## 4 Results and Validation  
### Effect of Numerical Aperture (NA) on FCS Measurements  
It is expected that the FCS performance will depend on the choice of objective lens. In order to employ FCS in combination with imaging MPM, it would be preferable to use objective lenses with large field-of-view and long working distances but at the cost of reducing the NA. Thus, in this study the performance of two different objective lenses were investigated under similar conditions and using the same experimental system. Figure 3 shows the effect of numerical aperture on FCS measurements obtained from two different microscope objectives, 63\(\times\) (NA1.2, working distance 0.28 mm) and 40\(\times\) (NA 0.8, working distance 1.8 mm) for measurements of a Rhodamine B solution at 10 nM concentration. It is evident from the figure that FCS curves can be acquired using both the 63\(\times\) and 40\(\times\) objective lenses; however, the amplitude of the correlation function is higher for the 63\(\times\) objective, i.e., \(\mathrm{G(0)=0.6}\); compared to \(\mathrm{G(0)=0.035}\) for the 40\(\times\) objective. This is expected as a higher NA will generate a smaller excitation volume, thereby comprising less molecules giving rise to a higher \(\mathrm{G(0)}\) value according to Equation (3) in theory section. This consequently means that the versatility for utilizing the long working distance 40\(\times\) objective lens is lower as autocorrelation amplitude will approach 0 if concentration increase further.  
Included in the figure, autocorrelation curves extracted directly from the SPCM64 software using so called multi-tau binning are presented (Figure 3a,d) and compared with the linear-tau algorithm with fixed binning in logarithmic (Figure 3b,e) and linear scale (Figure 3c,f). The multi-tau algorithm was found to distort the data fitting compared to using linear-tau binning. Therefore, linear-tau binning was preferred for fitting the data (see further analysis subsequent section) while multi-tau was used during acquisition for fast display and assessment of data quality.  
The results obtained from the data fitting show a shift in acquired diffusion times \(\tau_{\mathrm{D}}\) for different objective lenses. It is in the time range of 0.05 \(\pm\) 0.01 ms (average from 5 measurements) for 63\(\times\) objective lens and 15 \(\pm\) 6 ms (average from 5 measurements) for the 40\(\times\) objective lens. According to the theoretical size of the 2 PE volume, as described by others [2], the axial radius is 0.17 \(\upmu\)m for the 63\(\times\) objective lens (NA 1.2, \(\lambda\) = 800 nm) compared to 0.23 \(\upmu\)m for the 40\(\times\) lens (NA 0.8, \(\lambda\) = 800 nm). Through a theoretical argument (see Supplementary Material) based on previous published theory on the relation between the diffusion constant and diffusion time in the context of size of detection volume [22], the expected shift in diffusion time should be a factor 2 for the lower NA lens. However, in this case the shift is rather in the order of a factor 300. Because of this large discrepancy, the observed shift in observed diffusion time acquired for the two different objective lenses cannot simply be related to the size of focal volume and a further interpretation of this relationship will follow.