Fourier Transform Analysis

Abstract

The decay curves observed on excitation of a fluorescent system by a short light pulse are distorted as a result of both the finite duration of the excitation and the limited frequency response of the detection system. If both the fluorescent system and the apparatus behave linearly, the decay curve observed, F(t) is given by a convolution integral:

$$F\left( t \right) = {E^ * }\left( t \right) * f\left( t \right) * H\left( t \right)$$

(1)

where E^*(t) is the shape of the excitation light pulse, f(t) the (impulse) response function of the fluorescent system and H(t) the apparatus response function. Similarly the observed shape E(t) of the excitation light pulse E^*(t) is given by:

$$E\left( t \right) = {E^ * }\left( t \right) * H\left( t \right)$$

(2)

Since convolution is commutative, the observed fluorescence decay F(t) can be considered simply as the convolution of an effective apparatus function E(t) with the true fluorescence evolution, f(t).

$$F\left( t \right) = E\left( t \right) * f\left( t \right)$$

(3)

To observe fluorescence decay curves which are essentially undistorted a very narrow effective apparatus function E(t) is necessary. For the accurate determination of lifetimes which are on the order of or shorter than E(t), convolution or deconvolution techniques must be applied.