  We present a statistical study of the effects induced by substructures on the
deflection potential of dark matter halos in the strong lensing regime. This
investigation is based on the pertubative solution around the Einstein radius
(Alard 2007) in which all the information on the deflection potential is
specified by only a pair of one-dimensional functions on this ring. Using
direct comparison with ray-tracing solutions, we found that the iso-contours of
lensed images predicted by the pertubative solution is reproduced with a mean
error on their radial extension of less than 1% - in units of the Einstein
radius, for reasonable substructure masses. It demonstrates the efficiency of
the approximation to track possible signatures of substructures. We have
evaluated these two fields and studied their properties for different lens
configurations modelled either through massive dark matter halos from a
cosmological N-body simulation, or via toy models of Monte Carlo distribution
of substructures embedded in a triaxial Hernquist potential. As expected, the
angular power spectra of these two fields tend to have larger values for larger
harmonic numbers when substructures are accounted for and they can be
approximated by power-laws, whose values are fitted as a function of the
profile and the distribution of the substructures.
