Rapid Techniques for Correcting Nanosecond Kinetic Traces for Convolution Error

Abstract

In the analysis of data on fast transients, such as are taken in pulse radiolysis or flash photolysis, significant errors may be introduced by neglect of convolution of the chemical or photophysical transient with the excitation pulse shape and with the time response function of the detection apparatus. Due to the commutative and associative properties of the convolution integral, the observed signal S(t) in a linear system is given by

$$\text{S}(\text{t}) = \int\limits_o^t {\text{I}(\text{t}')\,\text{G}(\text{t - t}')\,\text{dt}' = \int\limits_\text{o}^\text{t} {\text{G}(\text{t}')} \,I(\text{t - t'})\,\text{dt}'}$$

(1)

where I(t) is the ideal signal that would be obtained with a pulse of negligible width and a detector of infinite bandwidth, and where G(t) is itself given by the convolution of the excitation pulse shape E(t) with the detector response function R(t):

$$\text{G}(\text{t}) = \int\limits_o^t {\text{E}(\text{t}')\,\text{R}(\text{t - t}')\,\text{dt}'}$$

(2)

G(t) is ordinarily measured directly by passing the excitation pulse (or a signal proportional to it) through the detector, although variations in the detector response function with wavelength may also have to be taken into account (1). The formalism applies equally to the photon counting technique and to single shot, analog experiments employing intense pulsed sources, photomultiplier tube detectors, and fast transient digitizers.