## 2 Structure Design, Fabrication and Topography Measurement  
### Design and Fabrication  
In general, the method of combining a SPP and a convex lens can be used to present the effect of generation convergent LG beams. But this poorly integrated system will limit its application in imaging micro-systems, so as to inspire us to integrate a miniaturized convex lens and a SPP leading to a convex spiral phaser. A \"Solidworks\" software is utilized to draw a schematic diagram of the SPP and the convex spiral phaser, as shown in Figure 1. Comparing Figure 1,b, it can be found that the top surface of the SPP is formed by a straight linear spiraling around a central-axis. The top surface of the convex spiral phaser is formed by a circular arc-spiraling around the central-axis. The ideal is to achieve a full beam convergence through this structure like a convex lens. The main design parameters are shown in Figure 2. In formula (1), \(n_{0}\) represents the refractive index of the surrounding medium, and \(n\) the refractive index of the material for constructing the structure mentioned above, and \(\lambda\) the wavelength of incident light beams, and \(l\) the TC. Among them, \(l\) is generally an integer. If \(H\) is not an integer corresponding to the wavelength, the phase of incident beams at each phase step will be discontinuous, so as to destroy the circular intensity distribution of the transmitted light [39; 40; 41; 42]. We set the central wavelength at \(\sim\)650 nm, where the refractive index of Nanoscribe IP-Dip is 1.545 and the \(l\) being 5, and then the parameter \(H=5.963\)\(\upmu\)m can be calculated. The value of the key parameters are shown in Table 1:  
\[l=\frac{H(n-n_{0})}{\lambda} \tag{1}\]  
From the parameter table, it can be seen that our processing accuracy is at the sub-micron level. In the past, due to the technological limitation, it is difficult to make a spiral phase plate with a relatively smooth surface, and generally a multi-level step spiral phase plate [43; 44; 45] is used instead. But with a rapid advancement of 3D printing technology, it becomes possible to sh  
Figure 1: Structural diagram: (**a**) a SPP and (**b**) a convex spiral phaser. phase plate through 3D manufacturing equipment according to the processing principle based on the two-photon absorption effect. In the actual processing, the photosensitive material of Nanoscribe IP-Dip is used, and the laser fabrication wavelength is ~780 nm, pulse width 120 fs, repetition frequency 80 MHz, peak power 6 kw, horizontal resolution ~200 nm, longitudinal resolution ~1 mm. First, the photosensitive material is spin-coated on a glass substrate, and then placed on a precision moving platform, and thus a focused laser beam is used to localized expose it. Generally, the two-photon absorption only occurs in a limited three-dimensional area near the focus of the 3D printer. So, the complete three-dimensional structure can be thoroughly exposed through moving the platform, and then the unexposed areas dissolved by acetone, and finally the residual solution air-dried with nitrogen. A \(5\times 5\) CSPA with a period of 25 mm and 30 mm is successfully fabricated, respectively. The manufacturing equipment (high-speed femtosecond laser three-dimensional direct writing system 1.0-HUST) and the fabricated sample are shown in Figure 3. From Figure 3b, we can see that because the material used for fabricating the sample is transparent in the visible wavelength region, the surface morphology of the sample cannot be directly observed. So, a laser confocal microscope is utilized to present the detailed structural character of the sample fabricated. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value (\(\mu\)m)** \\ \hline \(r\) & 200 \\ \(R\) & 20 \\ \(d\) & 2 \\ \(H\) & 5.963 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter-Value correspondence table. Figure 3: (a) Manufacturing equipment and (b) a sample fabricated.