Development of Gaussian Window Function for Precision Topography of Silicon-Wafer Surface Using Wavelength-Modulation Interferometry

• Published in: International journal of precision engineering and manufacturing Vol. 25; no. 12; pp. 2549 - 2561
• Main Authors: Bae, Juncheol, Kim, Yangjin, Ito, Yusuke, Sugita, Naohiko
• Format: Journal Article
• Language: English
• Published: Seoul Korean Society for Precision Engineering 01.12.2024; Springer Nature B.V
• Subjects: Algorithms; Diffraction patterns; Engineering; Errors; Fourier transforms; Gaussian window functions; Industrial and Production Engineering; Interferometry; Lasers; Materials Science; Measurement techniques; Modulation; Polynomials; Regression analysis; Regular Paper; Silicon; Silicon wafers; Window functions; Fizeau interferometer; Wavelength modulation; Gaussian window function; Characteristic polynomial; Silicon wafer; Surface profiling
• ISSN: 2234-7593; 2005-4602
• DOI: 10.1007/s12541-024-01134-0

Wavelength-modulation Fizeau interferometry is widely used to measure the surface shapes of silicon wafers. The target phases of the fringe patterns can be calculated using a phase-modulation algorithm. According to Surrel’s characteristic polynomial theory, flexible phase-modulation algorithms can be designed by adjusting the window function. 3 N   −  2, 4 N  − 3, 5 N  − 4, …, and 9 N  − 8 phase-modulation algorithms in the form of AN  − ( A  − 1) have already been developed based on the characteristic polynomial theory. The AN  − ( A  − 1) algorithm has powerful error-suppression capabilities, able to suppress up to ( A  −  2 )th-order nonlinear phase-modulation errors, up to ( N  − 2)th-order harmonic components and coupling errors between these errors. As the industry demands higher measurement precision, there is an increasing need to develop flexible phase-modulation algorithms with enhanced error-suppression capabilities. However, developing these algorithms is challenging because of the difficulty of deriving their polynomial window functions, which are expressed as powers of M -sample characteristic polynomials. In this study, a Gaussian window function is developed by generalizing the window functions of the AN  − ( A  − 1) phase-modulation algorithm via regression analysis. Finally, the surface of a 4-inch high-reflectivity silicon wafer is profiled using a wavelength-modulation Fizeau interferometer and 10 N  − 9 algorithms calculated using the Gaussian window function and a discrete Fourier transform.