High-accuracy eigenmode computation in slab waveguides with arbitrary material profiles: a transfer matrix approach

Accurately solving both guided and unguided complex modes for waveguides with arbitrary material profiles remains a significant challenge in computational photonics. Traditional analytical approaches to solving Maxwell's equations in optical waveguide analysis often involve cumbersome formulation and large numerical errors. In this work, an analytical unified mathematical framework is presented for one-dimensional waveguide problems based on the local higher-order transfer matrix method. This approach is inherently compatible with Perfectly Matched Layer boundary conditions. It is demonstrated that the method can seamlessly adapt to diverse boundary conditions, enabling comprehensive computation of guided modes, leaky modes, and Berenger modes-preserving both methodological elegance and computational efficiency. Furthermore, this paper presents a hybrid root-finding framework that efficiently computes complete complex roots. This solver achieves exceptional computational speed for nonlinear systems and is fundamentally generalizable to arbitrary nonlinear equations.