A New Approach of Eigenmodes for Varying Refractive-Index Profile’s Waveguides

For the modal computation of an open optical waveguide with a varying refractive-index profile, two perfectly matched layers are used to terminate the waveguide, and the refractive-index profile is approximated by a piecewise polynomial of degree two. Then, the corresponding Sturm-Liouville problem (eigenvalue problem) of the Helmholtz operator in each layer can be solved analytically by the Whittaker functions, and the analytical approximate dispersion equations are established for both TE and TM cases. The approximate solutions converge fast to the exact ones for the continuous refractive-index function as the maximum value of the subinterval sizes tends to zero. Numerical simulations show that high-precision modes may be obtained by Müller's method with suitable initial values. The validation of the proposed method is also verified by use of the finite difference method with a fine grid.