The exponential decay of fluorescence intensity is a fundamental concept in fluorescence spectroscopy and microscopy. It describes how the emitted light from a population of excited fluorophores decreases over time after the excitation source is removed.
The population of excited molecules will decay by a rate $\Gamma+k_{ nr }$ given by the equation 1 :
$$ \frac{d n(t)}{d t}=-\left(\Gamma+k_{nr}\right) n(t) $$
where $n(t)$ is the number of molecules excited at time $t$, $\Gamma$ is the rate of fluorescence emission and $k_{ nr }$ is the non-radiative decay rate (loss of energy by routes other than the emission of a photon). The fluorescence emission is a random event with each excited molecule having the same probability of emitting a photon at a given time point. This leads to an exponential loss of the excited state population with the population at any point in time after excitation given by
$$ n(t)=n_0 \exp (-t / \tau) $$
where $n_0$ is the initial number of molecules excited. In any of the methods used to determine the fluorescence lifetime we do not directly observe the number of excited molecules but rather the intensity of photon emission, which is clearly proportional to the number of excited molecules at any one time n(t). By using this fact and integrating equation 1 we can determine the intensity of light being emitted at any particular point after excitation given by
$$ I(t)=I_0 \exp (t / \tau) $$
where $I_0$ is the intensity at the time of the end of the excitation pulse. The lifetime $\tau$ is the inverse of the total decay rate containing the radiative and non-radiative terms. It is this exponential loss of fluorescence intensity that is then plotted based upon the detected photon numbers to provide the fluorescence lifetime image.
https://gist.github.com/viadean/1eb9c0c6cdd300d9a98a5756d322571e
Here is the plot of the exponential decay of fluorescence intensity over time. As expected, the intensity decreases exponentially, governed by the fluorescence lifetime $\tau$.