# The Inverse Optimization of an Optical Lithographic Source with a Hybrid Genetic Algorithm
## 1 Introduction
used a new adaptive  
Figure 1: Schematic representation of the optical lithographic imaging process \(r_{s}\) represents the effective radius of the illumination source, \(r_{p}\) is the radius of the pupil, and \(\theta_{max}\) is the maximal incident angle of the rays in the objective exit. operator to achieve a nonlinear adaptive adjustment for image segmentation via a GA [25]. [PERSON] et al. found a suitable membership function for a mining problem through a GA-based framework and applied the best-set membership function in fuzzy association rule mining [26]. As this is a global optimization method, it is more convenient to find the optimal condition of the merit function for a complex model because of its gradient-free features. Moreover, due to its inherent property of being based on a probabilistic rule, the more flexible search function of a GA can be implemented to decrease the influence of multiple parameters of a complex model to improve the optimization efficiency. Several studies have indicated that GAs have displayed considerable capacities for optimizing the imaging performance in optical lithography. [PERSON] et al. proposed that it is significantly effective to improve the imaging quality by using a GA to optimize the illumination geometries in optical lithography [27]. [PERSON] et al. applied a GA for optical proximity correction to compensate for the image distortion induced by the optical proximity effect [28]. [PERSON] et al. used a modified GA method to improve the process windows in inverse lithography for various common objectives, including the depth of focus, normalized image log slope, and process variability band [29]. Nevertheless, the multiple parameters of a complex model potentially cause the search range of an optimization model based on a GA to be reduced, thus causing premature convergence [30; 31]. Due to the scalability of GAs, it is possible to combine a GA with other algorithms to enhance the optimization capabilities and improve the convergence efficiency [24; 32; 33]. To achieve the optimal source shape in the imaging process of optical lithography for advanced applications, a hybrid GA can be considered as an effective approach for complex imaging models. In this study, a hybrid algorithm created by combining a GA with two decision methods is proposed to find the optimal illumination mode according to different feature patterns in order to improve the imaging performance in optical lithography. These decision methods are the iterative search approaches of the tabu search algorithm (TS) and particle swarm optimization algorithm (PSO). Because of the complex model of the optical lithographic imaging process, the convergence efficiency of GA-based SO methods is limited and can be excessively reduced. To overcome the search limitations, an update strategy using PSO can expand the search scope. The tabu list function from the TS can improve the optimization capabilities within a reasonable range. Furthermore, the combination of a crossover operation and mutation operation in the GA can ensure the diversification of the population for the SO model. The simulation results prove that the imaging performance in optical lithography can be significantly improved by utilizing an SO method with the proposed hybrid GA approach.