# Stitching-Based Resolution Enhancement in Wavefront Phase Measurement of Silicon Wafer Surfaces
## 1 Introduction  
The semiconductor industry plays a vital role in the manufacturing of various electronic devices, including mobile phones, computers, televisions, and radios. This industry relies on integrated circuits, transistors, and other components utilizing semiconductor materials [1]. In February 2022, the European Commission introduced the \"European Chips Act\", aiming to drive EUR 43 billion in investment policies for the EU semiconductor sector by 2030. With the rapid growth of vertical and horizontal chip applications, the semiconductor market is projected to double from current USD 550 billion to more than USD 1 trillion by 2030 [2]. The semiconductor industry is currently facing unprecedented challenges in reducing the size of devices, which requires the development of new materials and processes, particularly for sizes below 50 nm [3]. This necessitates advances in the device structure and contacts, including advanced gate dielectrics, source/drain structures, and potential adoption of new three-dimensional structures to enhance circuit performance. To implement these new technologies, tighter manufacturing tolerances, lower ownership costs, andbetter process control are all required. Furthermore, improving the predictability of device designs through computer-aided design tools is necessary in order to avoid unnecessary expenses in the manufacturing line. These efforts aim to increase device density and reduce costs by shrinking device sizes to meet the growing demand in the electronics market [4,5]. This trend aligns with [PERSON]'s law [6], an empirical observation stating that the number of transistors integrated into an integrated circuit doubles approximately every two years, enabling the development of advanced computing technologies over successive decades. Given the factors mentioned above, quality inspection becomes of paramount importance [7], particularly in the context of silicon wafers, which serve as the substrate for most integrated electronic circuits. In the semiconductor industry, high-purity silicon wafers are the preferred and most widely used material for integrated circuit manufacturing. Silicon wafers will continue to be the fundamental and essential material in the integrated circuit industry over the coming decades. These wafers possess high-quality characteristics, including faster current flow compared to other conductors and excellent mobility at high temperatures. As a result, silicon wafers have a wide range of applications not only in semiconductors but also in areas such as microlithography and microelectromechanical systems, for example tire pressure sensor systems [8]. Surface profile measurement is a fundamental parameter for assessing the performance of a silicon wafer. Measurement methods can be categorized into non-optical and optical techniques. The inductance method [9] relies on changes in inductance in a magnetic circuit, whereas the capacitance method [10,11] relies on changes in capacitance in a capacitor. Atomic force microscopy [12] (AFM) and scanning probe microscopy [13] are non-optical techniques used for displacement measurement at the atomic level. Scanning electron microscopy [14] (SEM) is an imaging technique that provides high-resolution images at the subatomic level. Optical methods are essential for more accurate and reliable surface profile measurements, as non-optical methods have limitations. Specifically, non-optical methods have limitations such as nonlinearity and narrow dynamic range. Therefore, these methods do not meet the requirements for surface profile measurements, necessitating the use of optical methods that enable more precise and reliable measurements [15]. A variety of optical methods can be used for measurement purposes.