## 1 Introduction  
The rapid advancements in high-end equipment fields, such as semiconductor manufacturing [1, 2], flat-panel display exposure, and wafer defect inspection, have placed increasing demands on the imaging accuracy and compensation capabilities of doubly telecentric projection optical systems [3, 4, 5, 6, 7, 8]. In systems that utilize large-size masks as object planes and large-size glass substrates as image planes, the structural characteristics of these components make them susceptible to deformation caused by gravity and clamping forces during scanning exposure. Figure 1 is a schematic diagram of image plane deformation. This deformation results in positional shifts and distortions in the projected patterns on the substrate, which significantly impacts the performance of the system [9, 10]. Therefore, effective compensation for deformations in both the object and image planes is essential for improving exposure accuracy.  
Previous studies have explored both mechanical and optical compensation methods to mitigate image plane deformation [11; 12; 13; 14]. Mechanical compensation involves adjusting the position of objects and image planes to restore conjugation and is widely used due to its simplicity and the absence of additional optical components [15]. However, this method can only provide average adjustments and lacks the precision required for high-end optical systems. In contrast, optical compensation introduces compensating elements (such as phase plates, adaptive optics elements, or prisms) into the optical path to dynamically correct image distortions [16]. Despite offering higher accuracy, existing optical compensation methods heavily rely on empirical fitting and numerical simulation rather than analytical models, limiting their predictability and effectiveness [17].  
Several researchers have investigated the use of prisms and phase compensation techniques in aberration correction and image stabilization. [PERSON] et al. analyzed the influence of a single wedge prism on optical aberrations [18; 19], while [PERSON] et al. developed aberration analysis and correction strategies for combinations of planar symmetric optical systems [20]. However, most prior work has focused on aberration correction in general optical systems rather than high-precision doubly telecentric projection optical systems, where large-scale image plane deformations present additional challenges. Moreover, existing optical compensation methods cannot fully quantify the relationship between prism adjustments and the resulting image plane deformation, leading to uncertainty in compensation accuracy.  
To address these challenges, this study proposes a novel high-precision dual-prism compensation method based on aberration theory, specifically designed to mitigate image plane deformation in doubly telecentric projection optical systems. Unlike previous approaches that rely on simulation-based fitting methods, this study establishes a comprehensive analytical compensation model, quantifies defocus, tilt, and aberration effects, and derives precise relationships between prism parameters and image corrections. The proposed method is validated through simulations, demonstrating significant improvements in image quality and providing theoretical support for future doubly telecentric projection optical system designs. This approach offers a more systematic, predictable, and analytically driven solution for high-precision image plane compensation, marking a substantial advancement over existing methods.