  Galaxy-scale strongly lensed systems have been shown to provide a unique
technique for exploring the underlying physics of dark matter at sub-galactic
scales. In the past, much attention was given to detecting and studying
individual haloes in a strong lens system. In addition to the subhaloes,
line-of-sight haloes contribute significantly to the small perturbations in
lensed images. In prior work, we demonstrated that these line-of-sight haloes
imprint a distinctive anisotropic signature and hence give rise to a detectable
non-zero parity-even quadrupole moment in the effective convergence field's
two-point correlation function. In this study, we show that these line-of-sight
haloes also produce a non-zero curl component of the effective deflection field
with a parity-odd quadrupole moment of the two-point function. These multipole
moments have the ability to statistically separate line-of-sight haloes from
dark matter substructure. In this paper, we examine how these multipole moments
evolve in the presence of warm dark matter and self-interacting dark matter in
terms of central density evolution and dark matter halo abundance. Importantly,
we show that these different multipole moments display exquisite sensitivity to
both the amplitude and the velocity dependence of the dark matter
self-interaction cross-section. Our approach opens the door for strong lensing
observations to probe dark matter self-interaction over a broad range of
relative velocities.
