Pockels phase modulator with electrode gap $g$, length $L$ and overlap $\Gamma$ between the applied field and the optical mode,

$$V_\pi = \frac{\lambda\, g}{n_e^3\, r_{33}\, \Gamma\, L}.$$

With $\lambda$ = 1550 nm, $n_e$ = 2.138, $r_{33}$ = 30.8 pm/V, $g$ = 10 µm, $\Gamma$ = 0.5 and $L$ = 2 cm, $V_\pi$ = 5.1 V and $V_\pi L$ = 10.3 V·cm, inside the 5–15 V·cm range of titanium-diffused waveguides. Driving that device to $\beta = 1$ rad takes a sinusoid of $V_\pi/\pi$ = 1.6 V peak. A push-pull Mach-Zehnder halves $V_\pi$, and thin-film lithium niobate reaches 1.5–3 V·cm because the electrodes sit a few micrometers apart. $V_\pi$ is measured from the transfer curve, maximum to adjacent minimum, or at microwave frequencies from the carrier-to-sideband ratio using the Bessel relations above.

Bandwidth, loss and extinction

A short electrode behaves as a lumped capacitor, and its bandwidth is limited by the RC time constant of that capacitance with the driver's source impedance. Longer devices use traveling-wave electrodes, in which the microwave propagates alongside the light; the bandwidth is then limited by the mismatch between the microwave and optical velocities and by the electrode's microwave loss, both of which accumulate with length. Lowering $V_\pi$ by lengthening the electrode therefore costs bandwidth.

Insertion loss of a packaged lithium niobate modulator, including fiber coupling, is typically 3–5 dB. The extinction ratio of a Mach-Zehnder device is set by the balance of its two arms and is typically 20–35 dB.

Applications

Mach-Zehnder and IQ modulators are the transmitters of coherent fiber links. Phase modulators produce the sidebands for Pound-Drever-Hall locking of a laser to a reference cavity and, driven hard or placed in a resonator, generate electro-optic frequency combs. Pockels cells switch cavity loss in Q-switched lasers and pick single pulses from mode-locked trains.

Pitfalls

Bias drift. The operating point of a lithium niobate modulator drifts over minutes to days, from charge migration in buffer layers and the pyroelectric effect, so most transmitters lock the bias with a low-frequency dither and feedback loop.

Photorefractive damage. At visible wavelengths and high intensity, undoped lithium niobate develops light-induced index changes; MgO doping reduces them.

Polarization. Lithium niobate devices use $r_{33}$ only for light polarized along the crystal $z$ axis; the orthogonal polarization sees $r_{13}$, about a third as large, so the input fiber is polarization maintaining.

Common questions

What is the difference between an electro-optic and an acousto-optic modulator?

An electro-optic modulator changes the index directly with a voltage and reaches tens to more than 100 gigahertz. An acousto-optic modulator diffracts light from a sound wave, shifts its frequency by the acoustic frequency (tens to hundreds of megahertz), and is limited in speed by the acoustic transit time across the beam.

Does an electro-optic modulator change the laser frequency?

A sinusoidal drive moves power from the carrier into sidebands; the carrier frequency stays fixed. A linear voltage ramp shifts the frequency by $\Delta f = (1/2\pi)\,d\phi/dt$, which is the basis of serrodyne frequency shifting.

References: A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); E. L. Wooten et al., "A review of lithium niobate modulators for fiber-optic communications systems," IEEE Journal of Selected Topics in Quantum Electronics 6, 69 (2000).