## 2 Concepts and Theory
### Autocorrelation Using TCSPC Data  
Time correlated single photon counting (TCSPC) is based on recording the time of arrival of single photons synchronized by a pulsed trigger. In connection to spectroscopic measurements, TCSPC is generally implemented for recording the fluorescence generated after exposure to laser pulses shorter than the lifetimes of the excited states. Since MPM employ ultrashort (~100 fs) laser pulses at high repetition rate (typically 80 MHz), the combination of MPM and TCSPC is practical [32; 33]. In TCSPC, the photon arrival times are recorded with respect to the excitation pulse (micro time) and start of the experiment (macro time). The main application of TCSPS in combination with MPM is fluorescence lifetime imaging (FLIM). It is achieved by constructing the histogram for photon arrival times [29; 30]. However, the TCSPC data in its raw form also convey information about the fluorescence fluctuations acquired from the excitation volume being related to the thermodynamic fluctuations of fluorophores--corresponding to the fundamental prerequisite in order to perform FCS.  
In TCSPC, fluorescence correlation curves are obtained by correlating macro time of the photons [31; 33]. The clock period is defined as the time between each trigger pulse and clock period is longer than the dead time of the detectors. Therefore, only one photon is detected per clock period. Since the resolution of macro time is typically in the order of 10-100 ns, the generated count vs. time vector N(t) will have a binary format with most of its entries being 0. This means that autocorrelating the data using discrete autocorrelation on this timescale, i.e.,  
\[\mathrm{G}(\tau)=\sum\mathrm{N}(\mathrm{t})\mathrm{N}(\mathrm{t}+\tau) \tag{4}\]  
will generate an autocorrelation function \(\mathrm{G}(\tau)\) primarily being 0. In order for \(\mathrm{G}(\tau)\) to carry relevant information which can be related to the molecular diffusion the count vs. time vector N(t) has to be binned to a relevant timescale. Most commonly multi-tau algorithms are implemented where the time binning incrementally increases with \(\tau\). The multi-tau algorithms are generally fast [34] and traditionally implemented in most of the hardware correlators. Therefore, many reports on FCS employ and present the data using the multi-tau format [14; 16].  
For the purpose of this study the multi-tau approach was deemed to generate distortion to the autocorrelation curve and therefore not utilized more than as instant guidance during data acquisition. Instead, we implement a method based on linear time binning for data analysis. Therefore, an algorithm that fixes the time binning for the whole dataset was implemented. In fact, the so called linear tau has been suggested preferably in literature [35; 36]. It should be noted that when increasing the binning time, the count vs. time vector N(t) is no longer binary. Instead, the number of photons counted for each time element increases above one and the values fluctuate around a mean value. Therefore, the count vector cannot directly be subject to autocorrelation. Going back to the phenomenological description of the method [13], the principle is to record the signal fluctuation from the thermodynamical mean. Therefore, the autocorrelation function for time-binned data is instead defined as [32]:  
\[\mathrm{G}(\tau)=\sum\delta\mathrm{N}(\mathrm{t})\delta\mathrm{N}(\mathrm{t}+\tau) \tag{5}\]  
where  
\[\delta\mathrm{N}(\mathrm{t})=\mathrm{N}(\mathrm{t})-\langle\mathrm{N}(\mathrm{ t})\rangle \tag{6}\]