Abstract
We develop an efficient numerical method for computing defect modes in two dimensional photonic crystals based on the Dirichletto- Neumann (DtN) maps of the defect and normal unit cells. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative. The frequencies of the defect modes are solved from a condition that a small matrix is singular. The size of the matrix is related to the number of points used to discretize the boundary of the defect cell. The matrix is obtained by solving a linear system involving only discrete points on the edges of the unit cells in a truncated domain.
Highlights
In a photonic crystal (PhC) [1] with a point defect, defect modes [2, 3, 4] may exist for some frequencies in bandgaps
The defect modes can be calculated in time domain using the finite difference time domain (FDTD) method [16, 17, 18, 19, 20] or finite element time domain method [21, 15]
We have developed an efficient numerical method for calculating defect modes for two-dimensional (2D) photonic crystals (PhCs) with defects
Summary
In a photonic crystal (PhC) [1] with a point defect, defect modes [2, 3, 4] may exist for some frequencies in bandgaps. The defects modes can be calculated by the plane wave expansion method using the supercell approach that assumes periodic boundary conditions on the boundary of the truncated domain [6, 7]. The Dirichlet-to-Neumann (DtN) maps of unit cells have been used to develop efficient numerical methods for analyzing photonic crystal problems [23, 24, 25, 26, 27, 28]. The DtN map allows us to reduce the computations to the edges of the unit cells for boundary value problems such as those related to finite PhCs [25, 26, 27].
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