Perturbative signature of substructures in strong gravitational lenses
  In the perturbative approach, substructures in the lens can be reduced to their effect on the two perturbative fields $f_1$ and $\frac{d f_0}{d\theta}$. A simple generic model of elliptical lens with a substructure situated near the critical radius is investigated in details. Analytical expressions are derived for each perturbative field, and basic properties are analyzed. The power spectrum of the fields is well approximated by a power-law, resulting in significant tails at high frequencies. Another feature of the perturbation by a substructure is that the ratio of the power spectrum at order $n$ of the 2 fields $R_n$ is nearly 1. The ratio $R_n \simeq 1$ is specific to substructures, for instance an higher order distortion ($n>2$) but with auto-similar isophotes will result in $R_n \propto \frac{1}{n^2}$. Finally, the problem of reconstructing the perturbative field is investigated. Local field model are implemented and fitted to maximize image similarity in the source plane. The non-linear optimization is greatly facilitated, since in the perturbative approach the circular source solution is always known. Examples of images distortions in the subcritical regime due to substructures are presented, and analyzed for different source shapes. Provided enough images and signal is available, the substructure field can be identified confidently. These results suggests that the perturbative method is an efficient tool to estimate the contribution of substructures to the mass distribution of lenses. 
C. Alard