Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 163
each displaying a different emission spectrum. In this case
the intensity decay at each wavelength (λ) is given by
(5.12)
where the values of α i (λ) represent the relative contribution
of the ith fluorophore at wavelength λ. The frequency
response is typically measured at several wavelengths
resulting in wavelength-dependent values of the phase
angle φ ω (λ) and the modulation m ω (λ). In this case the values
of N ω λ and D ω λ depend on the observation wavelength,
and are given by
N λ ω ∑ i
D λ ω ∑ i
(5.13)
(5.14)
The wavelength-dependent data sets can be used in a global
minimization of χ R 2:
χ 2 R 1
ν ∑ λ,ω
I(λ,t) ∑ αi(λ) e
i1
t/τi 1
ν ∑ λ,ω
α i(λ)ωτ 2 i
1 ω 2 τ 2 i / ∑ i
α i(λ)ωτ i
1 ω 2 τ 2 i / ∑ i
[ φ ω(λ) φ cω(λ)
δφ
α i (λ)τ i
α i (λ)τ i
[ m ω(λ) m cω(λ)
δm
(5.15)
where the sum now extends over the frequencies (ω) and
wavelengths (λ). Typically the values of τ i are assumed to
be independent of wavelength and are thus the global
parameters. The values of α i (λ) are usually different for
each data set, that is, are non-global parameters. The data
are normalized at each wavelength, allowing one of the
amplitudes at each wavelength to be fixed in the analysis.
It is important to estimate the range of parameter values
that are consistent with the data. As for TCSPC, the
asymptotic standard errors (ASEs) recovered from leastsquare
analysis do not provide a true estimate of the uncertainty,
but provide a significant underestimation of the
range of parameter values which is consistent with the data.
This effect is due to correlation between the parameters,
which is not considered in calculation of the asymptotic
standard errors. Algorithms are available to estimate the
upper and lower bounds of a parameter based on the extent
] 2
] 2
of correlation. 14–16 If the analysis is at the limits of resolution
we prefer to examine the χ R 2 surfaces. This is accomplished
just as for the time-domain data. Each parameter
value is varied around its best fit value, and the value of χ R 2
is minimized by adjustment of the remaining parameters.
The upper and lower limits for a parameter are taken as
those which result in an elevation of the F P value expected
for one standard deviation (P = 0.32) and the number of
degrees of freedom (Section 5.7.1).
5.2. FREQUENCY-DOMAIN INSTRUMENTATION
5.2.1. History of Phase-Modulation
Fluorometers
The use of phase-modulation methods for measurements of
fluorescence lifetimes has a long history. 17 The first lifetime
measurements were performed by Gaviola in 1926 using a
phase fluorometer. 18 The first suggestion that phase angle
measurements could be used for measuring fluorescence
lifetimes appears to have been made even earlier in 1921. 19
The use of phase delays to measure short time intervals
appears to have been suggested even earlier, in 1899. 20
Hence, the use of phase shifts for timing of rapid processes
has been recognized for 100 years. Since the pioneering
measurements by Gaviola 18 a large number of phase-modulation
instruments have been described. These include an
instrument by Duschinky in 1933, 21 and an instrument of
somewhat more advanced design described by Szymonowski
22 , on which many of Jablonski's early measurements
were performed. Phase fluorometers have been described
by many research groups. 23–53 The first generally useful
design appeared in 1969. 41 This instrument used a Debye-
Sears ultrasonic modulator 42–43 to obtain intensity-modulated
light from an arc lamp light source. The use of the
Debye-Sears modulator has been replaced by electrooptic
modulators in current FD instruments. However, an important
feature of this instrument 41 is the use of cross-correlation
detection (Section 5.11.2). The use of this radio frequency
mixing method simplified measurement of the
phase angles and modulation values at high frequencies,
and allowed measurement of the phase angle and modulation
with relatively slow timing electronics.
The primary technical factor limiting the development
of frequency-domain fluorometers was the inability to
obtain intensity-modulated light over a range of modulation
frequencies. Debye-Sears modulators are limited to operating
at the frequency of the crystal, or multiples thereof.