Synchronous Multisurface Interferometry Based on Rotational Invariance Under Nonfull-Period Sampling

Wavelength-tuning interferometry is essential instrument to measure the surface profile of transparent plates in semiconductor lithography field. The wavelength-tuning range of the laser is a key parameter in interferometry. An insufficient tuning range can cause the interference signal being sampled under non-full-period, resulting in the inability to separate the surface profile from the superimposed interferograms. To tackle these challenges, a synchronous multi-surface interferometry algorithm based on rotational invariance under non-full-period sampling (SMSI-RIns) is developed to measure the optical thickness variation, front, and rear surface profile of transparent plates in this paper. The proposed SMSI-RIns first identifies the harmonic components from the superimposed interferogram. To mitigate the nonlinear of phase demodulation error by the misalignment of the air gap between the test and the reference of the interferometer, the harmonic frequencies of the interference signals are determined using rotation operator and total least squares in the signal subspace. Then, an iterative framework for phase demodulation is developed to determine the initial phase of the surface profile and suppress the ripple caused by harmonic crosstalk in the reconstructed one. Numerical simulations incorporating multiple factors is performed and thousands of Monte Carlo simulations also confirm the robustness of the proposed SMSI-RIns under non-full-period. Experiments with different factors on transparent plates with a thickness of 1 mm are conducted to validate the effectiveness of the proposed SMSI-RIns. The standard deviations of peak-to-valley (PV) and root-mean-square (RMS) are better than 0.0091 λ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> and 0.45×10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> λ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> (λ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 632.8 nm) at 90%, 80%, 70%, 60%, and 50% of the full-period sampling, respectively. Moreover, the relative error between the PV and RMS of the proposed SMSI-RIns over 100 GHz wavelength-tuning range and the ZYGO VeriFire MST over 200 GHz for a transparent plate with a thickness of 0.6 mm is less than 5% and 2%.