Photobleaching

Photobleaching is an irreversible loss of fluorescent capacity caused by illumination. It changes the number of active fluorophores over experimental time; fluorescence lifetime concerns the delay after an individual excitation. Reversible dark states and ordinary return to the ground state are different processes.

In a stipulated population-loss model, let $T$ denote elapsed illumination time and $t$ denote delay after a pulse. If active population declines as $e^{-k_{\mathrm{b}}T}$ while surviving fluorophores retain the same decay rates, then $I(t;T)=A_0{e}^{-k_{\mathrm{b}}T}{e}^{-t/\tau }$. Brightness falls while $\tau$ stays fixed. This example does not assert that all bleaching follows exponential decay: chemical changes and selective loss from a mixture can also change the observed decay shape. The existing FRET photobleaching controls illustrate why deliberate bleaching requires careful interpretation.

Two initially active independent emitters with exponential lifetimes of rate $k$ have last-bleaching time $T=\max (T_1,T_2)$. Independence gives $P(T\le t)=(1-e^{-kt}{)}^2$ for $t\ge 0$. Differentiating proves the density in the title, and integrating its survival function gives $E[T]=3/(2k)$. The first event has rate $2k$, while the waiting time from first to second has rate $k$.