Boxcar averager (gated integrator)

A boxcar averager, or gated integrator, measures a weak repetitive pulsed signal by opening a time gate of adjustable width and delay after each trigger, integrating the detector output only while the gate is open, and averaging that integral over many shots. Gates range from about 100 ps to many microseconds, and the instrument suits low-duty-cycle sources: Q-switched or amplified lasers at 10 Hz to tens of kHz, fluorescence decays, pulsed laser-diode tests and photoacoustic signals. It is the pulsed counterpart of the lock-in amplifier, which is optimal for signals modulated near a 50% duty cycle but spends most of each period averaging noise when the signal occupies a tiny fraction of it.

How it works

Three settings define a measurement:

• Delay: the time from the trigger to the opening of the gate, set so the gate sits on the signal pulse.
• Gate width: how long the integrator is connected to the input.
• Averaging: the number of shots $N$ combined, either as a linear sum or as an exponential moving average whose time constant is expressed in shots.

The output is a slowly varying voltage proportional to the mean signal inside the gate. Scanning the delay slowly across the pulse traces out its waveform. Today the same operation is often done digitally: a high-speed digitizer records each shot and the gate, baseline window and averaging are applied in software, with more flexibility and the same noise arithmetic.

Signal-to-noise gain from gating

For white noise, integrating for a time $t$ collects a signal proportional to $t$ and a noise proportional to $\sqrt{t}$. A rectangular pulse of width $t_p$ repeating with period $T$ has duty cycle $D = t_p/T$. Integrating continuously over the whole period collects the same signal as a gate matched to the pulse but $\sqrt{T/t_p}$ times more noise, so gating improves the signal-to-noise ratio by

$$G = \frac{1}{\sqrt{D}}.$$

A 10 ns pulse at 1 kHz has $D = 10^{-5}$, so $G \approx 316$. Averaging $N$ shots then adds the usual factor $\sqrt{N}$ for uncorrelated noise: 1000 shots, one second at 1 kHz, gives a further 31.6. In the frequency domain the averaging sets the noise bandwidth, so doubling the SNR costs four times the integration time, the same trade as a lock-in time constant.

Choosing the gate

A gate wider than the pulse adds noise without signal; a narrower one throws signal away. For a rectangular pulse the best gate matches the pulse. For an exponential decay with time constant $\tau$, the ratio of collected signal to noise, proportional to $(1 - e^{-t_g/\tau})/\sqrt{t_g}$, peaks at a gate of about $1.26\,\tau$, which collects 71.5% of the decay. A short delay scan before each measurement shows where the pulse actually sits.

Baseline subtraction

Slow drifts, detector offsets and 1/f noise appear in every gate and are not reduced by gating. Two methods remove them:

• A second gate placed before the pulse (or well after it) measures the baseline on every shot, and the instrument outputs the difference.
• Shot-to-shot modulation: the excitation is blocked on alternate shots with a chopper or pump-probe shutter, and the boxcar subtracts "off" shots from "on" shots. This moves the measurement to half the repetition rate, above most drift.

Either subtraction combines two noisy readings, so the white-noise floor rises by $\sqrt{2}$; in exchange the result is free of offsets that would otherwise dominate.

Pitfalls

• Timing jitter and drift between the trigger and the optical pulse move the pulse within the gate. With a gate barely wider than the pulse, a jitter of a fraction of the gate width becomes amplitude noise.
• Overload outside the gate. Large out-of-gate signals, such as a pump pulse seen by the same detector, can saturate the input amplifier or detector and corrupt the following gate even though they are not integrated.
• Ground loops. The gate trigger often shares a return path with a pulsed driver, and pickup coupled into the gate looks like signal. Checking that the output reads zero with the light blocked, or with the source below threshold, catches it; the pulsed versus CW LIV article describes this check for laser-diode testing.

Common questions

When is a boxcar averager better than a lock-in amplifier?

When the signal is a short pulse at low repetition rate, roughly when the duty cycle is below a few percent. A lock-in works at the fundamental of the repetition rate and averages noise across the whole period; a boxcar ignores the time between pulses. At high duty cycle the two give similar results and the lock-in is simpler.

Why is it called a boxcar?

The gate is a rectangular (boxcar-shaped) weighting function in time. Integrating with equal weight over a window is also called a boxcar filter, whose frequency response is a sinc function.

Can a digital oscilloscope replace a boxcar averager?

For many measurements, yes: an oscilloscope or digitizer with waveform averaging and measurement gates performs the same operation. Dedicated boxcars still help at high repetition rates, where every shot must be integrated without dead time.

References: P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge University Press, 2015); W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed. (Springer, 2014).