A Rigorous Analysis of Electromagnetic Scattering from Gratings of Metallic Cylinders with Arbitrary Cross Section

摘要

Photonic crystals, which are composed of periodic arrangement of dielectric or metallic scatterers, have received much attention since a pioneering work by Yablonvitch. The periodic structures which consist of dielectric and metallic objects in particular arrangement, can be used as negative refractive index materials. The new materials are expected to apply as new devices in the both fields of microwaves and optical waves. The electromagnetic scattering and guidance in periodic structures has been an important subject in the field of microwaves and optical waves, and various analytical or numerical techniques have been developed during the past decades. These approaches have been reexamined or new approaches have been proposed to deal with the electromagnetic problems related to photonic crystals. Recently, extensive theoretical studies have been devoted to two-dimensional photonic crystals consisting of parallel circular cylinders, using the cylindrical harmonic expansion method, the Fourier modal method, the finite element method, the differential method, and a time domain technique. In this context, Kushta and Yasumoto have proposed an analytical approach using the cylindrical harmonic expansion technique combined with the lattice sums, T-matrix, and generalized reflection matrix. The method is very effective and yields quite accurate solutions when the periodic scatterers consist of circular cylinders. However, it must rely on a numerical technique for calculating the T-matrix when the cross section of scatterers is not circular one. In this paper, we shall present a rigorous method for two-dimensional photonic crystals composed of parallel metallic rods with arbitrary cross-section. Such structures are widely used in microwave region in order to improve antennas performance and as negative refractive index materials. The proposed method treats the photonic crystals as a multilayered system of metallic gratings. The metallic rods with arbitrary cross section in each grating layer are sliced into a stacking sequence of rectangular cross-section. First the reflection and transmission matrices for a metallic lamellar grating with rectangular cross-section are calculated using the mode-matching technique. The results are stacked to derive the reflection and transmission matrices for a single-layer grating of metallic rods with original cross-section. The generalized reflection and transmission matrices of the photonic crystals are obtained by concatenating those matrices, of each grating layer over the entire layered system. Then the reflection and transmission characteristics of the photonic crystals composed of metallic rods -with arbitrary cross section can be calculated by a simpler matrix operation. Numerical examples are presented to confirm the fast convergence and high accuracy of the proposed method.