How to Measure Electro-Optic Modulator Bandwidth: S21 Setup, Conventions and Limits

Scope

This article gives the procedure for measuring the small-signal electro-optic (EO) frequency response of an intensity or phase modulator and reading its bandwidth from it. It covers the measurement setup, calibration and de-embedding, the choice of bias and drive level, the two dB conventions that make published bandwidths hard to compare, and the physical limits (velocity mismatch, electrode loss and RC) that the measured curve can be checked against. The DC half-wave voltage, and the relation between $V_\pi$ and the EO response, are in Measuring Modulator Vπ and Bandwidth; the device physics is in the Mach-Zehnder modulator and modulation bandwidth entries.

What is measured

A small RF voltage at frequency $f$ applied to the modulator produces a small modulation of the optical intensity. The EO response $m(f)$ is the modulation amplitude per volt, normalized to its low-frequency value, so $m \to 1$ as $f \to 0$. A photodiode converts the intensity modulation back to an RF current proportional to $m(f)$, and a vector network analyzer (VNA) reads the ratio of that current to the drive as $S_{21}$. Because the VNA reports power ratios, the trace shows $20\log_{10}|m(f)|$ once every other element in the path has been removed.

Setup

Calibration and de-embedding

The measured trace is the sum, in dB, of every element between the VNA ports:

$$S_{21} \;=\; H_\text{mod} + H_\text{PD} + H_\text{path}.$$

The VNA's own calibration (coaxial standards, or on-wafer standards for probed devices) moves the reference planes to the modulator's RF input and the photodiode's RF output, removing the cables. The photodiode's response $H_\text{PD}$ must come from its own calibration, usually supplied with a lightwave component analyzer or measured by the heterodyne method in How to Measure Photodetector Bandwidth. An RF probe or connector between the calibration plane and the electrode, and any amplifier, are removed by their measured S-parameters. What remains is $20\log_{10}|m(f)|$ for the modulator alone.

Procedure

1. Set the optical path. Launch the laser into the modulator with the polarization set for maximum modulation, and confirm the output power at the photodiode is within its linear range.
2. Bias at quadrature. For a Mach-Zehnder, set the bias to the half-power point of the transfer curve, where the small-signal slope is largest and even-order distortion vanishes. For a phase modulator, convert phase to intensity first, with an interferometer or a dispersive element, or use an optical spectrum analyzer on the sidebands.
3. Set the drive level. Choose a VNA output power low enough that the response does not change when it is lowered by a few dB; a drive that is a significant fraction of $V_\pi$ compresses the low-frequency response and flattens the curve.
4. Calibrate the VNA to the reference planes, as above.
5. Sweep from the lowest frequency the bias tee passes to beyond the expected bandwidth, with an IF bandwidth narrow enough to keep the high-frequency end above the noise floor.
6. De-embed the photodiode and any remaining path, normalize to the low-frequency value, and record the curve.
7. Read the bandwidth at the chosen convention (next section), and keep the whole curve: ripple, a resonance or a slow low-frequency roll-off each point to a specific electrode or termination problem.

Electrical and optical 3 dB conventions

The VNA trace falls by 3 dB when $|m|$ has fallen to $1/\sqrt{2}$ of its low-frequency value. This electrical 3 dB bandwidth is the one read directly from the corrected $S_{21}$.

Some papers and datasheets instead quote an optical 3 dB bandwidth, defined by the modulation amplitude falling to one half, $10\log_{10}|m| = -3$ dB. On the VNA trace that is the 6 dB point. The same modulator therefore has two bandwidths differing by a factor that depends on the shape of its roll-off; for the shapes below, the optical 3 dB value is 1.36 times the electrical one when velocity mismatch dominates, and about 4.7 times when electrode loss growing as $\sqrt{f}$ dominates. A bandwidth quoted without its convention cannot be compared with another, and a report should give the convention and, preferably, both frequencies.

The drive needed for a full $\pi$ phase shift rises as the response falls, $V_\pi(f) = V_{\pi,\text{DC}}/|m(f)|$: $\sqrt{2}$ times the DC value at the electrical 3 dB point, and twice the DC value at the optical 3 dB point.

Checking the result against the limits

A traveling-wave modulator's response is limited by how well the RF wave keeps pace with the light and by how much the electrode attenuates the RF wave. Each limit gives a curve shape and a bandwidth that the measurement can be checked against.

Velocity mismatch. With a lossless electrode of length $L$ and a difference $\Delta n$ between the RF effective index and the optical group index, the response is

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

The electrical 3 dB point is at $u$ = 1.392 and the optical 3 dB point at $u$ = 1.895, so the bandwidth-length product is fixed by $\Delta n$ alone:

For a conventional bulk lithium niobate modulator with a simple coplanar electrode, the RF index is about 4.2 and the optical group index about 2.2; with $\Delta n$ = 2.0 the products are 6.6 GHz·cm (electrical) and 9.1 GHz·cm (optical), so a 2 cm device is limited to a few gigahertz. The large bandwidths of modern designs, including thin-film lithium niobate modulators, come from reducing $\Delta n$ toward zero, after which loss sets the limit.

Electrode loss. With velocities matched and an RF field attenuation of $\alpha$ per unit length, the response is

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

It reaches the electrical 3 dB point when the electrode's total RF loss is 6.4 dB, and the optical 3 dB point when that loss is 13.8 dB. Since conductor loss grows as $\sqrt{f}$, the frequency where the measured electrode $S_{21}$ (electrical transmission through the electrode, from the same VNA) reaches 6.4 dB is a direct estimate of the loss-limited EO bandwidth. Measuring the electrode's electrical $S_{21}$ and $S_{11}$ alongside the EO response is therefore routine: together they separate loss, mismatch and termination effects.

RC limit. A short lumped electrode, much shorter than the RF wavelength, behaves as a capacitance charged through the source resistance, and its bandwidth is $1/(2\pi R C)$ with $R$ the source and termination resistance seen by the capacitance.

Common failure modes

Photodiode response not removed. The measured bandwidth is the combination of both devices and reads low; the error is largest when the two bandwidths are similar.

Termination mismatch. Reflections from a poorly matched termination or probe produce ripple on the EO response with a period set by the electrode's round-trip delay; $S_{11}$ shows the same period.

Bias drift. Lithium niobate modulators drift from quadrature over minutes; a drift during the sweep appears as a slow tilt in the response. Re-check the bias before and after, or use a bias controller.

Low-frequency normalization. A modulator whose response rises or falls below a few hundred megahertz, from the bias tee or from slow photorefractive or charge effects, gives a different bandwidth depending on where the curve is normalized. State the normalization frequency.

Convention not stated. The most common reason two measurements of similar devices disagree by a large factor.

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); 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); D. Derickson (ed.), Fiber Optic Test and Measurement (Prentice Hall, 1998), chapter on lightwave component analysis.