# The Inverse Optimization of an Optical Lithographic Source with a Hybrid Genetic Algorithm
## 1 Introduction  
Optical lithography is a critical part of the manufacturing of very-large-scale integrated circuits (VLSIs), and it can be used to implement pattern transfer from a mask to a wafer with a reduction in the magnification by using the optical projection method [1; 2; 3]. The imaging process in optical lithography is shown in Figure 1. The rays emitted by an illumination source pass through a group of optical mirrors to form Kohler illumination. These rays irradiate the mask to generate diffracted light, which carries the feature information. An objective lens collects the diffracted light, which finally creates an image on the surface of a wafer. Because of the low-pass filtering effect of the objective lens, the high-frequency information of the feature pattern is lost, which causes aerial image to be distorted. Therefore, improving the imaging quality in optical lithography is essential in current advanced technology nodes. According to the Rayleigh criterion \(R=k_{1}\lambda/NA\), the lithographic resolution (\(R\)) can be absolutely improved to shrink the illumination wavelength (\(\lambda\)) and increase the numerical aperture (\(NA\)) of the objective lens, where \(k_{1}\) is the process factor [4; 5]. In advanced nodes, the loss of high-frequency information is becoming more and more serious, and this causes an obvious optical proximity effect (OPE) [6; 7]. These two approaches can no longer guarantee that the imaging quality requirements of continuously evolving technology will be met. Several resolution enhancement technologies (RETs) for the attainment of high-quality aerial image have been proposed [8; 9; 10]. In these methods, pixelated source optimization (SO), which is an inverse optimization method, can efficiently promote the performance in optical lithographic imaging [11; 12; 13]. This can be achieved with mathematical and algorithmic approaches, which can inversely optimize the intensity distribution of an optical lithographic source according to aerial image or resistance pattern. Freeform illuminations, which can be created with a set of micromirror arrays, have been developed to provide more degrees of freedom for regulating the illumination shape in optical lithography [11; 14; 15]. In a pixelated source model, changes in intensity of the pixels can indicate the mirror flip angle, where a pixel represents a unit mirror. To improve the imaging quality in optical lithography, various types of pixelated SO methods have been proposed; of these, gradient-based methods, such as gradient descent (GD) [16], conjugate gradient (CG) [17; 18], and steepest descent (SD) [19], have become the most significant. As local optimization algorithms, gradient-based methods have quick convergence, robustness, and the ability to search for the optimal solution [16; 20; 21]. These methods are not limited by the size of the optimized matrix, but only depend on the complexity of the gradient function. However, with the decrease in the critical dimensions of feature patterns toward smaller process nodes, the complexity of photo-resistance models is continuously expanded. It is significantly complicated to derive the gradient function of a photo-resistance model. Furthermore, the optimal solutions of complex models have an increased probability of falling into local optima when applying these methods to process the optimization of lithographic sources in advanced applications. To improving the imaging performance in optical lithography, it is not enough to address the optimization efficiency. The genetic algorithm (GA), which is a typical example of a heuristic approach, has been widely employed as a key tool for achieving the optimization of complex models in several fields, such as production scheduling, image processing, and data mining [22; 23]. [PERSON] et al. utilized a GA to solve a comprehensive optimization model for production scheduling and preventive maintenance planning [24]. [PERSON] et al.