Study On Non-iterative Bidirectional Beam Propagation Method Based On Scattering Operators

The paper studies non-iterative bidirectional beam propagation method based on scattering operators which is used in numerical analysis of optical waveguide with abrupt longitudinal refractive index.The paper first studies beam propagation method which is used in numerical analysis of one-way process of an optical waveguide with continuous longitudinal refractive index. In the process of numerical analysis of optical waveguides with abrupt longitudinal refractive index, light wave reflection occurs at the interface of refractive index abrupt change, and be decomposed into forward and backward component at any point of the optical waveguide. Two one-way Helmholtz equations can be derived combination of wave equation and boundary condition.Only positive Helmholtz equation is considered in beam propagation method, while the negative component is approximately zero. Both of one-way Helmholtz equations are considered in bidirectional beam propagation method, and non-iterative bidirectional beam propagation method based on scattering operator is derived through introduction of the scattering operators. Scattering operators can be divided into reflection operator and transmission operator, both of which include square root operator. Rational approximation of the square root operator is the key step of bidirectional beam propagation method, which determines accuracy and stability of the method. For the optical waveguide with significantly varied refractive index, the rational approximation of square root operator is particularly critical. Pade approximation is used to rational approximate square root operator in the paper, and the correlation between Pade approximation and non-iterative bidirectional beam propagation method based on scattering operators is studied.Ideally, the size of optical waveguide is infinite, and radiation mode propagats unlimitedly in cross-sectional direction. However, the size of optical waveguide is limited in practical application, which leads to reflection at the border of cross section. Reflected waves interfere with guided waves in guided-wave layer. Perfectly Matched Layer which completely absorbs any incoming electromagnetic waves is introduced in the paper. It is able to help to avoid reflection at the lateral border, and there is no waves propagating through the border. The paper concludes with three numerical analysis examples. The range of numerical instability and validity in non-iterative bidirectional beam propagation method based on scattering operators associated with Pade approximations, wavelength and refractive index is studied.