  Just like light, gravitational waves (GWs) are deflected and magnified by
gravitational fields as they propagate through the Universe. However, their low
frequency, phase coherence and feeble coupling to matter allow for distinct
lensing phenomena, such as diffraction and central images, that are challenging
to observe through electromagnetic sources. Here we explore how these phenomena
can be used to probe features of gravitational lenses. We focus on two variants
of the singular isothermal sphere, with 1) a variable slope of the matter
density and 2) a central core. We describe the imprints of these features in
the wave- and geometric-optics regimes, including the prospect of detecting
central images. We forecast the capacity of LISA and advanced LIGO to study
strongly lensed signals and measure the projected lens mass, impact parameter
and slope or core size. A broad range of lens masses allows all parameters to
be measured with precision up to $\sim 1/{\rm SNR}$, despite large
degeneracies. Thanks to wave-optics corrections, all parameters can be
measured, even when no central image forms. Although GWs are sensitive to
projected quantities, we compute the probability distribution of lens redshift,
virial mass and projection scale given a cosmology. As an application, we
consider the prospect of constraining self-interacting and ultra-light dark
matter, showing the regions of parameter space accessible to strongly-lensed
GWs. The distinct GW signatures will enable novel probes of fundamental physics
and astrophysics, including the properties of dark matter and the central
regions of galactic halos.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology