Principles of Fluorescence Spectroscopy

106 TIME-DOMAIN LIFETIME MEASUREMENTS
the instrument. The width of the IRF is due to both the characteristics
of the detector and the timing electronics. The
IRF in Figure 4.9 is quite narrow, about 60 ps wide, measured
as the full width of the half maximum intensity
(FWHM). The use of a logarithmic intensity scale exaggerates
the low-intensity regions of the profile. There is an
afterpulse about 2 ns after the main peak. Afterpulses are
observed with many PMTs. The instrument response function
shown in Figure 4.9 is rather good, and some PMTs
give far less ideal profiles. For instance, the profile in Figure
4.3 was measured with an end-on linear-focused PMT,
for which the afterpulses and long time tail are more significant.
However, even in this case (Figure 4.3) the number of
photons in the peak of the afterpulse is only about 0.05% of
the counts in the peak channel.
The measured intensity decay N(t k ) is shown as a histogram
of dots. The height of the dots on the y-axis represents
the number of photons that were detected within the
timing interval t k to t k + ∆t, where ∆t is the width of the timing
channel. In this case the peak channel, with the largest
number of counts, has recorded approximately 3000 photons.
On the log scale the decay is seen to be a straight line
suggesting a single decay time.
The last curve is the calculated data N c (t k ), which is
usually called the fitted function. This curve (solid) represents
a convolution of the IRF with the impulse response
function, which is the intensity decay law. The fitted function
is the time profile expected for a given intensity decay
when one considers the form of the IRF. The details of calculating
the convolution are described in the next section.
For a single exponential decay the lifetime is the value of τ
that provides the best match between the measured data
N(t k ) and the calculated time-dependent intensities N c (t k ).
For a multi-exponential decay (eq. 4.2) the analysis yields
the values of α i and τ i that are most consistent with the data.
4.3.3. Convolution Integral
It is important to understand why the measured intensity
decay is a convolution with the lamp function. The intensity
decay law or impulse response function I(t) is what
would be observed with δ-function excitation and a δ-function
for the instrument response. Equations 4.2, 4.12, and
4.13 are examples of impulse-response functions. Unfortunately,
it is not possible to directly measure the impulse
response function. Most instrument response functions are
0.5 to 2 ns wide. However, we can imagine the excitation
pulse to be a series of δ-functions with different amplitudes.
Figure 4.10. Convolution of an impulse response function I(t) with
three excitation pulses (top) or with a lamp profile L(t k ) to yield the
measured data N(t k ).
Each of these δ-functions excites an impulse response from
the sample, with an intensity proportional to the height of
the δ-function (Figure 4.10, top). The measured function
N(t k ) is the sum of all these exponential decays, starting
with different amplitudes and different times.
Mathematically, the concept of convolution can be
expressed as follows. 16 Each δ-function excitation is
assumed to excite an impulse response at time t k :
I k(t) L(t k) I(t t k)∆t(t t k)
(4.14)
The amplitude of the impulse response function excited at
time t k is proportional to the excitation intensity L(t k ) occurring
at the same time. The term (t – t k ) appears because the
impulse response is started at t = t k , and it is understood that