Principles of Fluorescence Spectroscopy

184 FREQUENCY-DOMAIN LIFETIME MEASUREMENTS
Figure 5.36. Lifetime χ R 2 surfaces for the three component mixtures
of In, AA, and 2-AP; dashed, 380 nm; solid, global fit at 360, 380,
400, 420, and 440 nm. From [116].
of χ R 2 are 1.71 and 1.81, respectively. At first glance it
seems that χ R 2 has increased for the three-exponential fit.
However, the increase in χ R 2 for the three-decay-time model
is a result of the larger number of variable parameters and
the smaller number of degrees of freedom. The value of χ 2 ,
which is the sum of the squared deviations, for these three
fits is 2006, 59.7, and 59.6, for the one-, two-, and threedecay-time
fits. Hence, the fit is not worse for the threedecay-time
fit, but is essentially equivalent to the twodecay-time
model.
Samples such as this three-component mixture are difficult
to analyze. In this case we know there are three decay
times, and the three decay times are correctly determined
by the analysis. However, obtaining the correct values
required that the starting parameter values are close to the
correct values. Otherwise, the program stopped at incorrect
values, apparently trapped in local χ R 2 minima. Additionally,
the χ R 2 surface is almost independent of lifetime, as
shown in Figure 5.36 for the data measured at 380 nm
(dashed). Without prior knowledge of the presence of three
decay times, it would be difficult to know whether to accept
the two or three decay time fit.
At this point in the analysis there is little reason for
proceeding further. If the information is not present in the
data, no amount of analysis will create new information.
One can either add new experimental data, or add information
by restricting parameters based on separate knowledge
about the sample. If one or more of the lifetimes are known,
these can be held constant during the least-square fit. Similarly,
one of the amplitudes could be fixed. However, the
best approach is to add new data and perform a global
analysis.
Figure 5.37. Frequency-domain intensity decays of the three-component
mixture observed at 360, 380, 400, 420, and 440 nm. The lines
are for the best fit to three global decay times and non-global amplitudes.
The values of χ R 2 for the one-, two-, and three-decay-time fit
are 109.8, 2.3, and 1.1, respectively. From [116].
The emission from the three-component mixture was
measured at five wavelengths: 360, 380, 400, 420 and 440
nm (Figure 5.37). At each wavelength each fluorophore displays
the same decay time, but a different fractional amplitude
based on its emission spectrum (Figure 5.38). Because
of the different amplitudes at each wavelength, the frequency
responses are wavelength dependent (Figure 5.37). The
Figure 5.38. Emission spectra of the three-component mixture of
indole, anthranilic acid, and 2-aminopurine. Also shown are the fractional
intensities recovered from global analysis of the frequencydomain
data. From [116].