  We report a methodology to determine the quality factor ($Q$) in
implementations of the so-called dielectric haloscope, a new concept of wavy
dark matter detector equipped with a multilayered resonator. An anechoic
chamber enables the observation of the resonance frequency and its amplitude
for an unlimited series of layers for the first time, which is conveniently
filtered. The frequency-normalized power enhancement measured in a Dark-photons
\& Axion-Like particles Interferometer (DALI) prototype is a few hundred per
layer over a sweep bandwidth of half a hundred MHz. In light of this result,
this scaled-down prototype is sensitive to axions saturating the local dark
matter density with a coupling to photons between
$g_{a\gamma\gamma}\gtrsim10^{-12}$ GeV$^{-1}$ and $g_{a\gamma\gamma}\gtrsim$
few $\times 10^{-14}$ GeV$^{-1}$ at frequencies of several dozens of GHz once
cooled down to the different working temperatures of the experiment and
immersed in magnetic fields ranging from 1 T to 10 T; while the sensitivity of
the full-scale DALI is projected at
$g_{a\gamma\gamma}\gtrsim\mathrm{few}\times10^{-15}$ GeV$^{-1}$ over the entire
25--250 {\mu}eV range since $Q\gtrsim10^4$ is expected.
