  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.
