基于Dirichlet-to-Neumann映射方法计算二维光子晶体在布里渊区中心处的狄拉克点

Computing Dirac Points of Two-dimensional Photonic Crystal at the Brillouin Zone Center by Dirichlet-to-Neumann Map Method

光子晶体狄拉克点是光子晶体能带结构在布里渊区上形成具有线性色散关系的双锥形结构的交点.通常狄拉克点出现在布里渊区的边界上,近年来研究发现光子晶体狄拉克点也可以出现在布里渊区的中心处,而此时的光子晶体具有非常独特的性能,可以被制造成零折射率材料.利用Dirichlet-to-Neumann(DtN)映射方法计算二维光子晶体在布里渊区中心处的狄拉克点.固定圆柱的半径,改变圆柱介质的介电常数,通过寻找能带结构在布里渊区中心处的交点,来确定光子晶体在布里渊区中心处的狄拉克点.利用DtN映射方法只需在单元晶格的边界离散,使得在每一步计算中,求解的都是较小矩阵的线性特征值问题.最后数值算例验证了利用DtN映射方法来计算二维光子晶体在布里渊区中心处的狄拉克点是有效的.

A Dirac point of a photonic crystal is the touching point of the double-cone structure which is formed by two bands with linear dispersion at this point.The Dirac points of photonic crystals always appear on the boundary of the Brillouin zone.Recently,it has been found that the Dirac points can also appear at the center of the Brillouin zone.Under the circumstance,the photonic crystals have unusual properties that can be used to design zero refractive index materials.In this paper,we use the Dirichlet-to-Neumann（DtN）map method to calculate the Dirac points at the center of the Brillouin zone for two-dimensional photonic crystals.We fix the radius of the cylinder and change the dielectric constant of the cylinder to search for the touching point of the band structure at the center of the Brillouin zone.By using the DtN map method,we only need to discretize on the edges of a unit cell.Therefore,we solve linear eigenvalue problems of small matrices in each step.Finally,we use numerical examples to illustrate the efficiency of the method.