Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 161
Figure 5.6. Relationship between the decay time and the useful range
of light modulation frequencies.
represented by the solid lines, with the expected decay
times (2.5 and 10 ns) and fractional intensities (f 1 = f 2 = 0.5)
being recovered from the least-squares analysis.
The range of modulation frequencies needed to recover
the intensity decay depends on the lifetimes. The useful
modulation frequencies are those where the phase angle is
frequency dependent, and there is still measurable modulation
(Figure 5.6). Most fluorophores display lifetimes near
10 ns, so that modulation frequencies are typically near
2–200 MHz. If the decay time is near 100 ps, higher modulation
frequencies near 2 GHz are needed. For longer
decay times of 1 to 10 :s the modulation frequencies can
range from 10 kHz to 1 MHz. As the modulation frequency
increases, the modulation of the emission decreases. Hence
it becomes more difficult to measure the phase angles as
they approach 90E.
5.1.1. Least-Squares Analysis of
Frequency-Domain Intensity Decays
The procedures used to analyze the frequency-domain data
are analogous to those used for TCSPC. The frequency-
domain data are usually analyzed by the method of nonlinear
least squares. 10–13 The measured data are compared with
values predicted from a model, and the parameters of the
model are varied to yield the minimum deviations from the
data. The phase and modulation values can be predicted for
any decay law. This is accomplished using sine and cosine
transforms of the intensity decay I(t):
N ω
D ω
(5.5)
(5.6)
where ω is the circular modulation frequency (2π times the
modulation frequency in Hz). The denominator J =
normalizes the expression for the total intensity
of the sample. These expressions are valued for any intensity
decay law, whether the decay is multi-exponential or
non-exponential. Non-exponential decay laws can be transformed
numerically. For a sum of exponentials the transforms
are12–13 ∞
I(t) dt
0
N ω ∑ i
D ω ∑ i
∞
0
∞
0
I(t) sin ωt dt
∞
0
I(t) cos ωt dt
∞
0
α iωτ 2 i
(1 ω 2 τ 2 i ) / ∑ i
α iτ i
I(t) dt
I(t) dt
(1 ω 2 τ 2 i ) / ∑ i
(5.7)
(5.8)
For a multi-exponential decay J = E i α i τ i , which is proportional
to the steady-state intensity of the sample. Because of
this normalization factor one can always fix one of the
amplitudes (α i or f i ) in the analysis of frequency-domain
data. The calculated frequency-dependent values of the
phase angle (φ cω ) and the demodulation (m cω ) are given by
tan φ cω N ω/D ω
m cω (N 2 ω D 2 ω) 1/2
α iτ i
α iτ i
(5.9)
(5.10)
In the least-squares analysis the parameters (α i and τ i ) are
varied to yield the best fit between the data and the calcu