Atomic Structure of Amorphous Optical Coatings of TiO₂-doped GeO₂ by Grazing-Incidence Total X-ray Scattering Measurements

Reducing coating thermal noise in future gravitational-wave detectors requires identifying the atomic motifs that control mechanical loss in amorphous optical coatings. We combine grazing-incidence X-ray pair distribution function measurements with atomic-structure modeling to study amorphous TiO$_2$-doped GeO$_2$ films over Ti cation concentrations from $\sim$11 % to $\sim$48 %, before and after annealing. The structural analysis reveals systematic composition- and annealing-dependent changes in short- and intermediate-range order. Increasing Ti content raises the average Ti coordination and promotes edge- and face-sharing polyhedral connections, while Ge remains predominantly fourfold coordinated. Annealing reduces these compact shared-polyhedron motifs and sharpens the first sharp diffraction peak, indicating a more relaxed intermediate-range network. Among the structural descriptors examined, the clearest correlation with the annealing-induced reduction in mechanical loss is the decrease in edge- and face-sharing polyhedra. These results connect composition, annealing, atomic structure, and mechanical dissipation in TiO$_2$-doped GeO$_2$, providing microscopic guidance for optimizing low-noise mirror coatings.