Recently, it has been shown that if the paraxial wave equation is modified such that fields travel imaginary distances, the field resulting from an imaginary-distance propagation is the fundamental mode of an optical waveguide. For the finite-difference beam-propagation method, we derive the factor by which each eigenmode is amplified during one propagation step. This amplification factor places limits on the inputs-step size, input field, effective index guess, and implicitness parameter-and gives clues on how to optimize the inputs. In particular, we identify and study two optimal sets of inputs, which can reduce computational time significantly. We can obtain the fundamental mode and its propagation constant within a few propagation steps.
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