## 1 Introduction  
With the continuous scaling in the feature size of lithography technology node, the deep ultraviolet (DUV) lithography steps into the ultra-low \(k_{1}\) regime [1]. As the feature's sizes on a mask are smaller than the scanner exposure wavelength, the 3D mask effects must be taken into account in computational lithography [2; 3]. The 3D mask effects are mainly subject to the morphology of the mask absorber including mask absorber thickness and absorber profile. Related research has indicated that mask absorber thickness (MAT) and the absorber sidewall angle (SWA) have a big impact on critical dimension (CD), best focus (BF), and depth of focus (DOF) [4; 5; 6; 7; 8; 9]. However, mask absorber errors are inevitably introduced by the imperfect etching process in mask fabrication and measurement [10; 11]. In addition, the MAT error and tapered SWA deformation of the absorber are increased by repeated mask cleaning process [12; 13]. Recently, [PERSON] et al. and [PERSON] et al. studied the impact of photomask uncertainties on computational lithography [4; 14; 15]. The results revealed that the variations of MAT and SWA cause significant CD errors (CDEs) that cannot be ignored. The research of [PERSON] et al. [16] and [PERSON] et al. [17] indicated that the SWA variations of the mask absorber also have great influence on the lithography patterning result. Therefore, it is very necessary to mitigate the impacts of mask absorber errors on lithography performance in computational lithography. Various advanced computational lithography techniques have been proposed to compensate for the 3D mask effects. [PERSON] et al. proposed a genetic algorithm-based pupil wavefront optimization (PWO) method to compensate for the wavefront phase error introduced by the mask topography effects [18]. [PERSON] et al. developed a combined source, mask, and projector pupil optimization (SMPO) procedure to maximize the common process window (PW) [19]. [PERSON] et al. presented a conjugate gradient method-based source mask optimization (SMO) approach incorporating pupil wavefront aberrations to improve the usable DOF by compensating for mask topography effects [20]. [PERSON] et al. developed an inverse PWO method to improve image fidelity by compensating for thick mask induced aberration including 37 Zernike terms [21]. However, the computational lithography techniques mentioned above are established for an ideal lithography system without taking into account of the impacts of random mask absorber errors on lithography imaging performance. To mitigate the impact of mask absorber errors on lithographic imaging performance, this paper develops a source, numerical aperture (NA), process parameters co-optimization (SNPCO) method for the DUV lithography system. To achieve this, an effective alternate optimization procedure based on a statistical method is built, which treats the mask absorber errors as random variables and takes their statistical properties into account. In our prior work, we had proposed a mask, process, and lithography-tool parameters co-optimization (MPLCO) method to improve the PW [22]. However, that method adopted a parametric source represented by only three parameters (including inner radius, outer radius, and opening angle) with limited flexibility, and it was established with a 2D mask model without considering the impacts of random mask absorber error on lithography performance. To solve this problem, we first use the Zernike polynomials to represent the source patterns in view of their capacity in fitting arbitrary form wavefront [23; 24; 25]. The even, symmetric Zernike polynomials are selected as the basis functions based on the symmetry of the DUV lithography illumination source. Thus, a freeform source is created by the linear superposition of these basis functions. In this way, the flexibility of the source pattern is improved, and the computational burden of a pixelated source along with the 3D mask model is avoided.