Precision of least-squares mono-exponential decay time estimates from optical measurements with both photon shot noise and background noise

Abstract

Equations are derived for the precision of least-squares estimates of mono-exponential decay times from optical measurements containing both photon shot noise and background noise. The special case of data containing only photon shot noise leads to the same equation as that derived previously by others using the maximum likelihood estimation method. It is shown that for a sufficiently large number of data points used in the fit, spanning up to approximately ten decay times, the lifetime estimate precision reaches a limit value being the inverse square root of the number of photons contained in the decay curve used for the estimate. Further, an extension to the general case where both types of noise are present is provided. Approximating photon statistical noise as normally distributed, the expressions for precision are shown to be valid even for very low signal-to-noise ratios. Thus, the theory provides a straightforward and useful tool to assess the lifetime estimate precision from a single decay measurement.