Impedance matching (50 Ω termination)

Impedance matching means terminating a transmission line in a load equal to its characteristic impedance $Z_0$, so that a wave traveling down the line is absorbed completely at the end. Coaxial cables, connectors, instruments and most high-speed circuit boards are built to $Z_0$ = 50 Ω. A load that differs reflects part of the wave back toward the source: a laser diode presenting 5 Ω reflects 67% of the incident power, while a 35 Ω modulator electrode reflects 3.1%. The reflected wave distorts the drive signal and causes ringing and ripple in the frequency response. Matching matters once the connection is no longer short compared with the distance an edge travels during its rise time, typically when its electrical length exceeds about a tenth of a wavelength at the highest frequency of interest; at 10 GHz a wavelength in PTFE-filled coax ($\varepsilon_r$ = 2.1) is 20.7 mm.

Reflection coefficient and return loss

A load $Z_L$ at the end of a line of impedance $Z_0$ reflects a fraction of the incident voltage given by

$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}.$$

The reflected power fraction is $|\Gamma|^2$, the return loss is $\mathrm{RL} = -20\log_{10}|\Gamma|$ in decibels, and the voltage standing-wave ratio is $(1+|\Gamma|)/(1-|\Gamma|)$. Some values on a 50 Ω line:

A return loss of 10 dB, a common minimum for a device port, means 10% of the power is reflected.

Measuring it: S11

The reflection coefficient of a device port is measured as $S_{11}$ with a vector network analyzer, referred to the 50 Ω reference impedance of the calibrated port; in that system $S_{11}$ is $\Gamma$ itself, and a plot of $|S_{11}|$ in dB is the negative of the return loss. Fixtures, probes and bias tees between the calibration plane and the chip add their own reflections and are removed by de-embedding. Time-domain reflectometry shows the same information against position along the line.

Laser diodes

Above threshold a laser diode presents a low impedance, its series resistance plus a small dynamic junction term, typically a few ohms. On a 50 Ω line that is a near-short. The common remedy is a resistor in series with the diode on the submount, chosen so that the sum is close to the line impedance: 45 Ω in series with a 5 Ω diode makes a 50 Ω load. The match comes at a price in drive current. For an incident voltage wave $V_i$ the load current is $2V_i/(Z_0 + Z_L)$, so the matched 50 Ω load draws $V_i/(50\ \Omega)$ against $V_i/(27.5\ \Omega)$ for the bare 5 Ω diode: 5.2 dB less modulation current, and 90% of the power delivered goes into the resistor. Drivers designed for 25 Ω lines or placed millimeters from the laser are the alternatives. A quarter-wave transformer of $\sqrt{Z_0 Z_L}$ = 15.8 Ω matches 5 Ω to 50 Ω without loss, but only over a narrow band.

Modulators and photodiodes

A traveling-wave electrode, such as that of a Mach-Zehnder modulator, is itself a transmission line. It is terminated at its far end in a resistor equal to its impedance so that the microwave drive is absorbed after it has traveled with the light, instead of reflecting back through the modulator. The drive power dissipated is about $V_{pp}^2/4R$ for a two-level signal, 45 mW for 3 V into 50 Ω. Electrode impedances somewhat below 50 Ω are common, especially in silicon designs, and leave a small mismatch to the driver: a 35 Ω electrode reflects 3.1% of the incident power. Lumped devices smaller than a tenth of a wavelength, such as electro-absorption modulators, are matched with a shunt resistor across the device, which also sets the RC time constant.

A photodiode is a current source with high output impedance. Loading it with 50 Ω at the far end of a cable is a matched termination for the cable; adding a 50 Ω back-termination at the diode absorbs reflections from the far end as well, at the cost of half the photocurrent.

Pitfalls

A mismatch at both ends of a line produces repeated reflections, which appear as ringing in time and as periodic ripple in the measured modulation bandwidth, with a period of $c/(2nL)$ for a line of length $L$ and index $n$. Bond wires add inductance that turns a good DC match into a poor one above a few gigahertz.

Common questions

Why is 50 Ω the standard?

For air-filled coax the minimum attenuation occurs near 77 Ω and the maximum power handling near 30 Ω; 50 Ω is a compromise between them that became the convention for RF and microwave test equipment.

What return loss is good enough?

For a test port or connector, return loss above about 20 dB (1% of power reflected) is considered good. Device inputs are often specified at 10 dB or better across their band.

References: D. M. Pozar, Microwave Engineering, 4th ed. (Wiley, 2012); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).