Abstract
Algorithms for effective modeling of optical propagation in three- dimensional waveguide structures are critical for the design of photonic devices. We present a three-dimensional (3-D) wide-angle beam propagation method (WA-BPM) using Hoekstra's scheme. A sparse matrix algebraic equation is formed and solved using iterative methods. The applicability, accuracy and effectiveness of our method are demonstrated by applying it to simulations of wide-angle beam propagation, along with a technique for shifting the simulation window to reduce the dimension of the numerical equation and a threshold technique to further ensure its convergence. These techniques can ensure the implementation of iterative methods for waveguide structures by relaxing the convergence problem, which will further enable us to develop higher-order 3-D WA-BPMs based on Padé approximant operators.
Highlights
The classical finite-difference beam propagation method (FD-BPM) [1,2,3,4,5] is a technique for calculating the electromagnetic field transmitted through materials with inhomogeneous refractive index profiles
The field on the following plane is calculated based on a numerical solution of the scalar Helmholtz equation using the Crank-Nicholson scheme [6], and solved using the highly efficient Thomas algorithm [6] along with transparent boundary conditions (TBC) [7]
Hadley has developed wide-angle beam propagation method (WA-BPM) using the Padé approximant operators and the multistep method, which are widely used for 2-D structures, which, the author has claimed to be restricted for 3-D structures due to the lack of efficient solvers [16, 17]
Summary
The classical finite-difference beam propagation method (FD-BPM) [1,2,3,4,5] is a technique for calculating the electromagnetic field transmitted through materials with inhomogeneous refractive index profiles. A 3-D wide-angle BPM was recently developed using the efficient Thomas algorithm, Hoekstra’s scheme and the splitting of the 3-D Fresnel wave equation into three 2-D wave equations [20]. Both this splitting method and the ADI method may cause splitting errors. The coefficients matrix of the algebraic equation of this method is not tridiagonal but large and sparse, and its dimension (number of elements) is usually on the order of >108 In this case, use of direct matrix equation solvers is prohibitively expensive and in some cases impossible even with the best available computing power. In order to overcome this problem, we introduce a simulation window shifting scheme to decrease the dimension of the equation and a threshold technique to reduce the volatility of field distribution b
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