Analysis of guided and leaky modes of planar optical waveguides using transparent boundary conditions

We present a numerical approach to compute and characterize both guided and leaky modes in a multilayer planar optical waveguide made of any lossy and dispersive materials. Usually, in numerical calculations based on finite element methods, perfectly matched layers (PMLs) are used to truncate the unbounded substrate and cover layers. However, it is difficult to make such PMLs transparent for both guided and leaky modes at the same time, and often, different or even contradictory PML parameters are required for these two types of modes. In contrast, the transparent boundary conditions (TBCs) that we use in this work can terminate the unbounded waveguide and, simultaneously, provide perfect transparency for the modes. In addition, this type of boundary condition does not contaminate the solutions with non-existent modes, such as PML modes. More importantly, the TBC approach yields the nonlinear eigenvalue solutions that can be intrinsically mapped to the parameter space of transverse wavenumbers in the claddings. This allows us to uniquely determine the power flow properties of all the calculated modes. A finite element Python package is developed to treat a variety of planar waveguides in a robust and systematic way.