Reflection and transmission properties of multilayered crossed arrays of circular cylinders

In this paper, we shall discuss a rigorous analytical approach for the three-dimensional electromagnetic scattering from layered crossed-arrays of circular cylinders. The cylinders in each layer are infinitely long and parallel, while the cylinder axes are rotated by 90° in each successive layer. The layered system consists of a stacking sequence of the two orthogonal arrays. First, the reflection and transmission matrices are derived for each of two arrays by employing the electric and magnetic fields parallel to the cylinder axes as the leading fields. The matrices are expressed in terms of the one-dimensional lattice sums, and the three-dimensional T-matrices of the isolated single cylinder for obliquely incident plane waves. Next, a coordinate transformation of the reflection and transmission matrices is introduced to one of the two arrays so that the multiple scattering process can be described using the same leading fields. Then the results for the two arrays are linked to obtain the reflection and transmission matrices of the crossed-array which constitutes a unit cell in the stacking sequence. Finally, the generalized reflection and transmission matrices for the layered crossed-arrays are calculated by concatenating those of the crossed-array successively over the repeated number of cells. Thus the three-dimensional electromagnetic scattering from the layered crossed-arrays can be calculated by a simpler matrix operation on the basis of the one-dimensional lattice sums and the three-dimensional T-matrix of the isolated single cylinder. The numerical examples demonstrate the existence of a complete bandgap in which unpolarized electromagnetic waves with arbitrary angle of incidence are completely reflected.