A beam propagation method based on the Galerkin method with Hermite-Gauss basis functions for studying optical field propagation in weakly guiding dielectric structures is described. The selected basis functions natur...A beam propagation method based on the Galerkin method with Hermite-Gauss basis functions for studying optical field propagation in weakly guiding dielectric structures is described. The selected basis functions naturally satisfy the required boundary conditions at infinity so that the boundary truncation is avoided. The paraxial propagation equation is converted into a set of first-order ordinary differential equations, which are solved by means of standard numerical library routines. Besides, the calculation is efficient due to its small resulted matrix. The evolution of the injected field and its normalized power along the propagation distance in an asymmetric slab waveguide and directional coupler are presented, and the solutions are good agreement with those obtained by finite difference BPM, which tests the validity of the present approach.展开更多
This work mainly focuses on the numerical simulation of the Fredholm integral equation of the second kind. Applying the idea of Gauss-Lobatto quadrature formula, a numerical method is developed. For the integral item,...This work mainly focuses on the numerical simulation of the Fredholm integral equation of the second kind. Applying the idea of Gauss-Lobatto quadrature formula, a numerical method is developed. For the integral item, we give an approximation with high precision. The existence condition of the solution for the Fredholm equation is given. Furthermore, the error analyses are presented. Finally, the numerical examples verify the theoretical analysis, and show the efficiency of the algorithm we discussed.展开更多
In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadratur...In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.展开更多
文摘A beam propagation method based on the Galerkin method with Hermite-Gauss basis functions for studying optical field propagation in weakly guiding dielectric structures is described. The selected basis functions naturally satisfy the required boundary conditions at infinity so that the boundary truncation is avoided. The paraxial propagation equation is converted into a set of first-order ordinary differential equations, which are solved by means of standard numerical library routines. Besides, the calculation is efficient due to its small resulted matrix. The evolution of the injected field and its normalized power along the propagation distance in an asymmetric slab waveguide and directional coupler are presented, and the solutions are good agreement with those obtained by finite difference BPM, which tests the validity of the present approach.
文摘This work mainly focuses on the numerical simulation of the Fredholm integral equation of the second kind. Applying the idea of Gauss-Lobatto quadrature formula, a numerical method is developed. For the integral item, we give an approximation with high precision. The existence condition of the solution for the Fredholm equation is given. Furthermore, the error analyses are presented. Finally, the numerical examples verify the theoretical analysis, and show the efficiency of the algorithm we discussed.
文摘In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.
