Formulation of Electromagnetic Wave Equations for Nonlinear Optical Waveguides
Nonlinear optical waveguides can be formulated in a linear way provided that an iteration scheme is incorporated to tackle the nonlinearity, as the nonlinear coefficients of known optical materials are very small. The homogeneous electromagnetic wave equation for the solution of optical waveguides can be formulated either in a vectorial form or in a scalar form as a weak-guidance approximation. The outcome is a weak formulation of the partial-differential equation that reduces to a matrix equation when the two finite sets of basis functions are substituted. A reduction method of constraint using Hermitian finite elements was reported for inhomogeneous waveguides where the reduction is performed on an element basis and consequently the sparsity of the matrices in the global matrix equation is retained. Lagrangian finite elements are derived from Lagrangian interpolation, where the coefficients of interpolation involve only the field components.