Three-Dimensional Finite-Element Mode-Solver for Nonlinear Periodic Optical Waveguides and Its Application to Photonic Crystal Waveguides

A rigorous definition of nonlinear parameter γ and effective area A_eff for photonic crystal optical waveguides

“…To avoid confusion, we call each method "Method A, B, C, D-1, D-2, E and F", and the descriptions and equations are summarized in Table 1. The 3D finite-element modesolver [19] developed in our laboratory is used. For Methods A, B, C, D-1, and D-2, the analysis can also be done by using commercial software.…”

“…We proposed a method to directly estimate the nonlinear parameter using the 3D-FEM for nonlinear periodic optical waveguides, based on the self-consistent algorithm [19]. The nonlinear parameter and effective area obtained by the nonlinear modal analysis are defined as…”

“…γ is mainly related to SPM, whereas some of the thirdorder nonlinear phenomena, including XPM, ThPA and FWM, can also be evaluated by γ [12,14,16,17]. There are two ways to evaluate γ theoretically: (i) calculate the nonlinear phase shift by directly solving the nonlinear Maxwell's equation self-consistently using the nonlinear mode solver [18,19] and (ii) calculate the effective area, A eff , of the "linear" optical waveguide and use the definition γ = k 0 n 2 /A eff [14], where k 0 is the free-space wavenumber and n 2 is the nonlinear refractive index. The method (i) seems to be physically more rigorous than method (ii) and has been used to evaluate the nonlinear phase shift of uniform and periodic optical waveguides.…”

“…We first describe the proposed and previous definitions of γ briefly in Section 2. Next, we compare the values of γ obtained by the proposed definition and conventional definitions for an optical fiber, a Si-nanowire waveguide, and dispersion engineered PC waveguides using the three-dimensional (3D) finite element method (FEM) [19]. From the numerical results presented in Section 3, it is demonstrated that γ values calculated by the proposed definition agree well with those obtained by the nonlinear mode solver and the experiment for all three waveguide structures (weakly guiding, strongly guiding, and strongly guiding periodic waveguides), showing the validity of the definition.…”

Plasmonic and photonic crystal applications of a vector solver for 2D Kerr nonlinear waveguides

“…In the low power regime, it has already been shown that, in the lossless case, the conventional way of computing γ nl [22] and equation ( 4) coincide [21,24]. Moreover, it was recently shown that the definition given by equation ( 4) is more general and valid even for high lossy waveguides, while the conventional definition γ nl ≈ k 0 n 2 /A ef f is not valid for lossy waveguides [23].…”