Muller Boundary Integral Equations for Mathematical Modeling of Guided Waves in Optical Fibers

The paper is devoted to the investigation of wave propagation along weakly guiding optical waveguides using computational methods. Based on Muller boundary integral equations, which are well-suited for numerical solutions, a characteristic equation for the complex-valued propagation constants is constructed. It is shown that the left-hand side of this equation is proportional to the product of the left-hand sides of the well-known characteristic equation for the original eigenvalue problem of the Helmholtz equation on the plane and a characteristic equation for the so-called ‘‘turned inside-out’’ eigenvalue problem. As a result, the eigenvalue set of the Muller boundary integral equations can be divided into two subsets: the true eigenvalues of the original problem and the fictitious eigenvalues of the turned inside-out problem. Theorems defining the conditions for the existence of solutions for both problems are proved. Thus, under these conditions, the Muller boundary integral equations method enables the search for true eigenvalues only, as confirmed by numerical experiments.