Single-Photon Counting Module
SPCM dead time correction, SNR, dark count effects, and afterpulsing analysis.
Incident Photons
3.272e+6 /s
Avalanche Rate R (photons + dark)
2.127e+6 /s
Recorded Rate m
2.032e+6 /s
4.47 % lost to dead time
Dead-Time Correction R/m = 1 + Rτ
1.0468
Afterpulse Rate m·P_ap
1.016e+4 /s
Total Output m(1 + P_ap)
2.042e+6 /s
SNR in 1 s (corrected)
1421.6
SNR without dead time or afterpulses
1458.4
Key Formulas
R = P·λ/(hc)·PDE + DCR, m = R / (1 + R·τ) [non-paralyzable]
Afterpulses: m·P_ap, first order (no afterpulses of afterpulses)
SNR = (R_s/R)·√(m·T) / √(1 + (1 + Rτ)²·P_ap(1 − P_ap)/(1 + P_ap)²)
The SNR is that of the photon rate after dead-time correction, with DCR and P_ap known and subtracted
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
A fixed non-paralyzable dead time acts on photon and dark avalanches, and afterpulses are first order. The SNR is that of the dead-time corrected photon rate over the gate, with the dark and afterpulse rates known. That SNR is statistical only: a dead-time error δτ biases R by Rτ·δτ/τ (0.43 % for the datasheet's 24 against 22 ns), and an afterpulse error δP by δP/(1 + P). Defaults: an Excelitas SPCM-AQRH-14 at 650 nm. Above about 5 Mc/s real modules need more correction than a fixed 22 ns gives: 1.16 against 1.12 at 5 Mc/s.
- Excelitas Technologies. SPCM-AQRH single-photon counting module, datasheet Rev 2023-1. PDE 65 % at 650 nm, 22 ns dead time (-1X), dark count ≤ 100 cps (-X4), afterpulsing 0.5 %, linearity correction factor 1.02 at 1 Mc/s and 1.16 at 5 Mc/s. excelitas.com
- Knoll G. F. (2010). Radiation Detection and Measurement, 4th ed. Wiley. Ch. 4, section VII: non-paralyzable m = n/(1 + nτ) and paralyzable m = n·e^(−nτ) dead-time models.
- Müller J. W. (1973). Dead-time problems. Nucl. Instrum. Methods 112, 47–57. Dead-time models and the statistics of the recorded counts. doi:10.1016/0029-554X(73)90773-8