Fractal characterization of surface complexity and anisotropic scratch evolution in sapphire polishing

A fractal analysis framework is developed to quantitatively characterize the multiscale complexity and anisotropy of polished sapphire surfaces. By integrating white-light interferometry with box-counting, power spectral density, and structure function methods, both global and local features are systematically assessed. Ultrasonic vibration-assisted polishing produces smoother, more homogeneous surfaces with lower roughness and fractal dimensions than conventional polishing. The three-dimensional evolution of individual curved scratches is reconstructed, revealing moderate fluctuations in valley depth, width, and area, and a decrease in fractal dimension from 1.16 to 1.08. These findings clarify the relationship between surface topography, fractal complexity, and tribological performance, providing a quantitative basis for optimizing advanced material surfaces.