## 2 Micro-Imaging System
### Analysis of the Micro-Imaging System  
To realize an accurate measurement of the nanoparticle 3D trajectories with the micro-imaging system in Figure 1, there are two aspects that need to be analyzed: (1) analysis of whether the nanoparticle image intensity distributions corresponding to different coplanar positions on the same object plane are identical, and (2) analysis of the relationship between the coplanar displacements of the nanoparticle on the object plane and the corresponding coplanar displacements of the image on the image plane. The former is the requirement to realize an axial displacement measurement of the nanoparticle along the Z-axis, and the latter is the necessity to realize a coplanar displacement measurement of the nanoparticle in the XY plane. The given parameters of the ray transfer matrix enumerated above are fixed values, except for n\({}_{\rm i}\), which depends on the measured specimen. The unknown distance parameters are constant values for a particular magnification, except for d\({}_{0}\), which is the position of the nanoparticle fluorescent probe along the Z-axis. Initially, we determined the distance parameters d\({}_{1}\), d\({}_{2}\), and d\({}_{3}\) for analysis of the micro-imaging system. To determine the distance parameters d\({}_{1}\), d\({}_{2}\), and d\({}_{3}\), each given parameter except n\({}_{\rm i}\) is assigned a value: n = 1.5, n\({}_{\rm a}\) = 1, R\({}_{\rm c}\) = 51.53 mm, \(\rm s_{0}\) = 0.17 mm, \(\rm s_{1}\) = 3 mm, and \(\rm s_{2}\) = 4.5 mm. If the emergent  
Figure 1: Optical configuration. light ray is parallel to the Z-axis, namely, \(X_{\rm e}=[X_{\rm e},\,y_{\rm e},\,0,\,0]^{\rm T}\), then \(-A_{\rm X}=W(1,\,1)=-1-444.44\times\rm d_{2}\)\(-444.44\times\rm d_{3}\) and \(-A_{\rm Y}=W(2,\,2)=-1-434.74\times\rm d_{2}\)\(-444.44\times(1-9.7\times\rm d_{3})\times\rm d_{2}\), where \(-A_{\rm X}\) and \(-A_{\rm Y}\) are the magnifications along the X- and Y-axis, respectively. If the emergent light ray vector is \(X_{\rm e}\) = [0, 0, \(a_{\rm e},\,b_{\rm e}\)]\({}^{\rm T}\), the corresponding incident light rays vector components \(x_{\rm i}\) and \(y_{\rm i}\) for ideal imaging are both zeros, no matter what the emergent light slopes \(a_{\rm e}\) and \(b_{\rm e}\) are. Thus, the equations \(0=W(1,\,3)=-A_{\rm X}\times(\rm d_{0}+n_{\rm i}\times\rm d_{1})+0.036\times n _{\rm i}\) and \(0=W(2,\,4)=-A_{\rm Y}\times(\rm d_{0}+n_{\rm i}\times\rm d_{1})+0.025\times n _{\rm i}\) should both be satisfied. Here, \((\rm d_{0}+n_{\rm i}\times\rm d_{1})\) is denoted as \(\rm d_{var}\), where \(\rm d_{var}=0.025\times n_{\rm i}/A_{\rm Y}=0.036\times n_{\rm i}/A_{\rm X}\) is the focal position on the optic axis at which a one-point light source can be ideally imaged. Making \(0.025\times n_{\rm i}/A_{\rm Y}=0.036\times n_{\rm i}/A_{\rm X}\) be true means the magnification ratio \(A_{\rm X}/A_{\rm Y}\) is 1.44; if the system is not capable of guaranteeing the magnification ratio, a one-point light source cannot be imaged to one point. For actual cases, the magnification ratio 1.44 is difficult to realize, and so a \(\rm d_{var}\) variation range between \(0.025\times n_{\rm i}/A_{\rm Y}\) and \(0.036\times n_{\rm i}/A_{\rm X}\) is the relatively ideal focus depth. Obtaining \(\rm d_{var}\) in the range means realizing the nanoparticle Z-axis position measurement for the micro-imaging system in Figure 1. Here, assuming \(-A_{\rm X}=-53.67\) and \(-A_{\rm Y}=-39.07\), then \(\rm d_{2}=42\) mm and \(\rm d_{3}=76.5\) mm; assuming \(n_{\rm i}=1.33\), then \(0.85\) mm \(<\rm d_{var}<0.89\) mm. Before practically measuring the nanoparticle 3D trajectories, the imaging spot intensity distribution (X- and Y-axis width ratio) of the adopted nanoparticle probes when they are located on different longitudinal object planes should be calibrated. For nanoparticle fluorescent probes, \(\rm d_{1}\) is constant and \(\rm d_{0}\) is varied to adjust \(\rm d_{var}\).