| With the development of science,periodic media are increasingly used in micro-nano technology in modern science and play an important role.The various propagation phenomenon of waves in periodic media can be described by using partial differential equations that satisfy different boundary conditions.In this paper,the analytic expression of the Dirichlet-to-Neumann operator of the second-order characteristic elliptic equation with symmetric periodic potential is rigorously derived,and then an algorithm for comput-ing the absorbing boundary condition of the Helmholtz equation in the periodic medium is proposed.This paper first proves the correctness of the analytic expression of a Dirichlet-to-Neumann operator proposed by Zheng Chunxiong[1].This operator can be applied to the second-order characteristic elliptic equation with sinusoidal potential as an absorbing boundary condition.Furthermore,it can be applied to the second-order characteristic elliptic equation with general symmetric periodic coefficient.In this paper,we also consider the Helmholtz equation in the semi-unbounded singly-periodic and doubly-periodic media.The corresponding Bloch modes which satisfy the quasi-periodic boundary conditions are investigated.We use the energy band structure of the medium to determine the wavenumber k of the equation.By using the properties of the Floquet multiplier?and other physical criteria,we can prove the existence of the symplectic orthogonal relation between the physical solutions and the anti-physical solutions.Moreover,we propose the idea that the Dirichlet value and the Neumann value of all the solutions on the boundary can be represented as a linear combination of those of the physical solutions on the same boundary.Thus,the absorbing boundary conditions of the Dirichlet-to-Neumann form are obtained.Finally,we apply the algorithm to the singly and doubly periodic medium,and carry out different numerical experiments to validate the effectiveness of the algorithm. |
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