Theoretical and ML-based method for thickness estimation of coatings on rough substrates using nondestructive SEM imaging

Quantifying the thickness of thin coatings on rough substrates is essential for process control but usually relies on destructive cross sections or techniques that assume laterally uniform films. We present a non destructive framework that estimates deposited thickness from paired top down SEM images acquired before and after coating, targeting films that are thin relative to the lateral spacing of surface features. In the analytic route, we treat coating as an effective geometric dilation of substrate features. For binarized images, the increase in bright area, normalized by the average of the pre and post deposition perimeters, yields a first thickness estimate. An overlap correction is introduced to account for feature coalescence after deposition, which would otherwise bias the estimate upward when asperities merge. Tests on synthetic shapes and numerically generated rough landscapes show that the corrected estimator reduces both bias and variance, with robustness improving for surfaces with larger correlation lengths. Application to experimental SEM pairs from coated rough substrates gives thickness values consistent with independent indications. Two machine learning modules complement the analytic approach. A supervised regressor that ingests before and after images together with difference channels predicts an effective dilation length calibrated to nanometers and reduces sensitivity to segmentation choices. A U-Net model performs an internal consistency check by reconstructing a plausible pre deposition image from a post deposition image and a scalar thickness, supporting the thickness as dilation interpretation for the explored coating and roughness ranges.