Photon counting

Photon counting is the regime of detection in which each absorbed photon produces a separate electrical pulse, and the measurement consists of counting those pulses, and often timing them, instead of integrating a photocurrent. It is used when light is too weak for an analog detector: a 1 pW beam at 1550 nm carries $P\lambda/(hc)$ = 7.8 × 10⁶ photons per second, and 1 fW at 532 nm only about 2700. The detectors that make it possible are those with enough internal gain that one photoelectron gives a pulse well above the electronic noise: the photomultiplier tube, the single-photon avalanche diode, the silicon photomultiplier and the superconducting nanowire detector.

How counting works

Each detector pulse passes a discriminator, a comparator set above the electronic noise, and the resulting logic pulses go to a counter, a multichannel scaler or a timing module. Counts are recorded per time bin, per position of a scan, or relative to a trigger, as in time-correlated single-photon counting and photon-counting time of flight.

Poisson statistics and signal-to-noise ratio

For a laser or any light with Poisson photon statistics, the number of counts $N$ in a window has a variance equal to its mean, so a measurement of 100 counts has a standard deviation of 10, a 10% uncertainty. With detection efficiency $\eta$, incident photon rate $R$, dark count rate $D$ and counting time $t$,

$$\text{SNR} = \frac{\eta R t}{\sqrt{(\eta R + D)\,t}}.$$

The only noise is the Poisson noise of the counts themselves, signal plus dark, which is the counting equivalent of shot noise. With $\eta$ = 0.25, $D$ = 100 counts/s and $t$ = 1 s, an SNR of 1 needs a signal of 10.5 detected counts per second, 42 incident photons per second, or 5.4 × 10⁻¹⁸ W at 1550 nm; an SNR of 10 in the same time needs 162 detected counts per second.

Dead time

After each detection the detector and its electronics are blind for a dead time $\tau$, typically tens of nanoseconds for SPADs and nanowire detectors and a few nanoseconds for photomultiplier counting electronics. For a non-paralyzable detector the measured rate $m$ is related to the true rate $n$ by

$$m = \frac{n}{1 + n\tau}, \qquad n = \frac{m}{1 - m\tau}.$$

With $\tau$ = 50 ns, a true rate of 10⁵ counts/s loses 0.5% of its counts, 10⁶ counts/s loses 4.8%, and 10⁷ counts/s is measured as 6.7 × 10⁶, a 33% loss; the measured rate cannot exceed $1/\tau$ = 2 × 10⁷ counts/s. The correction is accurate only while the loss is modest and the detector's dead time is fixed; afterpulsing, which adds correlated false counts shortly after each true one, is often suppressed by a hold-off that lengthens $\tau$. In timing measurements a related effect, pile-up, biases the histogram toward early photons, which is why TCSPC keeps the detection rate at 1–5% of the laser repetition rate, 0.8–4 × 10⁶ counts/s at 80 MHz.

When to count and when to integrate

An integrating detector, such as a photodiode with a transimpedance amplifier or a camera pixel, adds electronic noise independent of the signal: amplifier noise, or read noise of a few electrons per readout. When the expected number of photoelectrons per measurement is comparable to that noise, counting wins: 4 photons on a pixel with quantum efficiency 0.8 and 2 e⁻ read noise give an SNR of 1.2 by integration, against 1.8 for an ideal counter with the same efficiency and no dark counts. At high flux the comparison reverses, since dead time limits counting to tens of millions of events per second while a photodiode handles microwatts and above with only shot noise. Counting therefore suits fluorescence lifetime, single-molecule and quantum optics experiments, long-range lidar and astronomy.

Pitfalls

Dark counts, afterpulses and stray light all count like signal and must be measured with the source blocked under identical conditions. The detection efficiency of a counting system includes coupling, the detector's quantum efficiency and the discriminator's threshold, and it may fall at high rates as the detector recovers. Saturation is not obvious: a counter driven far beyond $1/\tau$ can report a rate that falls as the light increases, for a paralyzable detector, or flattens, for a non-paralyzable one.

Common questions

What is the difference between photon counting and photon-number resolution?

A counting detector registers that at least one photon arrived during its active time; two photons arriving within its response time give one click. Photon-number-resolving detectors, such as transition-edge sensors or arrays of SPADs viewed together, output a signal proportional to the number of photons in a pulse.

How many photons per second is a nanowatt?

At 1550 nm, 7.8 × 10⁹ photons per second; at 532 nm, 2.7 × 10⁹. Both are far above the rate a single counting detector can handle, so such signals are attenuated or measured with a photodiode.

Why does a photon counter saturate?

Every count is followed by a dead time during which further photons are missed, so the measured rate approaches $1/\tau$ as the light increases.

References: W. Becker, Advanced Time-Correlated Single Photon Counting Techniques (Springer, 2005); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); G. F. Knoll, Radiation Detection and Measurement, 4th ed. (Wiley, 2010); R. H. Hadfield, "Single-photon detectors for optical quantum information applications," Nature Photonics 3, 696 (2009).