Common-Mode Rejection in Balanced Detectors

Scope

This article covers the common-mode rejection ratio (CMRR) of a balanced detector: its definition, the physical mismatches that limit it, a formula for its frequency dependence with numbers, the noise benefit it delivers in a coherent receiver, and a procedure for measuring and trimming it. The balanced detection entry gives the receiver itself; relative intensity noise and shot noise are the two noise terms whose ratio CMRR controls.

Definition

A balanced detector subtracts the photocurrents of two photodiodes. A signal that arrives with opposite sign at the two diodes, the signal-LO beat, adds in the subtraction; a signal that arrives with the same sign at both, the local oscillator's intensity and its noise, cancels. CMRR is the measure of how well the second happens: the ratio of the receiver's response to a common-mode input, applied equally to both diodes, against its response to the same input applied to one diode alone,

$$\mathrm{CMRR}(f) = 20 \log_{10} \frac{|H_\mathrm{single}(f)|}{|H_\mathrm{common}(f)|},$$

expressed in decibels and, because the mismatches that limit it depend on frequency, quoted at a frequency or as a curve. A perfect receiver has infinite CMRR; commercial parts specify 25 to 40 dB at low frequency and less at the top of their bandwidth.

Where finite CMRR comes from

Let the two diodes have responsivities $R_1$ and $R_2$ and let the light reach the second diode later than the first by $\Delta\tau$ (a longer fiber lead, an asymmetric coupler, a photodiode further from the amplifier node). For a common-mode intensity modulation at frequency $f$, the difference current is

$$i_\mathrm{diff}(f) \propto R_1 - R_2\, e^{-j 2\pi f \Delta\tau} \approx \Delta R + j\, 2\pi f \Delta\tau\, R,$$

for small mismatches, so the residual common-mode response has two terms in quadrature: a frequency-independent amplitude mismatch $\Delta R/R$ and a delay mismatch that grows linearly with frequency. The CMRR is

$$\mathrm{CMRR}(f) = -20 \log_{10} \sqrt{\left(\frac{\Delta R}{R}\right)^2 + \left(2\pi f \Delta\tau\right)^2}.$$

The amplitude term includes everything that makes the two arms unequal in DC gain: photodiode responsivity, the coupler's split ratio, connector and lens losses, and the two transimpedance gains if the currents are subtracted after amplification. The delay term includes fiber length, free-space path, and any difference in the diodes' own transit times. A third contribution, mismatched bandwidth, behaves like a delay term at frequencies approaching the diodes' roll-off and is best treated as part of the measured curve rather than modelled.

The numbers are unforgiving at high frequency. A 1% amplitude mismatch alone gives a 40 dB floor, 2% gives 34 dB, and 5% gives 26 dB. A delay mismatch of 5 ps, which is one millimeter of fiber (4.9 ps/mm at $n = 1.47$), gives 50 dB at 100 MHz, 30 dB at 1 GHz, and 10 dB at 10 GHz; 20 ps, four millimeters, gives 38, 18, and effectively nothing. Combined, 1% and 5 ps give 29.6 dB at 1 GHz. A receiver specified at 30 dB out to a gigahertz therefore has its optical paths matched to about a millimeter, and a receiver meant for 10 GHz needs them matched to a tenth of that, which is why high-speed balanced receivers integrate both diodes on one chip fed by one on-chip coupler.

