A general relation, considering the interface conditions, between a sampled point and its nearby points is derived. Making use of the derived relation and the generalized Douglas scheme, the three-point formulas in the finite-difference modeling of step-index optical devices are extended to fourth order accuracy irrespective of the existence of the step-index interfaces. With numerical analysis and numerical assessment, several frequently used formulas are investigated.
Yih-Peng Chiou, Yen-Chung Chiang, and Hung-Chun Chang, "Improved Three-Point Formulas Considering the Interface Conditions in the Finite-Difference Analysis of Step-Index Optical Devices," J. Lightwave Technol. 18, 243- (2000)