Modification of the finite difference scheme for efficient analysis of thin lossy metallayers in optical devices

A method is proposed for the analysis of optical devices with lossy metal layer by including an analytical formulation into the finite difference expressions. Losses of TM0 mode in an electro-optic switch for wavelength 1523 nm, exhibiting a sharp maximum of 37 dB are computed by this method. The results show very good agreement with the data obtained by the Mode Matching Method and Finite Difference Beam Propagation Method.     

A finite difference beam propagation algorithm based on generalized transmission line equations

A wide angle beam propagation algorithm is presented, which is based on generalized transmission line (GTL) equations. Besides the discretization, no further approximation is introduced. In principle, a full vectorial analysis is possible, and anisotropic material can be taken into account. The algorithm has been verified for planar 2D-waveguide devices. The wide angle characteristic has been examined by determining the propagation of tilted Gaussian beams in homogeneous media. Further studies were performed for tilted waveguides and for taper structures. Results for the latter devices were compared with other methods showing a very good agreement.     

Modelling of self-aligned total internal reflection waveguide mirrors: an interlaboratory comparison

Results of modelling of light propagation in 45° self-aligned total internal reflection rib waveguide mirrors on InP substrate are compared. Six laboratories participated in the comparison with the following six modelling methods: the standard fast-Fouriertransform beam propagation method (BPM), the standard finite-difference (FD) BPM using the Crank-Nicholson scheme (two laboratories), the FD-BPM with the correction for the slowly varying envelope approximation, the method of lines, the eigenmode expansion and propagation method, and a simple method based on the field overlap. All the laboratories used the effective-index method to reduce the three-dimensional problem to two dimensions. The differences among the results obtained by different methods are briefly discussed and qualitatively compared to measured values.     

Determining reflections for beam propagation algorithms

Reflections occur whenever different waveguide sections are concatenated. In this paper, we will give approximate formulas for determining the reflection coefficient. Starting with generalized transmission line equations we will derive general formulas where suitable approximations are introduced. These approximations allow the determination of the reflections with low numerical effort. The expressions were implemented in an FD-BPM algorithm to determine the field distribution in waveguide devices. For an application in CAD systems the reflection coefficient of the fundamental mode was computed with the approximate formulas. Numerical results were obtained for various concatenations of different waveguide sections. A comparison with the method of lines, a semi-analytic algorithm, shows a good agreement.     

Publisher's Note

When the article 'Modification of the finite difference scheme for efficient analysis of thin lossy metal layers in optical devices' by O. Conradi, S. Helfert, and R. Pregla was first published in the Optical Waveguide special issue of Optical and Quantum Electronics (volume 30, nos. 5–6, pages 369–373) the authors' affiliation given was incorrect. The article is printed again in full with the original pagination and credit lines. A misprint in the Results section has been corrected as well: the refractive index of both silver layers is 0.14–j11.0 at a wavelength of = 1523 nm, and not 0.41–j11.0.     

Analysis of a deep waveguide Bragg grating

Spectral properties of a very deep Bragg grating operating in a first diffraction order on a single-mode planar waveguide have been studied theoretically. It is shown that the scattering loss can be low (a few percent), the reflectivity very high (over 90%), the reflection band is shifted against the Bragg wavelength toward the shorter wavelengths, and its spectral shape is very different from that of a shallow grating. Inside a reflection band, a part of the input optical power penetrates through the grating even if it is infinitely long. These properties are predicted by modelling using two independent computer codes based on different modelling methods, namely the bi-directional mode expansion and propagation method (BEP), and a method of lines (MoL). The first method is discussed in some detail here. The work has been performed within the framework of European Action COST 240.     

Bragg waveguide grating as a 1d photonic band gap structure: COST 268 modelling task

Modal reflection, transmission and loss of deeply etched Bragg waveguide gratings were modelled by six European laboratories using independently developed two-dimensional (2D) numerical codes based on four different methods, with very good mutual agreement. It was found that (rather weak) material dispersion of the SiO₂/Si₃N₄ system does not significantly affect the results. The existence of lossless Floquet-Bloch modes in deeply etched gratings was confirmed. Based on reliable numerical results, the physical origin of out-of-plane losses of 1D or 2D photonic band gap structures in slab waveguides is briefly discussed.     

Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task

Modal reflection, transmission and loss of deeply etched Bragg waveguide gratings were modelled by six European laboratories using independently developed two-dimensional (2D) numerical codes based on four different methods, with very good mutual agreement. It was found that (rather weak) material dispersion of the SiO₂/Si₃N₄ system does not significantly affect the results. The existence of lossless Floquet–Bloch modes in deeply etched gratings was confirmed. Based on reliable numerical results, the physical origin of out-of-plane losses of 1D or 2D photonic band gap structures in slab waveguides is briefly discussed.     

Novel subgridding technique for the analysis of optical devices

A novel subgridding algorithm for the Method of Lines is presented. It enables to refine a mesh in subgridded regions by any positive integer number. High-accuracy of the proposed algorithm is achieved by second-order finite difference operators. The interpolation of the missing field components at the interface coarse/fine grid is performed with second-order accuracy as well. Numerical results are provided to check the accuracy of the proposed formulation.