What it buys

The purpose of the rejection is to keep the local oscillator's intensity noise below the shot noise the local oscillator itself creates. Take 1 mW of LO on each diode at 0.9 A/W, so 0.9 mA of photocurrent per diode, and a laser with RIN of $-150$ dB/Hz. The RIN current noise on one diode is $10^{-15} \times (0.9\ \mathrm{mA})^2 = 8.1 \times 10^{-22}$ A²/Hz; the shot noise of that diode is $2 q I = 2.9 \times 10^{-22}$ A²/Hz. Single-ended, the RIN sits 4.5 dB above the shot noise and the receiver is RIN-limited. In the balanced receiver the shot noises of the two diodes add to $5.8 \times 10^{-22}$ A²/Hz while the RIN is suppressed by the CMRR: with 20 dB it falls to $8.1 \times 10^{-24}$, 18.5 dB below the shot noise; with 30 dB, to 28.5 dB below. The CMRR needed to put the RIN 10 dB under the shot noise in this example is 11.5 dB, which any balanced receiver meets at low frequency and which becomes the real specification at the receiver's upper frequency, where the delay term is eating the margin. The RIN measurement article covers the laser side of this number; the receiver sensitivity entry covers where the shot-noise limit lands once the RIN is out of the way.

Measuring CMRR

The measurement needs an intensity-modulated source applied to both diodes through the receiver's own coupler, a way to block one arm, and a spectrum analyzer.

1. Modulate the intensity of a laser at the frequency of interest, either by driving the laser current or with an external modulator, at a depth of a few percent. Feed it into the LO port of the receiver's coupler with the signal port dark.
2. Measure the receiver's output at the modulation frequency on an electrical spectrum analyzer with both diodes illuminated. This is the common-mode response.
3. Block the light to one diode, by disconnecting or attenuating one arm after the coupler, and measure the output at the same frequency. This is the single-diode response.
4. Compute $\mathrm{CMRR} = 20 \log_{10}(V_\mathrm{single}/V_\mathrm{common})$. Repeat across frequency to obtain the curve; it will be flat where the amplitude term dominates and fall at 20 dB per decade where the delay term takes over, and the corner between the two identifies which mismatch to fix.
5. Record the optical power per diode and the modulation depth with the result, since a saturated diode or a modulator driven into nonlinearity produces an apparent CMRR that is neither the receiver's nor repeatable.

Two shortcuts fail. Reading the DC balance (equal photocurrents) is not a CMRR measurement; equal DC currents say nothing about delay. And measuring with the modulation on the signal port instead of the LO port measures the differential response, which is the quantity the receiver is designed to pass, not the one it is designed to reject.

Trimming

Where the receiver allows it, the two mismatches are trimmed separately because they act separately. A variable attenuator in one arm trims the amplitude term and raises the low-frequency floor; a fiber delay line or a free-space path adjustment trims the delay term and raises the high-frequency CMRR. The order matters: trim delay at a high frequency where it dominates, then trim amplitude at a low frequency where it dominates, then re-check both, since the attenuator adds a small delay and the delay line adds a small loss. A trim is valid at one wavelength and one temperature; responsivity mismatch drifts with both, which is why receivers that must hold 40 dB use matched diodes on a common substrate with a shared thermal path rather than external trims.

Reading a specification

A CMRR figure without a frequency is incomplete. Ask at what frequency it was measured, whether with the receiver's own coupler or with a test splitter, and at what optical power. A part specified at 35 dB "typical" at 10 MHz may be 20 dB at 1 GHz, and for a coherent receiver whose LO RIN matters most at the low end of the signal band and whose delay mismatch matters most at the high end, the curve is the specification. The integrated receivers used in coherent transceivers, with four balanced pairs on one photonic chip, achieve their CMRR by construction and quote it across the signal band for that reason.

References: Y. Painchaud, M. Poulin, M. Morin and M. Têtu, "Performance of balanced detection in a coherent receiver," Opt. Express 17, 3659 (2009); K. Kikuchi, "Fundamentals of coherent optical fiber communications," J. Lightwave Technol. 34, 157 (2016); P. C. D. Hobbs, Building Electro-Optical Systems, 2nd ed. (Wiley, 2009), on balanced and auto-balanced receivers; Koheron, "Common mode rejection ratio measurement" (application note, 2017), for a single-frequency measurement example. All numbers above are computed from the stated mismatches, powers, and RIN.