# Investigation of Quadrate Fresnel Zone Plates Fabricated by Femtosecond Laser Direct Writing
## 2 Theoretical and Experimental Methods  
### Theoretical Approach  
Quadrate FZP refers to a zone plate whose outer contour is square and inner zone is straight, such as linear, orthogonal, and square. The L-FZPs, as a one-dimensional zone plate, have linear focusing. It can be used in a cylindrical lens. After 90\({}^{\circ}\) rotation, the L-FZPs overlap the original zone plate vertically, and the O-FZPs, also known as two-dimensional zone plates, can be obtained [18]. Then, the S-FZPs are formed by adjusting the length of O-FZPs by extending it from inside to outside [19; 20]. As the most important and complex structure in quadratic FZPs, the S-FZP comprises transparent and opaque zones like circular FZPs.  
Figure 1 shows the schematic diagrams of S-FZP and circular FZP. \(l_{1}\) represents the edge to center distance, which substantially affects the diffraction performance. The half side length of \(n_{th}\) zone is \(l_{n}\), and \(r_{n}\) is the radius of circular FZP. Several investigations have focused on the relationship between \(r_{n}\) and \(l_{n}\), and, proposed different theories on determining the half side length of the S-FZP when it is converted from the circular FZP. in the first one, [PERSON] et al. [7] indicated that the area of S-FZP should be the same as circular FZP, as \((2l_{n})^{2}=\pi r_{n}^{2}\). Thus, \(l_{n}\approx 0.886\pi_{n}\). The second, [PERSON] indicated that optimized diffraction performance of polygonal FZP depends on the number of sides \(s\)[21], and the determination of \(l_{n}\) is related to \(s\), as \(l_{n}=r_{n}/\sqrt{1+\frac{\tan^{2}(\pi/s)}{4}}\). When the polygon is square, then \(s=4\), and \(l_{n}\approx 0.894\pi_{n}\). The third, [PERSON] et al. [8] proposed that different half side lengths can be obtained via the angle of the \(l_{n}\) and \(r_{n}\) (shown in Figure 1). When \(\theta_{1}=\pi/4\), \(l_{n}=r_{n}\cos\theta_{1}=0.707\tau_{n}\); when \(\theta_{2}=\pi/8\), \(l_{n}=r_{n}\cos\theta_{2}=0.924r_{n}\); and when \(\theta_{3}=0\), \(l_{n}=r_{n}\cos\theta_{3}=r_{n}\). The diffraction performance primarily determines the transfer from circular FZPs to S-FZPs.  
For circular FZPs, focal length \(f\) is \(f^{2}+r_{n}^{2}=(f+n\lambda/2)^{2}\), Where \(\lambda\) is the wavelength of incident light and \(n\) is the number of zones. So the expression for \(r_{n}\) can be obtained as \(r_{n}=\sqrt{n\lambda f+n^{2}\lambda^{2}/4}\), where \(n^{2}\lambda^{2}/4\) represents the spherical aberration. When \(f\gg n\lambda/2\), then \(r_{n}\approx\sqrt{n\lambda f}\). Similarly, when \(n=1\), \(f=r_{1}^{2}/\lambda\). However, for S-FZPs, this formula only applies to the rays emitted from the points at the junction of the square and the circle, and the rays from other positions are not considered. When converting the circular FZP into the S-FZP, the position of the focal plane should be kept constant so that the focal length does not change. In this article, \(l_{n}\) is set as \(l_{n}=r_{n}=\sqrt{n\lambda f}\) proposed by [PERSON] et al. [8] as the transformation law. Hence, the diffraction performance of quadrate FZPs was verified by simulations and experiments presented in the next section.  
Moreover, horizontal or vertical phase shifting can obtain the phase shift FZP (PS-FZP). Its diffraction pattern shows new characteristics with the change of structure. Figure 2 presents three quadrate FZPs and the corresponding PS-FZPs. In Figure 2a,d, the center moves to the right and left sides, with the total width of 2\(L\) staying the same. The ratio between 2\(a\) and 2\(L\) is the phase shift parameter \(a\), that is \(\alpha=a/L,\ (-1<\alpha<1)\). The phase shift O-FZP (PO-FZP) and phase shift S-FZP (PS-FZP) can be obtained using the same phase-shift method.