  Relative astrometric shifts between multiply lensed images provide a valuable
tool to investigate haloes in the intergalactic space. In strong lens systems
in which a single lens plays the primary role in producing multiple images, the
gravitational force exerted by line-of-sight (LOS) haloes can slightly change
the relative positions of multiply lensed images produced by the dominant lens.
In such cases, a LOS halo positioned sufficiently far from the dominant lens
along the LOS can create a pattern in the reduced deflection angle that
corresponds to the B-mode (magnetic or divergence-free mode). By measuring both
the B-mode and E-mode (electric or rotation-free mode), we can determine the
LOS distance ratios, as well as the 'bare' convergence and shear perturbations
in the absence of the dominant lens. However, scale variations in the distance
ratio lead to mass-sheet transformations in the background lens plane,
introducing some uncertainty in the distance ratio estimation. This uncertainty
can be significantly reduced by measuring the time delays between the lensed
images. Additionally, if we obtain the redshift values of both the dominant and
perturbing haloes, along with the time delays between the multiply lensed
images that are affected by the haloes, the B-mode can break the degeneracy
related to mass-sheet transformations in both the foreground and background
lens planes. Therefore, measuring the astrometric lensing B-mode has the
potential to substantially decrease the uncertainty in determining the Hubble
constant.

Cosmology and Nongalactic Astrophysics