Calculations of the resonances of open systems can often be challenging compared to calculations of the bound states of closed systems. A new transparent boundary method is proposed to target these resonances, allowing for highly accurate calculations using smaller computational domains than are achievable with traditional boundary methods. The domain is separated into two regions: one for the interior of the system and one for the environment. Expressions are first found for the resonances in the exterior region, which are then used to establish boundary conditions for a numerical treatment of the interior region. This results in a nonlinear eigenvalue problem, which we resolve using a simple iteration procedure.
The proposed method is first applied to calculate resonances for the Stark system, using two implementations with different assumptions and formulations to achieve highly accurate solutions for the Stark resonances of atomic hydrogen. The simple iteration procedure is shown to be sufficient to resolve the nonlinear nature of the eigenvalue problem for these scalar systems. The proposed method is also used to perform first-of-their-kind three-dimensional calculations of the resonances of molecular oxygen, which could not be achieved using existing methods.
The method is then applied to the calculation of leaky modes in optical waveguides, with specific interest given to the guidance of high-energy optical pulses in atmosphere. Radially-symmetric and generalized implementations are presented, demonstrating that the simple iteration procedure performs well for the calculation of vector fields. The proposed method can be used in the future to enable 3D+1 simulations of nonlinear pulse propagation in long-range plasma waveguides.