## 5 Experiments and Results  
An experiment was used to verify the ability of the proposed method to obtain the 3D PSF. A standard 150 nm diameter fluorescent nanoparticle (Fluorescence Polystyrene, excitation peak: 540 nm; emission peak: 580 nm) was fixed in the cell of the proposed PSF measurement setup (Figure 2). The objective lens' nominal magnification was 60X (apochromatism objective 0.75 NA) using a commercial microscope, which needed to be calibrated.  
The entire 3D PSF evaluation procedure (Section 4) was run to obtain the results of Figure 7 and Table 1. The focal object plane was used as the original position along the Z-axis, and it was demonstrated that the obtained PSF fitting line closely matched the original data (Figure 7). The cubic polynomial coefficients for fitting the 3D PSF were obtained as detailed in Section 3.3 (Table 1). For the consistency of presentation, here, the mentioned paraxial region was actually its corresponding ideal imaging region on the image plane. The intensity distribution of the nanoparticle's image and its PSF intensity distribution on the original Z-axis position are shown in Figure 8.  
Figure 7: The obtained 3D PSF (**a**), magnification (**b**), and paraxial region (**c**).  
In order to test the results of the 3D PSF evaluation, 100 and 200 nm diameter nanoparticles were measured on the three perpendicular object planes (the Z-axis positions of 0.5, 0, and \(-0.5\)\(\upmu\)m) along the Z-axis (the optical axis of the commercial microscope), as well as the three obtained PSFs and magnifications corresponding to the three perpendicular object planes. We compared the FWHMs of the measured and calculated intensity distributions of the 100 and 200 nm diameter nanoparticles (the calculated intensity distributions could be determined by Equation (5)). The FWHM differences between the measured FWHMs and the calculated FWHMs on the three perpendicular object planes demonstrated that the differences clearly defined the paraxial region (Figure 9). The plane position was the distance from the center of the paraxial region. The FWHM differences in the paraxial region (the red line) were almost all below 5 nm (Figure 9). The FWHM differences became bigger with an increase in the deviation from the paraxial region (the black line; Figure 9). This verified that the obtained 3D PSF, magnification, and paraxial region (Figure 7) were correct and also demonstrated the validity of the proposed PSF evaluation method to calibrate the object plane that was perpendicular to the optical axis.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & \(a\) & \(b\) & \(c\) & \(\tau_{Io}^{2}\) \\ \hline Z-axis positive direction & \(8.11\times 10^{-30}\) & \(-9.86\times 10^{-30}\) & \(-1.11\times 10^{-16}\) & \(2.08\times 10^{-14}\) \\ Z-axis negative direction & \(1.14\times 10^{-28}\) & \(-1.64\times 10^{-28}\) & \(1.12\times 10^{-16}\) & \(2.08\times 10^{-14}\) \\ \hline \end{tabular}
\end{table}
Table 1: Cubic polynomial coefficients of fitting \(\mathbf{Z}_{P}\) and \(\tau_{\mathbf{IZ}}\).  
Figure 8: Intensity distribution of the nanoparticle’s image (**a**) and its PSF’s intensity distribution in the X–Y (**b**) and X–Z (or Y–Z) (**c**) planes.