Deep learning-driven virtual phase shifting for enhanced interferometric surface profiling

Abstract

Accurate surface measurement of silicon wafers is essential for various high-precision applications, where even sub-nanometer deviations can lead to critical performance degradation. Phase-shifting interferometry (PSI) is widely used for this task owing to its high accuracy. Among PSI methods, the classical 4-step phase-shifting algorithm (PSA) with π/2 shifts offers an optimal balance between accuracy and simplicity, effectively compensating for second harmonic distortion. However, in real-world experiments, ideal phase shifts are difficult to realize because of environmental disturbances, which result in phase-shift errors and coupling effects with harmonic distortions. To address this limitation, we propose a deep learning-based virtual phase shifter (DLVPS) that generates three additional interferograms with accurate phase shifts, which accounts for second harmonic components, from a single experimental interferogram. The original input interferogram and the three generated interferograms comprise a synthetic interferogram sequence that enables accurate phase retrieval. Experimental validation on silicon wafer profiling confirmed that the DLVPS can produce more accurate phase shifts than the classical experimental method, demonstrating its robustness against environmental disturbances. The evaluation of the retrieved phase exhibited phase error of 0.0725 and standard deviation of 0.0368 rad, indicating that the proposed method achieved both high phase retrieval accuracy and robust repeatability. Our approach utilizes deep learning to reduce reliance on complex hardware-based phase shifting, achieving stable and precise phase retrieval in the presence of environmental disturbances. Furthermore, by combining this with analytic phase retrieval, the proposed method overcomes the practical limitations of PSI while preserving the strengths of the established 4-step PSA.