2.4 Time-correlated single photon counting
2.4.1 Working principle
Time-correlated single photon counting (TCSPC) is a well-established and com- mon technique for fluorescence lifetimes measurements. It is a time-domain technique (as opposed to a frequency-domain technique). In TCSPC, the sample is excited by a pulsed light source, and the PL decay profile is re- constructed by registering the arrival times of many single photon events.
Measuring the time-dependent intensity profile requires the excitation of the sample by a short flash of light. In principle, one could attempt to measure the decay profile from a single excitation-emission cycle, but this is hindered by some practical problems. Firstly, ordinary electronic recorders often do not possess the required temporal resolution to resolve the shape of the decay profile of fast emitters (e.g. a fluorescent compound with a lifetime of 500 ps would require signal sampling with a 10-ps time step)[73]. Secondly, the PL signal may be too weak to create an analog voltage that represents the incoming photon flux. This would be especially so in the case of single molecule spectroscopy. TCSPC overcomes these two problems by extending the data collection over multiple excitation-emission cycles. The PL decay profile is reconstructed from the precise registration of arrival times of single photons collected over multiple cycles.
The experimental setup used in TCSPC is shown in figure 2.4. TCSPC requires a pulsed light source, for instance a pumped picosecond dye laser or Ti:sapphire laser. Since the turn of the century, also solid-state light sources such as pulsed laser diodes and pulsed light emitting diodes have found their introduction in the technique [74]. The excitation light pulse is split into two components, one to excite the sample and one to serve as a reference for the timing mechanism. After their detection, the arrival times of the excitation and emission pulses are accurately determined with a common fraction denominator (CFD).
At the heart of the timing mechanism lies a time-to-amplitude converter (TAC), which generates a voltage that increases linearly with time on the nano- second scale. In the normal TCSPC configuration, the TAC voltage ramp is started by the arrival of the excitation pulse. The TAC is occupied either until a STOP signal is generated by the first detection of a photon emitted by the sample, or until the TAC completes its sweep. However, running data collection in the normal configuration results in substantial deadtime, i.e. time during which the TAC is in operation and unable to respond to another signal [75]. The deadtime is minimized in the reverse configuration, in which the first photon from the sample serves as a START signal, while the excitation pulse serves as the STOP signal.
After amplification, the TAC voltage is digitized by an analog-to-digital con- verter (ADC). The resulting binary value is sent to the multi-channel analyser
Ş Ş ſƀ Ş ſƀ ŵ Ŝ
Figure 2.4 – Block diagram of a TCSPC system (reverse configuration).
(MCA) and provides the address of the channel (memory location) of which the content has to be increased by one. The channels are like the bins of a histogram, each one corresponding to a certain time interval. By the repetitive detection of single photons, a histogram of the emission decay is constructed.
Because only the first photon is observed in TCSPC, it is critical that the photon detection rate remains low (typically 1 photon per 100 excitation pulses) [74]. If the photon count rate is too high, the histogram will be biased towards shorter times and it will not represent the actual PL decay.
Figure 2.5 shows a typical example of TCSPC data. Three functions are shown: the measured decay N (ti), the instrument response function (IRF)
L(ti), and the calculated decay Nc(ti). These three functions exist only at dis-
crete times ti, because the photon arrival times are binned over a finite number
of channels by the MCA. The IRF represents the overall timing precision of the TCSPC system. For an ideal system the IRF is infinitely narrow. In reality, the IRF is broadened by the duration of the excitation light pulse, the timing accuracy of photon registration by the detectors, etc. The IRF can be deter- mined by tuning the emission monochromator to the excitation wavelength and measuring scattered excitation light (e.g. from a colloidal solution).
It is important to understand that the measured intensity decay N (ti) is
a convolution of the real intensity decay I(t) and the instrument response function L(ti). The measured intensity decay therefore takes on the shape of
the IRF. Mathematically, the convolution of two functions f and g, supported on [0, +∞[, is given by
2.4. TIME-CORRELATED SINGLE PHOTON COUNTING| 26 0 1 0 2 0 3 0 4 0 5 0 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 0 1 0 2 0 3 0 4 0 5 0 - 5 0 5 c a l c u l a t e d d e c a y N c( ti) N ( ti) C o u n ts / c h a n n e l L ( t i) d t i m e ( n s )
Figure 2.5 – TCSPC data for QDs dispersed in water. The top panel shows
the instrument response function (blue points), the PL decay curve of the QD sample (red points), and a three-component exponential fit to the PL decay curve (black line). The lower panel shows the fit residuals.
(f ⊗ g) (t) =
t
ˆ
0
f (t0)g(t − t0)dt0. (2.13) Applying this to our case, we have
N (t) = (L ⊗ I) (t) =
t
ˆ
0
L(t0)I(t − t0)dt0. (2.14) Because N (ti) and L(ti) exist only at discrete times, this is more appropriately
written as N (ti) = i X k=0 L(tk)I(ti− tk)∆t, (2.15)
with ∆t the channel width.
The task is now to determine the intensity decay I(t) that best matches the experimental data. One method to do this is by non-linear least squares (NLLS) analysis. In fact, NLLS assumes a model decay I(t) that is believed to describe the data. Let’s say that we expect our data to obey first order kinetics. Hence the decay is given by
I(t) = Ae−t/τ. (2.16)
The goal is to test whether this model is consistent with the data and if so, to obtain the set of parameter values that provides the best match between the measured decay N (ti) and the calculated decay Nc(ti). The problem is
solved by iterative reconvolution. Given a set of start values for the floating parameters (A and τ in this case), I(t) is convoluted with the IRF L(t), and the calculated decay Nc(ti) is compared with the measured decay N (ti). The
parameters are adjusted iteratively in order to minimize the goodness-of-fit parameter χ2, which is given by
χ2= n X i=1 [N (ti) − Nc(ti)] 2 N (ti) . (2.17)
Usually, the reduced value χ2
R is reported, because χ2depends on the num-
ber of data points.
χ2R= χ
2
n − p, (2.18)
where n is the number of data points and p the number of floating pa- rameters. The goodness-of-fit should also be judged visually by inspecting the residuals R(ti)=N (ti)-Nc(ti) (figure 2.5, lower panel). The residuals should be