Development of Multidomain Pseudospectral Mode Solvers for Optical Waveguides and Photonic Crystals

A new full-vectorial pseudospectral mode solver based on multidomain pseudospectral
methods for optical waveguides with arbitrary step-index profile is presented. Both Legendre and Chebyshev collocation methods are employed. The multidomain advantage helps in proper fulfillment of dielectric interface conditions, which is essential in achieving high numerical accuracy. Suitable multidomain division of the computational domain is performed to deal with general curved interfaces of the permittivity profile and field continuity conditions are carefully imposed across the dielectric interfaces. Therefore, a curvilinear coordinate mapping technique is introduced to perfectly deal with curved boundaries. Each contiguous subdomain is joined by intensionally imposing different types of boundary conditions to enhance the accuracy. Moreover, perfectly matched layer (PML) absorbing boundary conditions are incorporated into the model so that leaky
modes with complex propagation constants can be analyzed. The solver is applied to the calculation of guided modes on optical fibers, fused fiber couplers, D-shaped fibers, channel waveguides, rib waveguides, and photonic crystal fibers, and comparison with analytical results or reported ones based on other methods is made. It is demonstrated that numerical accuracy in the effective index up to the remarkable 10⣵76;10 order can be easily achieved.
The multidomain pseudospectral scheme is for the first time applied to the calculation of the band diagrams of two-dimensional photonic crystals with the inclusion of the required periodic boundary conditions, and is again shown to possess excellent numerical convergence behavior and accuracy. The proposed method shows uniformly excellent convergence characteristics for both the transverse-electric and transverse-magnetic waves in the analysis of di_erent structures. The analysis of a mini band gap with the normalized frequency gap width as small as on the order of 107 is also shown to demonstrate the extremely high accuracy of the proposed method.
A novel numerical calculation of chromatic dispersion coefficients of optical fibers including holy fibers is also proposed in this research using a procedure involving Chebyshev-Lagrange interpolation polynomials. Only numerically determined effective indices at several wavelengths are needed for obtaining the dispersion curve and no direct numerical differentiation of the effective refractive index is involved.