正入射时二维空气孔结构正方形和三角形光子晶体的带隙结构

 利用平面波展开法推导出光在光子晶体中传播时的本征方程及其矩阵表达形式，数值求解了圆形空气孔结构的正方形光子晶体和三角形结构光子晶体的色散曲线，给出了不同空气孔结构下的色散曲线。计算分析发现，在<11>方向上，二维正方形结构的光子晶体E波的第一带隙与H波的第一带隙完全分开。在<10>方向上，E波的第一带隙比H波略窄，两者的变化趋势也不尽相同。对E波来说，第一带隙随着空气孔径的增大而增宽；但对H波来说，第一带隙先增宽再变窄。与正方形结构光子晶体不同的是，二维三角形结构光子晶体在<11>方向的第一带隙边缘位置并不在M（1，1）点，而是出现在（0.5，0.5）点，其中心频率比<10>方向的小，但其带隙宽度较正方形结构的大。在<10>方向上，E波的第一带隙比H波的略宽，且两者的变化趋势完全相反。对E波来说，第一带隙先增宽再变窄；但对H波来说，第一带隙随着空气孔径的增大而增宽。

Band Structure of 2D Air-hole-type Square and Triangular Photonic Crystal Under Vertical Incidence

The eigenmatrix for the propagation of light in a photonic crystal is obtained by the plane-wave expansion method. Dispersion curves of photonic crystals of two dimensional square and triangular lattices composed of circular air cylinders are calculated numerically under vertical incidence, together with the dispersion curves of two dimensional photonic crystals with different radii of holes. It is shown that the first band gap of E wave of the two dimensional square lattice divides in direction <11>, and so does that of H wave; in direction <10>, the first band gap of E wave is a little smaller than that of H wave and becomes wider with the increase of the radius of the hole; on the other hand, that of H wave becomes wider first then decreases. The edge of the first band gap in direction <11> of two dimensional triangular lattice, different from that of the square lattice, is at point (0.5, 0.5) instead of at point (1, 1); the center frequency is smaller than that in direction <10> and the band gap is larger than that of the square lattice. In direction <10>, the first band gap of E wave is a little wider than that of H wave, and their variation trends are completely different: the first band gap of E wave becomes wider with the increase of the radius of the hole at the first and then decreases, but that of H wave is getting wider all the time.