二维光子晶体波导中慢光的Dirichlet-to-Neumann映射方法计算

Computing Slow Light in Two Dimensional Photonic Crystal Waveguide by Dirichlet-to-Neumann Map Method

利用Dirichlet-to-Neumann(DtN)映射方法计算了三角形排列的二维单线缺陷光子晶体波导管中的慢光.首先利用六边形单元晶格的(DtN)映射构建三角形排列的超级晶体的DtN映射,然后利用超级晶体的DtN映射把求解Bloch传播模问题转化为求较小矩阵的线性特征值问题.单元晶格的DtN映射把单元晶格边界上的波动场映射成边界上的法向导数.利用单元晶格的DtN映射避免了在单元晶格内部的离散,只需在边界上计算,减少了未知数个数,得到的矩阵较小,从而大大减少了计算量.不管介质是否是色散的,利用超级晶体的DtN映射建立的特征值问题都是线性的.数值算例表明用DtN映射方法来计算波导管中的慢光是非常有效的.

In this paper,we use the Dirichlet-to-Neumann（DtN）map method to compute the slow light of two dimensional（2D） single line defect（W1） photonic crystal waveguide（PhCW）,where the W1 PhCW is formed by removing one row of air holes of 2D triangular lattice photonic crystal constructed on silicon slab.By using the DtN map of a supercell,the Bloch propagation mode is calculated as a small linear eigenvalue problem on two boundaries of the supercell,where the DtN map of a supercell is constructed by the DtN map of hexagon unit cells and the eigenvalue is a function of the Bloch wave vector.The DtN map of a unit cell is an operator that maps the wave field on the boundary of the unit cell to its normal derivatives.Using the DtN map of a unit cell,we can avoid the discretization inside the unit cell and only calculate on the boundary.This gives rise to a significant reduction in the total number of unknowns.Even if the material is dispersive,we can also obtain a small linear eigenvalue problem.Finally,we use numerical examples to verify the effectiveness of the method.