Eigenvalue equations for solving full-vector modes of optical waveguides are formulated using Yee-mesh-based finite difference algorithms and incorporated with perfectly matched layer absorbing boundary conditions. The established method is thus able to calculate the complex propagation constants and the confinement losses of leaky waveguide modes. Proper matching of dielectric interface conditions through the Taylor series expansion of the fields is adopted in the formulation to achieve high numerical accuracy. The method is applied to the study of the holey fiber with triangular lattice, the two-core holey fiber, and the air-guiding photonic crystal fiber.

Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers.

A finite-difference frequency-domain method based on the Yee’s cell is utilized to analyze the band diagrams of two-dimensional photonic crystals with square or triangular lattice. The differential operator is replaced by the compact scheme and the index average scheme is introduced to deal with the curved dielectric interfaces in the unit cell. For the triangular lattice, the hexagonal unit cell is converted into a rectangular one for easier mesh generation. The band diagrams for both square and triangular lattices are obtained and the numerical convergence of computed eigen frequencies is examined and compared with other methods.

Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals.

High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces

A general relation considering an interface condition between a sampled point and its nearby points in the finite-difference analysis of optical waveguides, is derived. Our formulation is a full-vectorial one, and the interface can be oblique to the global coordinate axes or even can have curvature. Using the derived formulation, optical waveguide structures with arbitrary step-index profiles can be dealt with. The validity of the proposed method is examined by comparing the numerical calculation with the analytical solution of a step-index optical fiber. The method is also applied to the analysis of fused fiber-optic couplers and compared with prior analysis using the surface integral equation method.

Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles

New full-vectorial optical waveguide eigenmode solvers using pseudospectral frequency-domain (PSFD) formulations for optical waveguides with arbitrary step-index profile are presented. Both Legendre and Chebyshev collocation methods are considered in the formulation. By applying Legendre-Lagrange or Chebyshev-Lagrange interpolating polynomials to the approximation of spatial derivatives at collocation points, the Helmholtz equations for the transverse-electric or transverse-magnetic components are converted into a matrix eigenvalue equation which is then solved for the eigenmodes by the shift inverse power method. Suitable multidomain division of the computational domain is arranged to deal with general curved interfaces of the refractive-index profile together with a curvilinear mapping technique for each subdomain so that field continuity conditions can be carefully imposed across the dielectric interfaces, which is essential in achieving high numerical accuracy. The solver is applied to the optical fiber for the assessment of its numerical performance, to the classical benchmark rib waveguide for comparing with existing high-accuracy results, and to the fused fiber structure for demonstrating its robustness in calculating the form birefingence.

Full-Vectorial Optical Waveguide Mode Solvers Using Multidomain Pseudospectral Frequency-Domain (PSFD) Formulations

A general relation, considering the interface conditions, between a sampled point and its nearby points is derived. Making use of the derived relation and the generalized Douglas scheme, the three point formulas in the finite-difference modeling of step-index optical devices are extended to fourth order accuracy irrespective of the existence of the step-index interfaces. With numerical analysis and numerical assessment, several frequently used formulas are investigated.

Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices

With the help of an improved finite-difference (FD) formulation, we investigate the field behaviors near the corners of simple dielectric waveguides and the propagation characteristics of a slant-faceted polarization converter. The formulation is full-vectorial, and it takes into consideration discontinuities of fields and their derivatives across the abrupt interfaces. Hence, the limitations in conventional FD formulation are alleviated. In the first investigation, each corner is replaced with a tiny arc rather than a really sharp wedge, and nonuniform grids are adopted. Singularity-like behavior of the electric fields emerge as the arc becomes smaller without specific treatment such as quasi-static approximation. Convergent results are obtained in the numerical analysis as compared with results from the finite-element method. In the second investigation, field behaviors across the slanted facet are incorporated in the formulation, and hence the staircase approximation in conventional FD formulation is removed to get better modeling of the full-vectorial properties.

Finite-Difference Modeling of Dielectric Waveguides With Corners and Slanted Facets

An improved finite-difference frequency-domain method based on Taylor series expansion, local coordinate trans- formation, and boundary condition matching is developed for analyzing the band diagrams of 2-D photonic crystals. This newly proposed scheme can deal with piecewise homogeneous structures with curved dielectric interfaces. It also shows much better per- formance in modeling structures of highly index contrast ratio. We have verified this improved scheme with frequently-used plane-wave expansion method through the calculation of the band diagrams of a 2-D photonic crystal with square lattice. We also compare the scheme with the staircase approximation. We further improve the convergence behavior by adopting high-order terms. The result of this adopting is also demonstrated.

Finite-Difference Frequency-Domain Analysis of 2-D Photonic Crystals With Curved

An improved finite-difference frequency-domain method based on Taylor series expansion, local coordinate transformation, and boundary condition matching is developed for analyzing the band diagrams of 2-D photonic crystals. This newly proposed scheme can deal with piecewise homogeneous structures with curved dielectric interfaces. It also shows much better performance in modeling structures of highly index contrast ratio. We have verified this improved scheme with frequently-used plane-wave expansion method through the calculation of the band diagrams of a 2-D photonic crystal with square lattice. We also compare the scheme with the staircase approximation. We further improve the convergence behavior by adopting high-order terms. The result of this adopting is also demonstrated.

Yen-Chung Chiang

Papers

Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices

Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles

Finite-Difference Modeling of Dielectric Waveguides With Corners and Slanted Facets

Finite-Difference Frequency-Domain Analysis of 2-D Photonic Crystals With Curved

Finite-Difference Frequency-Domain Analysis of 2-D Photonic Crystals With Curved Dielectric Interfaces