## 2 Micro-Imaging System
### Analysis of the Micro-Imaging System
The size changes with d\({}_{\rm var}\), namely the nanoparticle image intensity distribution according to the object plane variation changes; furthermore, the presented sizes of the entrance pupil are identical for d\({}_{\rm var}\) during calibration and measurement as long as the values of d\({}_{\rm var}\) are the same. Thus, regarding the first aspect of analysis, it is proven that the nanoparticle imaging spot intensity distribution corresponding to different positions on same object plane do not vary for an object. For actual measurements, the entrance pupil size cannot be obtained; rather, what is obtained from the imaging plane is the X- and Y-axis width ratio of the point source imaging. Figure 4 shows that the X- and Y-axis width ratios of the point source imaging also changes with d\({}_{\rm var}\), and so the X- and Y-axis width ratio can be used to mark different nanoparticle image intensity distributions along the Z-axis; thus, it can measure the nanoparticles displacement along the Z-axis. To analyze the relationship between the coplanar displacement of the nanoparticle on the object plane and the corresponding coplanar displacement of the imaging spot on the image plane (the second aspect to be analyzed), it must first be determined how to calculate the center of the point source image. Figure 3: Radius of the entrance pupil determined using the four central rays for different d\({}_{\rm var}\) and coplanar shift values \(\Delta\) of the point source: (**a**) variation of d\({}_{\rm var}\) with d\({}_{0}\), and (**b**) variation of d\({}_{\rm var}\) with d\({}_{1}\). When the point source is located on the Z-axis, the center of its image is at the intersection of the CCD and the Z-axis, which is also the intersection of the angular bisector of the right and left marginal rays (or upper and lower marginal rays) and the CCD. When a point source moves along the X-axis, the symmetrical relationship of the upper and lower marginal rays to the CCD in space is invariant, whereas the symmetrical relationship of the right and left marginal rays in space is undermined and the center of the image is shifted only along the X-axis, which is the intersection point between the angular bisector of the right and left marginal rays and the CCD. Similarly, for a point source moving along the Y-axis, the center of the image is the intersection point of the angular bisector of the upper and lower marginal rays and the CCD. Figures 5 and 6 show the center position measurement errors of the point source when it moves along the X (\(\Delta_{\rm X}\)) and Y (\(\Delta_{\rm Y}\)) axes, respectively. These figures show that the results are identical for d\({}_{\rm var}\) during measurement and calibration for the same value of d\({}_{\rm var}\). The linear relationships between the measurement errors and the actual coplanar displacement of the point source are evident for different d\({}_{\rm var}\)'s. The slopes of the linear plane displacement measurement errors are also linearly related to d\({}_{\rm var}\), as shown in Figure 7.  
From the second requested analysis aspect stated above, the (fluorescent nanoparticle's) actual coplanar displacement of the point source on an object plane can be related to its measured displacement value based on the linear relationship shown in Figure 7:  
\[ \begin{array}{c}\rm{actual\,coplanar\,displacement}\\ =\frac{measurement\,coplanar\,displacement\,by\,CCD}{\left|magnification \right|\cdot\left(1+\text{slope\,of\,coplanar\,displacement\,measurement\,error} \right)}\end{array}  \tag{2}\]  
The slope of the coplanar displacement measurement error differs with d\({}_{\rm var}\), as seen in Figure 7; thus, the d\({}_{\rm var}\) measurement needs to be labeled when using Equation (2) to obtain the actual coplanar displacement. The d\({}_{\rm var}\) labeling process uses the longitudinal object planes calibration stated above.