Gain and Photon Transfer Curves

The shot noise standard deviation, in photon units, follows from the randomness of individual photon arrival times as described by Poisson statistics:

\[\sigma_{shot,p} = \sqrt{N_p \frac{e^{hc/(\lambda k_BT)}}{e^{hc/(\lambda k_BT)}-1}} \quad \mathrm{Janesick\: p.66}\]

For most practical applications, the temperatures and wavelengths are small enough to use an approximation:

\[\sigma_{shot,p} = \sqrt{N_p}\]

The value is converted from photon to electron units by applying the quantum yield, i.e. the number of liberated electrons per interacting photon:

\[\sigma_{shot,e} = \sqrt{\eta_i N_p}\]

The recorded signal in digital number (DN) units is

\[S_{DN} = \frac{\eta_i N_p}{G}\]

where G is the gain (e/DN). By substitution, σ_shot,e can be expressed in terms of the recorded signal:

\[\sigma_{shot,e} = \sqrt{S_{DN} G}\]

Finally, the shot noise standard deviation is expressed in terms of DN:

\[\sigma_{shot,DN} = \frac{\sigma_{shot,e}}{G} = \frac{\sqrt{S_{DN} G}}{G} = \frac{\sqrt{S_{DN}}}{\sqrt{G}}\]

Solving for G,

\[G = \frac{S_{DN}}{\sigma^2_{shot,DN}}\]

The gain can be visualized by plotting the means of the recorded signal against the variances. This is known as a photon transfer curve. Each point in the plot should have a different number of detected photons. This is generally done by using a constant source and varying the exposure times. At each exposure time, a number of trials is measured in order to calculate the means and variances.