## 4 Experiments and Results  
Except for the coaxiality and magnification calibration in Section 3, the operations to implement the d\({}_{\mathrm{var}}\) labeling process and to obtain the slope of coplanar displacement measurement error were also needed before the proposed micro-imaging system measured the 3D trajectory of the adopted fluorescent nanoparticle. In the operations, it was necessary to confirm the Z-axis and coplanar displacements of the adopted nanoparticle probe, which was fixed between the cover glass and glass slide. In Figure 8, the multidimensional displacement stage provided precise Z-axis and coplanar displacements for the operations. Three fluorescent nanoparticles of diameters 50 nm, 100 nm, and 200 nm (fluorescence polystyrene, excitation peak: 540 nm, emission peak: 580 nm) were adopted. After being fixed between the cover glass and glass slide with the fixative (refractive index: 1.392), the d\({}_{\mathrm{var}}\) labeling process was implemented to build mapping relationships between the X- and Y-axis width ratios and the relative position along the Z-axis for the three fluorescent nanoparticles at three magnifications (M1: \(-A_{\mathrm{X}}=-112.31\) and \(-A_{\mathrm{Y}}=-67.82\); M2: \(-A_{\mathrm{X}}=-53.67\) and \(-A_{\mathrm{Y}}=-39.07\); M3: \(-A_{\mathrm{X}}=-37.85\) and \(-A_{\mathrm{Y}}=-30.04\)), resulting in nine experiment results. Figure 9a-c show the mapping relationships between the X- and Y-axis width ratios and the relative position along the Z-axis for fluorescent nanoparticles of three diameters, where the relative zero position was the position at which the width ratio was one for each case. Evidently, these mapping relationships could be used to obtain the Z-axis displacement of the three diameters of fluorescent nanoparticle in their actual measurement operations. After calibrating these mapping relationships, the relationships between the measurement errors and the actual coplanar displacement for each of the three diameters of fluorescent nanoparticles must be determined for realizing the coplanar displacement measurement according to Equation (2). Figure 10a-f shows the relationship between the slope of the coplanar displacement measurement errors of five corresponding relative positions along the Z-axis for three diameters of a fluorescent nanoparticle. In Figure 10, the slopes corresponding to each of the five positions of each magnification can be seen to lie close to the fitted slope obtained by the micro-imaging system analysis. The intermediate magnification, M2, performed better than the other two magnifications, demonstrating that improper magnification has an influence on the measurement accuracy, where high magnification could cause noise to increase, and low magnification made the collected fluorescence nanoparticle images insufficiently accurate, both causing an inaccurate measurement. The degree of fit was used as a criterion to choose an appropriate magnification for the adopted fluorescent nanoparticle during the sample measurement. Figure 9: X- and Y-axis width ratio of three diameters of fluorescent nanoparticles versus their relative position, along with the Z-axis, at magnifications M1 (**a**), M2 (**b**), and M3 (**c**). After the above operations, the three fluorescent nanoparticles were dissolved in a silicone oil (refractive index: 1.392) to be the three test solutions. We synthesized the results of Figures 9 and 10, and Equation (2) to realize the nine cases of the fluorescent nanoparticle 3D tracking measurements and to validate the performance of the proposed micro-imaging system. The diffusion constant \(D\) of the fluorescent nanoparticle had a relationship with its 3D MSD: MSD = \(6 Dt+\sigma\), where \(\sigma\) was the offset at time zero, accounting for the positional accuracy, and \(t\) was the time delay. Figure 11 shows the relationships between the MSD in each case and the corresponding diffusion constant \(D\).