Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps

Abstract

An efficient numerical method is developed for modal analysis of twodimensional photonic crystal waveguides. Using the Dirichlet-to-Neumann (DtN) map of the supercell, the waveguide modes are solved from an eigenvalue problem formulated on two boundaries of the supercell, leading to significantly smaller matrices when it is discretized. The eigenvalue problem is linear even when the medium is dispersive. The DtN map of a domain is an operator that maps the wave field on the boundary of the domain to the normal derivative of the field. The DtN map of the supercell can be efficiently calculated by merging the DtN maps of the ordinary and defect unit cells.

Efficient Numerical Modeling of Photonic Crystal Heterostructure Devices

“…Since then, the method has been extended to analyze PhC waveguides [36], microcavities [37], and general PhC devices [33]. In the following, we give a brief outline of the DtN-map method for analyzing PhC waveguides.…”

Dirichlet-to-Neumann Map Method with Boundary Cells for Photonic Crystals Devices

“…Using the DtN maps of the unit cells, we can reduce various mathematical problems for PhCs to smaller problems on the edges of the unit cells, avoiding the interiors of the unit cells completely. In earlier works, the DtN map technique has been applied to eigenvalue problems such as band structures [14,16], waveguide modes [17] and cavity modes [18], and it has also been applied to boundary value problems involving PhCs of finite thickness [15,19,20] and PhC devices with an infinite PhC background [21,22]. In this paper, we extend the DtN map method to PhC devices in a finite PhC surrounded by a homogeneous medium.…”

Efficient numerical method for analyzing photonic crystal slab waveguides

“…Recently, some efficient numerical methods based on the Dirichlet-to-Neumann (DtN) maps of the unit cells have been developed to analyze waveguides in ideal 2D PhCs which have an invariant direction [10,11]. The DtN map of a unit cell is an operator that maps the wave fields to their normal derivatives on the cell boundary, and it can be approximated by a small matrix.…”

“…The DtN map of a unit cell is an operator that maps the wave fields to their normal derivatives on the cell boundary, and it can be approximated by a small matrix. Similar to the methods based on the scattering matrices [8,9], the DtN map method solves eigenvalue problems for fixed frequencies, and it gives either a linear eigenvalue problem with relatively small matrices [10] or a nonlinear eigenvalue problem with even smaller matrices [11]. Notice that the DtN map method was first used to calculate transmission and reflection spectra for PhCs which are finite in one periodic direction [12].…”