Misalignment measurement using circular grating moiré fringes via high-order phase harmonics

High-accuracy misalignment measurement with a large range remains challenging in lithography. Although overlapping circular gratings generate isotropic moiré fringes capable of two-dimensional measurement, their complex phase variations limit accuracy and practical applications. We propose a novel displacement measurement method using circular gratings that eliminates the prerequisite of knowing reference centers. This method incorporates previously neglected high-order harmonics, thereby enhancing measurement accuracy. First, the intensity of moiré fringes is modeled as a polynomial amplitude-modulated sinusoidal signal, whose coefficients enable efficient phase extraction via linear least-squares algorithms. Then, maximum likelihood estimation integrates high-order angular harmonic amplitudes of the phase measured at multiple origin positions, yielding precise and robust displacement measurements. The results demonstrate that incorporation of the second harmonic significantly improves performance over linear methods based on windowed Fourier transform and wavelet transform, achieving sub-

experimental accuracy and sub-

theoretical accuracy. The flexible parameter selection enables a trade-off between precision and robustness across diverse grating periods and sizes. This framework expands the versatility and applicability of circular gratings, demonstrating their substantial potential for misalignment measurement.