EMCCD Gain Calculator
EMCCD gain: Robbins–Hadwen excess noise of an N-stage register, effective read noise and SNR with and without EM gain.
Output Signal
1000 e⁻
Effective Read Noise
0.500 e⁻
SNR with EM
2.24
SNR at G = 1
0.20
Same EM amplifier and read noise
EM Benefit vs G = 1
×11.20
Same EM amplifier at unity gain
SNR, Conventional Output
1.47
EM output ×1.52 of it
Crossover Signal
36.7 e⁻
EM output wins below, conventional above
ENF² (F²)
1.9750
Gain per Stage
0.765 %
SNR_EM = S / √(F²·(S + CIC) + (σ_read/G)²)
F² = 2(G − 1)·G^(−(N+1)/N) + 1/G (Robbins & Hadwen 2003)
G = (1 + p)^N; F² → 2 − 1/G for N → ∞, F² = 1 at G = 1
SNR_conv = S / √(S + CIC + σ_conv²); crossover S* = (σ_conv² − (σ_read/G)²)/(F² − 1) − CIC
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
Robbins–Hadwen F² for N stages that each multiply an electron with probability p, G = (1 + p)^N; it equals the Galton–Watson result for a binomial stage. Clock-induced charge is counted at the input although charge made inside the register sees less gain. Photon-counting (threshold) readout, which removes the factor F² ≈ 2 at low flux, and register saturation are not modelled. The often-quoted F² = 2 − 1/G is the N → ∞ limit; it puts a false SNR optimum at G = 2σ²/(S + CIC), 500× at the defaults. "SNR at G = 1" uses the same EM amplifier; the conventional output (F² = 1, its own read noise) wins above the crossover signal (σ_conv² − (σ_EM/G)²)/(F² − 1) − CIC. Both outputs see the same CIC: the gain-register share is absent at the conventional output, negligible next to σ_conv².
- Robbins M. S., Hadwen B. J. (2003). The noise performance of electron multiplying charge-coupled devices. IEEE Trans. Electron Devices 50, 1227–1232. F² = 2(G − 1)·G^(−(N+1)/N) + 1/G. doi:10.1109/TED.2003.813462
- Basden A. G., Haniff C. A., Mackay C. D. (2003). Photon counting strategies with low-light-level CCDs. Mon. Not. R. Astron. Soc. 345, 985–991. Thresholding removes the multiplication noise at low flux. doi:10.1046/j.1365-8711.2003.07020.x