Figure 1.
Schematic of the sources of noise during the photon measurement.
Figure 2.
Histogram of the experimentally observed image counts from a dark image.
The data was taken with an EM gain of 300.
Figure 3.
Removal of systematic contributions.
Colour coded, experimentally observed image counts from a dark image. The strip on the far right shows the colour code assignment, where black corresponds to the lowest observed image value and white corresponds to the highest observed image value. From left to right: the original, uncorrected dark image; the image of systematic contributions and the dark image after contributions have been subtracted.
Table 1.
Table of mathematical symbols.
Figure 4.
Simulation of the EM register composed of 536 stages with the Tubbs model, gamma distribution and normal distributions fitted.
The parameters for the distributions were calculated from the Poisson distribution (50,000 samples). Gain is estimated as sample mean divided by the number of input electrons, the parameters for the normal distribution are sample mean and sample standard deviation. (A) 1 input electron, (B) 2 input electrons, (C) 60 input electrons. The probability to create a new electron for each existing electron per multiplication stage is 1%. That yielded overall gains between 206 and 208. Both simulations, the Tubbs model and the gamma distribution are very similar in all cases. For a high number of input electrons, the similarity of the normal distribution to the simulated data is also high. However, EMCCDs are used to measure low intensities or single photons.
Figure 5.
Difference of the Tubbs model and the gamma distribution for low gain settings.
The EM register models are fitted to simulated data (A) 2 input electrons, (B) 15 input electrons. The probability to create a new electron per existing electron in a multiplication bin is 0.5%. That yielded an overall gain of 14.48. Sample number 250,000. For two input electrons the Tubbs model fits the data slightly better than the gamma distribution. However, EMCCDs are usually operated with much higher EM gain values.
Figure 6.
Sampling of the distribution of output electrons of an EM register for 5 photons.
The simulation shows the effect of low light intensities in the model. A large number of bins is chosen to emphasise the spike at 0 output electrons. The spike is the result of the assumption that zero input electrons will always yield zero output electrons. Other parameters: 90% quantum efficiency and 0.02 electrons spurious charge per pixel. Sample number 100,000.
Figure 7.
Mean-variance plot for the A/D factor estimation.
Each dot represents mean and variance of the intensity of an individual pixel for 60 frames across a single data set. The values of 9 data sets are shown which appear as “blobs” in the image. Each data set was taken with a different but constant light intensity. The data shown in red was taken with 3 MHz readout rate and the data shown in black was taken with 10 MHz.
Figure 8.
Mean-variance plot for the EM gain estimation.
Mean and variance were calculated from 60 values per pixel. Shown are data sets taken with an EM gain setting of 150 (blue), 200 (green) and 250 (red). The black lines indicate the fitted linear model. Each data set contains 932970 points; 7500 randomly sampled data points are shown.
Figure 9.
Maximum likelihood estimation for spurious charge (electrons/pixel) and readout noise (electrons).
Shown are the samples of the likelihood function (see Material and Methods) of readout noise and spurious charge for a set of dark images. The images for (A) and (B) were taken with 3 MHz and 10 MHz readout rates respectively. The manufacturer gives ca. 53 electrons readout noise for the settings used for (A) and ca. 32 electrons readout noise for the settings used for (B).
Figure 10.
Comparison of the density of stacks of white light images with the model density.
Each panel shows intensity density of a data set of 100 frames and 3 optical channels (continuous lines). The data sets were taken with different intensities; (A) low intensity, (B) medium intensity and (C) high intensity. The model densities are drawn with dashed lines. The common gain estimate is 175.9. The difference in intensities for different channels of the same data set is caused by the splitter optics. Panel (D) shows the comparison of the density of the image counts with the density of the CCD model and the density of a convolution of a gamma distribution with a normal distribution. The shape parameter of the gamma distribution is half the number of input electrons of the EM register, while the scale parameter is twice the estimated EM gain. The estimated number of average input electrons is 8.7. For higher light intensities, the densities become more similar.