Principles of Fluorescence Spectroscopy

160 FREQUENCY-DOMAIN LIFETIME MEASUREMENTS
Figure 5.5. Simulated frequency-domain data for single- (top) and
double- (bottom) exponential decays. The phase angle increases and
the modulation decreases with increasing modulation frequency. The
datapoints indicate the simulated data. Top: The solid lines show the
best fits to a single decay time. Bottom: The dashed and solid lines
show the best single- and double-exponential fits, respectively.
ns long. In this case the molecules excited at the peak of the
intensity continue to emit during the entire modulation
cycle. This results in an averaging of the decay across the
peaks and valleys of the excitation. This averaging results in
the phase shift and decrease in modulation of the emission.
At an intermediate modulation frequency of 25 MHz some
averaging occurs, but to a lesser extent than at 250 MHz.
In FD measurements the phase angle and modulation
are measured over a wide range of frequencies. These data
are called the frequency response of the sample. The characteristic
features of the frequency response of a sample are
illustrated in Figure 5.5 (top) for a single exponential decay.
As the light modulation frequency is increased the phase
angle increases from 0 to 90E. At first glance the 90E phase
angle limit is counterintuitive. For a time delay of the type
available from an optical delay line, the phase shift can
exceed 90E and reach any arbitrary value. For a single exponential
or multi-exponential decay, the maximum phase
angle is 90E. Hence the phase angle displayed by any sample
is some fraction of 90E, independent of the modulation
frequency. Only under special circumstances can the phase
angle exceed 90E (Chapter 17).
The modulation of the emission also depends on the
modulation frequency of the incident light. As the frequency
increases the modulation decreases from 1.0 to 0. The
modulation of the emission is zero when the frequency is
much larger than the emission rate. In presenting frequency-domain
data, the modulation frequency on the x-axis
(Figure 5.5) is usually described in cycles/s (Hz or MHz).
The circular modulation frequency (ω = 2π x Hz) in radians/s
is used for calculations.
The shape of the frequency response is determined by
the number of decay times displayed by the sample. If the
decay is a single exponential (Figure 5.5, top), the frequency
response is simple. One can use the phase angle or modulation
at any frequency to calculate the lifetime. For a single-exponential
decay, the phase and modulation are related
to the decay time (τ) by
and
tan φ ω ωτ
m ω (1 ω 2 τ 2 ) 1/2
(5.3)
(5.4)
The derivation of eqs. 5.3 and 5.4 is given in Section 5.11.
For the 10-ns decay time, the phase shift at 20 MHz is
51.5E, and the emission is demodulated by a factor of 0.62
relative to the excitation. At a modulation frequency of 100
MHz the phase angle increases to 81E, and the modulation
decreases to 0.16. Most samples of interest display more
than one decay time. In this case the lifetimes calculated
from the value of φ ω or m ω , measured at a particular frequency,
are only apparent values and are the result of a
complex weighting of various components in the emission
(Section 5.10). For such samples it is necessary to measure
the phase and modulation values over the widest possible
range of modulation frequencies.
The frequency response has a different shape for a
multi-exponential decay (Figure 5.5, bottom). In this simulation
the assumed decay times are 2.5 and 10 ns. The shape
of the frequency response is used to determine the form of
the intensity decay. This is generally accomplished using
nonlinear least-squares procedures. 10–13 The fitting procedure
is illustrated by the solid and dashed lines in Figure
5.5. For the single-exponential decays shown in the top half
of the figure, it is possible to obtain a good match between
the data (!) and the curves calculated using the singleexponential
model (solid line). For a double-exponential
decay, as shown in the bottom half of the figure, the data
cannot be matched using a single-decay time fit, represented
by the dashed lines. However, the complex frequency
response is accounted for by the double-exponential model,