# Multi-Objective Defocus Robust Source and Mask Optimization Using Sensitive Penalty
## 1 Introduction  
With the shrink in critical dimension (CD), the impact of the optical proximity effect (OPE) become obvious, it causes distortion in the exposure pattern and reduction of pattern fidelity and contrast, so that it must be corrected effectively. Besides, continuous shrinking also allows for control of the defocus in lithography to become increasingly stringent. In actual lithography processes, defocus is always uncertain at the wafer level, because of the unevenness of the wafer surface [1]. In addition, aberrations, thermal aberrations [2], thermal mask effects [3], and thick mask effects [4] all inevitably cause best-focus plane constant shift and further increase the OPE in off-focus conditions. Meanwhile, the continuous shrinkage of technology nodes has promoted the introduction of resolution enhancement technology (RET). Conventional RET, such as optical proximity correction (OPC) and source and mask optimization (SMO), generate the best exposure conditions by optimizing mask or simultaneously optimizing source and mask, respectively [5; 6]. However, general RET methods assume the lithography system is ideal, which only compensates the OPE in the nominal condition [7; 8; 9]. But, with CD shrinkage, it results in the imaging quality becoming more sensitive to defocus,thus requiring RET methods corresponding to off-focus conditions. Therefore, to tackle the focus variation, an analytical defocus expansion function was derived to predict the defocus aerial image in an inverse lithography technology (ILT) framework [10], so that a variational lithography model (VLIM) is derived to take into account exposure dose and focus variations [11]. Meanwhile, we previously proposed the source-mask-numerical aperture (NA) co-optimization (SMNO) method to extend the depth of defocus (DOF) by fine-tuning the NA [12; 13], but it inevitably sacrificed resolution due to the reduction of NA. In addition, [PERSON] et al. [14] also studied SMO methods to improve the pattern fidelity in the case of an assigned defocus plane which operated at 100 nm defocusing, and our subsequent works have drawn on this approach [15; 16], but it is hard to ensure global fidelity in different defocus variations. Thus, [PERSON] et al. [17; 18; 19; 20; 21] proposed statistical variations-based OPC and SMO methods to improve global fidelity at different defocus variations. Unfortunately, the above methods found it hard to improve the uniformity of exposure patterns within the DOF, so that the optimized system still had relatively high sensitivity to defocusing. To eliminate changeable and uncertain defocusing due to the presence of multiple factors effectively, it requires the reduction of defocusing sensitivity for optimized systems. Therefore, to the best of our ability to minimize the defocusing sensitivity, we propose a defocus robust SMO (DRSMO) approach that is driven by a new multi-objective optimization strategy. In our method, the total cost function is composed of expectation fidelity and expectation sensitivity in different defocus disturbances. In addition, the expectation sensitivity penalty function is introduced into an inverse optimization framework for the first time to constrain the uniformity of aerial images to defocus in different disturbances, so that more robust source and mask are designed through this optimization. Compared to conventional SMO approaches, the simulation results confirmed that DRSMO can further reduce defocus sensitivity and improve process robustness. The defocus sensitivity factor \(s_{\beta}\) maximally decreased 63.5% compared to conventional SMO, and the DOF corresponding to EL = 5% maximally increased 54.5%, EL maximally increased from 4.5% to 10.5% as well. It means that larger exposure tolerances were in the actual lithography process via DRSMO. In addition, this paper also discusses the established optimization problem which was solved by the stochastic gradient descent (SGD) algorithm and mini-batch gradient descent (MBGD) algorithm.