Differentiable forward modeling and inverse lithography for two-photon lithography: application to voxel-by-voxel, diffractive optical element, and spatial light modulator systems

BackgroundTwo-photon polymerization (TPP) enables additive manufacturing with sub-micron resolution, but fabrication accuracy is strongly limited by nonlinear exposure dynamics, radical transport, and chemical proximity effects, especially in parallelized writing setups. Simplified threshold models neglect the resist response, whereas compact models often omit key physical and time-dependent processes. Moreover, such models are typically non-differentiable, which prevents their use in solving inverse design problems.AimWe address these challenges by developing and validating a time-dependent, differentiable forward model of the TPP process and by demonstrating how it enables optimization approaches for inverse problems across different exposure conditions.ApproachWe implement the forward model in PyTorch, combining a precomputed point spread function, radical generation, oxygen quenching, diffusion, polymerization kinetics, and resist development. Based on this model, we establish two optimization strategies: (i) a gradient-based point-by-point backpropagation method to optimize power along scan paths and (ii) a hybrid approach where a neural network predicts an initial exposure strategy that is refined by the physics-based model. These methods are applied to three experimental configurations: point-by-point writing, diffractive optical element (DOE)-based parallel exposure, and spatial light modulator (SLM) projection.ResultsWe demonstrate that the forward model, combined with inverse optimization, derives exposure distributions directly from target structures and improves fabrication accuracy in all three cases. In point-by-point writing, backpropagation adjusts power to control feature height and connectivity in three-dimensional structures. In DOE-based parallelization, the framework predicts line merging and height loss at the start and end of lines caused by oxygen inhibition and compensates for them through local dose control. In SLM projection with a low-contrast resist, the U-Net-assisted workflow produces optimized exposure distributions that enable the fabrication of microlens arrays and freeform features consistent with the design targets within the material’s resolution limits.ConclusionsThe presented time-dependent, differentiable forward model enables gradient-based optimization of exposure conditions across different TPP setups. This provides a scalable tool for dose correction and lays the foundation for physics-based inverse lithography in TPP.