The Calculation Of Wave Propagation In Symmetric Waveguide With A Curved Interface

In this paper, we mainly deal with the wave propagation in the symmetric three layers slab optical waveguide. We can obtain the wave equation from the Maxwell Equation. By Fourier Transformation, the Helmholtz Equation is obtained.The original two dimension Helmholtz Equation is:where(x, z) satisfies : -∞ < x < +∞, 0 < z < +∞ , and their interface is a curve.First, the original problem has a unbounded region, so we can't use the numerical method to solve it. So we introduce the perfectly matched layers(PML), changing the unbounded problem to the bounded problem. By using the complex transformation , the equation is changed, and the area is —H < x < +H,0 < z < +∞.The problem we discussed has a curve interface. It will have big error if we use the traditional method as the staircase approximation. We propose a local orthogonal transformation. Local orthogonal transformation designs the function z|^ = f(x, z) , x|^= g(x, z) which satisfies the orthogonal condition and boundary condition, then let u = WV, we have the equation after transformation . We disperse at the x|^ direction with the eigenfunction, having Aφ = λφ, where A is a complex tridiagonal matrix and φ is the corresponding eigenvetor of A.And then, we use a Multi-Generalized Rayleigh Quotient Iteration method to compute the eigenvalues and the eigenvectors, and use the One-way method to compute the propagation, by comparing with the Marching method. At the computation of the propagation, Propagating modes and Leaky modes is very important, especially the Propagating modes. We found that eigenvalues maybe leave out by the Rayleigh Quotient Iteration method. So we make a small partition when we use the Multi-Generalized Rayleigh Quotient Iteration method, so that the difference between the initial value and the terminal value is small, it will ensure the convergence and decrease the eigenvalues left out.At last, the numerical examples verify the feasibility and validity of the method in this paper.