Modeling of alignment errors induced by secondary reflection stray light and suppression based on angular spectrum expansion

Secondary reflection stray light is a major source of measurement errors in wafer alignment systems. Conventional approaches mitigate its impact by reducing the longitudinal coherence length of the light source. However, the quantitative relationship between coherence length and alignment error is not yet fully understood. Furthermore, traditional coherence control methods typically broaden the spectral width, which induces chromatic dispersion and degrades the signal-to-noise ratio. Existing stray light models predominantly assume a single stray light beam, neglecting the cumulative interference effects arising from multiple stray beams and their partial coherence superposition. This study first establishes a “signal light–multiple stray light–complex coherence degree” superposition model that incorporates the statistical characteristics of stray light number, optical path differences, and phase distributions. By integrating partial coherence theory with Monte Carlo analysis, the model enables a quantitative mapping between longitudinal coherence length and alignment measurement error, thereby providing a theoretical foundation for coherence length optimization. Second, we propose a novel method for tuning the longitudinal coherence length via angular spectrum expansion, which achieves coherence reduction without increasing spectral bandwidth. The constraints imposed by diffraction order overlap are also analyzed. Experimental results demonstrate that, for an initial laser with a 3 m coherence length and 0.1 pm spectral width, angular spectrum expansion reduces the coherence length to 195 µm while preserving the original spectral width. When optical element positions within the alignment system are deliberately perturbed, the standard deviation of alignment error decreases from 1.7 nm to 1.4 nm, and the variation range of alignment results narrows from 7.8 nm to 1.9 nm, confirming effective suppression of stray light-induced errors. This work fills a critical gap in coherence-related research for wafer alignment systems and provides both a theoretical framework and a practical approach for tuning longitudinal coherence length.