Abstract
<jats:p>Estimating the thickness of exfoliated two-dimensional flakes from ordinary optical micrographs is a foundational task in 2D-material fabrication. Current learning systems use thin-film optics only offline, as a synthetic-data generator for networks that classify thickness into discrete bins, and devote substantial machinery to closing the resulting gap between simulated and real imaging conditions. We instead place a differentiable transfer-matrix model of the full optics-to-color pipeline inside the inference loop, recovering thickness as a continuous variable by profile-likelihood continuation jointly with the imaging nuisances (oxide thickness, sensor gains, illuminant spectrum). The bare substrate present in every micrograph then acts as an in-frame calibration target, replacing domain adaptation with self-calibration and requiring no training data. On held-out simulated scenes with adversarially misspecified illuminants, median absolute error is 0.104 nm for graphene and 0.154 nm for MoS2 with zero retraining, since supporting a new material is one dispersion table; on simulated flakes of physically realistic integer-layer thickness it counts layers with 82.2% exact and 100% within-one-layer accuracy and resolves monolayers to 0.015 nm, a few percent of one atomic layer. These are simulation figures; the corresponding real-data evidence is the recovered interlayer spacing reported below, not a per-flake accuracy. Cram´er–Rao analysis establishes what is extractable from three color channels, shows that the problem is unidentifiable without priors, and predicts the residual failure domain, which we characterize as joint thickness–nuisance ridges rather than classical color metamers and address with two inexpensive acquisition protocols, evaluated on simulated scenes. We also show that the objective’s numerical aperture must be modeled: a normal-incidence approximation is harmless at NA 0.55 but destroys thin-regime accuracy at NA 0.9. On real micrographs from an external laboratory (the MaskTerial graphene substrate-variance series, n=200 held-out flakes each) a restricted mode using no labeled training data attains 95.3% exact layer agreement at n=600 and recovers graphene’s interlayer spacing in statistical agreement with its independently known value, along with each dataset’s effective substrate thickness, from the images alone. Code and experiments are released.</jats:p>