Comparing Machine Learning and Physics-Based Nanoparticle Geometry Determinations Using Far-Field Spectral Properties

Anisotropic metal nanostructures exhibit polarization-dependent light scattering. This property has been widely exploited to determine the geometries of subwavelength structures using far-field microscopy. Here, we explore the use of variational autoencoders (VAEs) to determine the geometries of gold nanorods (NRs) such as in-plane orientation and aspect ratio under linearly polarized dark-field illumination in an optical microscope. We input polarized dark-field scattering spectra and electron microscopy images into a dual-branch multimodal VAE with a single shared latent space trained on paired spectra-image data using a learnable linear adapter. We achieve prediction of Au NRs using only polarized dark-field scattering spectra input. We determine geometrical parameters of orientational angle and aspect ratio quantitatively via both dual-VAE and physics-based analysis. We show that orientational angle prediction by dual-VAE performs well with only a small (∼300 particle) training set, yielding a mean absolute error (MAE) of 14.4° and a concordance correlation coefficient (CCC) of 0.95. This performance is only marginally worse than the physics-based cos(2θ) fitting approach between the scattering intensity and the polarizing angle, which achieves an MAE of 8.78° and a CCC of 0.99. Aspect ratio determination is also similar for the dual-VAE and physics-based fitting comparison (MAE of 0.21 vs 0.23 and a CCC of 0.53 vs 0.68). We show that the spectra-to-structure inference route of dual-VAE achieves high reconstruction accuracy when referenced against well-established physics approach and effectively handles images and spectral data sets, suggesting a general recipe for inverse nano-optical problems requiring both structure and optical information.