How photons become electrons and preserve the Poisson distribution

The Poisson distribution

The Poisson distribution describes the statistics of a series of 'events'. For light an event is the arrival of a photon.

The defining property of a Poisson distribution is that the chance of an event is the same at each small interval of time.

Suppose that for each small unit of time,

the probability that a photon will arrive is constant, ρ.

duration  = 0.1;             % Exposure duration
deltaT    = 1e-4;            % A millisecond
rho       = 1e-2;            % Chance of a photon in deltaT
T = deltaT:deltaT:duration;  % Time samples over 0.1 sec

Time series

We can simulate the probably of an event happening or not, in each instance, using the binomial distribution. Here is a plot with a line at each moments when there is (1) or is not (0) a photon.

photons = binornd(1,rho,length(T),1);

iePlot(T,photons);

xlabel('Time (sec)'); ylabel('Photon Event'); grid on

fprintf('There were %d photons in this sample.\n',sum(photons));

There were 4 photons in this sample.

Now, run many samples and make a histogram of the counts

nSamples = 1000;

cnt = zeros(1,nSamples);

for ii=1:nSamples

photons = binornd(1,rho,length(T),1);

cnt(ii) = sum(photons);

end

mu = mean(cnt);

sigma = std(cnt);

ieFigure;

% Create the histogram

h = histogram(cnt, 'Normalization', 'probability');

h.EdgeColor = 'none';

h.FaceColor = [0.3 0.3 1];

hold on; % Prevent the next plot from overwriting the histogram

% Calculate the Probability Mass Function (PMF) of the Poisson distribution

xmin = min(cnt);

xmax = max(cnt);

x = xmin:xmax;

lambda = mu;

f = (lambda.^x .* exp(-lambda)) ./ factorial(x);

plot(x, f, 'k-', 'LineWidth', 2); % 'r-' for a red solid line

% Calculate the probability density function (PDF) of the Normal distribution

x = linspace(min(cnt), max(cnt), 100);

f = exp(-(x - mu).^2 / (2 * sigma^2)) / (sigma * sqrt(2 * pi));

plot(x, f, 'r:', 'LineWidth', 2); % 'r-' for a red solid line

% Add labels and title

xlabel('Value');

ylabel('Probability');

% title('Histogram with Normal Distribution Overlay');

% legend('Histogram', 'Normal Distribution');

hold off; % Allow subsequent plots to overwrite this one

The mean and variance are equal in the Poisson

fprintf('Mean %.3f Variance %0.3f\n',mean(cnt),var(cnt));
Mean 9.945 Variance 9.862

Photon to electron conversion

At the sensor, an event is when a photon creates an electron. For a CMOS or CCD image sensor, each photon turns into one electron or no electrons.

Suppose the probability of a photon generating an electron is ϵ. Let's set epsilon to one half, as an example.

epsilon = 0.3;

The probability of conversion is the same whenever the photon arrives. Here is the probability of the conversion, which is also a Poisson distribution.

eConversion = binornd(1,epsilon,length(T),1);

Each photon may become an electron

The probability is the same in each little time interval, and thus the whole calculation is Poisson.

electrons = photons .* eConversion;

Here is a plot of the electrons. This is a subset of the times when there is a photon

iePlot(T,electrons);

xlabel('Time (sec)'); ylabel('Electrons'); grid on

Equivalent calculation

We do not have to do the calculation in two steps. The calculation is the same distribution if we combine the two probabilities.

electrons2 = binornd(1,rho*epsilon,length(T),1);

iePlot(T,electrons2);

xlabel('Time (sec)');

ylabel('Electrons'); grid on

fprintf('There were %d electrons\n',sum(electrons2));

There were 2 electrons

Counts

These should be close to the Poisson distribution, which is close to the Normal except when the Normal goes negative.