  Measuring the losses arising from different materials and interfaces is
crucial to improving the coherence of superconducting quantum circuits.
Although this has been of interest for a long time, current studies can either
only provide bounds to those losses, or require several devices for a complete
characterization. In this work, we introduce a method to measure the microwave
losses of materials and interfaces with a single multi-mode superconducting
resonator. We demonstrate a formalism for analyzing the loss sensitivity of
multi-mode systems and discuss the design strategies of multi-mode resonators
for material loss studies. We present two types of multi-mode superconducting
resonators for the study of bulk superconductors: the forky
whispering-gallery-mode resonator (FWGMR) and the ellipsoidal cavity. We use
these resonators to measure the surface dielectric, conductor, and seam losses
of high-purity (5N5) aluminum and aluminum alloy (6061), as well as how they
are affected by chemical etching, diamond turning, and thin-film coating. We
find that chemical etching and diamond turning reduce both the surface
dielectric and conductive losses of high-purity aluminum, but provide no
appreciable improvement to the seam. Coating the surfaces of diamond-turned
aluminum alloys with e-beam evaporated or sputtered aluminum thin-films
significantly reduces all three losses under study. In addition, we study the
effect of chemical etching on the surface of high-purity aluminum using
transmission electron microscopy (TEM) and find that the chemical etching
process creates a thinner and more uniform oxide layer, consistent with the
observed improvement in the surface dielectric loss.
