## 1 Introduction
When using a binary photo-mask (typically a thin layer of chrome patterned on a quartz substrate), the 3D microstructures are obtained through a careful control of the relative movement between the laser-projected photo-mask pattern and the sample under a specific firing sequence of laser pulses. This technique, known as contour mask scanning [7; 8; 9; 10; 11; 12; 13; 14; 15], results in the formation of a 3D machined profile through the superposition of multiple 2D machined patterns appropriately distributed at different locations and orientations. In general, achieving a specific 3D machined profile requires a careful design of both the photo-mask pattern and the scanning path. When applying this method to realize arrayed 3D microstructures, the scanning path usually has the form of straight lines along two or three directions. Such a technique is known as laser dragging and has found widespread  
Figure 1: A system diagram of an excimer laser micromachining system based on image projection for carrying out both 2D and 3D micromachining. use for the patterning of rectangular and hexagonal arrays of microlenses with various aspherical surface profiles [16; 17]. In 3D excimer laser micromachining systems of the second type, the ablation process is performed using a grey-scale or half-tone mask [18; 19; 20; 21]. These masks locally vary (i.e., modulate) the transmission coefficient of the laser light incident on the mask surface such that the distribution of the laser fluence on the sample can be directly controlled. Thus, through an appropriate design of the projected laser fluence and laser machining rate, the distribution of the machining depth on the sample can be varied as required to realize micro-features with different 3D profiles. However, in implementing such systems, designing suitable transmission-varying masks poses a significant challenge. In practice, obtaining a true grey-scale mask is very difficult, so most experimental systems utilize half-tone masks, namely binary masks containing arrayed micro-patterns with feature sizes smaller than the resolution of the optical imaging system. By locally adjusting the feature size, the projected laser fluence can be effectively altered point by point, thereby achieving the goal of 3D micromachining. Compared to the contour mask scanning method, half-tone mask methods are both more direct and more straightforward. Furthermore, they enable the fabrication of 3D micro-structures with complicated surface profiles, which are often difficult (or even impossible) to achieve using mask contour scanning techniques. However, these performance advantages come at the expense of a greater cost since such masks typically have tiny characteristic feature sizes less than 1 \(\upmu\)m.  
Regardless of the type of mask and/or method used to perform image-projected laser machining, the optical diffraction effect plays a critical role in determining the final 3D micromachining outcome. In the contour mask scanning method, diffraction occurs at the rim of the open aperture in the photo-mask pattern, and results in the formation of a transition zone at the perimeter of the projected machining area on the sample surface. The machining depth within this transition zone is, strictly speaking, no longer digital or binary, but has the form of a slope. In practice, this optical diffraction effect is very important in achieving an analog 3D machined profile in contour mask scanning methods since it reduces the surface step size between adjacent laser shots and improves the machined surface roughness as a result. When using a half-tone mask, the optical diffraction phenomenon is even more important since the 3D laser machining process is based intrinsically on the optical diffraction effect. However, even though several authors have provided basic guidelines for half-tone mask designs, the literature lacks the feasibility of exploiting optical diffraction effect to achieve complex-geometry 3D microstructures.