Research And Application Of High-order Symplectic Compact Finite-different Time-domain Method

Research on the analytical method of eigenvalue problem of guided wave system is an important topic in the computational electromagnetic.Nnumerous numerical calculation methods of solving this kind of eigenvalue problem include Moments Method(MOM),Finite Element Method(FEM),Finite Difference Method(FD)and so on.The Finite Difference Time Domain(FDTD)method,as a 3-D full-wave numerical algorithm in the time domain,has the advantages of simple form,convenient modeling and wide comprehension.However,it takes plenty of computing resources and time in solving various eigenvalue problems of complex media(dispersion,anisotropy,left-handed media,etc.)or complex structures(non-uniform,multi-scale,etc.).Therefore,it is an urgent problem to study the efficient time-domain algorithm for solving the eigenvalue problem of guided wave structures.Based on higher-order symplectic integrator temporal algorithm for energy conservation and compact spatial algorithm for dimensional reduction,a novel high-order symplectic compact FDTD algorithm is proposed in this dissertation;basic theory and application of the new algorithm are also studied.Firstly,based on the traditional FDTD algorithm,the basic framework of the high-order SC-FDTD algorithm is constructed.Secondly,key techniques of this new algorithm are improved.Finally,the dual-dispersion model is established to realize the efficient and high-precision solution and application for electromagnetic field eigenvalue problem of complex media and complex structure,which therefore provides theoretical support for the design and optimization of photoelectric devices.The main work and innovation are as following:1.For widely used time domain numerical algorithms,a new high symplectic compact differential FDTD algorithm is proposed.This algorithm establishes a new high-order spatiotemporal matching evolution matrix in which the fast evolution matrix of spatial dimension reduction and the Hamiltonian matrix of temporal symplectic integral are combined.Then,the appropriate propagation constant is selected into the matrix and vector wave function expansion.Therefore,the Maxwell’s equation discretization framework of the novel SC-FDTD(4,4)algorithm is obtained to simulate the typical waveguide structure.Simulation results indicate that this algorithm not only guarantees simulation precision,but also ensures the high efficiency of simulation,which is attributed to the theory combination of the fast spatial algorithm and stable temporal algorithm.2.The present research mainly focuses on the novel algorithm optimization of the various key technologies that contain numerical stability,numerical dispersion and absorption boundary conditions,etc.By comparisons with other high-order time-domain algorithms,the advantages of the novel algorithm in terms of long time simulation,energy conservation and numerical accuracy were verified.For instance,with the same accuracy of time,the SCFDTD(4,4)algorithm can better improve the global dispersion error effectively.Meanwhile,the proposed algorithm can be made nearly independent of the maximum limitation of the Courant Friedrich Levy law,which largely enhances the computational efficiency without improving precision by using excessive fine subdivision.By analyzing various key technologies,the propagation constant and difference coefficient can be gradually selected to construct an optimum spatio-temporal framework.3.The application of the typical electromagnetic simulation of guided wave system is realized at the practical level.The new algorithm is used to analyze the eigenvalues of optical waveguide structures,such as,dielectric optical waveguides with complex cross sections and photonic crystal fiber.Through numerical experiments on long-term simulation,satisfying numerical results can be obtained in terms of on efficiency,numerical stability and precision.Thus,the advantages of the SC-FDTD(4,4)algorithm are further demonstrated.4.The single(double)dispersion model of SC-FDTD(4,4)algorithm for the double dispersive materials are well established.By introducing the Drude and Lorentz model for fitting the electromagnetic parameters and based on the auxiliary differential equation as well as the matrix splitting of symplectic compact integrator,the scheme of single(double)dispersion model is constructed with rigorous formula derivation.The model characteristics of hybrid plasma waveguide with single(double)dispersion model are studied.The stimulation process avoids the complexity of electromagnetic parameters changing with frequency.Meanwhile,the new dispersion model can save considerable computer resources.The research results not only verify the effectiveness of the model,but also provide efficient theoretical guidance for the design and optimization of optical devices at the nanometer scale.