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A three-level finite difference beam propagation formulation

The authors consider beam propagation in a linear and lossless parallel dielectric waveguide, extending the Crank-Nicholson method to a three-level finite difference scheme based on the Helmholtz wave equation. Stability analysis indicates that the three-level scheme is stable with small oscillatory behavior. However, simulation results exhibit conditional stability and the bigger the propagation step size, the more oscillation results.