## 3 Core Technologies and Algorithms
### Paraxial Region and Object Plane Perpendicularity Determination
Fitting with a quartic polynomial curve to obtain the green line, the second vertex position is split into difference data, where the two divided parts are fitted with logistic curves. The fitting model is:  
\[L(s)=\frac{K}{1+R\cdot e^{-UAs}} \tag{2}\]  
where \(K\), \(R\), and \(U\) are the undetermined coefficients that need to be confirmed through fitting; \(L\) is a dependent variable that is, in this case, the difference between the two FWHMs; and \(s\) is an  
Figure 5: Fitting the two divisions of the nanoparticle’s intensity distribution with a Gaussian function. independent variable that is, in this case, the linear position when the nanoparticle is moved linearly on the measured object plane. The logistic curve fittings of the front and the rear division parts are represented by the red and blue lines, respectively (Figure 6b). The position values of the red line's second inflection point and the blue line's first inflection point are \((\ln(R_{\text{red}})-1.317)/U_{\text{red}}\) and \((\ln(R_{\text{blue}})-1.317)/U_{\text{blue}}\), respectively. The red line's peak point and the blue line's peak point are \(\ln(R_{\text{red}})/U_{\text{red}}\) and \(\ln(R_{\text{blue}})/U_{\text{blue}}\), respectively. The line segment between the \((\ln(R_{\text{red}})-1.317)/U_{\text{red}}\) and \((\ln(R_{\text{blue}})-1.317)/U_{\text{blue}}\) points is chosen. The lowest of the difference values on the \((2\ln(R_{\text{red}})-1.317)/(2U_{\text{red}})\) and \((2\ln(R_{\text{blue}})-1.317)/(2U_{\text{blue}})\) positions is the threshold value. If the difference values of the positions within the chosen line segment are all below the threshold value, then the chosen line segment is the linear region within the ideal imaging region of the measured object plane, and two boundary points of the chosen line segment are determined. When two of the chosen line segments are obtained, then the four boundary points of the two chosen lines determine the ideal imaging region of the measured object plane (Figure 6a). To obtain one object plane that is perpendicular to the optical axis and its paraxial region, the following steps should be followed:  
1. Keep the measured object plane's space attitude for the pitching and Z-axis position constant. Roll the measured object plane in a small range of \(-Angle\) to \(+Angle\). 2. After each rolling operation, apply the polynomial combining logistic curve fitting method stated above to determine the ideal imaging region of the measured object plane. In the ideal imaging region, choose the Gaussian function's FWHM for one spot image as a reference object (here, the spot imaging nearest to the center of the ideal imaging region was chosen). Compare it with the Gaussian function's FWHM for any other function in the two chosen lines within the ideal imaging region, obtain the difference's mean square root (RMS) for each compared pair, record the  
Figure 6: (**a**) Imaging of a nanoparticle moving on the measured object plane; (**b**) polynomial combining the logistic curve fitting method to judge the ideal imaging region of the measured object plane. average of these RMS values for the current rolling space attitude, and then choose the measured object plane corresponding to the smallest average of all rolling space attitudes. 3. Keep the rolling and Z-axis position of the obtained object plane space attitude constant. Pitch the object plane in a small range and carry out the same operations in steps 1-2. The chosen object plane on this step is one object plane that is perpendicular to the optical axis. Then, again run the polynomial combining logistic curve fitting method on the perpendicular object plane to confirm its ideal imaging region. The paraxial region on some perpendicular object plane has its corresponding ideal imaging region on the image plane (Figure 4). Thus, the confirmation of this ideal imaging region is equal to the confirmation of the paraxial region of the perpendicular object plane.