Principles of Fluorescence Spectroscopy

PRINCIPLES OF FLUORESCENCE SPECTROSCOPY 801
where q is the quantum efficiency for detection of the emitted
photons, σ is the cross-section for absorption, and Q is
the quantum yield for emission of the fluorophore. The
brightness is the number of photons per second for a single
fluorophore observed for a given set of optical conditions.
Unlike the quantum yield, the brightness is not a molecular
property of the fluorophore, but depends on the precise
optical conditions including the light intensity, light collection
efficiency of the instrument, and the counting efficiency
of the detector. The measured intensity from the sample,
typically in kHz, is given by the integral of the fluorophore
concentration over the observed volume:
F(t) B CEF(r)I(r)C(r,t)dV
(24.7)
In this expression CEF(r) is the collection efficiency function
of the instrument as a function of position (r). The integral
extends over the entire observed space. The position r
is more properly described as a vector (r) but for simplicity
we assume r is a vector. The excited fluorophores will be
distributed in a three-dimensional volume, in the x-y plane
and along the optical z-axis. I(r) is the excitation intensity
at each position r, and C(r,t) is the distribution of fluorophores.
Equation 24.7 looks complex, but it has a simple
meaning. The intensity depends on the concentration and
spatial distribution of the excitation and detection efficiency,
and the brightness of the fluorophores. The observed
intensity also depends on the excited and observed volumes,
which is accounted for by the integral. It is usually
not necessary to consider these factors separately, so that
the instrument is described as having a detection profile:
p(r) CEF(r)I(r) MDE(r)
(24.8)
which is also referred to as the molecular detection efficiency
MDE(r).
Figure 24.4 shows a typical FCS instrument. A laser is
focused on the sample. The emission is selected with a
dichroic filter. The out-of-focus light is rejected with a pinhole,
which is typically large enough to pass all light from
a region slightly larger than the illuminated spot. 10 The
intensity profile of the focused laser is assumed to be
Gaussian. For this configuration, the brightness profile can
be approximated by a three-dimensional Gaussian (Figure
24.5):
p(r) I 0 exp2(x 2 y 2 )/s 2 exp(2z 2 /u 2 )
(24.9)
Figure 24.4. Typical instrumentation for FCS. Revised from [17].
The surface of the volume is not sharply defined. The radius
s and half-length u refer to distances at which the profile
decreases to e –2 of its maximal value I 0 .
Equation 24.7 gives the time-dependent intensity. We
now need to calculate the autocorrelation function due to
concentration fluctuations throughout the observed volume.
This function is more complex than for a fluorophore diffusing
in and out of a volume with a sharp boundary (Figure
24.1). As the fluorophore diffuses it enters regions
where the count rate per fluorophore is higher or lower
Figure 24.5. Ellipsoidal observed volume with focused single photon
excitation and confocal detection.