Goal oriented adaptive finite element method for precise simulation of optical components

Abstract

Adaptive finite elements are the method of choice for accurate simulations of optical components. However as shown recently by Bienstman et al. many finite element mode solvers fail to compute the propagation constant's imaginary part of a leaky waveguide with sufficient accuracy. In this paper we show that with a special goal oriented error estimator for capturing radiation losses this problem is overcome.

Finite Element Simulation of Radiation Losses in Photonic Crystal Fibers

“…Computational results agree well with experimental transmission spectra measured in [5]. In the context of topics related to the DFG priority programme photonic crystals, we have applied adaptive finite element algorithms to a variety of nano-optical problems [8,9,10,11,12,13,14,6,15].…”

“…These residuals are computed for each patch and only patches with the largest residuals are refined. More details about the mathematical formulation and implementation can be found in [15]. A closer look to ρ(E h ; ·) which is also used for field-energy based adaptive refinement will be given in the next section.…”

Advanced FEM analysis of optical waveguides: algorithms and applications

“…[11][12][13] The results which are presented here are a summary of our work published in. 14,15 The computational domain and triangulation of a HCPCF was already shown in Fig. 2(a) and (b).…”

“…It can be shown from Maxwell's equations that the imaginary part of k z is proportional to the power flux of the electric field across the boundary Γ of the computational domain. 15 This is a quantity one often wants to minimize in application. We will quantify the radiation losses by the imaginary part of the effective refractive index (10).…”

Finite element simulation of radiation losses in photonic crystal fibers

“…it is goal-oriented. The target functional depends on the solution of the electric field E. Since we are interested in radiation losses this will be the imaginary part of the propagation constant [15]. From Maxwell's equations applied to a waveguide structure one can derive the following expression for the imaginary part of the propagation constant:…”