# Investigation of Quadrate Fresnel Zone Plates Fabricated by Femtosecond Laser Direct Writing
## 3 Results and Discussion
### Simulation and Experimental Study on L-FZPs, O-FZPs and S-FZPs  
Figure 8a-d shows that the diffraction properties of quadratic FZPs and circular FZP are simulated and compared. All the energy of circular FZP is concentrated at the focus, without any other diffraction pattern. For quadratic FZPs, the light path from the region to the focus is different as the beam passes through at different locations, and therefore there are other diffraction patterns. Figure 8a-c shows simulation diagrams of L-FZP, O-FZP and S-FZP. The edge-to-center distances of the three FZPs are 80 \(\upmu\)m, the numbers of zones are 29, and the main focal lengths are 8 mm. The diffraction pattern of L-PZP is straight, with the highest energy at the central line and a symmetric pattern of lines on both sides, with decreasing energy. The diffraction pattern of O-FZP is crossed dashed lines because O-FZP is composed of vertically crossed L-FZPs. O-FZP is segmented into several narrow linear FZPs, thus forming a series of independent and disconnected focal points arranged in a straight line.  
Figure 8: The focal point pattern of FZPs: (**a**) L-FZP; (**b**) S-FZP; (**c**) O-FZP; (**d**) circular FZP (C-FZP), and the corresponding contrast diagram of focal point energy of FZP: (**e**) Unnormalized graph; (**f**) normalized graph.  
Figure 7: Contrast image of focal point morphology of S-FZPs with different numbers of zones.  
Figure 8e,f compares the focal point energy of the FZPs. It can be seen from Figure 8e that the peak energy of circular FZP is the highest. Similarly, the peak value of quadrate FZPs is lower because the energy of quadrate FZPs is dispersed to the cross line. Figure 8f compares the energy of the three FZPs after normalization. Compared with the circular, linear and orthogonal FZPs, the energy of S-FZP is distributed more on the cross line, and the cross line has the largest energy ratio. The crossline diffraction pattern of S-FZP is clear, which is suitable for the aiming system and other applications for cruciform diffraction pattern.  
The machining path design and direct writing machining of quadrate FZPs on sapphire material are carried out by a femtosecond laser experimental system. The laser power is 10 mW, and the 3D platform moves at a speed of 1000 \(\upmu\)m/s. Because quadrate FZPs have no curve and radian, the femtosecond laser machining of quadrate FZPs has a simple moving path and less parallel operation. The processing time of 29 zones S-FZP is about 12 min, O-FZP is 8 min, and L-FZP is only 4 min. However, the processing time of the circular FZP is 90 min, which is much longer than that of the quadrate FZPs. Femtosecond laser processing is non-hot melt processing and high precision, so after femtosecond laser fabrication, quadrate FZPs are processed with clear lines (Figure 9a-c). The number of zones of quadrate FZPs is 29, and the edge-to-center distance is 80 \(\upmu\)m. The experimental values are consistent with the theoretical design values.  
A diffraction testing system tests the diffraction performance of three FZPs. Figure 9d-f are experimental diffraction patterns of the focal plane, and Figure 9g-i are diffraction patterns simulated by ZEMAX. In comparison, the experimental diffraction pattern and simulation diffraction pattern are consistent. The model and analysis method of the simulation is correct and reliable. The main focal length measured in the experiment is 8 mm, which confirms the equation of S-FZP main focal length, i.e., \(f_{\text{Q-FZP}}=l_{1}^{2}/\lambda\).  
Figure 9: The fabricated morphology of **(a)** L-FZPs **(b)** O-FZPs **(c)** S-FZPs; the diffractive performance of corresponding FZPs: **(d-f)**: experimental patterns; **(g-i)**: simulation patterns.