Modal fields calculation using the finite difference beam propagation method

Abstract

Abstract-A method is described to construct modal fields for an arbitrary one-or two-dimensional refractive index structure. An arbitrary starting field is propagated along a complex axis using the slowly varying envelope approximation (SVEA). By choosing suitable values for the step-size, one mode is maximally increased in amplitude on propagating, until convergence has been obtained. For the calculation of the next mode, the mode just found is filtered out, and the procedure starts again. The method is tested for one-dimensional refractive index structures, both for nonabsorbing and for absorbing structures, and is shown to give fast convergence.

Efficient interface conditions for the semi-vectorial finite-difference beam propagation method

“…3 that there are fluctuations in the convergence, especially for the zeroorder EICs. Inspection of the EIC equations (27), (28) (zero-order) and (17) to (25) suggests that the fluctuations might be caused by the fact that the value of e will vary with variation of the grid spacing. A good test, therefore, would be a calculation with the interfaces midway between the grid points (i.e.…”

“…They used the FTBPM. This idea has been applied to the FDBPM using the power method for two-dimensional cross-sections [14][15][16] or the inverse iteration method (IIM) for one-dimensional cross-sections [17][18][19]. In this paper we apply the IIM to two-dimensional cross-sections, and the matrix equation will be solved using the alternating direction implicit (ADI) method [20][21][22].…”

Numerical solution method of nonlinear guided modes with a finite difference complex axis beam propagation method

“…In this subsection, the method for treating the linear case [17] is briefly summarized. We restrict ourselves to the twodimensional case and TE-polarization.…”

“…The idea is that the guided mode with the highest effective index gets the maximum increase in amplitude during propagation. They use as propagation scheme the Fourier Transform Beam Propagation Method (mBPM) [ 141-[ 161. In an earlier paper [17], we described the application the Finite Difference Beam Propagation Method (FDBPM) [18]- [20] for solving the modal field equation. The advantage of the FDBPM is its fast convergence.…”

Efficient semivectorial mode solvers

“…The inverse iteration method [28] was used to find the modes. The method works for lossless and absorbing structures [29] as well as for electric field intensity-dependent refractive-index profiles [30], [31]. The FDBPM gives much more accurate results for large index contrasts than the FTBPM (see, e.g., [32]- [34]), and, in contrast to the FTBPM, the continuity relations at steps in the refractive index may be incorporated in the FDBPM.…”