A bstract-A vector finite element method with the high-order mixed.lnterpolation·type triangular elements is described tor the analysis ot optical waveguiding problems. It is a combination at linear edge elements tor transverse components of the electrk or magnetic field and quadl"8.lic nodal elements ror the axial one. The use ot mixed-interpolation.type elements provides a diret:t salulion ror propagation constants and avoids spurious solutions. This approach can yield more accurate results compared with the conventional approach using the lowest order mlxedInterpolation-type elements., namely, constant edge elements and linear nodal elements. The accuracy ot this approach is investi· gated by calculating the propagation characteristics or optical rib waveguides. Results obtained tor both E~ and SW polarizations are validated using benchmark results produced by established methods,IFFERENT types of the vector fin ite element method (VFEM) have been developed for the analysis of optical waveguiding problems. Of the various formu lations. the VFEM using fu ll vector electric or magnetic field is quite suitable for a wide range of practical complicated problems (1]-f I 3,!. This approach has been widely used for various optical waveguiding structures and recently has been utilized as the optical waveguide solver of CAD paCkages [14]. The most serious problem associated with this approach is the appearance of spurious solutions. The penalty function method [11-( I 4] has been used to cure this problem, but in this technique an arbitrary positive constant, called the penalty coefficient. is involved and the accuracy of solutions depends on its magnitude. Furthermore. in the ful! vectorial fonnulat ion the propagation constant is fi rst given as an input datum, and subsequently the operating wavelength is obtained as a solution. There is another serious problem in the full vectorial approach. As was made clear by Birman [15] and Binnan and Solomyak [16], such an approach is quite difficult for dealing with comer singulari ties and interface singularities so long as the conventional Lagrange interpolation polyno- , but the accuracy of the fi nite element analysis using the lowest order eleme nts is. in general. insufficient.In this paper, in order to provide more accurate numerical solutions and faster convergence in applications. a vector finite element method with the high-order mixed-interpolation-type triangular elements is form ulated in detai l. It is a combination of linear edge elements for transverse components of the electric or magnetic fie ld and quadratic nodal (conventional Lagrange) elements for the axial one. This approach can yield more accurate results compared with the conventional approach using the lowest order elements. The accuracy of this approach is investigated by calculati ng the propagation characteristics of optical rib waveguides. Resul ts obtained for both £'" and E" polarizations are validated using benchmark results produced by established methods.

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We extend topology optimization method with function-expansion-based refractive index distribution to optimization for three-dimensional optical circuits, in which a refractive index distribution in a design region is expressed by an expansion with some analytical functions. Three-branch optical waveguides have been optimized as numerical examples. Equally branching three-branch waveguides are achieved using our method. A limitation of topology optimization in two dimensions and dependency of initial structure are also shown.