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

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. 3N  −  2, 4N − 3, 5N − 4, …, and 9N − 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 10N − 9 algorithms calculated using the Gaussian window function and a discrete Fourier transform.