# Stitching-Based Resolution Enhancement in Wavefront Phase Measurement of Silicon Wafer Surfaces
## 1 Introduction
These methods include confocal microscopy [16], which improves resolution and contrast by blocking out-of-focus light; structured light microscopy [17], which uses spatially structured light patterns to generate super-resolution images; and optical interferometry [18], which involves interference of light beams to obtain measurement data. Interferometry is particularly useful in silicon wafer metrology, but faces challenges related to mechanical drift and reflective surface interference [19,20]. Additionally, image analysis techniques such as the Shack-Hartmann [21] and Makyoh [22,23] methods are employed for phase front measurement in silicon wafers, but have limitations in lateral and depth resolution. Factors such as light source quality and wafer uniformity can also impact measurement precision. In general, optical methods offer valuable solutions for precise and reliable measurements in various applications. The present research focuses on studying the wavefront phase measurement technique to determine the surface profile of silicon wafers [24]. This technique relies on capturing images in two conjugate planes of the lens. Through this method, two defocused images were obtained and used to reconstruct the phase of the silicon wafer, and consequently of its surface, by estimating the first derivative of the phase. It should be noted that this measurement technique allows for the acquisition of approximately 8 million pixels, resulting in a high lateral resolution of 65 \(\upmu\)m and a depth resolution of less than 1 nm. Furthermore, the phase front measurement technique is noninvasive, which means that it does not require physical contact with the sample under study, making it particularly useful for analyzing fragile or delicate materials [25; 26]. However, the technique has limitations in terms of resolution. The resolution is determined by the camera resolution used to capture the images in the two conjugate planes as well as by the optical system employed. For measuring the shape of a silicon wafer with a lateral resolution of 65 \(\upmu\)m, a high-resolution camera with dimensions of 6388 \(\times\) 6388 pixels and a pixel size of 3.78 \(\upmu\)m is required. This translates to the need for a 30 cm field of view to cover the entire wafer surface. Achieving such high resolution entails significant costs due to the requirement for nonstandard lens sizes. To improve lateral resolution during measurement of the nanotopography and roughness of silicon wafers, a redesign of the optical system for measurement of the surface profile is proposed using the stitching technique [27]. The stitching technique involves combining multiple images into a single panoramic image. Overlapping images of the same sample are used to generate a panoramic image with higher resolution and a larger field of view than could be achieved with a single image. The stitching technique has previously been used with interferometry to characterize X-ray mirrors [28; 29; 30; 31]. It has also been applied to measure the wafer profile with interferometry, achieving a lateral resolution of 40 \(\upmu\)m, a field of view of 40 \(\times\) 40 mm\({}^{2}\), and a measurement time of 3-4 min for a wafer of 300 mm [32]. Another application of interferometry stitching was demonstrated by [PERSON] et al. [33], who presented a tool for measuring 300 mm wafer nanotopography using white-light interferometry. They achieved a lateral resolution of 150 \(\upmu\)m and a depth resolution of 10 nm obtained using an 85 \(\times\) 85 mm\({}^{2}\) field of view, allowing for a throughput of approximately twenty wafers per hour. The proposed new design of the optical system allows for improved lateral resolution with values below 10 \(\upmu\)m in the measurement of nanotopography and roughness of silicon wafers (see Figure 1). This leads to more precise measurements of the wafer's surface shape.