Principles of Fluorescence Spectroscopy

194 FREQUENCY-DOMAIN LIFETIME MEASUREMENTS
Using these identities yields
∞
0
exp (t’/τ) cos ω(t t’) dt’
τ
√1 ω2τ2 { cos ωt
√1 ω2 ωt sin ωt
2 τ √1 ω2τ2 }
τ
√1 ω2 cos (ωt φ)
2 τ
Equation 5.58 was obtained using
cos φ (1 ω 2 τ 2 ) 1/2
tan φ
sin φ
cos φ
Hence, the time dependent intensity is given by
b
I(t) I0τ { a
√1 ω2 cos (ωt φ)}
2 τ
(5.56)
(5.57)
(5.58)
(5.59)
(5.60)
(5.61)
This expression shows that the emission is demodulated by
a factor (1 + ω 2 τ 2 ) –1/2 relative to the excitation and that the
emission is delayed by an angle φ relative to the excitation.
5.11.2. Cross-Correlation Detection
The use of cross-correlation detection transforms the highfrequency
emission to a low-frequency signal while preserving
the meaning of the phase and modulation values.
This can be seen by considering the nature of the signals.
The high-frequency time-dependent intensity is given by
I(t) I 01 m cos (ωt φ)
(5.62)
This signal is multiplied by the sinusoidal gain modulation
of the detector: 138
G(t) G 01 m c cos (ω ct φ c)
(5.63)
where G 0 is the average value of the function, and m c , ω c ,
and φ c are the modulation, frequency, and phase of the
cross-correlation signal. Multiplication of eqs. 5.62 to 5.63
yields
S(t) N 0G 01 m cos(ωt φ) m c cos(ω ct φ c)
mm c cos(ωt φ) cos(ω ct φ c)
Using trigonometric identities the last term becomes
mm c
2 cos(∆ωt ∆φ) cos(ω ct ωt ∆φ)
(5.64)
(5.65)
where ∆ω = ω c – ω and ∆φ = φ c – φ. The frequencies ω c and
ω typically differ by only a small amount. Hence eq. 5.64
contains a constant term plus terms with frequencies, ω, ω c ,
ω + ω c , and ∆ω. The ∆ω term contains the phase and modulation
information. In the electronic filtering process the
constant term and terms in ω, ω c , and ω + ω c all contribute
to average intensity, and the term ∆ω determines the phase
and amplitude of the low-frequency modulated emission.
The presence of the phase and modulation information in
the low-frequency signal can also be seen by integration of
eqs. 5.62 and 5.63 over one measurement cycle. 41
5.12. PHASE-SENSITIVE EMISSION SPECTRA
The frequency-domain method also allows several other
types of measurement that can be useful in special circumstances.
One method is measurement of phase-sensitive
intensities and/or emission spectra. 140–143 In phase-sensitive
detection of fluorescence (PSDF) the measurements are
somewhat different than in frequency-domain fluorometers.
In PSDF the emission from the sample is analyzed with a
phase-sensitive detector, typically a lock-in amplifier. This
measurement procedure selectively attenuates the signal
from individual fluorophores on the basis of their fluorescence
lifetimes, or more precisely, their phase angles relative
to the phase of the detector. PSDF allows the emission
from any one species to be suppressed, or, more precisely,
the emission with any desired angle to be suppressed. Phase
suppression is accomplished when the phase of the detector
is 90E shifted from the phase angle of the emission. Then,
the resulting phase-sensitive emission spectrum represents
only the emission from the remaining fluorophores. For a
two-component mixture, suppression of the emission from