  Strong gravitational lensing has been identified as a promising astrophysical
probe to study the particle nature of dark matter. In this paper we present a
detailed study of the power spectrum of the projected mass density
(convergence) field of substructure in a Milky Way-sized halo. This power
spectrum has been suggested as a key observable that can be extracted from
strongly lensed images and yield important clues about the matter distribution
within the lens galaxy. We use two different $N$-body simulations from the
ETHOS framework: one with cold dark matter and another with self-interacting
dark matter and a cutoff in the initial power spectrum. Despite earlier works
that identified $ k \gtrsim 100$ kpc$^{-1}$ as the most promising scales to
learn about the particle nature of dark matter we find that even at lower
wavenumbers - which are actually within reach of observations in the near
future - we can gain important information about dark matter. Comparing the
amplitude and slope of the power spectrum on scales $0.1 \lesssim k/$kpc$^{-1}
\lesssim 10$ from lenses at different redshifts can help us distinguish between
cold dark matter and other exotic dark matter scenarios that alter the
abundance and central densities of subhalos. Furthermore, by considering the
contribution of different mass bins to the power spectrum we find that subhalos
in the mass range $10^7 - 10^8$ M$_{\odot}$ are on average the largest
contributors to the power spectrum signal on scales $2 \lesssim k/$kpc$^{-1}
\lesssim 15$, despite the numerous subhalos with masses $> 10^8$ M$_{\odot}$ in
a typical lens galaxy. Finally, by comparing the power spectra obtained from
the subhalo catalogs to those from the particle data in the simulation
snapshots we find that the seemingly-too-simple halo model is in fact a fairly
good approximation to the much more complex array of substructure in the lens.
