Traveling-wave electrode (velocity matching)

A traveling-wave electrode is a modulator electrode designed as a microwave transmission line running parallel to the optical waveguide. The drive signal enters at the optical input end, propagates along the line with the light and is absorbed in a termination at the far end. Each part of the optical wave sees the drive voltage that was present when it entered, provided the two waves travel at the same speed. This removes the RC limit of a lumped electrode, which a centimeter-long electrode could not otherwise escape, and lets a long electrode, and hence a low half-wave voltage, coexist with a bandwidth of tens of gigahertz. Lithium niobate, silicon and InP Mach-Zehnder modulators all use the design for electrodes longer than a few hundred micrometers.

Velocity mismatch

If the microwave effective index $n_m$ differs from the optical group index $n_g$, the light walks off the drive waveform and the phase accumulated along the electrode partly cancels at high frequency. For a lossless electrode of length $L$, with $\Delta n = |n_m - n_g|$, the normalized response is

$$|m(f)| = \left|\frac{\sin u}{u}\right|, \quad u = \frac{\pi f L\,\Delta n}{c}.$$

The electrical 3 dB point ($|m| = 1/\sqrt{2}$) is at $u$ = 1.392 and the optical 3 dB point ($|m| = 1/2$) at $u$ = 1.895, so the bandwidth-length product depends only on $\Delta n$:

$$f_{3\mathrm{dB,el}}\,L = \frac{1.392\,c}{\pi\,\Delta n}.$$

On bulk lithium niobate with a simple coplanar line, $n_m$ is about 4.2 and $n_g$ about 2.2. With $\Delta n$ = 2.0 the electrical bandwidth-length product is 6.6 GHz·cm, so a 2 cm electrode reaches only 3.3 GHz. Thick electrodes and a low-index buffer layer pull more of the microwave field into air and lower $n_m$; on thin-film lithium niobate the mismatch can be made small. With $\Delta n$ = 0.1 the product rises to 133 GHz·cm, and loss becomes the limit.

Microwave loss

With the velocities matched and a field attenuation $\alpha$ per unit length, the response is

$$|m(f)| = \frac{1 - e^{-\alpha L}}{\alpha L}.$$

It falls to the electrical 3 dB point when the total electrode loss reaches 6.4 dB, and to the optical 3 dB point at 13.8 dB. Conductor loss from the skin effect grows as $\sqrt{f}$. For an assumed loss of 0.5 dB/(cm·√GHz) on a 2 cm electrode, the total loss is 1.0 dB × √(f/GHz), which reaches 6.4 dB at 41 GHz. Lengthening the electrode lowers $V_\pi$ in proportion to $1/L$ but lowers the loss-limited bandwidth as $1/L^2$, which is a main trade-off in modulator design.

Impedance and termination

The line's characteristic impedance should match both the driver and the termination, usually 50 Ω. A mismatch at the input reflects part of the drive power, and the reflection coefficient follows the usual impedance matching relation; a 35 Ω line on a 50 Ω source reflects 18% of the field, a return loss of 15 dB. A mismatch at the far end sends a wave back toward the input. That wave travels against the light, so its effective mismatch is $n_m + n_g$ and its contribution is strongly low-passed, but it still produces ripple in the response and changes the voltage seen at low frequency.

In silicon and InP modulators the p-n junction capacitance loads the line. The added capacitance per unit length raises the microwave index and lowers the impedance,

$$n_m = c\sqrt{L'(C' + C_j')}, \quad Z = \sqrt{\frac{L'}{C' + C_j'}},$$

where $L'$ and $C'$ are the unloaded line inductance and capacitance per unit length and $C_j'$ the junction capacitance per unit length. Designers segment the electrode or adjust the line geometry to bring $n_m$ back toward $n_g$, and loaded lines are often terminated below 50 Ω to match their lower impedance.

Measurement

The electro-optic response $S_{21}$ is measured with a lightwave component analyzer or a network analyzer and a calibrated photodiode, together with the electrode's electrical $S_{21}$ and $S_{11}$. The electrode loss at the frequency where the electro-optic response is 3 dB down, compared with 6.4 dB, shows whether loss or velocity mismatch limits the device. The procedure is set out in How to Measure Electro-Optic Modulator Bandwidth.

Common questions

When does a modulator need a traveling-wave electrode?

When the electrode is no longer short compared with the wavelength of the drive signal in the line. At 50 GHz with $n_m$ = 2.2 that wavelength is about 2.7 mm, so electrodes longer than a few hundred micrometers behave as transmission lines whether or not they are designed as such.

Why is the electrical 3 dB bandwidth lower than the optical one?

The detected electrical power is proportional to the square of the optical modulation amplitude, so a 3 dB drop in electrical power corresponds to a 1.5 dB drop in optical modulation. Datasheets should state which convention they use.

References: G. K. Gopalakrishnan, W. K. Burns, R. W. McElhanon, C. H. Bulmer and A. S. Greenblatt, "Performance and modeling of broadband LiNbO₃ traveling wave optical intensity modulators," Journal of Lightwave Technology 12, 1807 (1994); C. Wang et al., "Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages," Nature 562, 101 (2018); D. M. Pozar, Microwave Engineering, 4th ed. (Wiley, 2012); A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007).