Effective Medium Properties Of 2D Finite Photonic Crystal

Since the concept of "photonic crystal" was proposed separately by E.Yablonovitch and S.John in 1987, this special material has been paid much attention to and a lot of researches about photonic crystal have been made. In this thesis, the scattering matrix method is used to study the effective group velocity refractive index of 2D finite structure.The scattering matrix method is very effective for the studying of 2D finite structure. We start from the scattering matrix of one single dielectric and metallic cylinder, then study the scattering matrix of a line of cylinders, and then study the scattering matrices of finite and periodic structures. Based on the above work, the scattering matrix method is set up for studying the electromagnetic properties of 2D finite and periodic structures.First, we study the band structure of 2D periodic structure. The band structures of 2D photonic crystal composed of dielectric and metallic cylinders are plotted. The results show that: for square photonic crystal composed of dielectric cylinders, there exists a complete photonic band gap for the TM mode and a directional band gap for the TE mode; for square photonic crystal composed of metallic cylinders, there exists a cut-off frequency for the TM mode and a flat band for the TE mode. The TM mode of square photonic crystal composed of dielectric cylinders is studied. The results show that: as the dielectric constant increases, the two edges of the band gap decrease, and the bandwidth reaches a maximum and then it decreases slightly; as the radius of the cylinder increases, the two edges of the band gap decrease, and the bandwidth reaches a maximum and then it decreases.Second, we study the eigen mode and effective medium property of 2D finite periodic structure. Dispersion relations and transmission spectrums are studied and the results show that: the locations of the band edges in the dispersion relation will hardly change with the variation of the number of layers when there are enough layers; the larger the number of layers is, the larger the number of modes inside the band will be; the band edges correspond with the curves of the band structure; the effective group velocity refractive index will hardly change with the number of layers when there are enough ones; the effective refractive index is not the same in different bands and incident angles, that is to say, the structure is equivalent to a dispersive medium of isotropy.At last, we study the eigen mode of 2D cascaded finite periodic structure. The waveguide mode of the structures is studied and the results show that: there exist waveguide modes in the photonic band gap and the wider the waveguide is, the larger the number of waveguide modes will be; the larger the eigenfrequency of the waveguide mode is, the larger the quality factor will be; the thicker the wall of the waveguide is, the larger the quality factor will be. So we can conclude that the structure is in fact a good waveguide.