Research On The Dispersive Materials Using The Symplectic FDTD Method

The available solution to the physical model of the realistic problem has been promoted along with the computational ability due to developing in the software and hardware of the computer. Nowadays, it is a hot spot that the metamaterials are received special focus on property investigating and structure designing. Actually, metamaterials are belonged to the dispersive material and preserve the properties of the dispersive materials. To solve the problem and to design the structure containing the dispersive materials will give a useful reference to solve the metamaterials problems, so the job about the dispersive material has a conspicuous meaning.Finite-difference time domain (FDTD) method which is largely based on the computational resource has been widely used in the computational electromagnetic problem solution due to its simplicity, easily implementing and facility for parallel computing. However, the conventional scheme of the simulation algorithm has many drawbacks such as low accuracy, large dispersion error and low Courant stability number. Luckly, there are several methods related to temporal and spatial discrete scheme that are developed and have been used to enhance the algorithm efficiency.A special formulation and discrete scheme should be established according to different types of dispersive materials using FDTD method. Especially, when comes to a complex electromagnetic problem which composed of two or multiply types of dispersive materials, to find a unified formulation which is suitable for types of materials is most important. It’s also reasonable and valuable to provide a criterion on choosing a best simulation algorithm especially to give an accurate solution to the hot spot problem related to dispersive media. This criterion should be satisfied with some basic conditions such as a low dispersion error, large Courant stability number and especially a high numerical accuracy. Considering the symplectic scheme conserves the property of the symplectic transformation, so it is a high numerical precision and efficient algorithm for computational electromagnetic problems. Whether Maxwell’s equation describing the dispersive media problem can be written as a Hamiltonian system like the conventional materials is a promising project. This result can achieve a low dispersion error, high accuracy and high Courant stability number by using the symplectic integrator technique.To solve these aforementioned questions, the auxiliary difference equations (ADEs) are used to establish a unified formula that could solve a linear and nonlinear dispersive media system, then the split operator technique is employed to give an approximation to a matrix operator, the method of numerical dispersion error analysis is constructed to analyze numerical dispersion error of the symplectic and alternative direction implicit (ADI) scheme. A discrete scheme is also constructed to solve the dispersive material described by the Drude model using a dispersive split operator method. A high-order accurate split perfectly matched layers (SPMLs) should be constructed to terminate the computational area, and the SPML are derived with a split operator method which is based on the Drude model.Some contributed works and creative things are as following:(1) It is a unified method that can be used to solve and analyze the problem which involves the linear dispersive material and nonlinear material has been established. This idea is based on the ADE method which has an advantage to deal with a complex system with a complicated structure and many types of dispersive materials.(2) The Maxwell’s equation has been taken as a form of Hamiltonian function. The symplectic and ADI scheme can be derived by applying different split operator approximations to the temporal evolution matrix. The result of numerical dispersion analysis and maximum numerical stability number via Courant stability number can be obtained by using the plane wave expansion method. Then three types of the spatial difference approximation are employed as the numerical experiment to benchmark the theory and provide the best simulation algorithm.(3) The Maxwell’s equation which can be used to describe the wave propagating in the dispersive materials is written as a Hamiltonian function. The discrete scheme are constructed to simulate the Drude model and to solve the dispersive electromagnetic problem with the split operator method. The results deduced by the high order unified symplectic FDTD are compared to the ADE-FDTD method, and then the properties of the left handed materials (LHMs) such as perfect lens, negative refraction phenomenon and phase compensation effect are also explored.(4) The split operator corresponding to the lossy Drude model is also provided by symplectic integrator and the formulation and discrete scheme of the split perfect matched layer are also constructed.