# Investigation of Quadrate Fresnel Zone Plates Fabricated by Femtosecond Laser Direct Writing
## 3 Results and Discussion  
### Simulation Study on Diffraction Process of S-FZP  
The modeling and simulations of quadratic FZPs and PS-FZPs were conducted in ZEMAX (America, Zemax OpticStudio 17). Figure 4a shows the convergence process of the S-FZP for the beam (\(\lambda=800\text{ nm}\)). The edge-to-center distance was set to \(80\text{ }\upmu\text{m}\) and the number of zones to 29. So, the square side length 2L is \(862\text{ }\upmu\text{m}\). The simulated results showed that laser beams converge gradually from the FZP plane to the focal plane. At the distance of \(8\text{ mm}\) from the FZP plane, the energy convergence is concentrated chiefly with a prominent bright spot and cross line in the center. The linear convergence of energy causes this in the XY direction. After \(8\text{ mm}\), the energy diffuses again. Figure 4b shows the energy distribution along the \(Z\)-axis after the incident light passes through S-FZP. It can be seen that the highest energy is at \(8\text{ mm}\), which is the main focal length of S-FZP. Meanwhile, other peaks with decreasing energies appear at \(2.67\text{ mm}\), \(1.60\text{ mm}\), and \(1.14\text{ mm}\). These values are the sub-focal lengths of S-FZP at \(f/3\), \(f/5\) and \(f/7\). As shown in Figure 5, the process of diffraction pattern formation can be seen more clearly by superimposing the focal plane and S-FZP. Along the diagonal, S-FZP can be divided into two triangles of transverse zones and two triangles of longitudinal zones. As can be seen from the triangles of transverse zones, the middle part is a complete linear FZP, so a clear focusing line is formed at the center of the focal plane. The outer region can be segmented into linear FZPs with some zones missing. Because of the absence of the zones, the focal lines formed on the focal plane become scattered. As more zones are missing, less energy is enhanced by diffraction at the center. Therefore, the energy of the focusing line on the focal plane becomes lower and gradually disappears. The same is true of the triangular longitudinal zones. Thus, the center of the diffraction pattern of an S-FZP is a clear crosshair, and the periphery is a widened focusing line. Figure 3: Schematic diagram of the femtosecond laser direct writing experimental system. For circular FZP, with an inner radius of 80 \(\upmu\)m and the incident wavelength of 800 nm, the main focal length is 8 mm, and the sub-focuses are at 2.67 mm, 1.60 mm, and 1.14 mm, according to the equation \(f=r_{1}^{2}/\lambda\). This is consistent with the primary and sub-focal lengths of S-FZP with 80 \(\upmu\)m edge to center distance. Therefore, when using the third  
Figure 4: Simulation diagram of the energy distribution of S-FZP along the \(Z\)-axis: (**a**) the distribution of the cross-sectional energy; (**b**) energy profile along with the \(z\)-axis. Figure 5: Structure and diffraction pattern of S-FZP. transformation law of [PERSON] et al. [8], the focal plane of S-FZP is at the same position as that of the circular FZP. Similarly, S-FZP has the same focus points as circular FZP. Moreover, the focusing equation of S-FZP is \(f_{\text{Q-FZP}}=l_{1}^{2}/\lambda\). This not only makes clear the mode of transforming the circular FZP into S-FZP, but also gives the equations of S-FZP focal length directly, which is beneficial for the study of the diffraction performance of quadrate FZP. The number of zones is an essential parameter of FZPs, affecting focal point morphology and energy distribution [22]. Several S-FZPs with different zone numbers were modeled to investigate the effects of zone numbers, and the diffraction performance was obtained via ZEMAX simulation. Table 1 compares the diffraction properties of S-FZPs with varying numbers of the zone. The focal depth of FZP gradually decreases from 3 mm to 0.7 mm with increasing zone numbers. As the focal depth decreases, the focal energy gradually centralizes, and the peak power increases with numbers. Figure 6 compares diffraction patterns of focal planes of quadrate FZPs.