# Investigation of Quadrate Fresnel Zone Plates Fabricated by Femtosecond Laser Direct Writing
## 3 Results and Discussion
### Simulation and Experimental Study on PS-FZPs  
The diffraction pattern of phase-shifted quadrate FZPs is studied by simulation and experiment. In the simulation, the quadrate FZPs with 1.6 mm side length were shifted with phase shift parameter \(\alpha\) of 0.4. The diffraction pattern of PL-FZP is shown in Figure 10a. The design of single-line focusing becomes double-line focusing after phase shifting. Figure 10b presents the diffraction pattern of PO-FZP. Two parallel focusing lines on the X and Y direction form a double crossline diffraction pattern. The four points of the diffraction pattern are concentrated, and the diffraction line is intermittent and low in energy. Figure 10c shows the diffraction pattern of PS-FZP, which is also a double-cross line. Compared to the PO-FZP, its diffraction line width is more expansive, and the display is clearer. Three phase-shifted quadrate FZPs were fabricated using a femtosecond laser direct writing and placed in the diffraction performance test system. The diffraction patterns of each FZP are shown in Figure 10d-f. The diffraction pattern obtained by the experiment is consistent with the result of the ZEMAX simulation, and the focus of the double-cross lines is also realized.  
Moreover, square blockers on the four corners of the PS-FZP were introduced to form the phase-shifted block FZP (PB-FZP), as shown in Figure 11a. The side length of blockers is \(b\), and the transmittance is expressed by the transmittance parameter \(\beta\) as \(\beta=(2L-2b)/2L=1-b/L\), \(\beta\in(0,1)\). The PS-FZP can also be understood as the PB-FZP with \(\beta=1\). When the side length b of blockers is large, the transmittance parameter \(\beta\) is small. Figure 11b-d present the diffraction pattern of PB-FZP with different values of transmittance parameter \(\beta\). Blockers at the corners block the light, so the length of the diffraction line is shortened. The phase shift parameter \(\alpha=0.6\) remains unchanged, so the spacing between diffraction lines at four sides does not change. When the transmittance parameter \(\beta\) increases gradually from 0.36 to 0.74, the diffraction line's length also increases. The PB-FZP is processed by femtosecond laser direct writing (Figure 12a). The phase shift parameter \(\alpha\) is 0.6, and the transmittance parameter \(\beta\) is 0.5. As shown in Figure 12b, a clear and bright square pattern is observed in the experimental diffraction pattern, the same as the simulation diffraction pattern. The results demonstrate the fabrication precision of femtosecond laser direct writing and the validity of transformation law for circular FZP to quadrate FZP.  
Figure 10: Diffraction pattern of phase-shifted quadrate FZPs: (**a–c**) Simulated diffraction pattern; (**d–f**) experimental diffraction pattern.  
The given results proposed different values of phase shift parameter \(\alpha\) in X and Y directions and the corresponding transmittance parameter \(\beta\) to obtain the rectangular diffraction pattern. Figure 13a represents the fs-laser irradiated FZPs with rectangular shapes, and Figure 13b,c shows the diffraction performance and simulated pattern. All boundaries of rectangular FZP have a bright edge in the diffraction pattern, in keeping with the simulated results. The use of parameters from square FZPs to rectangular PB-FZP proved that the focusing equation \(f_{Q\text{-}FZP}=l_{1}^{2}/\lambda\) applies to all the quadratic Fresnel zone plates in designing and manufacturing.  
Figure 11: (**a**) Schematic diagram of PB-FZP; the diffraction pattern of PB-FZP with different values of transmittance parameter \(\beta\): (**b**) \(\beta=0.36\); (**c**) \(\beta=0.6\); (**d**) \(\beta=0.74\).  
Figure 12: (**a**) PB-FZP fabricated by femtosecond laser; (**b**) Experimental focal plane; (**c**) Simulated Focal plane.  
Figure 13: (**a**) the rectangular PB-FZPs processing by fs-laser; (**b**) corresponding diffraction performance; and (**c**) corresponding simulated results.