Statistical treatment of photon/electron counting: extending the linear dynamic range from the dark count rate to saturation

Abstract

An experimentally simple photon counting method is demonstrated providing 7 orders of magnitude in linear dynamic range (LDR) for a single photomultiplier tube (PMT) detector. In conventional photon/electron counting methods, the linear range is dictated by the agreement between the binomially distributed measurement of counted events and the underlying Poisson distribution of photons/electrons. By explicitly considering the log-normal probability distribution in voltage transients as a function of the number of photons present and the Poisson distribution of photons, observed counts for a given threshold can be related to the mean number of photons well beyond the conventional limit. Analytical expressions are derived relating counts and photons that extend the linear range to an average of ∼11 photons arriving simultaneously with a single threshold. These expressions can be evaluated numerically for multiple thresholds extending the linear range to the saturation point of the PMT. The peak voltage distributions are experimentally shown to follow a Poisson weighted sum of log-normal distributions that can all be derived from the single photoelectron voltage peak-height distribution. The LDR that results from this method is compared to conventional single photon counting (SPC) and to signal averaging by analog to digital conversion (ADC).

Figure 1

(Top) Representative calculated lognormal voltage distributions for n photon events based on the same initial parameters, M_L,1 and S_L,1. M_L,n and S_L,n are calculated using equations 10 and 11. (Bottom) The CPHD for various mean number of photons per trial. Each curve integrates to p instead of 1 because no voltage is generated for zero photons.

Figure 2

Example relating Poisson distribution of photons and binomial probability, p.

Figure 3

The signal to noise ratios (μ/σ) calculated by equations 15 and 16 (red dots) and theoretical maximum based on the Poisson distribution (blue dots) are compared for N=100,000 trials. The green vertical bars mark the p values that correspond to given means. The multiplier indicates the increase in noise relative to the theoretical limit.

Figure 4

Evaluation of p for different thresholds from equation 12 as a function of the mean number of photons. Different thresholds have sensitivity to different photon ranges. Overlap in these curves ensures continuous quantitation.

Figure 5

Second harmonic generation optical setup 1. Pulsed infrared laser 2. Glan polarizer 3. 1064nm zeroth order halfwave plate 4. visible blocking filter 5. KTP doubling crystal 6. IR blocking filter (KG3) 7. PMT.

Figure 6

Peak voltage distributions for two incident photon fluxes. The solid lines are the best fit theoretical Poisson weighted sum of lognormal distributions with parameters determined from single photon voltage distributions. The green points and pink line are rescaled by a factor of ten for visualization.

Figure 7

Comparison of observed counts (red triangles) and calculated photons (blue diamonds) outside of the Poisson counting range (λ>0.103) calculated by the single threshold method in eq. 15.

Figure 8

Comparison of conventional SPC (red triangles), signal averaging ADC (green squares) and calculated photons (blue diamonds) using multiple counting threshold settings. The ADC measurements were rescaled to overlay to account for the proportionality between photons and voltage.