Mask optimization approaches in optical lithography based on a vector imaging model

Recently, a set of gradient-based optical proximity correction (OPC) and phase-shifting mask (PSM) optimization methods has been developed to solve for the inverse lithography problem under scalar imaging models, which are only accurate for numerical apertures (NAs) of less than approximately 0.4. However, as lithography technology enters the 45 nm realm, immersion lithography systems with hyper-NA (NA > 1) are now extensively used in the semiconductor industry. For the hyper-NA lithography systems, the vector nature of the electromagnetic field must be taken into account, leading to the vector imaging models. Thus, the OPC and PSM optimization approaches developed under the scalar imaging models are inadequate to enhance the resolution in immersion lithography systems. This paper focuses on developing pixelated gradient-based OPC and PSM optimization algorithms under a vector imaging model. We first formulate the mask optimization framework, in which the imaging process of the optical lithography system is represented by an integrative and analytic vector imaging model. A gradient-based algorithm is then used to optimize the mask iteratively. Subsequently, a generalized wavelet penalty is proposed to keep a balance between the mask complexity and convergence errors. Finally, a set of methods is exploited to speed up the proposed algorithms.