Broadband measurement of Feibelman's quantum surface response functions

The Feibelman $d$-parameter, a mesoscopic complement to the local bulk permittivity, describes quantum optical surface responses for interfaces, including nonlocality, spill-in and-out, and surface-enabled Landau damping. It has been incorporated into the macroscopic Maxwellian framework for convenient modeling and understanding of nanoscale electromagnetic phenomena, calling for the compilation of a $d$-parameter database for interfaces of interest in nano-optics. However, accurate first-principles calculations of $d$-parameters face computational challenges, whereas existing measurements of $d$-parameters are scarce and restricted to narrow spectral windows. We demonstrate a general broadband ellipsometric approach to measure $d$-parameters at a gold--air interface across the visible--ultraviolet regimes. Gold is found to spill in and spill out at different frequencies. We also observe gold's Bennett mode, a surface-dipole resonance associated with a pole of the $d$-parameter, around 2.5 eV. Our measurements give rise to and are further validated by the passivity and Kramers--Kronig causality analysis of $d$-parameters. Our work advances the understanding of quantum surface response and may enable applications like enhanced electron field emission.