## 1 Introduction
The operator relating the external scattered field to the internal dielectric contrast can be expanded in a Taylor operatorial series. In particular, the scattered field depends on the contrast and on the internal field. The internal field, in turn, satisfies a Fredholm equation that can be expanded into the well-known Neumann series and then substituted into the external equation [23; 24]. Consideration of only the first term implies linearization, whereas keeping the first two terms of the series brings the quadratic approximation that was considered in the following. In this sense, the quadratic approximation represents the lower degree of non-linearity. From the physical point of view, linearization corresponds to considering each portion of the investigation domain as if it were standing alone in free space, whereas the quadratic approximation can be interpreted as keeping a mutual coupling between different parts of the investigation domain. Similarly, cubic, quartic, and higher order terms would consider \"multiple\" interactions, which are neglected in the present paper. There are many different scattering models that allow for the obtaining of algorithms suitable for the reconstruction of desired unknowns [25]. Focusing algorithms, such as backpropagation and backprojection, and approximations, such as the well-known Born, distorted Born, Rytov, and Kirchhoff, fall within the class of linear models of scattering [5]. Linearity allows for dealing with inverse problems in which the attainable resolution can be forecast, and well-assessed techniques against noise are at hand. However, linear models neglect mutual interactions inside the structure and result in the qualitative reconstruction of the region under investigation [26]. On the other hand, preserving the full non-linearity results in algorithms that usually require a huge computational effort, are often solved using non-deterministic procedures, and do not guarantee global convergence to the solution [26]. To avoid such drawbacks, strategies that provide a moderate degree of non-linearity are attractive and worth investigating [27]. In this framework, a quadratic approximation of the scattering phenomenon is demonstrated to be a good compromise between the full non-linear model and the linearized one [28; 29]. Additionally, a formulation that includes all the a priori information can be helpful when dealing with partially known structures. This approach has been considered in [9; 10] in which a linear approximation was adopted. Another element that must be considered is measurement configuration. Full-view illumination allows for the collection of more information than a limited aspect one. However, sometimes the structure is not accessible from all around, and only reflection-mode illumination, which is typical, for instance, of ground-penetrating radar applications, is allowed. The approach proposed here was already briefly presented in the conference paper [30]. In the present extended paper, the analytical details of the proposed model are explicitly derived, and a critical study of the operator is performed. Moreover, several new examples are presented, and the role of the threshold in the detection is shown. The problem was formulated in canonical two-dimensional geometry: metallic bars embedded into a lossless dielectric homogeneous medium, and in turn, embedded in free space. The electric field polarization was assumed to be parallel to the vertical bars, thus providing a scalar problem. An \"almost\" quadratic approximation of the problem was derived, which allowed for the achievement of, for the geometry at hand, a synthetic formulation where some of the involved coefficients can be evaluated analytically, thus reducing the computational burden. The data of the problem were the multi-monostatic scattered fields observed at different frequencies. The structure was tested using incident fields radiated by filamentary electric sources placed at different positions along a straight segment outside the investigated region.