The Scattering Matrix Method (SMM), introduced by **Maystre**** and co-workers in 1994**, is an analytical approach for the evaluation of the field scattered by a random set of parallel cylinders in free space. Basically, that approach relies on the representation of the electromagnetic properties of the scatterers in terms of scattering matrices linking the coefficients of a suitable representation basis for the incident and scattered field. By so doing, the scattering problem can be solved through a system of linear equations in which the right-hand sides take into account the kind of incident field (e.g., plane or cylindrical wave) and the matrix relating data-unknowns embeds the mutual interactions between scatterers. It is worth noting that the computational complexity of the SMM grows with the sum of the perimeters of the different scatterers (and not with the sum of their areas); hence, this analysis tool allows a large saving in the number of unknowns with respect to mesh-based solvers.

By taking profit of the SMM advantages listed above, LEMMA researchers investigated the possibility of dealing with i-Design of EM devices by a reformulation of the original tool. In particular, the SMM can be turned into a design tool by considering the scattered field as a (proper assigned) specification while the EM properties of the scatterers become the unknowns of the problem. Thanks to this inverse formulation of the SMM (i-SMM), the design of artificial material (AM) based devices can be pursued in a more efficient fashion.

The i-SMM has been used to design AM-based lens antennas [1] as well as EBG structures [2].

Recently, the LEMMA group has started to explore the peculiarity of the scattering matrices for scenarios involving scatterers in presence of discontinuities, in particular, PEC planar surface. As a step towards i-Design, the pertaining forward problem has been treated first, such to establish a fast still effective method to compute the field associated to each configuration of scatterers and a given (plane wave) incident field. The proposed approach [3],[4] still allows the adoption of the free-space scattering matrix of each scatterers. More in detail, the presence of the planar PEC is handled through “image matrices” and the contributions due to the PEC are incorporated in the linear system as field radiated by the fictitious scatterers plus the reflection of the impinging plane wave.

[1] R. Palmeri and T. Isernia, “Inverse design of artificial materials based lens antennas through the scattering matrix method,” Electronics, 9(4), p. 559, 2020. click here

[2] R. Palmeri and T. Isernia, “Inverse design of EBG waveguides through scattering matrices,” EPJ Applied Metamaterials, 7(10), 2020. click here

[3] G.M. Battaglia, R. Abdullin, T. Isernia, A.F. Morabito, L. Crocco, and R. Palmeri, “Scattering from cylinders parallel to a perfectly conductive plane: an efficient analysis tool”, *4th IEEE URSI Atlantic Radio Science Meeting (AT-RASC)*, 2024. click here

[4] R. Abdullin, G.M. Battaglia, A.F. Morabito, T. Isernia, L. Crocco, and R. Palmeri, “A Scattering Matrix Method for Parallel Cylinders Above a Planar Perfect Electric Conductor”, *IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (AP-S/URSI),* 2024. click here