## 4 Entire 3D PSF Evaluation Procedure  
Regardless of the object plane's perpendicularity, the ideal imaging region determination of the measured object plane (a polynomial combining the logistic curve fitting method), and the 3D PSF evaluation, it is necessary to measure the PSF calibration module's spatial orientation and position through the dual PSD-based unit. Combining the statements in Section 3, the total PSF evaluation procedure can be presented as the following steps:  
1. Place a fluorescent nanoparticle under the microscope for imaging. Adjust the fiber-coupled laser to make its unit direction vector \(<\)0, 0, -1\(>\) in the coordinates determined by the dual PSD-based unit. 2. Determine the object plane's perpendicularity and the paraxial region, as detailed in Section 3.2. There are a few points to note during this process: **i. ** The PSF calibration module's spatial orientation (rolling angle \(\alpha\) and pitching angle \(\beta\)) and the PSF calibration module's plane position both need to be confirmed. It is clear that the PSF calibration module's plane position assurance can be realized by the dual PSD-based unit's CCD (Figure 2). The PSF calibration module's spatial orientation can be confirmed by observing the variation of the PSF calibration module's spatial orientation. The unit direction vector of the light beam from the fiber-coupled laser becomes \(<\)\(m_{X}\), \(n_{Y}\), \(q_{Z}\)\(>\), which is obtained by the dual PSD-based unit. Therefore, according to the transformation of the rectangular coordinates in space, \(\alpha\) and \(\beta\) can be solved by the following matrix: \[\left(m_{X},\ n_{Y},\ q_{Z}\right)=\left(0,\ 0,\ -1\right)\left[ \begin{array}{ c c c}\cos\beta&0&-\sin\beta\\ \sin\alpha\sin\beta&\cos\alpha&\sin\alpha\cos\beta\\ \cos\alpha\sin\beta&-\sin\alpha&\cos\alpha\cos\beta\end{array} \right]\] (8) **ii. ** After each rolling (or pitching) operation in the procedure of obtaining one object plane that is perpendicular to the optical axis (Section 3.2), the measured object plane is the X-Y plane of the coordinates determined by the dual PSD-based unit, and the unwanted rolling angle and pitching angle caused by movement can also be obtained with Equation (8). The six-degree-freedom macro/micro displacement platform is able to compensate for unwanted spatial orientation according to the rolling and pitching angles that are needed to keep the PSF calibration module's spatial orientation constant during the process of determining the ideal imaging region of the measured object plane (step 2 in Section 3.2). **iii. ** After each rolling (or pitching) operation, the fiber-coupled laser must be adjusted to make its unit direction vector \(<\)0, 0, -1\(>\) again and to adjust the Z-position measurement CCD to make the light beam reflected by the semi-transparent mirror normal (namely, to make the unit direction vector of the light beam reflected by the semi-transparent mirror \(<\)0, 0, -1\(>\) in the coordinates determined by the dual PSD-based unit). Then, the PSF calibration module's Z-axis direction position in its own coordinates can be confirmed (Z-axis direction position variation can be measured) to make sure the PSF calibration module (fluorescent particle) can stably move on the measured object plane (the X-Y plane of the dual PSD-based unit's coordinates) and keep its longitudinal space position unchanged. After the two steps above, one object plane that is perpendicular to the optical axis is confirmed, and the Z-axis of the dual PSD-based unit's coordinates becomes the optical axis. The Z-position measurement CCD is adjusted to normally reflect the light beam by the semi-transparent mirror. The light is transmitted through the semi-transparent mirror incidents into the Z-position measurement CCD. Then, the PSF calibration module's (fluorescent particle's) Z-axis direction position (longitudinal space position along the optical axis) can be measured by the Z-position measurement's CCD. 4.