Principles of Fluorescence Spectroscopy

100 TIME-DOMAIN LIFETIME MEASUREMENTS
Figure 4.3. Comparison of time-domain (left) and frequency-domain (right) decay time measurements of N-acetyl-L-tryptophanamide (NATA). L(t k )
is the instrument response function.
These expressions can be used to calculate the phase
(τ φ ) and modulation (τ m ) lifetimes for the curves shown in
Figure 4.2 (Problem 4.1). If the intensity decay is a single
exponential, then eqs. 4.5 and 4.6 yield the correct lifetime.
If the intensity decay is multi- or non-exponential, then eqs.
4.5 and 4.6 yield apparent lifetimes that represent a complex
weighted average of the decay components.
4.1.3. Examples of Time-Domain and
Frequency-Domain Lifetimes
It is useful to understand the appearance of the time-domain
(TD) and the frequency-domain (FD) data. TD and FD data
are shown for the tryptophan derivative N-acetyl-L-tryptophanamide
(Figure 4.3). This tryptophan derivative (NATA)
is known to display a single exponential decay (Chapter
17). In the time domain (left) the data are presented as log
counts versus time. The data are presented as photon counts
because most such measurements are performed by singlephoton
counting. The plot of the log intensity versus time
for NATA is linear, which indicates the decay is a single
exponential. The noisy curve marked L(t k ) is the instrument
response function (IRF), which depends on the shape of the
excitation pulse and how this pulse is detected by the instrument.
This IRF is clearly not a δ-function, and much of the
art of lifetime measurements is accounting for this nonideal
response in analyzing the data.
Analysis of the time domain is accomplished mostly by
nonlinear least squares. 1–2 In this method one finds the lifetime
that results in the best fit between the measured data
and the data calculated for the assumed lifetime. Although
not separately visible in Figure 4.3 (left), the calculated
intensity decay for τ = 5.15 ns overlaps precisely with the
number of photons counted in each channel. The lower
panel of Figure 4.3 (left) shows the deviations between the
measured and calculated data, weighted by the standard
deviations of each measurement. For a good fit the deviations
are random, indicating the only source of difference is
the random error in the data.
Frequency-domain data for the same NATA sample are
shown in Figure 4.3 (right). The phase and modulation are
measured over a range of light modulation frequencies. As
the modulation frequency is increased the phase angle
increases from 0 to 90E, and the modulation decreases from
1 (100%) to 0 (0%). As for the time-domain data, the frequency-domain
data are also analyzed by nonlinear least
squares. The dots represent the data, and the solid line represents
the best fit with a single lifetime of 5.09 ns. As for
the TD data, the goodness-of-fit is judged by the differences
(deviations) between the data and the calculated curves. For
the FD data there are two observables—phase and modulation—so
there are two sets of deviations (lower panel). The
randomness of the deviations indicates that a single lifetime
is adequate to explain the data.