Abstract
An efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the unit cells is developed for accurate simulations of two-dimensional photonic crystal (PhC) devices in the frequency domain. 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 and it can be approximated by a small matrix. Using the DtN maps of the regular and defect unit cells, we can avoid computations in the interiors of the unit cells and calculate the wave field only on the edges. This gives rise to a significant reduction in the total number of unknowns. Reasonably accurate solutions can be obtained using 10 to 15 unknowns for each unit cell. In contrast, standard finite element, finite difference or plane wave expansion methods may require a few hundreds unknowns for each unit cell at the same level of accuracy. We illustrate our method by a number of examples, including waveguide bends, branches, microcavities coupled with waveguides, waveguides with stubs, etc.
Highlights
In recent years, photonic crystals (PhCs) [1,2,3] have been extensively studied both theoretically and experimentally, due to their unusual ability to control and manipulate light
The device is allowed to have a finite number of PhC waveguides that extend to infinity
Away from a finite domain, we have a few PhC waveguides that extend to infinity
Summary
Photonic crystals (PhCs) [1,2,3] have been extensively studied both theoretically and experimentally, due to their unusual ability to control and manipulate light. Even for two-dimensional (2D) problems, standard numerical methods for frequency domain formulations, such as the finite element method, often give rise to large linear systems that are complex, non-Hermitian, indefinite but sparse. The DtN maps of the unit cells have been used to develop efficient methods for computing band structures [21,22], waveguide modes [23], cavity modes [24] and transmission/reflection spectra [25,26,27] of finite PhCs. In this paper, the DtNmap method is extended to general boundary value problems for arbitrary 2D PhC devices in an infinite background PhC. The problems associated with PhC slabs are certainly very important, but they are not studied here
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