Stefan F. Helfert

Numerical techniques for modeling guided-wave photonic devices

Accurate modeling of photonic devices is essential for the development of new, higher performance optical components required by current and future high-bandwidth communications systems. This paper reviews several key techniques for such modeling, many of which are used in commercial design tools. These include several mode-solving techniques, the beam propagation method, the method of lines, and the finite-difference time-domain technique.

Efficient analysis of periodic structures

An algorithm which allows the analysis of optical periodic structures with a very large number of periods with minimum numerical problems is presented, For this purpose the stable impedance transfer is combined with Floquets theorem. Numerical results for a Bragg grating with up to 20000 periods are presented featuring the very moderate numerical effort.

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.

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.

The Method of Lines: A Versatile Tool for the Analysis of Waveguide Structures

Algorithms for the determination of eigenmodes and for the analysis of longitudinally varying structures are presented. The media may be anisotropic and the devices may consist of an arbitrary number of layers or waveguide sections, respectively. Starting with generalized transmission line equations, suitable expressions are derived which are implemented in the method of lines, a semianalytical algorithm. To ensure the numerical stability, a transformation of the reflection coefficient is performed. Numerical results are presented for a Bragg-grating with a very high number of periods, for an electro-optical switch, and for an anisotropic groove waveguide. The results prove stability of the algorithm, and comparisons with other methods show the accuracy.

Comprehensive modeling of vertical-cavity laser-diodes by the method of lines

A comprehensive numerical model for the analysis of vertical-surface emitting-lasers is presented. An optical, electrical, and thermal submodel are introduced. The complete analysis is based on the method of lines. The temperature distribution and the current density are calculated in the whole structure. The optical behavior is investigated with a full vectorial wave equation in cylindrical coordinates. Multimode effects are considered when calculating the optical output power. A rate equation for electrons and holes is used, which includes diffusion and recombination effects inside the quantum well. By combining all submodels a self consistent solution is found.

A comparison between different propagative schemes for the simulation of tapered step index slab waveguides

The performance and accuracy of a number of propagative algorithms are compared for the simulation of tapered high contrast step index slab waveguides. The considered methods include paraxial as well as nonparaxial formulations of optical field propagation. In particular attention is paid to the validity of the paraxial approximation. To test the internal consistency of the various methods the property of reciprocity is verified and it is shown that for the paraxial algorithms the reciprocity can only be fulfilled if the paraxial approximation of the power flux expression using the Poynting vector is considered. Finally, modeling results are compared with measured fiber coupling losses for an experimentally realized taper structure.

Numerical stable determination of Floquet-modes and the application to the computation of band structures

Periodic structures like Bragg-gratings are important components of optical circuits. The analysis of these devices can be done very efficiently by combining eigenmode propagation methods with Floquets theorem. A particular problem is the determination of the Floquet modes. Transfer matrix formulas are stable only in case of low losses and when the length of the periods is not too big. A stable method, also in the mentioned cases, is presented in this paper. Reflection coefficients are transformed from the output of a periodic segment to its input and the fields are computed in opposite direction. By this, the exponential increasing terms, which lead to the numerical problems are avoided. The formulas are applied to determine eigenmodes in various waveguide structures. Particular periodic structures are photonic crystals (PC), who have very promising features. For tailoring these PCs the knowledge of the band structure is required. With the Floquet modes that have been determined before this band structure has been calculated. A comparison with the literature showed a very good agreement.

Three-dimensional vectorial analysis of waveguide structures with the method of lines

The method of lines (MoL) a special eigenmode algorithm has been proven as an efficient tool for the analysis of waveguide structures in optics and microwaves. The electric and magnetic fields in the cross-section and their derivatives with respect to the cross-section coordinates are discretized with finite differences (FD) while analytic expressions are used in the direction of propagation. The numerical effort for analyzing three-dimensional structures with a two-dimensional discretization can be very high, particularly if vectorial characteristics have to be taken into account. In this paper we introduce a reduction of the eigenmode system to keep the effort moderate. Only a certain number of eigenmodes is determined with the Arnoldi algorithm. We will show then how the electric field distribution of the eigenmodes can be computed from the magnetic field and vice versa. To match the fields at the interfaces we introduce left eigenvectors which are the inverse of the field distributions. The formulas were applied to the analysis of a polarization converter consisting of a periodical perturbation of a waveguide structure. A rotation angle greater than 80° was determined.

Finite difference expressions for arbitrarily positioned dielectric steps in waveguide structures

The finite difference expressions for arbitrary positions of the dielectric steps in waveguide structures are derived and inserted into the beam propagation algorithm based on the method of lines (MoL). The accuracy of these expressions is tested by calculating the effective indexes of slab waveguides and comparing the results with analytic solutions. Tapered waveguides are analyzed as an application of this new approach. The propagation behavior of the field is presented and the modal power loss is compared with other methods.