= Fluorescence instrument response
{title2=$R(t)$}
A <fluorescence instrument response> is the distribution of recorded delays for an effectively prompt optical signal. Finite excitation-pulse width and timing variation in the <photodetector> and electronics contribute. With a normalized response $R$, intrinsic fluorescent <photon> rate $I$, and background rate $B$, the expected trace is
$$
M(t)=\int_{-\infty}^{\infty}R(u)I(t-u)\,du+B(t),\qquad
\int_{-\infty}^{\infty}R(u)\,du=1.
$$
Fit a candidate decay after this <convolution> with the measured response; treating the broadened leading edge as the intrinsic decay can bias <fluorescence lifetime>. The accessible technical note describes measuring prompt scattered excitation with a nonfluorescent scattering sample.<Time-Correlated Single Photon Counting (technical note v. 2.1) by Michael Wahl>[Michael Wahl, Time-Correlated Single Photon Counting, v. 2.1, p. 3: scattering-medium response measurement and iterative reconvolution]{ref}
\Image[/fluorescence-decay-instrument-convolution.png]
{title=Synthetic <fluorescence lifetime> decay broadened by an instrument response}
{id=fluorescence-decay-instrument-convolution}
{description=Original stipulated model, not measurements: $I(t)=e^{-t/(4\,\mathrm{ns})}$ for $t\geq0$ and zero otherwise; $R$ is a normalized <Gaussian function> of <standard deviation> $0.6\,\mathrm{ns}$ centred at zero; the background is $B=0.025$ in the common signal units. The left panel compares intrinsic and convolved traces; the right shows how an unremoved background flattens the tail on a logarithmic scale. Negative delays come from the stipulated timing response about the pulse reference, not emission before excitation. Only the instrument-response curve is rescaled to unit peak for display.}
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