The Algorithms For Periodic Nonuniform Waveduides Based On DtN Maps

Computation of waveguides plays an important role in researches in integrated optics. Some M operator algorithms and the inverse of fundamental solution (IFS) operator algorithm based on Dirichlet-to-Neumann (DtN) maps are developed for periodic nonuniform waveguides structures of more applications in this paper. Compared with the traditional algorithms, these algorithms change the boundary value problems into the’initial’value problems of operators by introducing the DtN maps. Furthermore, the algorithms make the best of the waveguides’geometry structures and construct the marching schemes through introducing intermediate variables that only depend on the waveguide structures, which can reduce greatly the calculation amount for the peridic structures. Compared with the Chebyshev collocation method, the M operator algorithm approximates the 2nd order derivative through using a 4-order difference scheme with three points. It produces a block tridiagonal coefficient matrix and makes the method more efficient. And it decreases errors by taking the advantage of the original equation to deal with the boundary rather than numerical approximation. When the smoothness of refractive indexes of the media is not good enough, the method using the Riccati method to get the IFS operator is better. Those algorithms are developed to solve the non-homogeneous Helmholtz equation, but they also fit the homogenegous situations. Numerical examples show that those algorithms are efficient.