Non-self-adjointness of bent optical waveguide eigenvalue problem

In the literature, the mathematical problem of optical wave propagation in dielectric straight waveguides has been systematically studied as a self-adjoint eigenvalue problem with real eigenvalues. In terms of the underlying physics, such real eigenvalues meant no losses during the wave propagation. However, when the waveguides were bent, experiments showed that the wave propagation became lossy. In this paper, optical wave propagation in dielectric bent waveguides is mathematically analyzed. It is shown that the corresponding eigenvalue problem is a non-self-adjoint eigenvalue problem and has complex-valued eigenvalues. The imaginary part of the eigenvalues is a measure of loss. For large-bend radii, the eigenvalue problem for bent waveguides behaves as an eigenvalue problem for straight waveguides, and the complex-valued eigenvalues approach the real-valued eigenvalues of the straight waveguide problem. By expressing the bent waveguide eigenvalue operator as a sum of the self-adjoint operator and the non-self-adjoint operator, asymptotic behaviour of guided modes and their lossy nature are investigated.