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用平面波展开法研究了介质矩形柱二维正方格子光子晶体的能带结构,正方晶格的晶格常数为a,当介质矩形柱边长为0.3633a,相对介电常数为11.56时,光子晶体的能带在布里渊区的中心点获得一个三重简并态,包括在同一频率接触构成狄拉克锥的两个线性能带,一个穿过狄拉克点的平的能带,光子晶体等效为介电常数和磁导率都为零的零折射率材料。用有限元法仿真电磁波在光子晶体中的传输特性,仿真结果表明,狄拉克点频率的电磁波通过光子晶体时相位不发生变化,传输特性与零折射率材料相同,用这种光子晶体设计的凹透镜可以实现焦点的半峰全宽为波长λ的0.63倍,焦距为λ的4.12倍的亚波长聚焦。
The energy band structure of two-dimensional square lattice photonic crystals with rectangular lattice is studied by the plane wave expansion method. The lattice constant of the square lattice is a. When the edge length of the rectangular column of the medium is 0.3633a and the relative dielectric constant is 11.56, the photonic crystal Of the energy band to obtain a triple degenerate state at the center of the Brillouin zone, including contacting two linear bands of the Dirac cone at the same frequency, a flat energy band across the Dirac point, photonic crystal equivalent Zero-index material with zero dielectric constant and permeability. The finite element method is used to simulate the propagation characteristics of electromagnetic waves in photonic crystals. The simulation results show that the phase of Dirac point-frequency electromagnetic wave does not change when passing through the photonic crystal. The transmission characteristics are the same as that of the zero-refractive index material. The full width at half maximum of the focus can be achieved with a focal length of 4.63 times the wavelength of λ of 0.63 times the sub-wavelength of λ.