# Investigation of Quadrate Fresnel Zone Plates Fabricated by Femtosecond Laser Direct Writing
## 1 Introduction  
Fresnel zone plate (FZP) with a circular structure is the most widely used zone plate because it has better focusing performance [1; 2; 3; 4; 5]. However, the curved edges of the circular zone will cause problems in terahertz imaging and eye medicine [6]. In an array generator and integrated optical circuit, the circular FZPs cannot be 100% array because the pixel arrays are the proper rectangular shape. Therefore, FZP with a circular structure is not suitable for all application scenarios. According to the phase distribution, FZPs can be presented as linear, square, rectangular, or polygonal structures. In the micro/nanomanufacturing process, compared with the curved motion of circular FZP, the process of making linear and square FZP is simpler, with higher processing efficiency, fewer defects, and stronger process stability. Therefore, the quadrate FZPs have been proposed, and more and more researchers are paying attention to them.  
Generally, the classical circular FZPs have alternate dark and light cyclic zones. The realization of quadrate FZPs depends on the replacement of rings with lines. [PERSON] et al. [7] designed quadrate and circular Fresnel zone plate lenses (FZPL) with different numbers of zones, working in transmissive or reflective modes. The quadrate FZPLs were written on the backside of a silicon wafer by maintaining the area of their corresponding circular Fresnel zones. [PERSON] et al. [8] proposed an improved zoning law for designing a quadrate FZPL. The law resulted in a higher gain when the FZPL was used as an antenna element. It enhances the focusing properties of the square FZPL, which help collimate an incident plane wave. [PERSON] et al. [9; 10; 11] recently designed a series of FZPs with different structures. They presented a square light beam at the focal plane, which was made through lateral phase-shifted S-FZP bound to horizontal and vertical axes via two perpendicular windows of rectangular shape. Theoretically, a square-shaped light beam with desired side length was generated by adjusting the phase shift parameter and bounded width [12]. Theabove research was focused on the production of diffractive images. However, the detailed fabrication of the optical lens was ignored.  
FZPs can be fabricated using ultrafast laser ablation, lithography, nanoimprinting, electron beams, and reactive ion etching [3; 13]. In general, ultrafast laser refers to a pulse width of fewer than 10 ps, with several advantages, such as use in multi-material applications, high peak power density, and small heat-affected zone. Because of this, the ultrafast laser plays an important and extensive role in micro and nano-manufacturing [14]. [PERSON] et al. [15] reported a novel process to directly inscribe an FZPL on the cleaved facet of a stiff clad polymer fiber using a femtosecond pulsed laser in binary optical devices. The fabricated FZP has an exceptionally long focal length of over 600 m. [PERSON] et al. [16] proposed fabricating FZPs on the multimode fiber (MMF) tip by combining femtosecond laser direct writing with coating, polishing, and chemical etching. Twelve Fresnel zones were formed with diameters of 200 \(\upmu\)m, and obtained the minimum beam at 900 \(\upmu\)m from the bottom of FZP. [PERSON] et al. [17] investigated the photosensitivity to femtosecond laser irradiation of dedicated chalcogenide glass, and a prototype of Fresnel GRIN lens with a refractive index gradient was fabricated via femtosecond laser direct writing. In this paper, taking the advantages of the femtosecond laser in micromachining, designing and fabricating the linear FZP (L-FZP), orthogonal FZP (O-FZP), and square FZP (S-FZP) via femtosecond laser direct writing were proposed. Classical circular FZPs were introduced for the comparison of diffraction properties. Finally, several phase-shifted quadratic FZPs, a phase-shift block FZP (PB-FZP), and a rectangular PB-FZP were carried out to obtain the squarely focus pattern.