Abstract-Efficient nonuniform schemes, based on the generalized Douglas (GD) scheme, are developed for the finite-difference beam propagation method (FD-BPM). For a two-dimensional (2-D) problem, two methods are presented: a computational space method and a physical space method. In the former, the GD scheme is employed, after replacing a nonuniform grid in the physical space with a uniform one in the computational space. In the latter, the GD scheme is directly extended to a nonuniform grid in the physical space. We apply these two methods to paraxial and wide-angle FD-BPM's. The fourth-order accuracy is achieved in the transverse direction, provided that the grid growth factor between two adjacent grids is r = 1 + O(1x). For the paraxial BPM, the reduction in the truncation error is demonstrated through modal calculations of a graded-index waveguide using an imaginary distance procedure. For the wide-angle BPM, the propagating field in a tilted waveguide is analyzed to show the effectiveness of the present scheme. As an application of the physical space method, an adaptive grid is introduced into the multistep method.Index Terms-Finite-difference methods, optical beam propagation, optical waveguides.

Abstract-Nonconservation

Reaction of MCl 2 (cod) (M = Pd, Pt) with 1,8-bis[(diphenylphosphino)methyl]naphthalene (1,8-dpmn) gave MCl 2 (1,8-dpmn) 1 (M = Pd), 2 (M = Pt), as confirmed by X-ray analyses. Complex 1 reacted with xylyl isocyanide a (XylNC) in the presence of NaPF 6 to give [Pd(1,and 9 (M = Pt). The structures of 6b, 7b and 8b were confirmed by X-ray analyses. Reaction of [Pd 2 Cl 2 (XylNC) 4 ] with two equiv. of 1,8-dpmn in the presence of NaPF 6 gave 8a, whereas the reaction in the presence of NH 4 PF 6 produced 10a together with 7a. It was confirmed by an X-ray analysis of 10a that the crystal lattice is constructed by two types of cations (3a and two 2,2,6,6-tetramethyl-4-piperidonium) and four PF 6 anions. Photochemical reaction of 6a or 8a in CH 2 Cl 2 led to cleavage of the Pd-Pd bond, giving [PdCl(XylNC) 3 ](PF 6 ) and [PdCl(1,8-dpmn)(XylNC)](PF 6 ).

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