# Three-Dimension Resolution Enhanced Microscopy Based on Parallel Detection
## 2 Methods
### Theoretical Analysis
Referred to 3D FED algorithm [9; 10; 15], 3D-PRFED can be described by  
\[\mathrm{I_{3D-PRFED}=I_{PRISM}-\gamma J_{3D-negative}} \tag{4}\]  
\[\mathrm{I_{3D-negative}=\sum_{i=1}^{n}S\otimes((PSF_{vortex}+PSF_{hollow}) \times(PSF_{em}\otimes D(x+d_{n})))} \tag{5}\]  
where \(\mathrm{I_{3D-PRFED}}\) is the image of 3D-PRFED, \(I_{PRISM}\) is the volume image of PRISM, \(\mathrm{I_{3D-negative}}\) is the volume image modulated by 3D negative PSF which is incoherently combined by lateral negative PSF and axial negative PSF, as in Figure 2. \(\gamma\) is the subtraction factor of two images, which would affect final resolution and effective information retention, \(S\) is the sample distribution function, \(PSF_{vortex}\) is the lateral negative PSF of excitation light modulated by vortex phase mask, \(PSF_{hollow}\) is the axial negative PSF of excitation light modulated by hollow phase mask, and \(PSF_{em}\) is the PSF of emission light. \(D(x+d_{n})\) is the same definition as equation 2. Normally this value would be set as 0.7, which is an empirical value and can be set based on the SNR of the result images to keep a balance between the best resolution and the minimum ghost images [7; 9; 10; 19]. Vortex phase mask can be used in generating lateral negative PSF which can enhance the transverse resolution but not improve axial resolution. Meanwhile, 0-\(\pi\) phase mask can generate axial negative PSF which can enhance the axial resolution but not improve transverse resolution. These two phase retardation functions can be described by [9; 10; 19; 20]. \[\Delta\theta_{xy}(r,\varphi)=\varphi,\ \varphi\in[0,2\pi] \tag{6}\]  
Figure 2: 3D vortex generation process. (**A**) The vortex pattern used for generating transverse vortex PSF, (**B**) The intensity distribution cross-section of a vortex PSF, (**C**) The hollow pattern used for hollow PSF, (**D**) The intensity distribution cross-section of a hollow PSF, (**E**) The intensity distribution cross-section of 3D negative PSF. \[\Delta\theta_{z}(r,\varphi)=\left\{ \begin{array}{ll}\pi,&r\leq r_{max}/\sqrt{2}\\ 0,&r>r_{max}/\sqrt{2}\end{array} \right. \tag{7}\]  
where \(\Delta\theta_{xy}\) is the vortex 0-2\(\pi\) mask and \(\Delta\theta_{z}\) is the 0-\(\pi\) mask, and r is the radius of the mask. According to Equations (6) and (7), vortex PSF and hollow PSF can be generated by phase modulation based on the vector diffraction model, as shown in Figure 2.  
3D negative PSF as in Figure 2E can be incoherently combined by a lateral negative focus as in Figure 2B and an axial negative focus as in Figure 2D, which can be used to enhance three-dimension resolution in 3D-PRFED. For a single excitation light beam, the p-polarization component of the excitation light can be modulated by 0-2\(\pi\) vortex phase mask as in Figure 2A to generate a lateral negative PSF as in Figure 2B and the s-polarization component of excitation light can be modulated by 0-\(\pi\) hollow phase mask as in Figure 2C to generated an axial negative PSF as in Figure 2D. For the sake of making sure that these two components can be incoherently combined at the focus plane, the required light path difference should be introduced into these two components to destroy their coherence. According to Debye integral [9; 21; 22; 23] and Equations (1)-(4), the simulation results in Figure 3 show us the PSF comparison between CLSM, PRISM, and 3D-PRFED. By measuring the lateral and axial full width half maximum (FWHM) of the equivalent PSF profile along the \(x\) axis and the \(z\) axis, as shown in Figure 3D,E, the transverse resolution in 3D-PRFED is 1.89 times better than CLSM and 1.5 times better than PRISM. Meanwhile, the axial resolution in 3D-PRFED is 1.68 times better than CLSM and 1.58 times better than PRISM, which indicates that 3D-PRFED can well improve transverse and axial resolution compared with CLSM and PRISM. Figure 3: The effective PSF comparison among CLSM, PRISM, and 3D-PRFED. (**A**) The front view of CLSM PSF, 0.9, 0.61, 0.5, 0.15 is the corresponding value of their contour plane.