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Numerical results are given to test the accuracy of the algorithm. In this paper, SFDTD combined with ADE method is presented for the lossy Lorentz–Drude model. Since the expansion of the matrix exponential in SFDTD is a nodus, there are no papers referring to the lossy Lorentz–Drude dispersive model. In paper, the SFDTD is presented to model the lossless Drude dispersive model. Overcoming the shortcomings mentioned above, the SFDTD and its improved algorithm, are proposed. So a high-order approximation in the temporal domain is highly required. Moreover, the nondissipative approach is very important over long time integration. And the numerical dispersion has serious drawbacks in view of computing long-range propagation. The aforementioned dispersive FDTD methods approximate the time derivative with a low-order approximation (no more than second-order). Piecewise linear recursive convolution (PLRC) and Z-transform approaches are also widely used. Dispersive models are usually written in the frequency domain, and the auxiliary differential equation (ADE) is used in order to make them consistent with time domain methods. Each of them has different properties and usage depending on the nature of applications and their requirements. For example, the permittivity of the metal approximates to one at a low frequency, but varies significantly at a high frequency.ĭifferent dispersive models are necessary to model dispersive materials, such as Debye, Drude, Lorentz, Lorentz–Drude and Cole–Cole. For a material to behave causally, the imaginary part of permittivity or permeability (loss) can be small but will not disappear at any frequencies according to the Kramers–Kronig relation. For some electromagnetic problems, permittivity can be considered a constant at some frequency ranges, which is actually an approximation. Actually, the permeability of most of the available materials in nature is equal to one except magnetic materials, while the permittivity is different for different materials. However, the conventional FDTD method requires additional treatments to simulate electromagnetic wave propagation in dispersive materials, due to the frequency-dependent permittivity or permeability of the materials.
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And as a grid-based on numerical technique, it also enables one to model objects with arbitrary material fillings, such as dispersive materials. Since it is a time-domain method, FDTD solutions can cover a wide frequency range with a single simulation run. The finite-difference time-domain (FDTD) method, as an efficient numerical analysis technique used for modeling computational electrodynamics, has been shown to be one of the most widely used methods in electromagnetic simulations. Numerical results for a more realistic structure, the simulation of periodic arrays of silver split-ring resonators (SRRs) using the Drude dispersion model, are also included, and the results agree well with those obtained by the finite element method (FEM). Focusing action of the matched left-handed materials (LHMs) slab is also achieved as the second example in the two-dimensional space.
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An excellent agreement is achieved between the SFDTD-calculated and exact theoretical results when calculating the transmission coefficient in simulation of metal films. Moreover, the present algorithm can also be applicable to the lossy Drude and Lorentz dispersive model in a straightforward manner. With the rigorous and artful formula derivation, the detailed formulations are provided. The algorithm is applied to the lossy Lorentz–Drude dispersive model. A high-order symplectic finite-difference time-domain (SFDTD) algorithm, based on the matrix splitting, the symplectic integrator propagator and the auxiliary differential equation (ADE) technique, is presented.